Argumentation in Artificial Intelligence.pdf
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This document provides a foreword for a book on argumentation in artificial intelligence. It discusses how argumentation is an important part of human interaction and intelligence. It notes that argumentation involves both reasoning within individual minds and interaction between multiple agents. The foreword argues that studying argumentation can help unite different fields like logic, mathematics, computation, and studies of human behavior. It presents the book as bringing these various strands together and establishing argumentation as a new field at the intersection of artificial and natural intelligence.
![Chapter 1
Argumentation Theory: A Very Short
Introduction
Douglas Walton
1 Introduction
Since the time of the ancient Greek philosophers and rhetoricians, argumentation
theorists have searched for the requirements that make an argument correct, by some
appropriate standard of proof, by examining the errors of reasoning we make when
we try to use arguments. These errors have long been called fallacies, and the logic
textbooks have for over 2000 years tried to help students to identify these fallacies,
and to deal with them when they are encountered. The problem was that deduc-
tive logic did not seem to be much use for this purpose, and there seemed to be no
other obvious formal structure that could usefully be applied to them. The radical
approach taken by Hamblin [7] was to refashion the concept of an argument to think
of it not just as an arbitrarily designated set of propositions, but as a move one party
makes in a dialog to offer premises that may be acceptable to another party who
doubts the conclusion of the argument. Just after Hamblin’s time a school of thought
called informal logic grew up that wanted to take a new practical approach to teach-
ing students skills of critical thinking by going beyond deductive logic to seek other
methods for analyzing and evaluating arguments. Around the same time, an interdis-
ciplinary group of scholars associated with the term ‘argumentation,’ coming from
fields like speech communication, joined with the informal logic group to help build
up such practical methods and apply them to real examples of argumentation [9].
The methods that have been developed so far are still in a process of rapid evolu-
tion. More recently, improvements in them have been due to some computer scien-
tists joining the group, and to collaborative research efforts between argumentation
theorists and computer scientists. Another recent development has been the adop-
tion of argumentation models and techniques to fields in artificial intelligence, like
multi-agent systems and artificial intelligence for legal reasoning. In a short paper, it
is not possible to survey all these developments. The best that can be done is to offer
Douglas Walton
University of Windsor, e-mail: dwalton@uwindsor.ca
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 1
DOI 10.1007/978-0-387-98197-0 1, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-15-320.jpg)
![2 Douglas Walton
an introduction to some of the basic concepts and methods of argumentation theory
as they have evolved to the present point, and to briefly indicate some problems and
limitations in them.
2 Arguments and Argumentation
There are four tasks undertaken by argumentation, or informal logic, as it is also
often called: identification, analysis, evaluation and invention. The task of identifi-
cation is to identify the premises and conclusion of an argument as found in a text
of discourse. A part of this task is to determine whether a given argument found in
a text fits a known form of argument called an argumentation scheme (more about
schemes below). The task of analysis is to find implicit premises or conclusions in
an argument that need to be made explicit in order to properly evaluate the argu-
ment. Arguments of the kind found in natural language texts of discourse tend to
leave some premises, or in some instances the conclusion, implicit. An argument
containing such missing assumptions is traditionally called an enthymeme. The task
of evaluation is to determine whether an argument is weak or strong by general cri-
teria that can be applied to it. The task of invention is to construct new arguments
that can be used to prove a specific conclusion. Historically, recent work has mainly
been directed to the first three tasks, but there has been a tradition of attempting
to address the fourth task from time to time, based on the tradition of Aristotelian
topics [1, ch. 8].
There are differences in the literature in argumentation theory on how to define
an argument. Some definitions are more minimal while others are more inclusive.
We start here with a minimal definition, however, that will fit the introduction to
the elements of argumentation presented below. An argument is a set of statements
(propositions), made up of three parts, a conclusion, a set of premises, and an infer-
ence from the premises to the conclusion. An argument can be supported by other
arguments, or it can be attacked by other arguments, and by raising critical questions
about it.
Argument diagramming is one of the most important tools currently in use to
assist with the tasks of analyzing and evaluating arguments. An argument diagram
is essentially a box and arrow representation of an argument where the boxes con-
tain propositions that are nodes in a graph structure and where arrows are drawn
from nodes to other nodes representing inferences. At least this is the most common
style of representation. Another style growing in popularity is the diagram where
the nodes represent arguments and the boxes represent premises and conclusions of
these arguments. The distinction between a linked argument and a convergent ar-
gument is important in argumentation theory. A linked argument is one where the
premises work together to support the conclusion, whereas in a convergent argument
each premise represents a separate reason that supports the conclusion. Arguments
fitting the form of an argumentation scheme are linked because all of the premises
are needed to adequately support the conclusion. Here is an example of a convergent](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-16-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 3
Fig. 1.1 Example of an Argument Diagram
argument: gold is malleable; it can be easily made into jewelry, and my metallurgy
textbook says it is malleable. In the example shown in Figure 1.1, two linked argu-
ments are combined in a chain of reasoning (called a serial argument).
The linked argument on the right at the bottom has a colored border and the label
Argument from Expert Opinion is shown in a matching color at the top of the con-
clusion. This label represents a type of argument called an argumentation scheme.
Figure 1.1 was drawn with argument diagramming tool called Araucaria [14].
It assists an argument analyst using a point-and-click interface, which is saved in
an Argument Markup Language based on XML [14]. The user inserts the text into
Arauacaria, loads each premise or conclusion into a text box, and then inserts ar-
rows showing which premises support which conclusions. As illustrated above, she
can also can insert implicit premises or conclusions and label them. The output is
an argument diagram that appears on the screen that can be added to, exported or
printed (http://araucaria.computing.dundee.ac.uk/).
The other kind of format for representing arguments using visualization tools
is shown in the screen shot in Figure 1.2. According to this way of represent-
ing the structure of the argument, the premises and conclusions appear as state-
ments in the text boxes, while the nodes represent the arguments. Information about
the argumentation scheme, and other information as well, is contained in a node
(http://carneades.berlios.de/downloads/).
Both arguments pictured in Figure 1.2 are linked. Convergent arguments are rep-
resented as separate arguments. Another chapter in this book (see Chapter 12) shows
how Carneades represents different proof standards of the kinds indicated on the
lower right of the screen shot.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-17-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 5
about it. By this dialog process of pitting the one argument against the other, the
weaknesses in each argument are revealed, and it is shown which of the two argu-
ments is the stronger.1
To fill out the minimal definition enough to make it useful for the account of
argumentation in the paper, however, some pragmatic elements need to be added,
that indicate how arguments are used in a dialog between two (in the simplest case)
parties. Argumentation is a chain of arguments, where the conclusion of one infer-
ence is a premise in the next one. There can be hypothetical arguments, where the
premises are merely assumptions. But generally, the purpose of using an argument
in a dialog is to settle some disputed (unsettled) issue between two parties. In the
speech act of putting forward an argument, one party in the dialog has the aim of
trying to get the other party to accept the conclusion by offering reasons why he
should accept it, expressed in the premises. This contrasts with the purpose of using
an explanation, where one party has the aim of trying to get the other party to under-
stand some proposition that is accepted as true by both parties. The key difference
is that in an argument, the proposition at issue (the conclusion) is doubted by the
one party, while in an explanation, the proposition to be explained is not in doubt by
either party. It is assumed to represent a factual event.
3 Argument Attack and Refutation
One way to attack an argument is to ask an appropriate critical question that raises
doubt about the acceptability of the argument. When this happens, the argument
temporarily defaults until the proponent can respond appropriately to the critical
question. Another way to attack an argument is to question one of the premises. A
third way to attack an argument is to put forward counter-argument that opposes
the original argument, meaning that the conclusion of the opposing argument is the
opposite (negation) of the conclusion of the original argument. There are other ways
to attack an argument as well [10]. For example, one might argue that the premises
are not relevant to the conclusion, or that the argument is not relevant in relation to
the issue that is supposedly being discussed. One might also argue that the original
argument commits a logical fallacy, like the fallacy of begging the question (argu-
ing in a circle by taking for granted as a premise the very proposition that is to be
proved). However, the three first ways cited above of attacking an argument are es-
pecially important for helping us to understand the notion of argument refutation. A
refutation of an argument is an opposed argument that attacks the original argument
and defeats it.
A simple way to represent a sequence of argumentation in the dialogical style is
to let the nodes in a graph represent arguments and the arrows represent attacks on
1 This approach has been neglected for a long time in the history of logic, but it is not new. Ci-
cero, based on the work of his Greek predecessors in the later Platonic Academy, Arcesilaus and
Carneades, adopted the method of dialectical inquiry that, by arguing for and against competing
views, reveals the one that is the more plausible [16, p. 4].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-19-320.jpg)
![6 Douglas Walton
arguments [5]. In this kind of argument representation, one argument is shown as
attacking another. In this example, argument A1 attacks both A2 and A3. A2 attacks
A6. A6 attacks A7, and A7 attacks A6. A3 attacks A4, and so forth.
Fig. 1.3 Example of a Dung-style Argument Representation
Notice that arguments can attack each other. A6 attacks A7 and A7 also attacks
A6. An example [4, p. 23] is the following pair of arguments.
Richard is a Quaker and Quakers are pacifists, so he is a pacifist. Richard is a
Republican and Republicans are not pacifists, so he is a not a pacifist.
In Dung’s system, the notions of argument attack are undefined primitives, but the
system can be used to model criteria of argument acceptability. One such criterion
is the view that an argument should be accepted only if every attack on it is attacked
by an acceptable argument [3, p. 3].
There is general (but not universal) agreement in argumentation studies that there
are three standards by which the success of the inference from the premises to the
conclusion can be evaluated. This agreement is generally taken to mean that there
are three kinds of arguments: deductive, inductive, and defeasible arguments of a
kind widely thought not to be inductive (Bayesian) in nature. This third class in-
cludes arguments like ‘Birds fly; Tweety is a bird; therefore Tweety flies,’ where
exceptions, like ‘Tweety has a broken wing’ are not known in advance and cannot
be anticipated statistically. Many of the most common arguments in legal reasoning
and everyday conversational argumentation that are of special interest to argumen-
tation theorists fall into this class. An example would be an argument from expert
opinion of this sort: Experts are generally right about things in their domain of ex-
pertise; Dr. Blast is an expert in domain D, Dr. Blast asserts that A, A is in D;
therefore an inference can be drawn that A is acceptable, subject to default if any
reasonable arguments to the contrary or critical questions are raised. Arguments of
this sort are important, for example in legal reasoning, but before the advent of ar-
gumentation theory, useful logical tools to identify, analyze and evaluate them were
not available.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-20-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 7
4 Argumentation Schemes
Argumentation schemes are abstract argument forms commonly used in everyday
conversational argumentation, and other contexts, notably legal and scientific argu-
mentation. Most of the schemes that are of central interest in argumentation theory
are forms of plausible reasoning that do not fit into the traditional deductive and
inductive argument forms. Some of the most common schemes are: argument from
witness testimony, argument from expert opinion, argument from popular opinion,
argument from example, argument from analogy, practical reasoning (from goal to
action), argument from verbal classification, argument from sign, argument from
sunk costs, argument from appearance, argument from ignorance, argument from
cause to effect, abductive reasoning, argument from consequences, argument from
alternatives, argument from pity, argument from commitment, ad hominem argu-
ment, argument from bias, slippery slope argument, and argument from precedent.
Each scheme has a set of critical questions matching the scheme and such a set rep-
resents standard ways of critically probing into an argument to find aspects of it that
are open criticism.
A good example of a scheme is the one for argument from expert opinion, also
called appeal to expert opinion in logic textbooks. In this scheme [1, p. 310], A is a
proposition, E is an expert, and D is a domain of knowledge.
Major Premise: Source E is an expert in subject domain S containing proposition
A.
Minor Premise: E asserts that proposition A is true (false).
Conclusion: A is true (false).
The form of argument in this scheme could be expressed in a modus ponens for-
mat where the major (first) premise is a universal conditional: If an expert says that A
is true, A is true; expert E says that A is true; therefore A is true. The major premise,
for practical purposes, however, is best seen as not being an absolute universal gen-
eralization of the kind familiar in deductive logic. It is best seen as a defeasible
generalization, and the argument is defeasible, subject to the asking of critical ques-
tions. If the respondent asks any one of the following six critical questions [1, p.
310], the proponent must give an appropriate reply or the argument defaults.
CQ1: Expertise Question. How credible is E as an expert source?
CQ2: Field Question. Is E an expert in the field that A is in?
CQ3: Opinion Question. What did E assert that implies A?
CQ4: Trustworthiness Question. Is E personally reliable as a source?
CQ5: Consistency Question. Is A consistent with what other experts assert?
CQ6: Backup Evidence Question. Is E’s assertion based on evidence?
Some other examples of schemes will be introduced when we come to study the
example of extended argumentation in section 2.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-21-320.jpg)
![10 Douglas Walton
Bob (4): The government has a responsibility to protect its citizens, but it also has
a responsibility to defend their freedom of choice. Banning smoking would be
an intrusion into citizens’ freedom of choice.
Ann (5): Smoking is not a matter of freedom of choice. Nicotine is an addictive
drug. Studies have shown that once smokers have begun smoking, they become
addicted to nicotine. Once they become addicted they are no longer free to choose
not to smoke.
Bob (6): Governments should not stop citizens from doing things that can be ex-
tremely harmful to their health. It is legal to eat lots of fatty foods or drink alcohol
excessively, and it makes no sense for governments to try to ban these activities.
Examining Ann’s first argument, it is fairly straightforward to put it in a format
showing that it has two premises and a conclusion.
Premise: Governments should protect its citizens from harm.
Premise: Smoking tobacco is extremely harmful to the smoker’s health.
Conclusion: Therefore governments should ban smoking.
This argument looks to be an instance of the argumentation scheme for argument
from negative consequences [1, p. 332]. The reason it offers to support its conclusion
that governments should ban smoking is that smoking has negative consequences.
An implicit premise is that being extremely harmful to health is a negative conse-
quence, but we ignore this complication for the moment.2
Scheme for Argument from Negative Consequences
Premise: If A is brought about, then bad consequences will occur.
Conclusion: Therefore A should not be brought about.
The reason is that a premise in the argument claims that the practice of smoking
tobacco has harmful (bad) consequences, and for this reason the conclusion advo-
cates something that would make it so that smoking is no longer brought about.
However there is another argumentation scheme, one closely related to argument
from negative consequences, that could also (even more usefully) be applied to this
argument. It is called practical reasoning. The simplest version of this scheme, called
practical inference in [1, p. 323] is cited below with its matching set of critical
questions.
Scheme for Practical Inference
Major Premise: I have a goal G.
Minor Premise: Carrying out this action A is a means to realize G.
Conclusion: Therefore, I ought (practically speaking) to carry out this action A.
Critical Questions for Practical Inference
CQ1: What other goals do I have that should be considered that might conflict
with G?
2 To more fully analyze the argument we could apply a more complex scheme called value-based
practical reasoning [2].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-24-320.jpg)
![14 Douglas Walton
Type of Dialog Initial Situation Participant’s Goal Goal of Dialog
Persuasion Conflict of Opinions Persuade Other Party Resolve or Clarify Issue
Inquiry Need to Have Proof Find and Verify Evi-
dence
Prove (Disprove) Hy-
pothesis
Negotiation Conflict of Interests Get What You Most
Want
Reasonable Settlement
Both Can Live With
Information-
Seeking
Need Information Acquire or Give Infor-
mation
Exchange Information
Deliberation Dilemma or Practical
Choice
Co-ordinate Goals and
Actions
Decide Best Available
Course of Action
Eristic Personal Conflict Verbally Hit Out at Op-
ponent
Reveal Deeper Basis of
Conflict
Table 1.1 Six Basic Types of Dialog
on that thesis or arguing for an opposed thesis. These tasks are set at the opening
stage, and remain in place until the closing stage, when one party or the other fulfils
its burden of persuasion. The proponent has a burden of persuasion to prove (by a
set standard of proof) the proposition that is designated in advance as her ultimate
thesis.3 The respondent’s role is to cast doubt on the proponent’s attempts to succeed
in achieving such proof. The best known normative model of the persuasion type of
dialog in the argumentation literature is the critical discussion [17]. It is not a formal
model, but it has a set of procedural rules that define it as a normative structure for
rational argumentation.
The goal of a persuasion dialog is to reveal the strongest arguments on both sides
by pitting one against the other to resolve the initial conflict posed at the opening
stage. Each side tries to carry out its task of proving its ultimate thesis to the standard
required to produce an argument stronger than the one produced by the other side.
This burden of persuasion, as it is called [13], is set at the opening stage. Meeting
one’s burden of persuasion is determined by coming up with a strong enough argu-
ment using a chain of argumentation in which individual arguments in the chain are
of the proper sort. To say that they are of the proper sort means that they fit argumen-
tation schemes appropriate for the dialog. ‘Winning’ means producing an argument
that is strong enough to discharge the burden of persuasion set at the opening stage.
In a deliberation dialog, the goal is for the participants to arrive at a decision on
what to do, given the need to take action. McBurney, Hitchcock and Parsons [11]
set out a formal model of deliberation dialog in which participants make proposals
and counter-proposals on what to do. In this model (p. 95), the need to take action
is expressed in the form of a governing question like, “How should we respond
to the prospect of global warming?” Deliberation dialog may be contrasted with
persuasion dialog.
3 The notions of burden of persuasion and burden of proof have recently been subject to investiga-
tion [6, 13]. Here we have adopted the view that in a persuasion dialog, the burden of persuasion
is set at the opening stage, while a burden of proof can also shift from one side to the other during
the argumentation stage.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-28-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 15
In the model of [11, p. 100], a deliberation dialog consists of an opening stage, a
closing stage, and six other stages making up the argumentation stage.
Open: In this stage a governing question is raised about what is to be done. A gov-
erning question, like ‘Where shall we go for dinner this evening?,’ is a question
that expresses a need for action in a given set of circumstances.
Inform: This stage includes discussion of desirable goals, constraints on possible
actions that may be considered, evaluation of proposals, and consideration of
relevant facts.
Propose: Proposals cite possible action-options relevant to the governing question
Consider: this stage concerns commenting on proposals from various perspec-
tives.
Revise: goals, constraints, perspectives, and action-options can be revised in light
of comments presented and information gathering as well as fact-checking.
Recommend: an option for action can be recommended for acceptance or non-
acceptance by each participant.
Confirm: a participant can confirm acceptance of the recommended option, and
all participants must do so before the dialog terminates.
Close: The termination of the dialog.
The initial situation of deliberation is the need for action arising out of a choice
between two or more alternative courses of action that are possible in a given situ-
ation. The ultimate goal of deliberation dialog is for the participants to collectively
decide on what is the best available course of action for them to take. An important
property of deliberation dialog is that an action-option that is optimal for the group
considered as a whole may not be optimal from the perspective of an individual
participant [11, p. 98].
Both deliberation and persuasion dialogs can be about actions, and common
forms of argument like practical reasoning and argument from consequences are
often used in both types of dialog. There is no burden of persuasion in a delibera-
tion dialog. Argumentation in deliberation is primarily a matter of supporting one’s
own proposal for its chosen action-option, and critiquing the other party’s proposal
for its chosen action-option. At the concluding stage one’s proposal needs to be
abandoned in favor of the opposed one if the reasons given against it are strong
enough to show that the opposed proposal is better to solve the problem set at the
opening stage. Deliberation dialog is also different from negotiation dialog, which
centrally deals with competing interests set at the opening stage. In a deliberation
dialog, the participants evaluate proposed courses of action according to standards
that may often be contrary to their personal interests.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-29-320.jpg)
![16 Douglas Walton
8 Dialectical Shifts
In dialectical shifts of the kind analyzed in [18, pp. 100-116], an argument starts out
as being framed in one kind of dialog, but as the chain of argumentation proceeds,
it needs to be framed in a different type of dialog. Here is an example.
Example 1.2 ((The Dam)). In a debate in a legislative assembly the decision to be
made is whether to pass a bill to install a new dam. Arguments are put forward
by both sides. One side argues that such a dam will cost too much, and will have
bad ecological consequences. The other side argues that the dam is badly needed to
produce energy. A lot of facts about the specifics of the dam and the area around it
are needed to reasonably evaluate these opposed arguments. The assembly calls in
experts in hydraulics engineering, ecology, economics and agriculture, to testify on
these matters.
Once the testimony starts, there has been a dialectical shift from the original
deliberation dialog to an information-seeking dialog that goes into issues like what
the ecological consequences of installing the dam would be. But this shift is not
a bad thing, if the information provided by the testimony is helpful in aiding the
legislative assembly to arrive at an informed and intelligent decision on how to vote.
If this is so, the goal of the first dialog, the deliberation, is supported by the advent
of the second dialog, the information-seeking interval. A constructive type of shift
of this sort is classified as an embedding [18, p. 102], meaning that the advent of
the second dialog helps the first type of dialog along toward it goal. An embedding
underlies what can be called a constructive or licit shift.
Other dialectical shifts are illicit, meaning that the advent of the second dia-
log interferes with the proper progress of the first toward reaching its goal [18, p.
107]. Wells and Reed [19] constructed two formal dialectical systems to help judge
whether a dialectical shift from a persuasion dialog to a negotiation dialog is licit
or illicit. In their model, when a participant is engaged in a persuasion dialog, and
proposes to shift to a different type of dialog, he must make a request to ask if the
shift is acceptable to the other party. The other party has the option of insisting on
continuing with the initial dialog or agreeing to shift to the new type. Wells and
Reed have designed dialog rules to allow for a licit shift from persuasion to negoti-
ation. Their model is especially useful in studying cases where threats are used as
arguments. This type of argument, called the argumentum ad baculum in logic, has
traditionally been classified as a fallacy, presumably because making threat to the
other party is not a relevant move in a persuasion dialog. What one is supposed to
do in a persuasion dialog is to offer evidence to support one’s contention, and mak-
ing a threat does not do this, even though it may give the recipient of the threat a
prudential reason to at least appear to go along the claim that the other party wants
him to accept.
The study of dialectical shifts is important in the study of informal fallacies, or
common errors of reasoning, of a kind studied in logic textbooks since the time of
Aristotle. A good example is provided in the next section.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-30-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 17
9 Fallacious Arguments from Negative Consequences
Argument from consequences (argumentum ad consequentiam) is an interesting fal-
lacy that can be found in logic textbooks used to help students acquire critical think-
ing skills. The following example is quoted from Rescher [15, p. 82].
Example 1.3 ((The Mexican War)). The United States had justice on its side in wag-
ing the Mexican war of 1848. To question this is unpatriotic, and would give comfort
to our enemies by promoting the cause of defeatism.
The argument from consequences in this case was classified as a fallacy for the
reason that is not relevant to the issue supposedly being discussed. Rescher (p. 82)
wrote that “logically speaking,” it can be “entirely irrelevant that certain undesirable
consequences might derive from the rejection of a thesis, or certain benefits accrue
from its acceptance.” It can be conjectured from the example that the context is a
persuasion dialog in which the conflict of opinions is the issue of which country had
justice on its side in the Mexican war of 1848. This issue is a historical or ethical
one, and prudential deliberation about whether questioning whether the U.S. had
justice on its side would give comfort to our enemies is not relevant to resolving it.
We can analyze what has happened by saying that there has been a dialectical shift
at the point where the one side argues that questioning that the U.S. was in the right
would promote defeatism.
Notice that in this case there is nothing wrong in principle with using argumen-
tation from negative consequences. As shown above, argument from the negative
consequences is a legitimate argumentation scheme and any argument that fits this
scheme is a reasonable argument in the sense that if the premises are acceptable,
then subject to defeasible reasoning if new circumstances come to be known, the
conclusion is acceptable as well. It’s not the argument itself that is fallacious, or
structurally wrong as an inference. The problem is the context of dialog in which
these instances of argumentation from negative consequences have been used. Such
an argument would be perfectly appropriate if the issue set at the opening stage
was how to make a decision about how to best support the diplomatic interests of
the United States. However, notice that the first sentence of the example states very
clearly what the ultimate thesis to be proved is: “The United States had justice on
its side in waging the Mexican war of 1848.” The way that this thesis is supposed
to be proved is by giving the other side reasons to come to accept it is true. Hence
it seems reasonable to conjecture that the framework of the discussion is that of a
persuasion dialog.
Rescher (1969, 82) classified the Mexican War example as an instance of ar-
gument from negative consequences that commits a fallacy of relevance. But what
exactly is relevance? How is it to be defined? It can be defined by determining what
type of dialog an argument in a given case supposedly belongs to, and then deter-
mining what the issue to be resolved is by determining what the goal of the dialog is.
The goal is set at the opening stage. If during the argumentation stage, the argumen-
tation strays off onto a different path away from the proper kind of argumentation
needed to fulfill this goal, a fallacy of relevance may have been committed. Based](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-31-320.jpg)
![18 Douglas Walton
on this analysis, it can be said that a fallacy of relevance has been committed in
the Mexican War example. The dialectical shift to the prudential issue leads to a
different type of dialog, a deliberation that interferes with the progress of the origi-
nal persuasion dialog. The shift distracts the reader of the argument by introducing
another issue, whether arguing this way is unpatriotic, and would give comfort to
enemies by promoting the cause of defeatism. That may be more pressing, and it
may indeed be true that arguing in this way would have brought about the negative
consequences of giving comfort to enemies in promoting the cause of defeatism.
Still, even though this argument from negative consequences might be quite reason-
able, framed in the context of the deliberation, it is not useful to fulfill the burden of
persuasion necessary to resolve the original conflict of opinions.
10 Relevance and Fallacies
Many of the traditional informal fallacies in logic are classified under the heading
of fallacies of relevance [8]. In such cases, the argument may be a reasonable one
that is a valid inference based on premises that can be supported, but the problem is
that the argument is not relevant. One kind of fallacy of irrelevance, as shown in the
Mexican War example above, is the type of case where there has been a dialectical
shift from one type of dialog to another. However, there is also another type of
fallacy of relevance, where there is no dialectical shift, but there still is a failure
to fulfill the burden of persuasion. In this kind of fallacy, which is very common,
the arguer stays within the same type of dialog, but nevertheless fails to prove the
conclusion he is supposed to prove and instead goes off in a different direction.
The notion of relevance of argumentation can only be properly understood and
analyzed by drawing a distinction between the opening stage of a dialog, where the
burden of persuasion is set, and the argumentation stage, where arguments, linked
into chains of argumentation, are brought forward by both sides. In a persuasion
dialog, the burden of persuasion is set at the opening stage. Let’s say, for example,
that the issue being discussed is whether one type of light bulb lasts longer than an-
other. The proponent claims that one type of bulb lasts longer than another. She has
the burden of persuasion to prove that by bringing forward arguments that support
it. The respondent takes the stance of doubting the proponent’s claim. He does not
have the burden of persuasion. His role is to cast doubt on the proponent’s attempts
to prove her claim. He can do this by bringing forward arguments that attack the
claim that one type of bulb lasts longer than another. Suppose, however, that during
the argumentation stage, he wanders off to different topic by arguing that the one
type of bulb is manufactured by a company has done bad things that have led to neg-
ative consequences. This may be an emotionally exciting argument, and the claim
made in it may even be accurate, but the problem is that it is irrelevant to the issue
set at the opening stage.
This species of fallacy of relevance is called the red herring fallacy. It occurs
where an arguer wanders off the point in a discussion, and directs a chain of argu-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-32-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 19
mentation towards proving some conclusion other than the one he is supposed to
prove, as determined at the opening stage. The following example is a classic case
of this type of fallacy cited in logics textbook [8].
Example 1.4 ((The Light Bulb)). The Consumers Digest reports that GE light bulbs
last longer than Sylvania bulbs.4 But do you realize that GE is this country’s major
manufacturer of nuclear weapons? The social cost of GE’s irresponsible behavior
has been tremendous. Among other things, we are left with thousands of tons of
nuclear waste with nowhere to put it. Obviously, the Consumers Digest is wrong.
In the first sentence of the example, the arguer states the claim that he is supposed
to prove (or attack) as his ultimate probandum in the discussion. He is supposed to
be attacking the claim reported in the Consumers Digest that GE light bulbs last
longer than Sylvania bulbs. How does he do this? He launches into a chain of argu-
mentation, starting with the assertion that GE is this country’s major manufacturer
of nuclear weapons. This makes GE sound very bad, and it would be an emotionally
exciting issue to raise. He follows up the statement with another one to the effect
that the social cost of GE’s irresponsible behavior has been tremendous. This is an-
other serious allegation that would rouse the emotions of readers. Finally he uses
argumentation from negative consequences by asserting that because of GE’s irre-
sponsible behavior, we are left with thousands of tons of nuclear waste with nowhere
to put it. This line of argumentation is a colorful and accusatory distraction. It di-
verts the attention of the reader, who might easily fail to recall that the real issue is
whether GE light bulbs last longer than Sylvania bulbs. Nothing in all the allega-
tions made about GE’s allegedly responsible behavior carries any probative weight
for the purpose of providing evidence against the claim reported in the Consumers
Digest that GE light bulbs last longer than Sylvania bulbs.
In the red herring fallacy the argumentation is directed along a path of argumen-
tation other than one leading to proving the conclusion to be proved. The chain of
argumentation goes off in a direction that is exciting and distracting for the audience
to whom the argument was directed. The red herring fallacy becomes a problem
in cases where the argumentation moves away from the proper chain of argument
leading to the conclusion to be proved. Sometimes the path leads to the wrong con-
clusion (one other than the one that is supposed to be proved), but in other cases it
goes nowhere. The general pattern of this type of fallacy is displayed in Figure 1.7.
Such a distraction may be harmless if there is plenty of time for discussion. But
it can be a serious problem if there is not, because the real issue is not discussed.
According to the burden of persuasion, the line of argumentation has as its end point
a specific conclusion that needs to be proved. And if the argumentation moves away,
it may not do this.
4 In the 1994 edition (p. 127), the first sentence of the light bulb example is, “The Consumers
Digest reports that Sylvania light bulbs last longer than GE bulbs.” The example makes more sense
if the two light bulb manufacturers names are reversed, and so I have presented the light bulb
example this way.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-33-320.jpg)
![1 Argumentation Theory: A Very Short Introduction 21
arguer’s commitments are, and what should properly be taken to constitute common
knowledge, in specific cases where we are examining a text of discourse to find
implicit statements. While the field has helped to develop objective methods for
collecting evidence to deal with these problems in analyzing arguments, much work
remains to be done in making them more precise.
Of all the types of dialog, the one that has been most carefully and systematically
studied is persuasion dialog, and there are formal systems of persuasion dialog [12].
Just recently, deliberation dialog also has come to be formally modeled [11, p. 100].
There is an abundance of literature on negotiation, and there are a software tools for
assisting with negotiation argumentation. Comparatively less work is noticeable on
information-seeking dialog and on the inquiry model of dialog. There is a scattering
of work on eristic dialog, but there appears to be no formal model of this type of
dialog that has been put forward, or at least that is well known in the argumentation
literature. The notion of dialectical shift needs much more work. In particular, what
kinds of evidence are useful in helping an argument analyst to determine when a
dialectical shift has taken place during the sequence of argumentation in discourse
is a good topic for study.
The concepts of burden of proof and presumption are also central to the study
of argumentation in different types of dialog. Space has prevented much discussion
of these important topics, but the recent work that has been done on them raises
some general questions that would be good topics for research. This work [6, 13]
suggests that what is called the burden of persuasion is set at the opening stage of a
persuasion dialog, and that this burden affects how a different kind of burden, often
called the burden of production in law, shifts back and forth during the argumen-
tation stage. Drawing this distinction is extremely helpful for understanding how
a persuasion dialog should work, and more generally, it helps us to grasp how the
critical questions should work as attacks on an argumentation scheme. But is there
some comparable notion of burden of proof in the other types of dialog, for exam-
ple in deliberation dialog? This unanswered question points a direction for future
research in argumentation.
References
1. D. Walton, C. Reed and F. Macagno. Argumentation Schemes. Cambridge University Press,
Cambridge, UK, 2008.
2. T. J. M. Bench-Capon. Persuasion in practical argument using value-based argumentation
frameworks. Logic and Computation, 13:429–448, 2003.
3. T. J. M. Bench-Capon and P. E. Dunne. Argumentation and dialogue in artificial intelligence,
IJCAI 2005 tutorial notes. Technical report, Department of Computer Science, University of
Liverpool, 2005.
4. P. Besnard and A. Hunter. Elements of Argumentation. MIT Press, Cambridge MA, USA,
2008.
5. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic
reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–358,
1995.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-35-320.jpg)
![Chapter 2
Semantics of Abstract Argument Systems
Pietro Baroni and Massimiliano Giacomin
1 Abstract Argument Systems
An abstract argument system or argumentation framework, as introduced in a semi-
nal paper by Dung [13], is simply a pair A,R consisting of a set A whose elements
are called arguments and of a binary relation R on A called attack relation. The set
A may be finite or infinite in general, however, given the introductory purpose of this
chapter, we will restrict the presentation to the case of finite sets of arguments. An
argumentation framework has an obvious representation as a directed graph where
nodes are arguments and edges are drawn from attacking to attacked arguments. A
simple example of argumentation framework AF2.1 = {a,b},{(b,a)} is shown in
Figure 2.1.
While the word argument may recall several intuitive meanings, like the ones of
“line of reasoning leading from some premise to a conclusion” or of “utterance in
a dispute”, abstract argument systems are not (even implicitly or indirectly) bound
to any of them: an abstract argument is not assumed to have any specific structure
but, roughly speaking, an argument is anything that may attack or be attacked by
another argument. Accordingly, the argumentation framework depicted in Figure
2.1 is suitable to represent many different situations. For instance, in a context of
reasoning about weather, argument a may be associated with the inferential step
b a
Fig. 2.1 AF2.1: a simple argumentation framework
Pietro Baroni
Dip. di Elettronica per l’Automazione, University of Brescia, Via Branze 38, 25123 Brescia, Italy
e-mail: baroni@ing.unibs.it
Massimiliano Giacomin
Dip. di Elettronica per l’Automazione, University of Brescia, Via Branze 38, 25123 Brescia, Italy
e-mail: giacomin@ing.unibs.it
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 25
DOI 10.1007/978-0-387-98197-0 2, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-39-320.jpg)
![26 Pietro Baroni and Massimiliano Giacomin
“Tomorrow will rain because the national weather forecast says so”, while b with
“Tomorrow will not rain because the regional weather forecast says so”. In a legal
dispute a might be associated with a prosecutor’s statement “The suspect is guilty
because an eyewitness, Mr. Smith, says so” while b with the defense reply “Mr.
Smith is notoriously alcohol-addicted and it is proved that he was completely drunk
the night of the crime, therefore his testimony should not be taken into account”.
In a marriage arrangement setting (corresponding to the game-theoretic formulation
of stable marriage problem [13]), a may be associated with “The marriage between
Alice and John”, while b with “The marriage between Barbara and John”. Similarly,
the attack relation has no specific meaning: again at a rough level, if an argument
b attacks another argument a, this means that if b holds then a can not hold or,
putting it in other words, that the evaluation of the status of b affects the evaluation
of the status of a. In the weather example, two intrinsically conflicting conclusions
are confronted and the attack relation corresponds to the fact that one conclusion is
preferred to the other, e.g. because the regional forecast is considered more reliable
than the national one (this kind of attack is often called rebut in the literature). In
the legal dispute, the fact that Mr. Smith was drunk is not incompatible per se with
the fact that the suspect is guilty, however it affects the reason why s/he should be
believed to be guilty (this kind of attack has sometimes been called undercut in the
literature, but the use of this term has not been uniform). In the stable marriage
problem, the attack from b to a may simply be due to the fact that John prefers
Barbara to Alice.
Abstracting away from the structure and meaning of arguments and attacks en-
ables the study of properties which are independent of any specific aspect, and, as
such, are relevant to any context that can be captured by the very terse formalization
of abstract argument systems. As a counterpart of this generality, abstract argument
systems feature a limited expressiveness and can hardly be adopted as a modeling
formalism directly usable in application contexts. In fact the gap between a prac-
tical problem and its representation as an abstract argument system is patently too
wide and requires to be filled by a less abstract formalism, dealing in particular with
the construction of arguments and the conditions for an argument to attack another
one. For instance, in a given formalism, attack may be identified at a purely syntac-
tic level (e.g. the conclusion of an argument being the negation of the conclusion
of another one). In another formalism, attack may not be a purely syntactic notion
(e.g. the conclusions of two arguments being incompatible because they give rise to
contradiction through some strict logical deduction). Only after the attack relation
is defined in these “concrete” terms, it is appropriate to derive its abstract represen-
tation and exploit it for the analysis of general properties.
2 Abstract Argumentation Semantics
Given that arguments may attack each other, it is clearly the case that they can not
stand all together and their status is subject to evaluation. In particular what we are
usually interested in is the justification state of arguments: while this notion will](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-40-320.jpg)
![2 Semantics of Abstract Argument Systems 29
argumentation framework belongs to DS. Further, an important terminological con-
vention concerning the cardinality of the set of extensions (labellings) is worth intro-
ducing. If a semantics S always prescribes exactly one extension (labelling) for any
argumentation framework belonging to DS then S is said to belong to the unique-
status (or single-status) approach, otherwise it is said to belong to the multiple-status
approach.
While the adoption of the labelling or extension-based style is a matter of subjec-
tive preference by the proponent(s) of a given semantics, a natural question concerns
the expressiveness of the two styles. It is immediate to observe that any extension-
based definition can be equivalently expressed in a simple labelling-based formula-
tion, where a set of two labels is adopted (let say L = {in,out}) corresponding to
extension membership. On the other hand, an arbitrary labelling can not in general
be formulated in terms of extensions. It is however a matter of fact that labellings
considered in the literature typically include a label called in which naturally cor-
responds to extension membership, while other labels correspond to a partition of
other arguments, easily derivable from extension membership and the attack rela-
tion. As a consequence, equivalent extension-based formulations of labelling-based
semantics are typically available. Given this fact and the definitely prevailing cus-
tom of adopting the extension-based style in the literature, this chapter will focus on
extension-based semantics.
3 Principles for extension-based semantics
While alternative semantics proposals differ from each other by the specific notion
of extension they endorse, one may wonder whether there are reasonable general
properties which are shared by all (or, at least, most) existing semantics and can be
regarded as evaluation principles for new semantics. We discuss these principles in
the following, distinguishing between properties of individual extensions and prop-
erties of the whole set of extensions. The reader may refer to [2] for a more extensive
analysis of this issue.
The first basic requirement for any extension E corresponds to the idea that E is a
set of arguments which “can stand together”. Consequently, if an argument a attacks
another argument b, then a and b can not be included together in an extension. This
corresponds to the following conflict-free principle, which, as to our knowledge, is
satisfied by all existing semantics in the literature.
Definition 2.1. Given an argumentation framework AF = A,R, a set S ⊆ A is
conflict-free, denoted as cf(S), if and only if ∄a,b ∈ S such that aRb. A semantics
S satisfies the conflict-free principle if and only if ∀AF ∈ DS, ∀E ∈ ES(AF) E is
conflict-free.
A further requirement corresponds to the idea that an extension is a set of ar-
guments which “can stand on its own”, namely is able to withstand the attacks it
receives from other arguments by “replying” with other attacks. Formally, this has
a counterpart in the property of admissibility, which lies at the heart of all seman-
tics discussed in [13] and is shared by many more recent proposals. The definition](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-43-320.jpg)
![32 Pietro Baroni and Massimiliano Giacomin
In words, the intersection of any extension prescribed by S for AF with an
unattacked set S is equal to one of the extensions prescribed by S for the re-
striction of AF to S, and vice versa. Referring to the example of Figure 2.4,
U
S(AF2.4) = {{a,b},{a,b,c}}. Then, the restriction of AF2.4 to its unattacked set
{a,b} coincides with AF2.2 shown in Figure 2.2. A hypothetical semantics S1 such
that ES1 (AF2.4) = {{a,c},{b}} and ES1 (AF2.2) = {{a},{b}} would satisfy the di-
rectionality principle in this case. On the other hand, a hypothetical semantics S2
such that ES2 (AF2.4) = {{a,c}} and ES2 (AF2.2) = {{a},{b}} would not.
4 The notion of justification state
At a first level, the justification state of an argument a can be conceived in terms
of its extension membership. A basic classification encompasses only two possible
states for an argument, namely justified or not justified. In this respect, two alterna-
tive types of justification, namely skeptical and credulous can be considered.
Definition 2.12. Given a semantics S and an argumentation framework AF ∈ DS,
an argument a is:
• skeptically justified if and only if ∀E ∈ ES(AF) a ∈ E;
• credulously justified if and only if ∃E ∈ ES(AF) a ∈ E.
Clearly the two notions coincide for unique-status approaches, while, in general,
credulous justification includes skeptical justification. To refine this relatively rough
classification, a consolidated tradition considers three justification states.
Definition 2.13. [18] Given a semantics S and an argumentation framework AF ∈
DS, an argument a is:
• justified if and only if ∀E ∈ ES(AF), a ∈ E (this is clearly the “strongest” possible
level of justification, corresponding to skeptical justification);
• defensible if and only if ∃E1,E2 ∈ ES(AF) : a ∈ E1,a /
∈ E2 (this is a weaker
level of justification, corresponding to arguments which are credulously but not
skeptically justified);
• overruled if and only if ∀E ∈ ES(AF), a /
∈ E (in this case a can not be justified
in any way and should be rejected).
Some remarks are worth about the above classification, which is largely (and
sometimes implicitly) adopted in the literature: on one hand, while (two) differ-
ent levels of justified arguments are encompassed, no distinctions are drawn among
rejected arguments; on the other hand, the attack relation plays no role in the deriva-
tion of justification state from extensions. These points are addressed by a different
classification of justification states introduced by Pollock [16] in the context of a
unique-status approach. Again, three cases are possible for an argument a:
• a belongs to the (unique) extension E: then it is justified, or, using Pollock’s
terminology, undefeated;](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-46-320.jpg)
![2 Semantics of Abstract Argument Systems 33
• a does not belong to the (unique) extension E and is attacked by (some member
of) E: then, using Pollock’s terminology, it is defeated outright, corresponding to
a strong form of rejection;
• a does not belong to the (unique) extension E but does not receive attacks from
E: then it is provisionally defeated, corresponding to a weaker form of rejection.
Both these classifications of justification states are unsatisfactory in some respect.
It is however possible to combine the intuitions underlying both of them, obtaining
a systematic classification of seven possible justification states [6]. As a starting
point, considering the relationship between an argument a and a specific extension
E, three main situations3, as in Pollock’s classification, can be envisaged:
• a is in E, denoted as in(a,E), if a ∈ E;
• a is definitely out from E, denoted as do(a,E), if a /
∈ E ∧ERa;
• a is provisionally out from E, denoted as po(a,E), if a /
∈ E ∧¬ERa.
Then, taking into account the possible existence of multiple extensions, an argu-
ment can be in any of the above three states with respect to all, some or none of
the extensions. This gives rise to 27 hypothetical combinations. It is however easy
to see that some of them are impossible: for instance, if an argument is in a given
state with respect to all extensions this clearly excludes that it is in another state
with respect to any extension. Directly applying this kind of considerations, seven
possible justification states emerge.
Definition 2.14. Given an argumentation framework AF = A,R and a non-empty
set of extensions E the possible justification states of an argument a ∈ A according
to E are defined by the following mutually exclusive conditions:
• ∀E ∈ E in(a,E), denoted as JSI;
• ∀E ∈ E do(a,E), denoted as JSD;
• ∀E ∈ E po(a,E), denoted as JSP;
• ∃E ∈ E : do(a,E), ∃E ∈ E : po(a,E), and ∄E ∈ E : in(a,E), denoted as JSDP;
• ∃E ∈ E : in(a,E), ∃E ∈ E : po(a,E), and ∄E ∈ E : do(a,E), denoted as JSIP;
• ∃E ∈ E : in(a,E), ∃E ∈ E : do(a,E), and ∄E ∈ E : po(a,E), denoted as JSID;
• ∃E ∈ E : in(a,E), ∃E ∈ E : do(a,E), and ∃E ∈ E : po(a,E), denoted as JSIDP.
Correspondences with “traditional” definitions of justification states are easily
drawn. An argument is skeptically justified if and only if it is in the JSI state, while
credulous justification corresponds to the disjunction of the states JSI, JSIP, JSID,
and JSIDP. As to Definition 2.13, the state of justified corresponds to JSI, the state of
overruled to the disjunction of JSD, JSP, and JSDP, while the state of defensible to
the disjunction of JSIP, JSID, and JSIDP. Turning to Pollock’s classification, it is easy
to see that in the case of a unique-status semantics only JSI, JSD and JSP may hold,
which correspond to the state of undefeated, defeated outright and provisionally
defeated, respectively. Other meaningful ways of defining aggregated justification
states are investigated in [6].
3 The case a ∈ E ∧ERa is prevented by the conflict-free principle.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-47-320.jpg)
![34 Pietro Baroni and Massimiliano Giacomin
5 A review of extension-based argumentation semantics
Turning from general notions to actual approaches, we now examine several argu-
mentation semantics proposed in the literature. From a historical point of view, it is
possible to distinguish between:
• four “traditional” semantics, considered in Dung’s original paper [13], namely
complete, grounded, stable, and preferred semantics;
• subsequent proposals introduced by various authors in the literature, often to
overcome some limitation or improve some undesired behavior of a traditional
approach: we consider stage, semi-stable, ideal, CF2, and prudent semantics.
It is important to note that while the definitions of the semantics we will de-
scribe are formulated in the context of purely abstract argumentation frameworks,
the underlying intuitions have commonalities with other (somewhat more concrete)
formalizations in related contexts. In fact, as already mentioned, one of the main
results of Dung’s paper is showing that abstract argumentation frameworks are able
to capture the properties of a large variety of more specific formalisms. This means
that it is possible to define mappings from entities defined in a more specific formal-
ism into an argumentation framework. In [13] mappings of this kind are provided for
stable marriage problems, default theories in Reiter’s default logic, logic programs
with negation as failure, and Pollock’s theory of defeasible reasoning.
Let AF = A,R be an argumentation framework obtained through a mapping
from a more specific formalism (e.g. a logic program). We have now two ways of
deriving a “meaningful” set of subsets of A: on one hand, we can apply a purely
abstract semantics S to AF, obtaining the relevant set of extensions ES(AF). On the
other hand, we can start from a meaningful concept in the underlying formalism
(e.g. the set of models of a logic program) and then map it into a set M of subsets
of A (continuing the example, by deriving the set of arguments corresponding to
each model). An important question then arises: can interesting relationships be
identified between ES(AF) and M, given a suitable choice of the semantics S? A
strikingly affirmative answer is provided in [13]: it is shown that properly selecting
S among the traditional grounded, preferred, and stable semantics one obtains sets
of extensions ES(AF) which coincide with the sets of arguments corresponding to
meaningful notions in the formalisms mentioned above. Specific indications about
these coincidences will be given in the review of individual semantics. At a general
level, they confirm that abstract argumentation semantics is a powerful analysis tool,
able to focus on essential properties of a variety of formalisms and to shed light on
their (possibly hidden) significant common features.
5.1 Complete semantics
We start our review by the notion of complete extension, as it lies at the heart of all
traditional Dung’s semantics. Actually the notion of complete extension is not asso-
ciated with a notion of complete semantics in [13], but the term complete semantics
has subsequently gained acceptance in the literature and will be used in the present](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-48-320.jpg)
![36 Pietro Baroni and Massimiliano Giacomin
[2]. As the examples above abundantly show, CO does not satisfy I-maximality,
since a complete extension E1 may be a proper subset of a complete extension E2.
5.2 Grounded semantics
Grounded semantics, denoted as GR, is a traditional unique-status approach whose
formulation in argumentation-based reasoning (see e.g. the one proposed by Pollock
in [16]) predates the following quite technical definition given in [13].
Definition 2.16. The grounded extension of an argumentation framework AF, de-
noted as GE(AF), is the least fixed point of its characteristic function FAF .
The underlying and preexisting informal intuition is however rather simple and
can be directly put in relationship with some notions we already discussed in the
context of complete semantics. The basic idea is that the (unique) grounded exten-
sion can be built incrementally starting from the initial unattacked arguments. Then
the arguments attacked by them can be suppressed, resulting in a modified argumen-
tation framework where, possibly, the set of initial arguments is larger. In turn the
arguments attacked by the “new” initial arguments can be suppressed, and so on.
The process stops when no new initial arguments arise after a deletion step: the set
of all initial arguments identified so far is the grounded extension. An example with
a graphical illustration of this incremental process is given in Figure 2.6, resulting
in EGR(AF) = {{a,c}}. To put it in other words, the grounded extension includes
those and only those arguments whose defense is “rooted” in initial arguments (see
[2] for a formal treatment of this notion, called strong defense). If there are no initial
arguments the grounded extension is the empty set.
The reader should have noticed that the above construction corresponds to the
iterated application of FAF (/
0) already met in previous section: the set of arguments
obtained up to each step i above coincides with Fi
AF (/
0) and the process stops when
Fi
AF (/
0) = Fi+1
AF (/
0) (i.e. a fixed point of FAF is reached). As a counterpart to this
intuitive correspondence, it is proved in [13] that:
i) the grounded extension is the least (wrt. set inclusion) complete extension: for
any AF GE(AF) ∈ ECO(AF) and ∀E ∈ ECO(AF) GE(AF) ⊆ E;
ii) for any finite (and, more generally, finitary [13]) AF GE(AF) = ∪i=1,...,∞Fi
AF (/
0).
Given the above explanations it should be now immediate to see that GE(AF2.1) =
{b}, GE(AF2.3) = {b,c}, GE(AF2.2) = GE(AF2.4) = GE(AF2.5) = /
0.
c d
b f
a e
c d f
a e
c f
a e
Fig. 2.6 AF2.6: an example to illustrate grounded semantics](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-50-320.jpg)
![2 Semantics of Abstract Argument Systems 37
Besides the relationship with Pollock’s approach [16], the grounded extension
has been put in correspondence in [13] with the well-founded semantics of logic pro-
grams [20]. Turning to general semantics properties it is immediate to see that GR
is universally defined and satisfies admissibility, reinstatement and I-maximality. It
is proved in [2] that GR also satisfies directionality.
5.3 Stable semantics
Stable semantics, denoted as S
T, relies on a very simple (and easy to formalize)
intuition: an extension should be able to attack all arguments not included in it. This
leads to the notion of stable extension [13].
Definition 2.17. Given an argumentation framework AF = A,R, a set E ⊆ A is a
stable extension of AF if and only if E is conflict-free and ∀a ∈ A, a /
∈ E ⇒ ERa.
By definition, any stable extension E is also a complete extension (thus in par-
ticular GE(AF) ⊆ E) and a maximal conflict-free set of AF. Referring to the ex-
amples seen so far, identifying stable extensions is straightforward: ES
T(AF2.1) =
{b}, ES
T(AF2.2) = {{a},{b}}, ES
T(AF2.3) = {b,c}, ES
T(AF2.4) = {{a,c},{b}},
ES
T(AF2.5) = {{a,c},{a,d},{b,d}}, ES
T(AF2.6) = {{a,c,e},{a,c, f}}.
The simple intuition underlying stable semantics has significant counterparts in
several contexts: it is proved in [13] that stable extensions can be put in correspon-
dence with solutions of cooperative n-person games, solutions of the stable mar-
riage problem, extensions of Reiter’s default logic [19], and stable models of logic
programs [15]. Stable semantics however has also a significant drawback: it is not
universally defined as there are argumentation frameworks where no stable exten-
sions exist. A simple example is provided in Figure 2.7: no conflict-free set is able
to attack all other arguments in this case. While it has sometimes been claimed
by supporters of stable semantics that situations where stable extensions do not
exist are “pathological” in some sense, it has been shown in [13] that perfectly
reasonable problems may be formalized with argumentation frameworks such that
ES
T(AF) = /
0.
As to general properties, it is easy to see that I-maximality is enforced by defi-
nition and, since stable extensions are a subset of complete extensions, also admis-
sibility and reinstatement are satisfied. S
T is not directional, due to the fact that it is
not universally defined. To see this, consider the example in Figure 2.8: we have that
ES
T(AF2.8) = {{b}}, however, considering the unattacked set S = {a,b}, we have
ES
T(AF2.8↓S) = {{a},{b}}. Therefore A
ES
T(AF2.8,S) = {{b}} = ES
T(AF2.8↓S).
a b
c
Fig. 2.7 AF2.7: a three-length cycle](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-51-320.jpg)
![38 Pietro Baroni and Massimiliano Giacomin
5.4 Preferred semantics
The “aggressive” requirement that an extension must attack anything outside it may
be relaxed by requiring that an extension is as large as possible and able to defend
itself from attacks. This is captured by the notion of preferred extension [13].
Definition 2.18. Given an argumentation framework AF = A,R, a set E ⊆ A is a
preferred extension of AF if and only if E is a maximal (wrt. set inclusion) element
of A
S(AF).
According to this definition every preferred extension E is also a complete ex-
tension (entailing that GE(AF) ⊆ E); indeed, preferred extensions may be equiv-
alently defined as maximal complete extensions. It follows that preferred seman-
tics, denoted as PR, is universally defined (as complete semantics is) and that,
taking into account the treatment of the examples in Section 5.1, EPR(AF2.1) =
{{b}}, EPR(AF2.2)={{a},{b}},EPR(AF2.3)={{b,c}},EPR(AF2.4)={{a,c},{b}},
EPR(AF2.5) = {{a,c},{a,d},{b,d}}, and EPR(AF2.6) = {{a,c,e},{a,c, f}}. The
reader may notice that in all these examples the set of preferred extensions coincides
with the set of stable extensions. In general any stable extension is also a preferred
extension but not vice versa. In the example of Figure 2.7 we have EPR(AF2.7) = {/
0}
while ES
T(AF2.7) = /
0. In the example of Figure 2.8 there are two preferred exten-
sions, namely {a}, {b}, but only one of them, namely {b}, is also stable.
The intuition underlying preferred semantics has a correspondence with pref-
erential semantics of logic programs [12]. As to general semantics properties it is
immediate to see that PR satisfies I-maximality, admissibility, reinstatement, while
it is proved in [2] that it is directional. Given these facts, PR has often been regarded
as the most satisfactory semantics in the context of Dung’s framework.
5.5 Stage and semi-stable semantics
Stage [21] and semi-stable [8] semantics, denoted respectively as S
TA and S
S
T, are
based on the idea of prescribing the maximization not only of the arguments in-
cluded in an extension but also of those attacked by it.
Definition 2.19. Given an argumentation framework AF = A,R and a set E ⊆ A
the range of E is defined as E ∪ E+, where E+ {a ∈ A : ERa}. E is a stage
extension if and only if E is a conflict-free set with maximal (wrt. set inclusion)
range. E is a semi-stable extension if and only if E is a complete extension with
maximal (wrt. set inclusion) range.
As evident from Definition 2.19, the two semantics differ in the requirement on
the sets whose range is maximal: being conflict-free for stage semantics, complete
b a
c
Fig. 2.8 AF2.8: S
T is not directional](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-52-320.jpg)
![2 Semantics of Abstract Argument Systems 39
extensions (or equivalently, admissible sets) for semi-stable semantics. Both S
TA
and S
S
T are clearly universally defined (differently from stable semantics), while co-
inciding with stable semantics (differently from preferred semantics) when stable
extensions exist. In fact, for any stable extension E it holds that E ∪ E+ = A. It
follows that any complete extension (conflict-free set) which is not stable does not
satisfy the range maximization requirement for argumentation frameworks where
stable extensions exist, hence the coincidence of S
T and S
S
T (S
TA) in these cases.
If stable extensions do not exist, S
S
T selects anyway as extensions some complete
extensions and the maximization requirement restricts the choice to preferred ex-
tensions: ∀AF ES
S
T(AF) ⊆ EPR(AF). On the other hand, S
TA does not necessarily
select admissible sets as extensions. Figure 2.9 shows an argumentation framework
where S
S
T and S
TA agree and do not coincide neither with S
T nor with PR: in fact
ES
T(AF2.9) = /
0, EPR(AF2.9) = {{a},{b}}, ES
S
T(AF2.9) = ES
TA(AF2.9) = {{b}}. On
the other hand, in the case of Figure 2.7 EPR(AF2.7) = ES
S
T(AF2.7) = {/
0} while
ES
TA(AF2.7) = {{a},{b},{c}}.
It is easy to see that both S
S
T and S
TA satisfy I-maximality, while only S
S
T satisfies
admissibility and reinstatement. S
S
T and S
TA do not satisfy directionality, as S
T does
not.
5.6 Ideal semantics
Ideal semantics [14], denoted as I
D, provides a unique-status approach allowing the
acceptance of a set of arguments possibly larger than in the case of GR.
Definition 2.20. Given an argumentation framework AF = A,R a set S ⊆ A is
ideal if and only if S is admissible and ∀E ∈ EPR(AF) S ⊆ E. The ideal extension,
denoted as ID(AF), is the maximal (wrt. set inclusion) ideal set.
The definition of ideal set prescribes admissibility and skeptical justification un-
der preferred semantics. From Section 5.2 we know that the grounded extension sat-
isfies both requirements and is therefore an ideal set (this in particular implies that
ideal semantics is universally defined). Ideal sets strictly larger than the grounded
extension may exist, as shown by the example in Figure 2.10 where it holds that
EPR(AF2.10) = {{a,d},{b,d}}: in this case the grounded extension is empty while
ID(AF2.10) = {d}.
By definition I
D satisfies I-maximality and admissibility. It is proved in [2] that
I
D also satisfies reinstatement and directionality.
d
b a
c
Fig. 2.9 AF2.9: S
S
T may not coincide with P
R or S
T](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-53-320.jpg)
![40 Pietro Baroni and Massimiliano Giacomin
5.7 CF2 semantics
CF2 semantics is defined in the frame of the SCC-recursive scheme [7] which is a
general pattern for the definition of argumentation semantics based on the graph-
theoretical notion of strongly connected components (SCCs) of an argumentation
framework, i.e. the equivalence classes of arguments under the relation of mutual
reachability via attack links. CF2 semantics can be roughly regarded as selecting
its extensions among the maximal conflict-free sets of AF, on the basis of some
topological requirements related to the decomposition of AF into strongly connected
components. Examining in detail the definition of CF2 semantics is beyond the
scope of this chapter: the interested reader may refer to [7].
Definition 2.21. Given an argumentation framework AF = A,R, a set E ⊆ A is
an extension of CF2 semantics, i.e. E ∈ ECF2(AF), if and only if
• E ∈ MCF(AF) if |SCCSAF | = 1
• ∀S ∈ SCCSAF (E ∩S) ∈ ECF2(AF↓UPAF (S,E)) otherwise
where MCF(AF) denotes the set of maximal conflict-free sets of AF, SCCSAF
denotes the set of strongly connected components of AF, and, for any E,S ⊆ A,
UPAF (S,E) = {a ∈ S | ∄b ∈ E : b /
∈ S,bRa}.
The underlying idea consists in relying only on I-maximality and the conflict-
free principle within a unique strongly connected component S, where all argu-
ments (if more than one) both receive and deliver at least an attack. In particular
it turns out that when AF consists of exactly one strongly connected component,
the set of extensions prescribed by CF2 semantics exactly coincides with the set of
maximal conflict-free sets of AF. This yields a uniform multiple-status treatment of
cycles, which is not achieved by other semantics. In fact, it is easy to see that for
any argumentation framework AF consisting of an even-length attack cycle it holds
that ES
T(AF) = EPR(AF) = ES
S
T(AF) = {/
0}. For instance we already know that
ES
T(AF2.2) = EPR(AF2.2) = ES
S
T(AF2.2) = MCF(AF2.2) = {{a},{b}}. Similarly, in
the case of a four-length cycle two extensions arise, each consisting of two argu-
ments. On the other hand, for any argumentation framework AF consisting of an
b
a
c d
Fig. 2.10 AF2.10: an example to illustrate ideal semantics
a
b
c e
d
Fig. 2.11 AF2.11: an example to illustrate CF2 semantics](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-54-320.jpg)
![2 Semantics of Abstract Argument Systems 41
odd-length attack cycle it holds that ES
T(AF) = /
0 and EPR(AF) = ES
S
T(AF) = {/
0}.
This disuniformity in the treatment of cycles gives rise to counterintuitive behav-
iors in some simple examples and has therefore been regarded as problematic in
the literature [17]. Clearly CF2 semantics is able to overcome this limitation since
ECF2(AF) = MCF(AF) for any argumentation framework AF consisting of an at-
tack cycle, independently of its length. For instance, ECF2(AF2.7) = {{a},{b},{c}}.
In a generic argumentation framework, CF2 extensions may be computed fol-
lowing the decomposition of AF into strongly connected components, which, due
to a well-known graph-theoretical property, can be partially ordered according to the
attack relation. Consider the example in Figure 2.11, which consists of two SCCs,
namely S1 = {a,b,c} and S2 = {d,e}. According to Definition 2.21 it must be the
case that E ∩ S1 is a maximal conflict-free set of AF2.11↓{a,b,c}. Thus we get three
possible starting points for the construction of extensions, namely {a}, {b}, {c}.
It has then to be noted that in any of these three cases d is attacked. Therefore in
all cases UPAF (S2,E) consists of the only argument e, yielding E ∩S2 = {e} as the
only possibility. In summary ECF2(AF2.11) = {{a,e},{b,e},{c,e}}. It can be seen
that in all other examples considered so far CF2 extensions coincide with preferred
extensions.
As to general properties, CF2 semantics is I-maximal, since its extensions are
maximal conflict-free sets of AF, directional [2], and universally defined, since, for
any AF, MCF(AF) = /
0. On the other hand, the “desired” treatment of odd-length
cycles entails that CF2 semantics gives up the traditional properties of admissibil-
ity and reinstatement as shown, for instance, by the example of Figure 2.7 where
no extension is an admissible set and all violate the reinstatement property. It is
proven however in [2] that the departure from the notion of reinstatement is not
radical, since CF2 semantics satisfies two weaker versions of this property called
CF-reinstatement and weak reinstatement. As another confirmation that CF2 se-
mantics has significant relationships with traditional semantics it can be recalled
that any preferred extension is included in a CF2 extension, any stable extension is
a CF2 extension and the grounded extension is included in any CF2 extension.
5.8 Prudent semantics
The family of prudent semantics [10] is introduced by considering a more extensive
notion of attack in the context of traditional semantics. In particular, an argument a
indirectly attacks an argument b if there is an odd-length attack path from a to b.
For instance, in Figure 2.5 a indirectly attacks d, while in Figure 2.6 a indirectly
attacks d and f, b indirectly attacks e, and c indirectly attacks f. The odd-length
path need not be the shortest path and may include cycles: in Figure 2.10 both a
and b indirectly attack d, while in Figure 2.11 a indirectly attacks e. A set S of
arguments is free of indirect conflicts, denoted as icf(S), if ∄a,b ∈ S such that a
indirectly attacks b. The prudent version of the admissibility property and of several
traditional notions of extension can then be defined.
Definition 2.22. Given an argumentation framework AF = A,R, a set of argu-
ments S ⊆ A is p(rudent)-admissible if and only if ∀a ∈ S a is acceptable wrt. S](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-55-320.jpg)
![2 Semantics of Abstract Argument Systems 43
frameworks, there are also argumentation frameworks where the sets of extensions
prescribed by different semantics coincide, as evident from several of the examples
seen above. Topological properties of argumentation frameworks providing suffi-
cient conditions for semantics agreement have been investigated in Dung’s original
paper: for instance the absence of attack cycles is sufficient to ensure agreement
among grounded, stable and preferred semantics, while the absence of odd-length
attack cycles is sufficient to ensure agreement between stable and preferred seman-
tics. Subsequent results concerning topological families of argumentation frame-
works relevant to agreement properties are reported in [11, 1], while a systematic
set-theoretical analysis of agreement classes is provided in [5].
As to the notion of skepticism, it has often been used in informal ways to dis-
cuss semantics behavior, e.g. by observing that a semantics is “more skeptical” than
another one. Intuitively, a semantics is more skeptical than another if it makes less
committed choices about the justification of the arguments: a skeptical behavior
tends to leave arguments in an “undecided” justification state, while a non-skeptical
behavior corresponds to more “resolute” choices about acceptance or rejection of
arguments. The issue of formalizing skepticism relations between sets of extension
in terms of set theoretical properties and then comparing semantics according to
skepticism has been first addressed in [6] and subsequently developed in [4].
Turning finally back to general properties of argumentation semantics, some of
them, like admissibility and reinstatement, have regularly been considered in the lit-
erature [18], while the task of systematically defining a set of criteria for semantics
evaluation and comparison has been undertaken only recently. Besides the princi-
ples we have explicitly discussed in Section 3, two families of adequacy properties
(based on skepticism relations) are introduced in [2], where it is shown that, consid-
ering also these properties, none of the literature semantics discussed in this chapter
is able to comply with all the desirable criteria. A novel semantics able to satisfy all
of them has been proposed in [3]. At a different abstraction level, where argument
structure and construction are explicitly dealt with, general rationality postulates for
argumentation systems have been introduced in [9]. Exploring the definition of gen-
eral principles for argumentation at different abstraction levels, investigating their
relationships and analyzing their suitability for different application domains appear
to be open and fruitful research directions. A related research issue concerns the
identification of a generic definition scheme able to encompass into a unifying view
a large variety of semantics. In this perspective, it is shown in [7] that all traditional
semantics adhere to the parametric SCC-recursive scheme, where they differ sim-
ply by a base function which only specifies the sets of extensions of argumentation
frameworks consisting of a single SCC.
The above mentioned results suggest that though there is a wide corpus of lit-
erature on abstract argumentation semantics, providing a rich variety of alternative
approaches, the field is far from being “mature” and there is still large room for
investigating both fundamental theoretical issues and their potential impact on prac-
tical applications.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-57-320.jpg)
![Chapter 3
Abstract Argumentation and Values
Trevor Bench-Capon and Katie Atkinson
1 Introduction
Abstract argumentation frameworks, as described in Chapter 2, are directed towards
determining whether a claim that some statement is true can be coherently main-
tained in the context of a set of conflicting arguments. For example, if we use
preferred semantics, that an argument is a member of all preferred extensions es-
tablishes that its claim must be accepted as true, and membership of at least one
preferred extension shows that the claim is at least tenable. In consequence, that
admissible sets of arguments are conflict free is an important requirement under all
the various semantics.
For many common cases of argument, however, this is not appropriate: two ar-
guments can conflict, and yet both be accepted. For an example suppose that Trevor
and Katie need to travel to Paris for a conference. Trevor offers the argument “we
should travel by plane because it is quickest”. Katie replies with the argument “we
should travel by train because it is much pleasanter”. Trevor and Katie may con-
tinue to disagree as to how to travel, but they cannot deny each other’s arguments.
The conclusion will be something like “we should travel by train because it is much
pleasanter, even though travelling by plane is quicker”. The point concerns what
Searle [24] calls direction of fit. For matters of truth and falsity, we are trying to
fit what we believe to the way the world actually is. In contrast, when we consider
what we should do we are trying to fit the world to the way we would like it to
be. Moreover, because people may have different preferences, values, interests and
aspirations, people may rationally choose different options: if Katie prefers comfort
Trevor Bench-Capon
Department of Computer Science, University of Liverpool, UK, e-mail: tbc@liverpool.ac.
uk
Katie Atkinson
Department of Computer Science, University of Liverpool, UK, e-mail: katie@liverpool.
ac.uk
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 45
DOI 10.1007/978-0-387-98197-0 3, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-59-320.jpg)
![46 Trevor Bench-Capon and Katie Atkinson
to speed she will rationally choose the train, but this does not mean that Trevor can-
not rationally choose the plane if he prefers speed to comfort. We will return to this
example throughout this chapter.
Within standard abstract argumentation frameworks one approach to recognising
the importance of direction of fit [22] is to require sceptical acceptance for epis-
temic arguments but only credulous acceptance for practical arguments. This does
successfully model the existence of a choice with respect to practical arguments,
but it does not motivate the choice, nor does it allow us to predict choices on the
basis of choices made in the past. Value based argumentation frameworks (VAFs)
[6], described in this chapter, are an attempt to address issues about the rational
justification of choices systematically.
Value based justification of choices is common in many important areas: in poli-
tics where specific policies are typically justified in terms of the values they promote,
and where politicians’ values are advanced as reasons to vote for them; in law, where
differences in legal jurisdictions and decisions over time can be explained in terms
of the values of the societies in which the judgements are made [11]; in matters
of morality where individual and group ethical perspectives play a crucial role in
reasoning and action [2]; as well as more everyday examples, such as given above.
In this chapter we will first give some philosophical background, in particular
introducing the notion of audience, and some of the features that we require from
practical reasoning. Section 3 will discuss the nature of values in more detail, in
particular the distinction between values and goals. Section 4 will introduce the
formal machinery of Value Based Argumentation Frameworks, and discuss some
of their more important properties. Section 5 describes some applications of value
based argumentation. Section 6 discusses some recent developments, and section 7
concludes the chapter with a summary.
2 Audiences
One of the first people to stress the importance of the audience in determining
whether an argument is persuasive or not was Chaim Perelman [20], [19]:
“If men oppose each other concerning a decision to be taken, it is not because they commit
some error of logic or calculation. They discuss apropos the applicable rule, the ends to
be considered, the meaning to be given to values, the interpretation and characterisation of
facts.” [[19] p.150, italics ours].
A similar point was made by John Searle [24]:
“Assume universally valid and accepted standards of rationality, assume perfectly rational
agents operating with perfect information, and you will find that rational disagreement will
still occur; because, for example, the rational agents are likely to have different and incon-
sistent values and interests, each of which may be rationally acceptable.” [[24], xv]
Both Perelman and Searle recognise that there may be complete agreement on
facts, logic, which arguments are valid, which arguments attack one another and the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-60-320.jpg)
![48 Trevor Bench-Capon and Katie Atkinson
founded. Values are used in the sense of fundamental social or personal goods that
are desirable in themselves, and should never be confused with any numeric measure
of the strength, certainty or probability of an argument. Liberty, Equality and Fra-
ternity, the values of the French Revolution, are paradigmatic examples of values.
Values are widely recognised as the basis for persuasive argument. For example, the
National Forensic League, which conducts debating competitions throughout the
USA uses the “Lincoln-Douglas” (LD) debate format which is based on the notion
of a clash of values. In an LD debate the resolution forces each side to take on com-
peting values and argue about which one is supreme. For example, if the resolution
is, “Resolved: An oppressive government is better than no government at all,” the
affirmative side might value “order” and the negative side might value “freedom”.
Such a debate would revolve around whether order is more valuable than freedom.
In the original debate between Abraham Lincoln and Stephen Douglas on which LD
debates are based, Douglas championed the rights of states to legislate for their par-
ticular circumstances, whereas Lincoln argued on the basis that there were certain
inviolable human rights that all states had to respect, even though this constrained
state autonomy.
But what is the role of values in practical reasoning? Historically the basis for
treatments of practical reasoning has been the practical syllogism, first discussed by
Aristotle. A standard modern statement is given in [16]:
K1 I’m to be in London at 4.15
If I catch the 2.30, I’ll be in London at 4.15
So, I’ll catch the 2.30.
The first premise is a statement of some desired state of affairs, the second an
action which would bring about that state of affairs, and the conclusion is that the
action should be performed. There are, however, problems with the formulation: it is
abductive rather than deductive, does not consider alternative, possibly better, ways
of achieving the desired state of affairs, or possibly undesirable side effects of the
action. Walton [26] addresses these issues by regarding the practical syllogism as an
argumentation scheme, which he calls the sufficient condition scheme for practical
reasoning, which provides a presumptive reason to perform the action, but which
can be critiqued on the basis of alternatives and undesirable consequences. He states
the sufficient condition practical reasoning scheme as:
W2 G is a goal for agent a
Doing action A is sufficient for agent a to carry out goal G
Therefore agent a ought to do action A.
This, however, still does not explain why G is a goal for the agent, nor indi-
cate how important bringing about G is to the agent. Neither K1 nor W1 make any
mention of values: rather that the agent has certain values is implicit in calling the
desired state of affairs a “goal” for that agent. Accordingly, to make this role of
values explicit, Walton’s scheme was developed by Atkinson and her colleagues [4]
into the more elaborated scheme:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-62-320.jpg)
![3 Abstract Argumentation and Values 49
A1 In the circumstances R, we should perform action A
to achieve new circumstances S, which will realise some goal G
which will promote some value V.
What this scheme does in particular is to distinguish three aspects which are
conflated into the notion of goal in K1 and W1. These aspects are: the state of affairs
which will result from the action; the goal, which is those aspects of the new state
of affairs for the sake of which the action is performed; and the value, which is the
reason why the agent desires the goal. Making these distinctions opens up several
distinct types of alternative to the recommended action. We may perform a different
action to realise the same state of affairs; we may act so as to bring about a different
state of affairs which realises the same goal; or we may realise a different goal which
promotes the same value. Alternatively, since the state of affairs potentially realises
several goals, we can justify the action in terms of promoting a different value. In
coming to agreement this last possibility may be of particular importance: we may
want to promote different values, and so agree to perform the action on the basis of
different arguments. Our contention is that, in the spirit of the notion of audience
developed in section 2, what is important, what is the appropriate comparison for
choosing between alternatives, is the value.
In order to see the distinction between a goal and a value, consider again Trevor
and Katie’s journey to Paris. The goal is to be in Paris for the conference, and this
is not in dispute: the dispute is how that goal should be realised and turns on the
values promoted by the different methods of travel. What is important is not the
state reached, but the way in which the transition is made.
The style of argumentation represented by A1 has been formalised in [1] in terms
of a particular style of transition system, Alternating Action Based Transition sys-
tems (AATS) [27]. An AATS consists of a set of states and a set of agents and the
transitions between the states are in terms of the joint actions of the agents, that is,
actions composed from the actions available to the agents individually. In terms of
A1 the circumstances R and S are represented by the states of the system, the goal
G is realised if G holds in S (of course, G may hold in several of the states), and
the action is the particular agent’s component of a joint action which is a transition
from R to S. The value labels the transition, indicating that it is the movement from
R to S using that particular transition that promotes the value. A fragment of the
AATS for Trevor and Katie’s travel dilemma is shown in Figure 3.1, t/kt/p is the
action of Trevor/Katie travelling by train/plane, C/St/k means that Comfort/Speed is
promoted in respect of Trevor/Katie, 00 that both are in Liverpool and 11 that both
are in Paris.
00 11
ttkt +Ct +Ck
tpkt +St +Ck
tpkp +St +Sk
ttkp +Ct +Sk
Fig. 3.1 AATS for travel to Paris example](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-63-320.jpg)
![50 Trevor Bench-Capon and Katie Atkinson
Although there is only one destination state, each of the four potential ways of
reaching it promotes different values, and hence give rise to different arguments in
their favour. Which arguments will succeed will depend on the preferences between
the values of Comfort and Speed of the two agents concerned.
Essentially then, in this problem there will be a number of possible audiences,
depending on how the values are ordered. Suppose that Trevor values his own speed
over his own comfort and Katie her comfort over her speed, and that neither consider
values promoted in respect of the other. Then Trevor will choose to go by plane and
Katie by train. Here the agents can choose independently, as their values are affected
only by their own actions: in later sections we will introduce a third value which
requires them to consider what the other intends to do also.
The basic idea underlying Value Based Argumentation Frameworks is that it is
possible to associate practical arguments with values, and that in order to determine
which arguments are acceptable we need to consider the audience to which they are
addressed, characterised in terms of an ordering on the values involved. We need,
however, to recognise that not all the arguments relevant to a practical decision will
be practical arguments. For example, if there is a train strike (or it is a UK Bank
Holiday when there are often no trains from Liverpool), the argument that the train
should be used cannot be accepted no matter how great the audience preference is
for Comfort over Speed. In order to recognise that such epistemic arguments con-
strain choice, such arguments are associated with the value Truth, and all audiences
are obliged to rank Truth above all other values. In the next section we will give a
formal presentation of Value Based Argumentation Frameworks.
4 Value Based Argumentation Frameworks
We present the Value Based Framework as an extension of Dung’s original Argu-
mentation Framework [14], defined in Chapter 2 of this book. We do this by extend-
ing the standard pair to a 5 tuple.
Definition 3.1. A value-based argumentation framework (VAF) is a 5-tuple:
VAF = A, R, V, val, P
where A is a finite set of arguments, R is an irreflexive binary relation on A (i.e.
A, R is a standard AF), V is a non-empty set of values, val is a function which
maps from elements of A to elements of V and P is the set of possible audiences
(i.e. total orders on V). We say that an argument a relates to value v if accepting A
promotes or defends v: the value in question is given by val(a). For every a ∈ A,
val(a) ∈ V.
When the VAF is considered by a particular audience, the ordering of values is
fixed. We may therefore define an Audience Specific VAF (AVAF) as:
Definition 3.2. An audience specific value-based argumentation framework (AVAF)
is a 5-tuple: VAFa = A, R, V, val, Valprefa](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-64-320.jpg)
![3 Abstract Argumentation and Values 51
where A, R, V and val are as for a VAF, a is an audience, a ∈ P, and Valprefa
is a preference relation (transitive, irreflexive and asymmetric) Valprefa ⊆ V x V,
reflecting the value preferences of audience a. The AVAF relates to the VAF in that
A, R, V and val are identical, and Valpref is the set of preferences derivable from
the ordering a ∈ P in the VAF.
Our purpose in introducing VAFs is to allow us to distinguish between one ar-
gument attacking another, and that attack succeeding, so that the attacked argument
may or may not be defeated. Whether the attack succeeds depends on the value or-
der of the audience considering the VAF. We therefore define the notion of defeat
for an audience:
Definition 3.3. An argument A ∈ AF defeatsa an argument B ∈ AF for audience a
if and only if both R(A,B) and not (val(B),val(A)) ∈ Valprefa.
We can now define the various notions relating to the status of arguments:
Definition 3.4. An argument a ∈ A is acceptable-to-audience-a (acceptablea) with
respect to set of arguments S, (acceptablea(A,S)) if:
(∀ x)((x ∈ A defeatsa(x,A)) → (∃ y)((y ∈ S) defeatsa(y,x))).
Definition 3.5. A set S of arguments is conflict-free-for-audience-a if:
(∀ x) (∀ y)((x ∈ S y ∈ S) → (¬ R(x,y) ∨ valpref(val(y),val(x)) ∈ Valprefa))).
Definition 3.6. A conflict-free-for-audience-a set of arguments S is admissible-for-
an-audience-a if: (∀ x)(x ∈ S → acceptablea(x,S)).
Definition 3.7. A set of arguments S in a value-based argumentation framework
VAF is a preferred extension for-audience-a (preferreda) if it is a maximal (with
respect to set inclusion) admissible-for-audience-a subset of A.
Now for a given choice of value preferences valprefa we are able to construct an
AF equivalent to the AVAF, by removing from R those attacks which fail because
they are faced with a superior value.
Thus for any AVAF, vafa = A, R, V, val, Valprefa there is a corresponding
AF, afa = A, defeats, such that an element of R, R(x,y) is an element of defeats
if and only if defeatsa(x,y). The preferred extension of afa will contain the same
arguments as vafa, the preferred extension for audience a of the VAF. Note that if
vafa does not contain any cycles in which all arguments pertain to the same value,
afa will contain no cycles, since the cycle will be broken at the point at which the
attack is from an inferior value to a superior one. Hence both afa and vafa will have
a unique, non-empty, preferred extension for such cases. A proof is given in [6].
Moreover, since the AF derived from an AVAF contains no cycles, the grounded
extension coincides with the preferred extension for this audience, and so there is a
straightforward polynomial time algorithm to compute it, also given in [6]. For the
moment we will restrict consideration to VAFs which do not contain any cycles in a
single value.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-65-320.jpg)
![52 Trevor Bench-Capon and Katie Atkinson
For such VAFs, the notions of sceptical and credulous acceptance do not apply,
since any given audience will accept only a single preferred extension. These pre-
ferred extensions may, and typically will, however, differ from audience to audience.
We may therefore introduce two useful notions, objective acceptance, arguments
which are acceptable to all audiences irrespective of their particular value order,
and subjective acceptance, arguments which can be accepted by audiences with the
appropriate value order.
Definition 3.8. Objective Acceptance. Given a VAF, A, R, V, val, P an argument
a ∈ A is objectively acceptable if and only if for all p ∈ P, a is in every preferredp.
Definition 3.9. Subjective Acceptance. Given a VAF, A, R, V, val, P an argu-
ment a ∈ A is subjectively acceptable if and only if for some p ∈ P, a is in some
preferredp.
An argument which is neither objectively nor subjectively acceptable (such as
one attacked by an objectively acceptable argument with the same value) is said to
be indefensible.
All arguments which are not attacked will, of course, be objectively acceptable.
Otherwise objective acceptance typically arises from cycles in two or more values.
For example, consider a three cycle in two values, say two arguments with V1 and
one with V2. The argument with V2 will either resist the attack on it when it is
preferred to V1, or, when V1 is preferred, fail to defeat the argument it attacks which
will, in consequence, be available to defeat its attacker. Thus the argument in V2
will be objectively acceptable, and both the arguments with V1 will be subjectively
acceptable. For a more elaborate example consider Figure 3.2.
a
blue
b
c
d
red
red
blue
e
f
g
h
red
red
blue
blue
Fig. 3.2 VAF with values red and blue
There will be two preferred extensions, according to whether red blue, or blue
red. If red blue, the preferred extension will be {e,g,a,b}, and if blue red,
{e,g,d,b}. Now e and g and b are objectively acceptable, but d, which would have
been objectively acceptable if e had not attacked d, is only subjectively acceptable
(when blue red), and a, which is indefensible if d is not attacked, is also subjec-
tively acceptable (when red blue). Arguments c, f and h are indefensible. Results
characterising the structures which give rise to objective acceptance are given in [6].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-66-320.jpg)
![3 Abstract Argumentation and Values 55
A1
Kt
Ck
A2
Kp
Sk
A6
KpTp
B
A5
KtTt
B
A4
Tp
St
A3
Tt
Ct
Fig. 3.5 AF for Combined Audience
We can use this VAF to illustrate the algorithm for finding the Preferred Exten-
sion given in [6]. First we include the arguments with no attacker: in this case A1.
A1 attacks A2 and A6 and so they are excluded. Now A5 has no attacker and so it
is included. A5 excludes A4, leaving A3 without an attacker, and so A3 is included
to give the preferred extension of the combined audience as {A1,A3,A5}.
This case is straightforward, because the combined audience yields a single pre-
ferred extension. The same is true if Trevor preferred S to B, the combined order
being shown in Figure 3.4(b). This would cause A5,A4 to be replaced in R by
A4,A5. Now both A1 and A4 are not attacked, and so they defeat the remaining
arguments yielding the preferred extension {A1,A4}. This is possible: they simply
agree to travel separately by their preferred means.
More complicated is the situation where Katie prefers B to C, so that the merged
order is as shown in Figure 3.4(c), and A6,A1 replaces A1,A6 in R. Now
there is no longer any argument which has no attackers, and the algorithm must
be applied twice; first including A5 and then including A6, so that are two pre-
ferred extensions, {A1,A3,A5}, and {A2,A4,A6}, both of which are acceptable to
them both. Now, since Katie will lean towards the former and Trevor the latter they
must find a way to decide between Ck and St. This might depend on who had the
strongest opinions, or who is the more altruistic or conciliatory. Alternatively one
person might change their preferences: if Katie moved back to her original ordering
of CBS, Trevor would either have to decide to prefer S to B or to agree to travel
by train. This possibility shows how preferences can emerge from the reasoning
process: although initially Katie might express a preference for B to C, and Trevor
for B to S, when the consequences are realised she may decide that C is actually
more important than B, and he may decide S is more important than B.
5 Example Applications
As noted in Section 1, reasoning with values is common to many application do-
mains. In previous work [2, 3, 5, 9] we have shown how the application of abstract
argumentation with values can be applied to problems in law, medicine, ethics and](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-69-320.jpg)
![56 Trevor Bench-Capon and Katie Atkinson
e-democracy, and we will briefly discuss these applications here. We begin by con-
sidering legal reasoning with values.
5.1 Law
Reasoning with legal cases has often been viewed as a decision being deduced about
a particular case through the application of a set of rules, given the facts of the case,
e.g. [25]. However, the facts of cases are not set in stone as they can be open to
interpretation from different lawyers. Additionally, the rules used to reach decisions
are defeasible by their nature and many are derived from precedent cases, so they
too may be open to interpretation. Thus, within the AI and Law literature it has
been recognised that when considering arguments in legal cases, the purposes of
the law – the values intended to be promoted or upheld through the application
of the law – must be represented and accounted for, e.g. [8] [21]. In the literature
on legal case-based reasoning the issue was first brought to attention in Berman
and Hafner’s seminal paper on the topic [8] arguing that legal case-based reasoning
needs to recognise teleological as well as factual aspects. This is so since the law
is not composed arbitrarily, rather it is constructed to serve social ends, so when
conflicts in the application of rules occur in legal cases they can be resolved more
effectively by considering the purposes of these rules and their relative applicability
to the particular case in question. This enables preferences amongst purposes to
be revealed, and then the argument can be presented appropriately to the audience
through an appeal to the social values that the argument promotes or defends.
In order to demonstrate how the values of the law can be represented and rea-
soned about within a case, we have previously presented a reconstruction [3] of a
famous case in property law by simulating the opinion and dissent in that case. The
case is that of Pierson vs Post1 which concerned a dispute about ownership of a
hunted fox. The said fox was being pursued by Post who was hunting with hounds
on unoccupied waste-land. Whilst Post was in pursuit of the fox another man, Pier-
son, came along and intercepted the chase, killing and carrying off the fox. Central
to the arguments considered in the case was whether ownership of a wild animal
can be attributed through mere pursuit. However, there are numerous other argu-
ments that need to be considered which draw out the emphasis placed on the values
considered within the case.
Firstly, the value ‘public benefit’ was considered as it was argued that fox hunting
is of benefit to the public because it assists farmers, so it should be encouraged by
giving the sportsman such as Post protection of the law. There are of course counter
arguments to this based on the humane treatment of animals. Furthermore, there are
arguments concerning consideration of public benefit based on the desire to punish
malicious behaviour as allegedly shown by Pierson in intercepting the fox that he
could see Post was chasing.
1 3 Cai R 1752 Am Dec 264 (Supreme Court of New York, 1805)](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-70-320.jpg)
![3 Abstract Argumentation and Values 57
Secondly, there were arguments set forth about the need for the law to be clear:
in attributing ownership without bodily possession this would encourage a climate
of litigation based on similar claims related to pursuit alone.
Thirdly, the value of ‘economic benefit’ was considered in relation to the protec-
tion of property rights where the claimant is engaged in a profitable enterprise.
Given the facts of the case and the values stated above that have been recognised
as pertinent to the reasoning in the case, the argument scheme for practical reasoning
can be applied to generate the competing arguments about who to decide for in the
case. Once generated these arguments can be organised into a VAF and evaluated in
the usual manner. In the actual case the court found for Pierson, thus holding that
clarity was more important than the values promoted by finding for Post. Preference
orderings of values that led to this decision are reflected in our full representation
of the case, which can be found in [3]. Explicitly representing the values promoted
by the arguments put forward in the case helps to clarify the justifications for the
arguments advanced and ground those justifications within the purposes that law is
intended to capture and uphold.
5.2 Medicine
A second example scenario that has been considered in terms of value-based ar-
gumentation is one concerning a system for reasoning about the medical treatment
of a patient [5]. Decision making in this domain often requires consideration of
a wide range of options, some of which may conflict, and may also be uncertain.
Thus, value-based argumentation can play a role in supporting the decision making
process in this domain.
The scenario modelled in [5] illustrates a running example of a patient whose
health is threatened by blood clotting. In deciding which particular treatment to ad-
minister to the patient there are a number of policies and concerns that affect the
decision, and each must be given its due weight. In the computational model of the
scenario a number of different perspectives are represented that are given as val-
ues of individual agents. The arguments and subsequent conclusions drawn by the
individual agents are then adjudicated by a central agent which comes to a deci-
sion based on an evaluation of the competing arguments. Concerning the individual
agents’ values, these represent perspectives such as: the treatment of the patient
based on general medical policy; the safety of the patient concerning knowledge of
contraindications of the various drugs; the efficacy of the treatments in reference to
specific medical knowledge; and, the cost of the different treatments available.
Given the above agent perspectives (and others that we do not detail here), the
practical reasoning argument scheme can be used to generate arguments about
which drug should be used to treat a particular patient. These arguments can be
critiqued by agents other than those that generated the recommendation, based on
their individual knowledge, through the posing of the appropriate critical questions.
This may lead to different agents recommending different treatments, one of which](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-71-320.jpg)
![58 Trevor Bench-Capon and Katie Atkinson
must be chosen. In order to decide between the competing choices, the arguments
justifying each are organised into a VAF and evaluated according to the preference
given over the values represented by the individual agents. For example, it may be
the case that the treatment agent recommends a particular drug that is known to be
highly effective (since no critique from the efficacy agent indicates otherwise) and
has no contraindications (according to the safety agent), yet the cost agent has an
argument that the drug cannot be used on monetary grounds. The question then is
whether treatment is to be preferred to cost (which may be the case if there are no
suitable alternatives are identified). Resolution of this issue will be determined by
the central adjudicating agent who provides the value ordering to decide upon the
winning argument and subsequent treatment recommended, in accordance with the
policy of the relevant health authority at the time.
Whilst a key motivation for the example application described above was the
representation of the different perspectives within the situation, there are other ad-
vantages worthy of note. Firstly, the reasoning involved in medical scenarios is often
highly context dependant and relative to specific individuals so there is a high de-
gree of uncertainty. Thus any ordering of preferences must take the specific context
into account and the argumentation based approach enables this. Secondly, the ar-
gumentation element is effected inside a single agent and the information that it
uses is distributed across different information sources, which need not themselves
consider every eventuality, and play no part in the evaluation. This simplifies their
construction and facilitates their reuse in other applications. Finally, the critiques
that are posed against putative solutions are made only as and when they can affect
the evaluation status of arguments already advanced. This means that all reasoning
undertaken is potentially relevant to the solution.
5.3 Moral Reasoning
The running example that we have presented in this paper concerning travel to a
conference is represented in terms of an AATS. We now turn to briefly discussing
another example scenario, concerning moral reasoning, that has been modelled in
these terms.
The scenario is a particular ethical dilemma discussed by Coleman [12] and
Christie [11], amongst others, and it involves two agents, called Hal and Carla, both
of whom are diabetic. The situation is that Hal, through no fault of his own, has
lost his supply of insulin and urgently needs to take some to stay alive. Hal is aware
that Carla has some insulin kept in her house, but Hal does not have permission to
enter Carla’s house. The question is whether Hal is justified in breaking into Carla’s
house and taking her insulin in order to save his life. By taking Carla’s insulin, Hal
may be putting her life in jeopardy, since she will come to need that insulin herself.
One possible response is that if Hal has money, he can compensate Carla so that her
insulin can be replaced before she needs it. Alternatively if Hal has no money but
Carla does, she can replace her insulin herself, since her need is not immediately](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-72-320.jpg)
![3 Abstract Argumentation and Values 59
life threatening. There is, however, a serious problem if neither have money, since
in that case Carla’s life is really under threat. Coleman argued that Hal may take the
insulin to save his life, but should compensate Carla. Christie’s argument against
this was that even if Hal had no money and was unable to compensate Carla he
would still be justified in taking the insulin by his immediate necessity, since no one
should die because of poverty.
In [2] we have represented this scenario in terms of an AATS and considered the
arguments that can be generated concerning how the agents could justifiably act.
Following our methodology, we take the arguments generated and organise them
into a VAF to see the attack relations between them and evaluate them in accor-
dance with the particular value preference orderings. An interesting point that can
be taken from this particular example concerns the nuances between different ‘lev-
els’ of morality that can be drawn out by distinguishing the individual agents within
the value orderings. For example, prudential reasoning takes account of the different
agents, with the reasoning agent preferring values relating to itself, whereas strict
moral reasoning ignores the individual agents and treats the values equally. For ex-
ample, in the insulin scenario two values are recognised: life, which is demoted
when Hal or Carla ceases to be alive, and freedom, which is demoted when Hal or
Carla ceases to have money. Thus, an agent may rank life over freedom, but within
this value ordering it may discriminate between agents; for example, the agent may
place equal value on its own and another’s life, or it may be that it prefers its own
life to another’s (or vice versa). This leads to distinctions such as selfish agents who
prefer their own interests above all those of other agents, and noble agents whose
values are ordered, but within a value the agent prefers another’s interests.
In addition to the AATS representation set out in [2], simulations have also been
run, which are reported in [10], that confirm the reasoning as set out.
5.4 e-Democracy
The final application area that we discuss is an e-Democracy setting whose focus is
more on the support given by value based argumentation within a system to facilitate
the collection and analysis of human arguments within political debates.
The application is presented as a discussion forum named Parmenides whose
underlying structure is based upon the practical reasoning argument scheme and the
latest version of the system is described in [9]. The system is intended as a forum
by which the government is able to present policy proposals to the public so users
can submit their opinions on the justification presented for the particular policy. The
justification for action is structured in the form of the practical reasoning argument
scheme, though this imposed structure is hidden from the user. Within a particular
topic of debate, a justification upholding a proposed government action is presented
to users of the system in the form of the argument scheme. Users are then led in a
structured fashion through a series of web pages that pose the appropriate critical
questions to determine which parts of the justification the users agree or disagree](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-73-320.jpg)
![60 Trevor Bench-Capon and Katie Atkinson
with (the circumstances, the action, the consequences or the value). Users are not
aware (and have no need to be aware) of the underlying structure for argument
representation but it is, nevertheless, imposed on the information they submit. This
enables the collection of information which is structured in a clear and unambiguous
fashion from a system which does not require users to gain specialist knowledge
before being able to use it.
In addition to collecting arguments, Parmenides also has analysis facilities that
make use of AFs. All the information that the users submit through the system
is stored in a back-end database. This information is then organised into an argu-
mentation framework to show the attacking arguments between the positions ex-
pressed. Associated with the arguments in the AF is statistical information con-
cerning a breakdown of support for the arguments, i.e. the number of users agree-
ing/disagreeing with a particular element of the justification. Thus, arguments can
be assessed by considering which ones are the most controversial to the users.
The Parmenides system is intended to overcome some of the problems faced
by existing discussion forum formats, such as unstructured blogs and e-petitions. In
such systems where there is no structuring of the information, it is undoubtedly very
difficult for the policy maker to adequately address each person’s concerns since he
or she is not aware of users’ specific reasons for disagreeing. Furthermore, it may
be difficult to recognise agreement and disagreement between multiple user replies.
In contrast, the structure imposed by Parmenides allows the administrator of the
system to see exactly which particular part of the argument is disagreed with by the
majority of users, e.g. arguments based on a description of the circumstances, or
arguments based on a disagreement about the importance of promoting a particular
value. Identifying these different sources of disagreement allows the policy maker
to see why his policy is disliked, so he may be able to better respond to the criticisms
made, or indeed change the policy. In particular, it can indicate whether the values
motivating the policy are shared by the respondents.
Parmenides has been tested on a number of different political debates, including:
the UK debate about banning fox hunting2; the justification for the 2003 war in Iraq;
and, a debate about the proposal to increase the number of speed cameras on UK
roads. Work on the Parmenides system is ongoing to further extend its representation
facilities, through the use of schemes additional to the practical reasoning scheme,
and to further extend the facilities for analysing the arguments through the use of
argumentation frameworks.
6 Developments of Value Based Argumentation
In this section we will mention some developments of Value Based Argumentation.
In [13] there is an interesting exploration of the relation between neural networks,
in particular neural-symbolic learning systems, and value based argumentation sys-
2 For this particular debate on the system see:
http://cgi.csc.liv.ac.uk/∼parmenides/foxhunting/](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-74-320.jpg)
![3 Abstract Argumentation and Values 61
tems, including an extensive discussion of the insulin example described in the last
section. In [15] there is a formal generalisation of VAFs to allow for arguments that
promote multiple values, and in which preferences among values can be specified
in various ways. In [7] a method is given to determine which audiences can accept a
particular set of arguments. Here, however, we will look in detail only at the appli-
cation of Modgil’s extended argumentation frameworks (EAF) [17] to VAFs. For a
preliminary exploration of the relation between EAFs and VAFs see [18].
The core idea of EAFs is, like VAFs, to enable a distinction between an argu-
ment attacking an argument, and an argument defeating another argument. Whereas
VAFs, however, rely on a comparison of properties of the arguments concerned,
EAFs achieve this in an entirely abstract manner by allowing arguments to attack
not only other arguments, but also attacks. EAFs thus enable arguments to resist an
attack for a number of reasons. In VAFs arguments resist attacks solely in virtue of
a preference between the values concerned. This enables VAFs to be rewritten as
standards AFs, by introducing some auxiliary arguments to articulate the notion of
an attack on an attack. These auxiliary arguments represent the status of arguments,
value preferences, and arguments representing particular audiences. Suppose we
have a VAF with two arguments, A and B which attack one another. A is associated
with value V1 and B with Value V2. A will be defeated if B defeats it, and B will
be defeated if A defeats it. Defeat is only possible if the attacking argument is not
defeated, and if the value of A is not preferred to that of B. Thus the attack on the
attack of A on B in an EAF becomes an attack on the argument that A defeats B.
This enables us to represent a VAF as a standard AF, with preferred extensions
depending on the choices made regarding value preferences. We can extend the AF
to include audiences as well. Suppose Audience X prefers V1 to V2 and Audience
Y prefers V2 to V1. This can be shown as in Figure 3.6.
A is
V1
justified
V1 V2
V2 V1
is X
Audience
Audience
is Y
B
A
defeats
B is
defeated
B is
justified
V2
B
A
defeats
A is
defeated
Fig. 3.6 AF representing VAF with audiences
The rewriting of VAFs in this way is shown to be sound and complete with re-
spect to EAFs in [18]. When we rewrite VAFs in this way, subjective acceptance
in the VAF is equivalent to credulous acceptance in the rewritten AF, and objective
acceptance in the VAF is equivalent to sceptical acceptance in the rewritten AF. This](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-75-320.jpg)
![64 Trevor Bench-Capon and Katie Atkinson
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action based alternating transition systems. Artificial Intelligence, 171(10–15):855–874, 2007.
2. K. Atkinson and T. Bench-Capon. Addressing moral problems through practical reasoning.
Journal of Applied Logic, 6(2):135–151, 2008.
3. K. Atkinson, T. Bench-Capon, and P. McBurney. Arguing about cases as practical reasoning.
In Proc. of the Tenth International Conference on Artificial Intelligence and Law (ICAIL ’05),
pages 35–44, 2005. ACM Press.
4. K. Atkinson, T. Bench-Capon, and P. McBurney. Computational representation of practical
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19. C. Perelman. Justice, Law, and Argument. D. Reidel Publishing Company, Dordrecht, 1980.
20. C. Perelman and L. Olbrechts-Tyteca. The New Rhetoric: A Treatise on Argumentation. Uni-
versity of Notre Dame Press, Notre Dame, IN, USA, 1969.
21. H. Prakken. An exercise in formalising teleological reasoning. In Proc. of the Thirteenth
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![Chapter 4
Bipolar abstract argumentation systems
Claudette Cayrol and Marie-Christine Lagasquie-Schiex
1 Introduction
In most existing argumentation systems, only one kind of interaction is considered
between arguments. It is the so-called attack relation. However, recent studies on
argumentation [23, 34, 35, 4] have shown that another kind of interaction may exist
between the arguments. Indeed, an argument can attack another argument, but it can
also support another one. This suggests a notion of bipolarity, i.e. the existence of
two independent kinds of information which have a diametrically opposed nature
and which represent repellent forces.
Bipolarity has been widely studied in different domains such as knowledge and
preference representation [10, 31, 25, 6]. Indeed, in [6] two kinds of preferences
are distinguished: the positive preferences representing what the agent really wants,
and the negative ones referring to what the agent rejects. This distinction has been
supported by studies in cognitive psychology which have shown that the two kinds
of preferences are completely independent and are processed separately in the mind.
Another application where bipolarity is largely used is that of decision making.
In [3, 19], it has been argued that when making decision, one generally takes into
account some information in favour of the decisions and other pieces of information
against those decisions.
In [20], a nomenclature of three types of bipolarity has been proposed using par-
ticular characteristics like exclusivity (can a piece of information be at the same time
positive and negative), duality (can negative information be computed using posi-
tive information), exhaustivity (can information be neither positive, nor negative),
computation of positive and negative information on the same data, computation of
positive and negative information with the same process, existence of a consistency
constraint between positive and negative information.
IRIT-UPS, 118 route de Narbonne, 31062 Toulouse, France, e-mail: {ccayrol,lagasq}@
irit.fr
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 65
DOI 10.1007/978-0-387-98197-0 4, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-79-320.jpg)
![66 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
The first type of bipolarity proposed by [20] (symmetric univariate bipolarity)
expresses the fact that the negative feature is a reflection of the positive feature
(so, they are mutually exclusive and a single bipolar univariate scale is enough for
representing them).
The second one (dual bivariate bipolarity) expresses the fact that we need two
separate scales in order to represent both features, although they stem from the same
data (so, an information can be positive and negative at the same time and there is
no exclusivity). However a duality must exist between both features.
And the third one (heterogeneous bipolarity) expresses the fact that both features
do not stem from the same data though there is some minimal consistency require-
ment between both features.
In this chapter, we focus on the use of bipolarity in the particular domain of
argumentation. In all the disparate cases, an argumentation process follows different
steps: i) building the arguments and the interactions between them, ii) valuating
the arguments and accounting for their interactions or not, iii) finally selecting the
most acceptable arguments (or sets of arguments) and using them in order to draw a
conclusion or choose a decision. Bipolarity can appear under different forms in each
step of this process. In this chapter we are only concerned by the use of bipolarity
at the interaction level (a more complete study of bipolarity in each step of the
argumentation process is proposed in [4, 5]).
At this level, the main point is the definition of the interactions between argu-
ments. As already said, due for instance to the presence of inconsistency in knowl-
edge bases, arguments may be conflicting. Indeed, in all argumentation systems, an
attack relation is considered in order to capture the conflicts.
However, most logical theories of argumentation assume that: if an argument a1
attacks an argument a3 and a3 attacks an argument a2, then a1 supports a2. In this
case, the notion of support does not have to be formalized in a way really different
from the notion of attack. It is the case of the basic argumentation framework defined
by Dung, in which only one kind of interaction is explicitly represented by the attack
relation. In this context, the support of an argument a by another argument b can be
represented only if b defends a in the sense of [21]. So, support and attack are
dependent notions. It is a parsimonious strategy, but it is not a correct description
of the process of argumentation. Let us take several examples for illustrating the
difference between “defence” and “support”:
Ex. 1 We want to begin a hike. We prefer a sunny weather, then a sunny and cloudy
one, then a cloudy but not rainy weather, in this order. We will cancel the hike only
if the weather is rainy. But clouds could be a sign of rain. We look at the sky early
in the morning. It is cloudy. The following exchange of informal arguments occurs
between Tom, Ben and Dan:
t1 Today we have time, we begin a hike.
b The weather is cloudy, clouds are sign of rain, we had better cancel the hike.
t2 These clouds are early patches of mist, the day will be sunny, without clouds,
so the weather will be not cloudy (and we can begin the hike).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-80-320.jpg)
![4 Bipolar abstract argumentation systems 67
d These clouds are not early patches of mist, so the weather will be not sunny but
cloudy; however these clouds will not grow, so it will not rain (and we can begin
the hike).
In this exchange, we can identify the following path of conflicts between ar-
guments: argument d attacks argument t2 which attacks argument b which in turn
attacks argument t1. So, with Dung’s framework, argument t2 is a defender of ar-
gument t1, and argument d is a defeater of argument t1. Nevertheless, arguments
t2 and d support the hike project. So, the idea of a chain of arguments and counter-
arguments in which we just have to count the links and take the even one as defeaters
and the odd ones as supporters is an oversimplification. So, the notion of defence
proposed by [21] is not sufficient to represent support.
The following example also illustrates the need for a new kind of interaction
between arguments; the following arguments are exchanged during the meeting of
the editorial board of a newspaper:
Ex. 2
a: Assuming agreement and no right of censor, information I concerning X will
be published.
b1: X is the prime minister who may use the right of censor.
c0: We are in democracy and even a prime minister cannot use the right of censor.
c1: I believe that X has resigned. So, X is no longer the prime minister.
d: The resignation has been announced officially yesterday on TV Channel 1.
b2: I is private information so X denies publication.
e: I is an important information concerning X’s son.
c2: Any information concerning a prime minister is public information.
repetition of c1 and d: . . .
c3: But I is of national interest, so I cannot be considered as private information.
In this example, some conflits appear: for instance, b1 (resp. b2) is in conflict with
a. But we may also consider that the argument d given by an agent Ag1 supports the
argument c1 given by another agent Ag2. It is not only a “dialogue-like speech act”:
a new piece of information is really given and it is given after the production of the
argument c1. So taking d into account leads either to modify c1, or to find a more
intuitive solution for representing the interaction between d and c1. In this case, we
adopt an incremental point of view, considering that pieces of information given by
different agents enable them to provide more and more arguments. We do not want
to revise already advanced arguments. In contrast, we intend to represent as much
as possible all the kinds of interaction between these arguments.
The last example shows how a notion of support between two arguments can be
formalized with a logical representation of the structure of the arguments.
Ex. 3 A murder has been performed and the suspects are Liz, Mary and Peter. The
following pieces of information have been gathered:
The type of murder suggests us that the killer is a female. The killer is certainly
small. Liz is tall and Mary and Peter are small. The killer has long hair and uses a](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-81-320.jpg)
![68 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
lipstick. A witness claims that he saw the killer who was tall. Moreover, we are told
that the witness is short-sighted, so he is no more reliable.
We use the following propositional symbols: sm (the killer is small), fem (the
killer is a female), mary (the killer is Mary), lglip (the killer has long hair and uses
a lipstick), wit (the witness is reliable), bl (the witness is short-sighted).
Here, an argument takes the form of a set of premises which entails a conclusion.
So the following arguments can be formed: a1 in favour of mary (with premises
{sm, fem,(sm ∧ fem) → mary}), a2 in favour of ¬sm (with premises {wit,wit →
¬sm}), a3 in favour of ¬wit (with premises {bl,bl → ¬wit}), a4 in favour of fem
(with premises {lglip,lglip → fem}).
a3 attacks a2 which attacks a1. So a3 defends a1 against a2.
Moreover, a4 confirms the premise fem of a1. So, a4 supports a1 (in the sense that
a4 strengthens a1). Contrastedly, a3 defends a1 against a2 means that a3 weakens
the attack on a1 brought by a2. So, on one side, a1 gets a support and on the other
side a1 suffers a weakened attack.
The above examples show that the argumentation process uses arguments and
counter-arguments, support and attack relations, but not always in the same way.
The arguments which are available in a dynamic argumentation process rely upon
premises which are not always pieces of evidence. If we accept that a new fact can
undermine one of the premises (thus forming an attack), we must also accept that a
new fact can enforce, or confirm a premise (thus forming a support interaction).
Following all these remarks, and in order to formalise realistic examples, a more
powerful tool than the abstract argumentation framework proposed by Dung is
needed. In particular, we are interested in modelling situations where two inde-
pendent kinds of interactions are available: a positive and a negative one (see for
example in the medical domain the work [23]). So, following [23, 35], we present a
new argumentation framework: an abstract bipolar argumentation framework.
The chapter is organized as follows: Section 2 introduces the formal definitions of
an abstract Bipolar Argumentation Framework (BAF). Then, we consider the fun-
damental problem of determining which arguments (or sets of arguments) can be
considered as acceptable. The formal way to handle this problem is to define argu-
mentation semantics. Section 3 introduces extension-based acceptability semantics
for a BAF. These new semantics rely upon criteria which make explicitly use of both
support and attack relations. In Section 4, another way to define extension-based
semantics for a BAF is followed. First, a transformation of a BAF into a Dung’s
meta-argumentation framework is given. The support relation is used to form meta-
arguments (called coalitions) in such a way that at the meta-level only conflict in-
teractions may appear. Extensions of a BAF can then be defined from Dung’s ex-
tensions of the meta framework. Section 5 addresses the question of labelling-based
semantics in a BAF. Some labelling functions are proposed for a BAF. Section 6 is
devoted to the related issues and to some concluding remarks. Note that the proofs
of the properties given in this chapter can be found in the associated original papers.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-82-320.jpg)
![4 Bipolar abstract argumentation systems 69
2 Abstract bipolar frameworks
An abstract bipolar argumentation framework is an extension of the basic abstract
argumentation framework introduced by [21] in which a new kind of interaction
between arguments is represented by the support1 relation2. This new relation is
assumed to be totally independent of the attack relation (i.e. it is not defined using
the attack relation). So, we have a bipolar representation of the interactions between
arguments.
Def. 1 An abstract bipolar argumentation framework (BAF) A,Ratt,Rsup consists
of: a set A of arguments, a binary relation Ratt on A called the attack relation and
another binary relation Rsup on A called the support relation. These binary relations
must verify the following consistency constraint: Ratt ∩Rsup = ∅3.
Consider ai and aj ∈ A, aiRattaj (resp. aiRsupaj) means that ai attacks (resp. sup-
ports) aj. Let a ∈ A, Ratt
−
(a) (resp. Rsup
−
(a)) denotes the set of attackers (resp.
supporters) of a.
In the following, we assume that A represents the set of arguments proposed by
rational agents at a given time, so we will assume that A is finite.
A BAF can be represented by a directed graph Gb called the bipolar interaction
graph, with two kinds of edges, one for the attack relation (→) and another one for
the support relation (❀). See for instance the following representations:
For Ex. 1 For Ex. 2 For Ex. 3
(hiking project) (editorial meeting) (murder)
b // t1 t2
oo o/ o/ o/
ff
d
__
??
OO
O
O
O
c0 // b1
// a b2
oo e
oo o/ o/ o/
d //
/o
/o
/o c1
OO
// c2
??
~
~
~
~
~
~
~
~
c3
OO a3 // a2 // a1
a4
~
~
~
~
In the following, we abstract from the structure of the arguments and we consider
arbitrary independent relations Ratt and Rsup.
Def. 2 Let BAF = A,Ratt,Rsup be a bipolar argumentation framework and Gb be
the associated interaction graph. Let a, b ∈ A. A path from a to b in Gb is a sequence
(a1,...,an) of elements of A s.t. n ≥ 2, a = a1, b = an, a1Ra2, . . . , an−1Ran, with
R = Ratt or Rsup. Such a path has length n−1.
Note that if n = 2 and a = b then the path is a loop and if the relation R used in
the loop is Ratt then a is said self-attacking.
The use of bipolarity suggests new kinds of interaction between arguments: in
Ex. 2, the fact that d supports an attacker of b1 may be considered as a kind of
1 Note that the term “support” refers to a relation between two arguments and not a relation between
premises and conclusion, as in Toulmin [32].
2 If the support relation is removed, we retrieve Dung’s framework.
3 In the context of the argumentation, this consistency constraint is essential: it does not seem ratio-
nal to advance an argument which simultaneously attacks and supports the same other argument.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-83-320.jpg)
![70 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
negative interaction between d and b1, which is however weaker than a direct at-
tack. From a cautious point of view, such arguments cannot appear together in a
same extension. In order to address this problem, a new kind of attack has been
introduced [13, 14] which combines a sequence of supports with a direct attack.
Def. 3 Let a, b ∈ A. There is a sequence of supports for b by a (or for short a
supports b) iff there exists a sequence (a1,...,an) of elements of A s.t. n ≥ 2, a = a1,
b = an, a1Rsupa2, . . . , an−1Rsupan.
Def. 4 A supported attack for an argument b by an argument a is a sequence (a,x,b)
of arguments of A s.t. a supports x4 and xRattb.
In Ex. 2, there is a supported attack for b1 by d.
Then, taking into account attacks and sequences of supports leads to the following
definitions applying to sets of arguments:
Def. 5 Let S ⊆ A, let a ∈ A. S set-attacks a iff there exists a supported attack or a
direct attack for a from an element of S. S set-supports a iff there exists a sequence
of supports for a from an element of S.
The above definitions are illustrated on the following example:
Ex. 4 Consider the following graph: a //
/o
/o
/o b //
/o
/o
/o c // d j
g
@@
h
@@
i //
/o
/o
/o e
OO
f
OO
O
O
O
// k
OO
In this graph, the paths a−b−c−d and i−c correspond to supported attacks.
The set {a,h} set-attacks d and b and set-supports b and c.
3 Extension-based semantics for acceptability
In Dung’s framework, the acceptability of an argument depends on its membership
to some sets, called acceptable sets or extensions. These extensions are characterised
by particular properties. It is a collective acceptability. Following Dung’s method-
ology, we propose characteristic properties that a set of arguments must satisfy in
order to be an output of the argumentation process, in a bipolar framework. We re-
call that such a set of arguments must be in some sense coherent and must enable to
win a dispute. Maximality for set-inclusion is also often required.
Considering a BAF A,Ratt,Rsup and using the notion of “set-attack” and “set-
support” given by Def. 5, we first investigate the notion of coherence, then we pro-
pose new semantics for acceptability in bipolar argumentation frameworks.
4 In the sense of Def. 3.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-84-320.jpg)
![4 Bipolar abstract argumentation systems 71
3.1 Managing the conflicts
In the basic argumentation framework, whatever the considered semantics, selected
acceptable sets of arguments are constrained to be coherent in the sense that they
must be conflict-free. In a bipolar argumentation framework, the concept of coher-
ence can be extended along two different lines:
• forbidding not only direct attacks but also supported attacks enforces a kind of
internal coherence: we do not accept a set S of arguments which set-attacks one
of its elements (this a generalization of Dung’s notion of conflict-free).
• extending the consistency constraint between support and attack relations leads
to define a kind of external coherence: we do not accept a set S of arguments
which set-attacks and set-supports the same argument.
Def. 6 Let S ⊆ A. S is +conflict-free5 iff ∃a,b ∈ S s.t. {a} set-attacks b.
S is safe6 iff ∃b ∈ A s.t. S set-attacks b and either S set-supports b, or b ∈ S.
Ex. 4 (cont’d) The set {h,b} is not +conflict-free (there is a direct attack). The set
{b,d} is not +conflict-free since d suffers a supported attack from b. Contrastedly,
{a,h} and {b, f} are +conflict-free.
The set {a,h} is not safe since a supports b and h attacks b. The set {b, f} is not safe
since d suffers a supported attack from b and f supports d. Contrastedly, {g,i,h} is
safe.
Note that the notion of safe set encompasses the notion of +conflict-free set:
Prop. 1 ([14]) Let S ⊆ A. If S is safe, then S is +conflict-free. If S is +conflict-free
and closed for Rsup then S is safe.
Ex. 4 (cont’d) The set {g,h,i,e} is +conflict-free and closed for Rsup. So it is safe.
3.2 New acceptability semantics
According to the methodology proposed by [21], two notions play an important
role in the definition of extension-based semantics: the notion of coherence, and the
notion of defence (that is for short attack against attack). In a BAF, several notions
of coherence, and two kinds of attack (direct and supported) are available. So several
extensions of the notion of defence could be proposed. However, we have chosen to
restrict to the classical defence, for the following reasons. First, the purpose of this
chapter is to present some principles governing bipolar frameworks, rather than an
exhaustive survey. Secondly, most of the works talking about bipolarity consider that
5 This notation means that checking if a set is +conflict-free needs to consider more conflicts than
with the basic notion of conflict-free suggested by Dung.
6 This definition is inspired by [35] and by the notion of a controversial argument given in [21].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-85-320.jpg)
![72 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
a support does not have the same strength as an attack. In that sense, an argument
can be considered as defended if and only if its direct attackers are directly attacked.
The above remark is illustrated by the following example: a1 // a2
~~
a3
oo o/ o/ o/
There is a supported attack for a1 by a3 and no attack for a3. However, a1 directly
attacks a2 and it seems sufficient to resinstate a1.
Let us recall the definition of defence given in [21].
Def. 7 Let S ⊆ A. Let a ∈ A. S defends a iff ∀b ∈ A, if bRatta then ∃c ∈ S s.t. cRattb.
In the following, the concept of admissibility is first extended. The idea is to
reinforce the coherence of the admissible sets. Then, extensions under the preferred
semantics will be defined as maximal (for ⊆) admissible sets of arguments.
Three different definitions for admissibility can be given, from the most general
one to the most specific one. First, a direct translation of Dung’s definition gives
the definition of d-admissibility (“d” means “in the sense of Dung”). Taking into
account external coherence leads to s(afe)-admissibility. Finally, external coherence
can be strengthened by requiring that an admissible set is closed for Rsup. So, we
obtain the definition of c(losed)-admissibility.
Def. 8 Let S ⊆ A.
S is d-admissible iff S is +conflict-free and defends all its elements.
S is s-admissible iff S is safe and defends all its elements.
S is c-admissible iff S is +conflict-free, closed for Rsup and defends all its elements.
From the above definitions, it follows that each c-admissible set is s-admissible,
and each s-admissible set is d-admissible.
Def. 9 A set S ⊆ A is a d-preferred (resp. s-preferred, c-preferred) extension iff
S is maximal for ⊆ (or for short ⊆-maximal) among the d-admissible (resp. s-
admissible, c-admissible) subsets of A.
Ex. 1 (cont’d) In this case, the three semantics give the same result: {d,t1} is the
unique d-preferred, s-preferred and c-preferred extension.
Ex. 4 (cont’d) The set {g,h,i,e, f,d, j} is the unique c-preferred extension.
Ex. 5 Consider the BAF represented by a ❀ b ← h. The set {a,h} is the unique
d-preferred extension. There are two s-preferred extensions {a} and {h}. And there
is only one c-preferred extension {h}.
One of the most important issues with regard to extensions concerns their ex-
istence. The existence of d-preferred (resp. s-preferred, c-preferred) extensions is
guaranteed since the empty set is d-admissible (resp. s-admissible, c-admissible),
and each d-admissible (resp. s-admissible, c-admissible) is included in a d-preferred
(resp. s-preferred, c-preferred) extension. Note that analogous definitions for admis-
sibility could be proposed using a stronger notion of defence (a stronger defence
would be defined for instance by replacing attack with set-attack in Def. 7).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-86-320.jpg)
![4 Bipolar abstract argumentation systems 73
Considering another well-known semantics, the stable semantics, nice results can
be obtained if we keep the basic definition of a stable extension, but replace attack
with set-attack. It is a straightforward way to extend the stable semantics in a BAF.
Def. 10 S is a stable extension iff S is +conflict-free and ∀a ∈ S, S set-attacks a.
In the following, we restrict to acyclic BAF, in the sense that the associated in-
teraction graph is acyclic. In Dung’s basic framework, it has been proved that, in
the case of an acyclic attack graph, there is always a unique stable (which is also
preferred) extension. So, Def. 10 ensures the existence of a unique stable extension
in an acyclic BAF7. However, the unique stable extension is not always safe.
Ex. 5 (cont’d) The set {a,h} is the unique stable extension, and it is not safe.
Indeed, the following result can be proved:
Prop. 2 ([14]) Let S be a stable extension. S is safe iff S is closed for Rsup.
The following results enable to characterize d-preferred, s-preferred and c-
preferred extensions when the BAF is acyclic:
Prop. 3 ([14]) Let S be the unique stable extension of an acyclic BAF.
1. S is also the unique d-preferred extension.
2. The s-preferred extensions and the c-preferred extensions are subsets of S.
3. Each s-preferred extension which is closed for Rsup is c-preferred.
4. If S is safe, then S is the unique c-preferred and the unique s-preferred extension.
5. If A is finite, each c-preferred extension is included in a s-preferred extension.
6. If S is not safe, the s-preferred extensions are the subsets of S which are ⊆-
maximal among the s-admissible sets.
7. If S is not safe, and A is finite, there is only one c-preferred extension.
Ex. 5 (cont’d) {h} is the only s-preferred extension which is also closed for Rsup.
So, {h} is the unique c-preferred extension.
Ex. 6 Consider the BAF represented by: a1 //
/o
/o
/o
A
A
A
A
A
A
A
A a2 //
/o
/o
/o b
c h
oo o/ o/ o/
OO
{a1,a2,h} is the only d-preferred extension. {a1,a2} and {h} are the only two
s-preferred extensions. None of them is closed for Rsup. ∅ is the unique c-preferred
extension. If we add an isolated argument a3 (for which no interaction exists with
the other available arguments), then we obtain: {a1,a2,a3,h} is the only d-preferred
extension. {a1,a2,a3} and {h,a3} are the only two s-preferred extensions. None of
them is closed, and {a3} is the unique c-preferred extension.
7 We instantiate Dung’s AF with the relation set-attacks and the resulting graph is still acyclic.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-87-320.jpg)
![74 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
The above discussion enables to draw the following conclusions. In the particular
case of an acyclic BAF, two semantics present nice features: the stable semantics and
the c-preferred semantics. If we are interested in internal coherence only, we will
have to determine the unique stable extension, which is also the unique d-preferred
extension. If we are interested in a more constrained concept of coherence, we will
compute the unique c-preferred extension.
4 Turning a bipolar framework into a Dung meta-framework
The extension-based acceptability semantics introduced in Section 3 rely upon cri-
teria which make explicitly use of support and attack relations, through the concept
of supported attack. Here, we follow another way to define extension-based seman-
tics for a BAF. First, a transformation of a BAF into a Dung’s meta-argumentation
framework is given. This meta-argumentation framework consists only of a set
of meta-arguments (called coalitions), and a conflict relation between these meta-
arguments. The attack relation of the initial BAF will appear only at the meta-level.
As a consequence, a meta-argument will gather arguments which are not in conflict.
The support relation of the initial BAF will not appear at the meta-level, but will
be used to gather arguments in a coalition. The idea is that a meta-argument makes
sense only if its members are somehow related by the support relation. So, the two
fundamental principles governing the definition of a coalition are: the Coherence
principle (there is no direct attack between two arguments of a same coalition) and
the Support principle (if two arguments belong to a same coalition, they must be
somehow, directly or indirectly, related by the support relation).
4.1 The concept of coalition
Consider BAF = A,Ratt,Rsup represented by the graph Gb. Gb
sup will denote the
partial graph representing the partial system A,Rsup8. AF will denote the partial
argumentation system A,Ratt associated with BAF and represented by the partial
graph denoted by Gb
att.
Def. 11 C ⊆ A is a coalition of BAF iff: (i) The subgraph of Gb
sup induced by C
is connected; (ii) C is conflict-free9 for AF; (iii) C is ⊆-maximal among the sets
satisfying (i) and (ii).
8 We consider that the reader knows the basic concepts of graph theory (chain, connexity,...). See
for instance [7] for a background on graph theory.
9 In the basic sense proposed by Dung.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-88-320.jpg)
![4 Bipolar abstract argumentation systems 75
Note that when Ratt is empty, the coalitions are exactly the connected compo-
nents10 of the partial graph Gb
sup.
Prop. 4 ([17]) An argument which is not self-attacking is in at least one coalition.
Ex. 7 Consider the BAF: a
O
O
O
c
oo
O
O
O
// e //
/o
/o
/o
O
O
O
f //
/o
/o
/o g //
/o
/o
/o h
b
@@
@
@
@
@
@
d i
hh
The coalitions are: C1 = {b,c,d}, C2 = {i}, C3 = {a,b}, C4 = {e, f,g,h}
The following result shows that coalitions can be restated in terms of connected
components of an appropriate subgraph. By the way, it gives a constructive way for
computing coalitions.
Prop. 5 ([17]) C ⊆ A is a coalition of BAF iff: (i) There exists S ⊆ A ⊆-maximal
conflict-free for AF s.t. C is a connected component of the subgraph of Gb
sup induced
by S and (ii) C is ⊆-maximal among the subsets of A satisfying (i).
Prop. 5 suggests a procedure for computing the coalitions of BAF:
Step 1: Consider AF and determine the maximal conflict-free sets for AF.
Step 2: For each set of arguments Si obtained at Step 1, determine the connected components of
the subgraph of Gb
sup induced by Si.
Step 3: Keep the ⊆-maximal sets obtained at Step 2.
The notion of conflict-free set is related to the notion of independent set:
Prop. 6 ([17]) Let S ⊆ A. S is conflict-free for AF iff S is an independent subset of
A in the graph Gb
att.
So, S is ⊆-maximal conflict-free for AF iff S is a ⊆-maximal independent set
of vertices in the graph Gb
att and Step 1 of the computational procedure consists in
determining all the ⊆-maximal independent subsets of Gb
att. Remark that the time
complexity of the best algorithms providing all the ⊆-maximal independent sets is
exponential. Note also that there exist several algorithms in the literature for finding
all the ⊆-maximal independent sets (see for instance the work of J.M. Nielsen [26]).
We also know that:
• For Step 2, a depth-first exploration of a graph provides the connected compo-
nents in linear time O(number of vertices + number of edges).
• And for Step 3, maximization with respect to ⊆ is also an exponential process.
10 Let G = (V,E) be a graph. Let S ⊆ V. S is a connected component of G iff the subgraph of G
induced by S is connected and there exists no S′ ⊆ V s.t. S ⊂ S′ and the subgraph of G induced by
S′ is connected.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-89-320.jpg)
![76 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
4.2 A meta-argumentation framework
Let C(A) denote the set of coalitions of BAF. We define a conflict relation on C(A)
as follows.
Def. 12 Let C1 and C2 be two coalitions of BAF. C1 C-attacks C2 iff there exists an
argument a1 in C1 and an argument a2 in C2 s.t. a1Ratta2.
It can be proved that:
Prop. 7 ([17]) Let C1 and C2 be two distinct coalitions of BAF. If C1 ∩C2 = ∅ then
C1 C-attacks C2 or C2 C-attacks C1.
So a new argumentation framework CAF = C(A),C-attacks can be defined, re-
ferred to as the coalition framework associated with BAF.
Ex. 7 (cont’d) In this example, CAF can be represented by (by abusing notations, →
represents the attack relation in BAF and also the C-attack relation in CAF):
C3 C1
oo // C4 C2
oo
Dung’s definitions apply to CAF, and it can be proved that:
Prop. 8 ([17]) Let {C1,...,Cp} be a finite set of distinct coalitions. {C1,...,Cp} is
conflict-free for CAF iff C1 ∪...∪Cp is conflict-free for AF.
So, CAF is a “meta-argumentation” framework with a set of “meta-arguments”
(the set of coalitions C(A)) and a “meta-attack” relation on these coalitions (the C-
attacks relation). A coalition gathers arguments which are close in some sense and
can be produced together. However, as coalitions may conflict, following Dung’s
methodology, preferred and stable extensions of CAF can be computed. Such exten-
sions will contain coalitions which are collectively acceptable. The last step consists
in gathering the elements of the coalitions of an extension of CAF. By this way, the
best groups of arguments (w.r.t. the given interaction relations) will be selected.
Def. 13 Let S ⊆ A. S is a Cp-extension (Cp means “Coalition-preferred”) of BAF
iff there exists {C1,...,Cp} a preferred extension of CAF s.t. S = C1 ∪...∪Cp.
S is a Cs-extension (Cs means “Coalition-stable”) of BAF iff there exists {C1,...,Cp}
a stable extension of CAF s.t. S = C1 ∪...∪Cp.
When the only preferred extension of CAF is the empty set, we define the empty
set as the unique Cp-extension of BAF.
Ex. 7 (cont’d) There is only one preferred extension of CAF, which is also stable:
{C1,C2}. So, S = {b,c,d,i} is the Cp-extension (and also the Cs-extension) of BAF.
Some nice properties of Dung’s basic framework are preserved:
• A BAF has always a (at least one) Cp-extension. It is a consequence of Def. 13.
• In contrast, there does not always exist a Cs-extension of BAF. The reason is that
there may be no stable extension of CAF.
• Each Cs-extension is also a Cp-extension. The converse is false.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-90-320.jpg)
![4 Bipolar abstract argumentation systems 77
• There cannot exist two Cp-extensions s.t. one strictly contains the other one. It
follows from Def. 11 and 13.
However, other properties are lost. A Cp-extension is not always admissible for
AF, and a Cs-extension is not always a stable extension of AF:
Ex. 8 Consider the BAF represented by: a // b c
oo o/ o/ o/ //
/o
/o
/o d // e
The coalitions are: C1 = {a}, C2 = {b,c,d}, C3 = {e}. And the associated CAF
can be represented by: C1
// C2
// C3
There is only one preferred extension of CAF, which is also stable: {C1,C3}. So,
S = {a,e} is the Cp-extension (and also the Cs-extension) of BAF. We have dRatte,
but a does not defend e against d (neither by a direct attack, nor by a supported
attack, though a attacks an element of the coalition which attacks e). So, S is not
admissible for AF. S does not contain c, but there is no attack (no supported attack)
of an element of S against c. So, S is not a stable extension of AF.
Note that a coalition is considered as a whole and its members cannot be used
separately in an attack process. Ex. 8 suggests that admissibility is lost due to the
size of the coalition {b,c,d}, and that it would be more fruitful to consider two
independent coalitions {c,b} and {c,d}. A new formalization of coalitions in terms
of conflict-free maximal support paths has been proposed in [17]. However, it does
not enable to recover Dung’s properties.
Note that the lost of admissibility in Dung’s sense is not surprising: admissibility
is lost because it takes into account “individual” attack and defence, whereas, with
meta-argumentation and coalitions, “collective” attack and defence are considered.
5 Labellings in bipolar frameworks
This section addresses the question of labelling-based semantics in a BAF. A
labelling-based semantics relies upon a set of labels and is defined by specifying
the criteria for assigning labels to arguments. An example of labelling-based seman-
tics in a basic argumentation framework is given by [22] with the robust semantics.
More generally, several approaches have been proposed for valuing the arguments
in a classical argumentation framework (for example [24, 29, 30, 8, 15, 1, 33]). In
some of them, the value of an argument depends on its interactions with the other ar-
guments; in other ones, it depends on an intrinsic strength of the argument. Besides,
Karacapilidis Papadias [23] have proposed an argumentation web-tool, named
HERMES, for decision making in the medical field. Taking into account notions of
attacks and supports between arguments, this system permits the expression and
the weighting of arguments. The basic elements of this system are: a solution (an
answer to the question which is discussed) and a position (expressing the support
for, or the opposition to a solution or to another position). HERMES can label the
solutions and the positions by the status “active” or “inactive”. At the end of the
discussion, the “active” (resp. “inactive”) solutions are accepted (resp. rejected). An](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-91-320.jpg)
![78 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
“active” solution is a recommended choice among the other solutions concerning
a same question. Different recursive labellings are proposed in HERMES. But, the
value of a position p depends only on the active positions which are linked to p in
the acyclic discussion graph, and the value of a position is always binary.
In this section, we propose a limited use of the notion of labelling-based seman-
tics for a BAF: we show how bipolar interactions can be used for defining valuations
over the set of arguments, i.e. functions which assign a value to each argument of the
BAF (a further step would be to use such a valuation in order to select arguments,
that is to completely define labelling-based semantics in a BAF, in an analogous
way as what has been done in [16] for basic argumentation frameworks).
The approach presented here (see [13]) has the following features: the valuation
process takes place before the selection process; the valuation process makes use of
a rich set of values and not only two as in HERMES (so, it is called a gradual valu-
ation); the value assigned to an argument takes into account all the direct attackers
and supporters of this argument (it is not the case in HERMES in which the value of
an argument only depends on the active positions); so it is called a local valuation.
This proposition extends the works [22, 8, 16] to bipolar argumentation frame-
works as defined in Section 2. It follows the same principles as those already de-
scribed in [15] augmented with new principles corresponding to the “support” in-
formation. So, the three underlying principles for a gradual interaction-based local
valuation are:
• P1: The value of an argument depends on the values of its direct attackers and of
its direct supporters.
• P2: If the quality of the support (resp. attack) increases then the value of the
argument increases (resp. decreases).
• P3: If the quantity of the supports (resp. attacks) increases then the quality of the
support (resp. attack) increases.
The value of an argument is obtained with the composition of several functions:
• one for aggregating the values of all the direct attackers; this function computes
the value of the “attack”
• one for aggregating the values of all the direct supporters; this function computes
the value of the “support”
• one for computing the effect of the attack and of the support on the value of the
argument.
In the respect of the previous principles, we assume that there exists a completely
ordered set V with a minimum element Vmin and a maximum element Vmax. The
following formal definition for a gradual local valuation can be given.
Def. 14 Let A,Ratt,Rsup be a bipolar argumentation framework. Let a ∈ A with
Ratt
−
(a) = {b1,...,bn} and Rsup
−
(a) = {c1,...,cp}.
A local gradual valuation on A,Ratt,Rsup is a function v : A → V s.t.:
v(a) = g(hsup(v(c1), . . . , v(cp)),hatt(v(b1), . . . , v(bn))) with](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-92-320.jpg)
![4 Bipolar abstract argumentation systems 79
the function hatt (resp. hsup): V∗ → Hatt (resp. V∗ → Hsup)11 valuing the quality
of the attack (resp. support) on a, and the function g: Hsup ×Hatt → V with g(x,y)
increasing on x and decreasing on y. The function h, h = hatt or hsup, must satisfy:
1. if xi ≥ x′
i then h(x1,...,xi ...,xn) ≥ h(x1,...,x′
i ...,xn),
2. h(x1,...,xn,xn+1) ≥ h(x1,...,xn),
3. h() = α ≤ h(x1,...,xn) ≤ β, for all x1,...,xn
12.
Def. 14 produces a generic local gradual valuation. Let us give two instances of
the generic definition, to illustrate the different principles.
• A first instance is defined by Hatt = Hsup = V = [−1,1] interval of the real
line, hatt(x1,...,xn) = hsup(x1,...,xn) = max(x1,...,xn), and g(x,y) = x−y
2 (so,
we have α = −1, β = 1 and g(α,α) = 0).
• Another one is defined by V = [−1,1] interval of the real line, Hatt = Hsup =
[0,∞[ interval of the real line, hatt(x1,...,xn) = hsup(x1,...,xn) = Σn
i=1
xi+1
2 , and
g(x,y) = 1
1+y − 1
1+x (so, we have α = 0, β = ∞ and g(α,α) = 0).
The following table shows the values computed with both instances on some
simple examples:
Example with 1st instance with 2nd instance
No attack, no support: a v(a) = 0 v(a) = 0
Direct attack: a // b v(b) = −0.5 v(b) = −0.33
Direct support: a //
/o
/o
/o b v(b) = 0.5 v(b) = 0.33
Defence: a // b // c v(c) = −0.25 v(c) = −0.25
Sequence of supports: a //
/o
/o
/o b //
/o
/o
/o c v(c) = 0.75 v(c) = 0.4
Supported attack: a //
/o
/o
/o b // c v(c) = −0.75 v(c) = −0.4
Ex. 1 (cont’d) With the first (resp. second) instance, v(t1) = 1
4 (resp. 37
154 ).
Ex. 5 (cont’d) With the first and the second instances, v(b) = 0. In this case, there
is a perfect equilibrium13 between support and attack.
A local gradual valuation defined as above satisfies the following properties [13]:
• If Ratt
−
(a) = Rsup
−
(a) = ∅ then v(a) = g(α,α).
• If Ratt
−
(a) = ∅ and Rsup
−
(a) = ∅ then v(a) = g(α,y) ≤ g(α,α) for y ≥ α.
• If Ratt
−
(a) = ∅ and Rsup
−
(a) = ∅ then v(a) = g(x,α) ≥ g(α,α) for x ≥ α.
11 V∗ denotes the set of the finite sequences of elements of V, including the empty sequence. Hatt
and Hsup are ordered sets.
12 So, α is the minimal value for an attack (resp. a support) – i.e. there is no attack (resp. no
support) –, and β is the maximal value for an attack (resp. a support).
13 Note that it is not necessarily the case, and an appropriate choice of the function g enables to
give more importance to the attack than to the support.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-93-320.jpg)
![80 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
And we have the following comparative scale14:
Vmin ≤ g(α,y) ≤ g(α,α) ≤ g(x,α) ≤ Vmax
(for y ≥ α) (for x ≥ α)
Moreover the valuation proposed in Def. 14 satisfies the principles P1 to P3
(see [13] for a more detailed discussion).
6 Related issues and conclusion
In this chapter, an extension of [21]’s abstract argumentation framework has been
proposed in order to take into account two kinds of interaction between arguments
modelled with a support relation and an attack relation. In this abstract BAF, two
issues have been considered:
• taking into account bipolarity for defining acceptability semantics: either by en-
forcing the coherence of the admissible sets, or by turning a BAF into a meta-
argumentation framework using the concept of coalition;
• taking into account bipolar interactions for proposing gradual labellings for the
arguments.
6.1 Related issues about acceptability and bipolarity
Deflog [35]: DEFLOG argumentation system enables to express a support or an at-
tack between sentences in the language, with a new sentence using specific connec-
tors (one for each kind of interaction). Examples of sentences (with → for the attack
relation and ❀ for the support relation) are: a, b, (a ❀ b), (a → b), (c ❀ (a ❀ b)),
(d → (a ❀ b)). In DEFLOG, the notions of sequence of supports and of supported
attacks can be retrieved but at the language level (between sentences). Moreover, the
notion of conflict-free set proposed in DEFLOG corresponds to the notion of safe set
(no sentence which is, at the same time, supported and attacked by the set).
DEFLOG enables to define the dialectical interpretations (or extensions) of a
given set of sentences S: an extension is built from a partition (J,D) of S such that J
is conflict-free and attacks the sentences of D.
Note that the attack relation and the support relation are explicitly expressed in
the sentences. So, one can have an extension of a set S s.t. some supported sentences
by J do not belong to S. DEFLOG extensions correspond to [21]’s stable extensions
for DEFLOG theories that do not go beyond the expressiveness of Dung’s argu-
mentation frameworks, and note that a Dung’s AF can always be expressed in DE-
FLOG. So in this precise sense, DEFLOG’s extensions are a faithful generalization
of Dung’s stable extensions, allowing more expressiveness. Moreover, [35] gives
also a faithful generalization of Dung’s preferred extensions.
14 Using this scale, the values ≤ (resp. ≥) to g(α,α) are considered as negative (resp. positive)
ones even if g(α,α) = 0.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-94-320.jpg)
![4 Bipolar abstract argumentation systems 81
Evidence-based argumentation [28]: In this work, the fundamental claim is that
an argument cannot be accepted unless it is supported by evidence. So, special ar-
guments are distinguished: the prima-facie arguments (which do not require any
support to stand).
Arguments may be acceptable only if they are supported (indirectly) by prima-
facie arguments. This is evidential support. Moreover, only supported arguments
may attack other arguments.
Then, the notion of defence is rather complex: A set of arguments S defends an
argument a if S provides evidential support for a and S invalidates each attack on
a (either by a direct attack on the attacker of a or by rendering this attack unsup-
ported).
Following our definitions, a BAF is an abstract framework, where arguments may
stand and attack with or without support. However, evidential reasoning as proposed
by [28] could also be handled in a BAF in the following way: Given X a set of
arguments (which are considered as prima-facie arguments in a given application),
a notion of evidential support can be defined via a sequence of supports from an
argument of X. Then, the notion of attack can be restricted so that attackers be
elements of X, or receive evidential support from X. Finally, instead of choosing the
classical definition for “S defends a” (as presented in Def. 7), it can be required first
that S provides support for a and secondly that for each supported attack on a, one
argument of the sequence of supports is directly attacked by S.
6.2 Related issues about coalitions of arguments
Another way for defining acceptability semantics in a bipolar framework is to turn
a bipolar argumentation framework into a meta-argumentation framework . This
transformation has the following characteristics: the support relation is used in or-
der to identify “coalitions” (sets of arguments which can be used together without
conflict and which are related by the support relation) and the attack relation is used
in order to identify conflicts between coalitions and then to define new acceptability
semantics as in Dung’s framework.
The concept of coalition has already been related to argumentation.
Collective argumentation framework [9, 27]: A collective argumentation frame-
work is an abstract framework where the initial data are a set of arguments and
a binary “attack” relation between sets of arguments. The key idea is the follow-
ing: a set of arguments can produce an attack against other arguments, which is not
reducible to attacks between particular arguments. That is in agreement with our
notion of coalition, since in our work, a coalition is considered as a whole and its
members cannot be used separately in an attack process. The proposal by Nielsen
and Parsons is similar to Bochman’s proposal. Both proposals take the attacks be-
tween sets of arguments as initial data, and define semantics by properties on subsets
of arguments. However, Nielsen and Parsons propose an abstract framework which
allows sets of arguments to attack single arguments only, and they stick as close](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-95-320.jpg)
![82 Claudette Cayrol and Marie-Christine Lagasquie-Schiex
as possible to the semantics provided by Dung. In contrast, Bochman departs from
Dung’s methodology and give new specific definitions for stable and admissible sets
of arguments. Our proposal essentially differs from collective argumentation in two
points. First, we keep exactly Dung’s construction for defining semantics, but we ap-
ply this construction in a meta-argumentation framework (the coalition framework).
The second main difference lies in the meaning of a coalition: we intend to gather
as many arguments as possible in a coalition, and a coalition cannot be broken in
the defence process.
Generation of coalition structures in MAS [18, 2]: In multi-agent systems
(MAS), the coalition formation is a process in which independent and autonomous
agents come together to act as a collective. A coalition structure (CS) is a partition of
the set of agents into coalitions. Each coalition has a value (the utility that the agents
in the coalition can jointly get minus the cost which this coalition induces for each
agent). So the value of a CS is obtained by aggregating the values of the different
coalitions in the structure. One of the main problems is to generate a preferred CS,
that is a structure which maximizes the global value. Recently, [2] has proposed an
abstract system where the initial data are a set of coalitions equipped with a conflict
relation. A preferred CS is a subset of coalitions which is conflict-free and defends
itself against attacks. Coalitions may conflict for instance if they are non-disjoint or
if they achieve a same task.
However, the generation of the coalitions is not studied in [2]. So, one perspective
is to apply our work to the formation of coalitions taking into account interactions
between the agents. Arguments represent agents in that case. Indeed, it is very im-
portant to put together agents which want to cooperate (“supports” relation) and to
avoid gathering agents who do not want to cooperate (“attacks” relation). Then, the
concept of Cp-extension provides a tool for selecting the best groups of agents (w.r.t.
the given interaction relations).
More generally, the work reported here is generic and takes place in abstract
frameworks, since no assumption is made on the nature of the arguments. Argu-
ments may have a logical structure such as a pair explanation, conclusion, may just
be positions advanced in a discussion, or may be agents interacting in a multi-agent
system. All that we need is the bipolar interaction graph describing how the argu-
ments under consideration are interrelated. We think that this generic work should
stimulate discussion across boundaries.
6.3 Related issues about valuation and bipolarity
Most works about valuations of arguments take place in the basic framework. Some
of them consider intrinsic valuations, which express to what extent an argument
increases the confidence in the statement it promotes. Other approaches consider
interaction-based valuations. These approaches usually differ in the set of values
which are available.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-96-320.jpg)
![4 Bipolar abstract argumentation systems 83
However, very few works have been interested in valuations which handle both
support and attack interactions. Most of these works have been developed for spe-
cific applications.
Medical applications: The most influential work has been proposed in HERMES
system [23]. But there is no graduality (only two possible values with HERMES),
and some parts of the interacting arguments are not taken into account for the com-
putation of the value. See in Section 5.
Valued maps of argumentations: The bipolar valuation in argumentation has been
used for a collective annotation of documents.
Collective annotation models supporting exchange through discussion threads. A
discussion thread is initiated by an annotation about a given document. Then, users
can reply with annotations which confirm or refute the previous ones. Annotations
are associated with a social validation which provides a synthetic view of the dis-
cussions. The purpose of this validation is to identify annotations which are globally
confirmed by the discussion thread. It can also take into account an intrinsic value
of the annotations.
In [11, 12] a discussion thread is modelled by a BAF. The set of arguments con-
tains the nodes of the thread. Pairs of the support (resp. attack) relation correspond
to replies in the thread of the confirm (resp. refute) type. The social validation of a
given annotation is computed with the local bipolar valuation. Moreover, the bipo-
lar valuation procedure has been slightly modified in order to take into account an
intrinsic value of each annotation.
References
1. L. Amgoud. Contribution à l’intégration des préférences dans le raisonnement argumentatif.
PhD thesis, Université Paul Sabatier, Toulouse, July 1999.
2. L. Amgoud. Towards a formal model for task allocation via coalition formation. In Proc. of
AAMAS, pages 1185–1186, 2005.
3. L. Amgoud, J.-F. Bonnefon, and H. Prade. An argumentation-based approach to multiple
criteria decision. In Proc. of ECSQARU, pages 269–280, 2005.
4. L. Amgoud, C. Cayrol, and M. Lagasquie-Schiex. On the bipolarity in argumentation frame-
works. In Proc. of the 10th NMR-UF workshop, pages 1–9, 2004.
5. L. Amgoud, C. Cayrol, M.-C. Lagasquie-Schiex, and P. Livet. On bipolarity in argumentation
frameworks. International Journal of Intelligent Systems, 23:1062–1093, 2008.
6. S. Benferhat, D. Dubois, S. Kaci, and H. Prade. Bipolar representation and fusion of prefer-
ences in the possibilistic logic framework. In Proc. of KR, pages 158–169, 2002.
7. C. Berge. Graphs and Hypergraphs. North-Holland Mathematical Library, 1973.
8. P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence,
128 (1-2):203–235, 2001.
9. A. Bochman. Collective argumentation and disjunctive programming. Journal of Logic and
Computation, 13 (3):405–428, 2003.
10. C. Boutilier. Towards a logic for qualitative decision theory. In Proc. of KR, pages 75–86,
1994.
11. G. Cabanac, M. Chevalier, C. Chrisment, and C. Julien. A social validation of collaborative
annotations on digital documents. In Proc. of IWAC, pages 31–40, 2005.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-97-320.jpg)
![5 Complexity of Abstract Argumentation 89
some C-hard, L′ : L′ ≤p
m L”. For the classes introduced in Section 2.2 we have the
following results of Cook [10] and Wrathall [45].
Theorem 5.1.
a. 3-SAT is NP–complete; 3-UNSAT is coNP–complete.
b. QSATΣ
k is Σ p
k –complete; QSATΠ
k is Π p
k –complete.
For the results discussed later in this Chapter with very few exceptions the complex-
ity classifications use reductions from 3-SAT or 3-UNSAT.
2.4 More Advanced Ideas
The topics outlined above provide sufficient background for the majority of com-
plexity analyses on decision problems for AFs. There are, however, a number of
developments – in particular in the related frameworks discussed in Section 4.1 –
which occur in more recent work on complexity of abstract argumentation: here
we briefly introduce the basic notions of oracle-based complexity classes and the
models proposed in work of Cook and Reckhow [11] in order to capture relative
complexity of proof systems.
Oracle computations, are defined in terms of the availability of a device (or ora-
cle) that at the cost of a single step in the algorithm provides the answer to a given
language membership query, e.g., oracle computations using 3-SAT may construct
a 3-CNF ϕ, query the 3-SAT oracle as to whether ϕ ∈ 3 − SAT with the answer
determining subsequent steps taken. This notion, when coupled with oracles for
NP-complete languages, gives rise to a range of complexity classes differentiated
by the particular restrictions placed on the manner in which such calls are made.
One important representative of such classes is the so-called difference class, Dp
formally defined as those languages, L whose members are the intersection of a
language L1 ∈ NP with a language L2 ∈ coNP: a language L ∈ Dp can thus be de-
cided by a polynomial time algorithm that is allowed to make at most two calls on
an NP oracle (which, by virtue of Thm 5.1(a) can be assumed to be 3-SAT): given
an instance x of L, test x ∈ L1 by forming the appropriate 3-SAT instance, τ1(x),
and calling the oracle; if the response is positive, then x ∈ L2 is tested in the same
way, forming the instance τ2(x) using a second oracle call to verify τ2(x) ∈ 3−SAT,
i.e. τ2(x) ∈ 3−UNSAT. More generally, the class of languages for which polynomial
time algorithms using at most f(|x|) calls on some oracle for a language complete in
a class C is denoted by PC[ f(|x|)] (so that Dp ⊆ PNP[2]). Where no restriction is placed
on the oracle invocation the notation PC
is used. As will be seen from the results
reviewed in Sect. 4.1, the algorithmic “base class” can be defined within arbitrary
complexity classes, not simply P: this leads to classes such as NPC
, etc.
The formalism from [11] has been adopted in order to relate the efficiency of
proof procedures for credulous reasoning in AFs to more widely known proof pro-
cedures in propositional logic, e.g., resolution, tableau-based, sequents, etc. The
model starts from an abstraction of “proof system” Π for a (coNP) language, L as](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-103-320.jpg)
![5 Complexity of Abstract Argumentation 91
3.1 Intractability Results in Preferred and Stable Semantics
We now turn to the problems defined in Table 5.1 with respect to the other extension
based semantics - Preferred and Stable - introduced in Dung [22]. Our main aim is
to outline the constructions of Dimopoulos and Torres [21] and Dunne and Bench-
Capon [28] from which the classifications shown in Table 5.2 result.
Table 5.2 Complexity of Decision Problems in Preferred (PR) and Stable Semantics (ST)
A VERPR coNP–complete [21]
B CAPR NP–complete [21]
C SAPR Π p
2 –complete [28]
D NEPR NP–complete [21]
E CAST NP–complete [21]
F SAST coNP–complete/Dp–complete See discussion below.
G EXST NP–complete attributed to Chvatal in [33]; also [16, 21, 32].
All of the lower bound results in Table 5.2 are obtained as variations on what we
shall refer to as the standard translation from 3-CNF formulae to AFs.
Definition 5.1. Given ϕ(z1,...,zn) a 3-CNF with clauses {C1,...,Cm} the AF,
Gϕ(Aϕ,Rϕ) constituting the standard translation from ϕ has
Aϕ = {ϕ}∪{C1,...,Cm}∪{z1,...,zn}∪{¬z1,...,¬zn}
Rϕ = {Cj,ϕ : 1 ≤ j ≤ m} ∪ {zi,¬zi,¬zi,zi : 1 ≤ i ≤ n}
∪ {yi,Cj : yi is a literal (i.e., zi or ¬zi) of the clause Cj}
The AF described in Defn 5.1 is, modulo some minor simplifications, identical to
that originally used in [21]. This framework provides an extremely versatile mecha-
nism that underpins almost all the complexity analyses of extension based semantics
in abstract argumentation frameworks.3 The basic form of the standard translation
suffices to establish (B) and (E) of Table 5.2, whereas (A) and (D) follow from quite
simple modifications to it.
For example consider the claim in Table 5.2(B) that CAPR is NP–complete. First
note that CAPR ∈ NP since we may use the set of all admissible subsets of A con-
taining x as witnesses to G,x ∈ CAPR. We may then use the standard translation
to prove 3-SAT ≤p
m CAPR: given ϕ form the instance Gϕ,ϕ of CAPR. If ϕ ∈ 3-SAT
then the literals instantiated to ⊤ by a satisfying assignment indicate a subset S of the
arguments {z1,...,zn,¬z1,...,¬zn} for which S ∪ {ϕ} is admissible. On the other
hand if Gϕ,ϕ ∈ CAPR then an admissible set containing ϕ must include a conflict-
free subset of literals arguments that collectively attack all of the clause arguments:
instantiating the corresponding literals to ⊤ produces a satisfying assignment of ϕ.
3 Exceptions are a specialized case of CAPR and a select number of reductions dealing with value-
based argumentation frameworks, cf. [25, Thm. 8(a), Thms. 23–25], and later in this chapter.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-105-320.jpg)
![92 Paul E. Dunne and Michael Wooldridge
Adapting the standard translation by adding a new argument, ψ that is attacked
by ϕ and attacks all of the literal arguments yields an AF, Hϕ for which (both) Hϕ ∈
NEPR and Hϕ ∈ EXST hold if and only if ϕ ∈ 3-SAT. Table 5.2(A) is an immediate
consequence of the former property using the special case of verifying if the empty
set is a preferred extension.
The discussion above accounts for all the cases given in Table 5.2 with the ex-
ceptions of SAPR and SAST . We first address the apparent ambiguity in the clas-
sification of SAST – Table 5.2(F). A further modification to the framework Hϕ
by which the new argument ψ now attacks every argument in Gϕ, gives an AF in
which ψ belongs to every stable extension if and only if ϕ ∈ 3-UNSAT so giving
a coNP-hardness lower bound.4 In principle, one appears to have a coNP method
via “A,R,x ∈ SAST ⇔ ∀ S ⊆ A (VERST (A,R,S) ⇒ (x ∈ S)”. There is,
however, a possible objection: G,x ∈ SAST even when G has no stable exten-
sion whatsoever, i.e., G ∈ EXST .5 In order to deal with this objection one might
require as a precondition of G,x ∈ SAST that G ∈ EXST leading to an easy Dp upper
bound: positive instances are characterised as those in CAST ∩{A,R,x : ∀ S ⊆
A (VERST (A,R,S) ⇒ (x ∈ S)}
The matching Dp–hardness lower bound provides another illustration of the flex-
ibility of the standard translation: instances ϕ1,ϕ2 of the canonical Dp–hard prob-
lem 3-SAT,3-UNSAT being transformed to an instance K,ψ2 of SAST . The con-
struction is illustrated in Fig. 5.1. We leave the reader to verify that this framework
satisfies both EXST and has ψ2 a member of every stable extension if and only if
ϕ1 ∈ 3-SAT and ϕ2 ∈ 3-UNSAT.6
α
Φ2
Φ1
Ψ2 attacks all arguments
Ψ1
Ψ2
Φ1
H
Φ2
H
Ψ1 attacks all arguments
Fig. 5.1 The reduction from ϕ1,ϕ2 ∈ 3-SAT,3-UNSAT to K,ψ2 ∈ SAST
4 In [28] the coNP-hardness result is attributed to Dimopolous and Torres who do not explicitly
consider this problem: the commentary of [28, p. 189] observes the lower bound follows from an
easy modification to a construction in [21].
5 Resulting in AFs for which A,R,x ∈ SAST and A,R,x ∈ CAST for every x ∈ A, i.e. every
argument is sceptically accepted but none credulously so.
6 The construction in Fig. 5.1 has not previously appeared in the literature. The issues concerning
precise formulations of SAST appear first to have been raised in [31].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-106-320.jpg)
![5 Complexity of Abstract Argumentation 93
Although the remaining case in Table 5.2 – Π p
2 -completeness of SAPR deals with
a, notionally, harder class of languages, again the lower bound follows by adapting
the standard translation: in this case to instances ϕ(y1,...,yn,z1,...,zn) of QSATΠ
2 .7
It is worth noting that the decision problem actually considered via this reduction,
in [28], is the so-called coherence property of AFs, i.e., whether G is such that
EPR(G) = EST (G): this problem is shown to be Π p
2 -complete with the classification
of SAPR an immediate consequence of the reduction used.
3.2 Dialogue and Relationships to Proof Complexity
The standard translation from 3-CNF (which easily generalises to arbitrary CNF)
gives rise to one concrete interpretation of argumentation process in terms of log-
ical proof: in particular, the concept of dialogue based procedures by which two
parties attempt to reach agreement on the (credulous) acceptability status of some
argument, will be discussed in Chapter 6. If one considers applying such procedures
to determining the status of ϕ (in the standard translation) then a demonstration
that ϕ is not admissible corresponds to a formal logical proof that the propositional
formula ¬ϕ is a tautology. There are, of course, a number of widely used and well-
studied proof mechanisms for propositional logic, the question of interest in terms
of complexity in argumentation, is what one can state about the efficiency of dia-
logue based argumentation processes: that is to say, using the comparative schema
proposed in [11], how does the use of dialogue approaches compare to other tech-
niques? This question has been examined with respect to one particular credulous
reasoning process: the two-party immediate response protocol (TPI) introduced in
[44]. Informally, TPI-dialogues involve two protagonists (PRO and CON) debating
the acceptability of a given argument x: PRO claiming x to be acceptable and CON
adopting the opposite stance. The dialogue is set in the context of an AF where each
player takes turns advancing arguments in A: PRO starts by putting forward x. A re-
quirement of the game is that, whenever possible to do so, a player must put forward
an argument that attacks the most recent argument put forward by their opponent:
where this is not possible the player must backtrack to a (specified) earlier point in
the discussion or concede. In [29] the number of moves required in this game when
played on the standard translation of an unsatisfiable CNF is considered. The TPI
procedure turns out to be equivalent to a standard propositional proof theory – the
so-called CUT-free Gentzen calculus [34] and, as a consequence of [42], there are
TPI-disputes requiring exponentially many moves in order to resolve the status of
particular arguments.
7 The reduction originally presented in [28] is not restricted to 3-CNF formulae but describes a
general translation from arbitrary propositional formulae over the logical basis {∧,∨,¬}.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-107-320.jpg)
![94 Paul E. Dunne and Michael Wooldridge
4 Complexity in Related Abstract Frameworks
As described in several chapters there are a number of abstract treatments of argu-
mentation that build on the basic structures and semantics proposed in Dung [22].
Among such are assumption-based frameworks (ABFs) discussed in Chapter 10; the
closely related deductive systems considered in Chapter 7; and the value-based ar-
gumentation frameworks (VAFs) whose elements have been presented in Chapter 3.
Our aim in this section is to review the range of complexity-theoretic results that
have been proved within these models. In general we will not give detailed defini-
tions of relevant ideas and refer the reader to the appropriate chapter for these.
4.1 Complexity in Assumption-based Argumentation
Assumption-based frameworks [5], can be interpreted as specific concrete interpre-
tations of abstract argumentation frameworks, i.e. as mechanisms for constructing
the structure A,R by generating arguments in A and attacks between these. This
approach starts from some deductive system – (L,R) in which L is a formal lan-
guage, e.g., well-formed propositional sentences, and R a set of inference rules of
the form α ← {α1,...,αn} describing the conclusions (α ∈ L) that are supported
by the premises {α1,...,αn} ⊆ L. Such systems have an associated derivability re-
lation, ⊢ : 2L → L; Δ ⊢ α holds whenever α may be obtained (via R) from Δ.
It should be noted that attention is restricted to theories in which the underlying
derivability relation is monotonic, i.e., (Δ ⊢ α) ⇒ (Δ′ ⊢ α) for any Δ′ ⊇ Δ.
The key elements added are assumption sets, A ⊆ L and the contrary function, –
which is a (total) mapping from α ∈ A to its contrary α ∈ L. In very simplified terms,
(sets of) assumptions define the basis for atomic arguments (in AFs), and the con-
trary mapping provides the reasons underpinning attacks between arguments. Just
as the semantics of “collection of acceptable arguments” in AFs is given by different
notions of extension, so too in ABFs the objects of interest are subsets of assump-
tions defining extensions. In total, viewing an argument as “a statement derivable
from some set of assumptions”, leads to the notion of attack between arguments
as (the argument) Δ ⊢ α attacks (the argument) Δ′ ⊢ β if Δ ⊢ γ for an assumption
γ ∈ Δ′. In this way we can take any basic semantics w.r.t. AFs, and define analogues
w.r.t. ABFs, e.g., a set of assumptions, Δ, is conflict-free if for every α ∈ Δ it is not
the case that Δ ⊢ α.
Similarly, one may formulate each of the decision problems of Table 5.1 in
ABF settings. There are, however, a number of important distinctions: as a result
complexity-theoretic treatments of ABFs use significantly different techniques to
those discussed in Section 3. In particular,
a. In AFs both arguments, A, and the attack relation, R, are specified explicitly. In
ABFs these are implicit and dependent on the underlying set of assumptions A
and the precise deductive theory embodied within (L,R).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-108-320.jpg)
![5 Complexity of Abstract Argumentation 95
b. The deductive system (L,R) is not limited to classical propositional logic with
the contrary being simply logical negation, e.g., (L,R) and – could be instantiated
in terms of a number of non-monotonic logics such as the default logic of [39].
Since the attack relation is defined between assumption sets, in principal one may
express any ABF (L,R),A,– as an AF, A,R: A = 2A, R = {Δ,Δ′ : Δ ⊢
γ for some γ ∈ Δ′}. There is, however, one complication: in practice not every
subset of A is of interest, only those that satisfy the technical requirement of be-
ing closed, i.e., Δ ⊆ A is closed if and only if α ∈ A Δ ⇒ ¬(Δ ⊢ α). While
this provides a starting point for algorithms and upper bound constructions, for
lower bounds such approaches yield little of benefit: |(L,R),A,–| is exponentially
smaller than the corresponding AF.
A detailed investigation of complexity-theoretic issues within ABFs has been pre-
sented in a series of papers by Dimopolous, Nebel and Toni [18, 19, 20]. Using the
notation LABF to distinguish ABF instantiations of decision problems L as presented
in Table 5.1 and LABF,LT with reference to different formal theories LT = (L,R), a
key element in exact complexity characterisations is the computational complexity
of the derivability relation for the underlying logic, i.e., the derivability problem
(DER) for the formal theory (L,R), has instances Δ,α – Δ ⊆ L, α ∈ L – accepted
if and only if Δ ⊢ α. For example, in standard propositional logic the derivability
problem is coNP–complete: Δ ⊢ α if and only if the formula α ∨ ∨ϕ∈Δ ¬ϕ is a
tautology.
Combining the notion of “oracle complexity classes” as described in Section 2.4
using oracles for DER(Δ,α) provides a generic approach to obtaining upper bounds
on the complexity of decision problems within ABFs. For example, consider the
decision problem VER
ABF,LT
ST of verifying that a given set of assumptions defines
a stable extension within (L,R),A,–, where the derivability problem for LT is in
some class C. In order to decide if Δ is accepted:
1. Check that Δ is closed.
2. Check that Δ is conflict-free, i.e., ∀ α ∈ Δ ¬(Δ ⊢ α ).
3. Check that Δ attacks every assumption α ∈ Δ, i.e., Δ ⊢ α for each α ∈ AΔ.
All of these stages can be carried out using |A| calls to an oracle for DER: (1) tests
(Δ,α) ∈ DER for α ∈ A Δ; (2) involves a further |Δ| calls; and (3) a final set of
|AΔ| calls. In consequence, VER
ABF,LT
ST ∈ PC
.
Concentrating on upper bounds for Table 5.1 within the most general settings8
the upper bounds obtained in [20] are stated in Table 5.3.
It may be noted that with the exception of upper bounds on stability related prob-
lems, those relating to preferred and admissible sets of assumptions are rather higher
than might be expected having allowed for the additional overhead associated with
calls to the DER oracle, e.g., CAADM ∈ NP whereas CAABF
ADM ∈ coNPNPC
rather than
8 That is to say, no specific properties of the underyling frameworks are assumed, e.g., the property
“flatness” described in [5].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-109-320.jpg)
![96 Paul E. Dunne and Michael Wooldridge
Table 5.3 Upper bounds for main decision problems in ABFs
Problem Semantics Instance (ABF) ABF bound (DER ∈ C) Instance (AF) AF bound
VERs Admissible (L,R),A,–, Δ ⊆ A coNPC
A,R, S ⊆ A P
VERs Preferred (L,R),A,–, Δ ⊆ A coNPNPC
A,R, S ⊆ A coNP
VERs Stable (L,R),A,–, Δ ⊆ A PC
A,R, S ⊆ A P
CAs Admissible (L,R),A,–, ϕ ∈ L NPNPC
A,R, x ∈ A NP
CAs Preferred (L,R),A,–, ϕ ∈ L NPNPC
A,R, x ∈ A NP
CAs Stable (L,R),A,–, ϕ ∈ L NPC
A,R, x ∈ A NP
SAs Preferred (L,R),A,–, ϕ ∈ L coNPNPNPC
A,R, x ∈ A Π p
2
SAs Stable (L,R),A,–, ϕ ∈ L NPC
A,R, x ∈ A coNP/Dp
NPC
. That the reasoning problems exhibit “higher than expected” complexity in
ABFs is not on account of the (additional) closure checking stage, despite the fact
this does not feature in corresponding AF algorithms.9 The increased complexity
arises from the nature of the attack relation, e.g. deciding G,x ∈ CAPR involves:
guess S ⊆ A, confirm that x ∈ S and S is conflict-free; check S attacks each y that at-
tacks S. Suppose, however, we consider the analogous version for CA
ABF,LT
PR : guess
Δ ⊆ A; confirm that Δ is closed, Δ ⊢ ϕ, and Δ does not attack itself; finally check
that any closed assumption set attacking Δ is itself attacked by Δ. This final stage
requires tests involving Δ and all other sets of assumptions, rather than (as is effec-
tively the case in AFs and suffices for stability) checking a property of Δ in relation
to single assumptions.
We conclude this overview of complexity in ABFs by noting that for the credu-
lous and sceptical reasoning variants, the classifications of Table 5.3 turn out to be
optimal for a wide range of instantiations of (L,R),A,– modelling non-classical
logics such as DL [39], AEL [38], etc. The typical approach to lower bound proofs,
e.g., as illustrated in the specific examples of DL and AEL, uses bounds on the com-
plexity of DER: the cases DL and AEL being coNP–complete. While for DL, one can
show CA
ABF,DL
PR ∈ NPNP = Σ p
2 , no reduction in the generic upper bound is possi-
ble for AEL: CA
ABF,AEL
PR ∈ NPNPNP
, i.e Σ p
3 . The Σ p
2 (resp. Σ p
3 ) hardness reductions
use instances of QSATΣ
2 (resp. QSATΣ
3 ) to define ABFs instantiated as DL (resp. AEL)
systems: detailed constructions may be found in [20].
4.2 Complexity in Value-based Argumentation Frameworks
We recall, from Chapter 3, that value-based argumentation frameworks (VAFs) aug-
ment the basic A,R abstraction of Dung’s AFs by introducing a finite set of values,
V, and a mapping η : A → V describing the abstract value, η(x) endorsed by x ∈ A,
so a VAF is described via a four tuple, G(V) = A,R,V,η. In VAFs the underlying
structures are the completely abstract frameworks of [22]: whereas ABFs provide
9 In fact this stage is redundant in a number of ABF models, e.g DL.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-110-320.jpg)
![5 Complexity of Abstract Argumentation 97
a basis for argument construction and attacks between arguments, the motivation
behind VAFs is to offer an explanatory mechanism accounting for choices between
distinct justifiable collections, S and T, which are not collectively acceptable, i.e., S
and T may be admissible under Dung’s semantics, however, S ∪T fails to be. Such
occurrences raise the question of the supporting reasons as to which of S or T is
adopted: as developed in Chapter 3, VAFs rationalize these choices in terms of value
orderings on V. Any commitment to a preference of vi ∈ V over vj ∈ V (written
vi ≻ vj) induces a simplification of A,R,V,η whereby every attack x,y ∈ R for
which η(x) = vj and η(y) = vi can be removed. Under the restrictions discussed
in Chapter 3, applying this refinement of R, any total ordering, α of V, will result
in an acyclic framework, G
(V)
α : as has been noted elsewhere such a framework will
have EGR(G
(V)
α ) = EPR(G
(V)
α ) = EST (G
(V)
α ). Value orderings thus motivate the two
principal decision problems that have been reviewed in algorithmic and complex-
ity studies of VAFs: Subjective Acceptance (SBA) and Objective Acceptance (OBA) .
Both take as an instance a VAF G(V) and argument x ∈ A.
G(V),x ∈ SBA ⇔ ∃ α a total ordering of V : CAPR(G
(V)
α ,x)
G(V),x ∈ OBA ⇔ ∀ α total orderings of V : CAPR(G
(V)
α ,x)
The acyclic form of G
(V)
α gives CAPR(G
(V)
α ,x) ∈ P hence SBA∈NP and OBA∈coNP.
Both bounds turn out to be exact, as shown in [30, 3]: these again use variants
of the standard translation from Defn 5.1. This may appear surprising given that
although the standard translation is well-suited to relating subsets (of arguments)
to instantiations of propositional variables, it is less clear how it could be applied
to deal with relating orderings of values to such instantiations. The device used in
[30] associates a “neutral” value with the formula and clause arguments in the VAF
defined from ϕ(Zn) and replaces the mutually attacking pairs {zi,¬zi} with a cycle
of four arguments pi → qi → ri → si → pi. Two arguments (pi and ri) are assigned
the value posi to promote “zi = ⊤ in a satisfying assignment of ϕ” while the others
(qi and si) are given the value negi in order to promote “zi = ⊥ in a satisfying
assignment of ϕ”. Although their definition and these classifications suggest that
SBA (resp. OBA) are closely related to CAPR (resp. SAPR for coherent AFs) , recent
work, discussed in Section 5 highlights several differences between the nature of
decision problems in VAFs and, what appear to be analogous problems in AFs.
5 Recent Developments
The computational complexity of the standard semantics (preferred, stable,
grounded) in AFs settings is, in the most general case, now well understood: ex-
act complexity bounds having been established for each of the canonical decision
problems given in Table 5.1 with respect to these semantics. There has, however,
continued to be extensive development of this aspect of the formal theory of ar-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-111-320.jpg)
![98 Paul E. Dunne and Michael Wooldridge
gumentation, driven by a number of reasons. Among these – and forming the top-
ics reviewed in this section – one has: the various proposals for novel extension-
based semantics, some of which have been discussed in Chapter 2, e.g., Ideal se-
mantics [23, 24], Semi-stable semantics [6, 7], Prudent semantics [12]. A second
consideration concerns the extent to which intractability issues may be alleviated by
constructing efficient algorithmic approaches applicable to AFs which are restricted
in some way, e.g., by analogy with the known tractable case of acyclic topologies.
5.1 Novel extension-based semantics and their complexity
In this section we outline recent treatments of complexity in three of the develop-
ments of Dung’s standard AF semantics: prudent, ideal, and semi-stable semantics.
We recall that the rationale underlying prudent semantics stems from the po-
tential problematic side effects that might eventuate by regarding as collectively
acceptable, arguments {x,y} for which x “indirectly attacks” y. An indirect attack
by x on y is present in A,R if “there exists a finite sequence x0,...,x2n+1 such that
(1) x = x0 and y = x2n+1 and (2) for each 0 ≤ i 2n, xi,xi+1 ∈ R” [22, p. 332].
It should be noted that this formulation, which we have quoted verbatim, presents
some ambiguity, which is significant from complexity-theoretic and semantic per-
spectives: it fails to distinguish “indirect attacks” in which no argument is repeated,
i.e., “simple paths”; from those in which arguments but not attacks may be repeated;
from those in which attacks may be repeated, e.g. the cases in Fig. 5.2.
x3
x1x4
x0
x2x5 x6x7 x7
x3
x0 x1x4 x5
x2
x3
x2
x1
x0
(a)
(b)
(c)
Fig. 5.2 Three possible forms of “indirect” attack – (a) Simple; (b) x1 = x4; (c) x1,x2 = x4,x5
The concept of conflict-free set from [22], is replaced under the prudent seman-
tics by that of prudently conflict-free set, i.e., one in which there is no indirect attack
between any two members. There are evident interpretative issues with cases (b) and](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-112-320.jpg)
![5 Complexity of Abstract Argumentation 99
(c) in Fig. 5.2, however, the most natural interpretation (where an indirect attack is
a simple path) has one significant computational drawback.
Fact 1 Given A,R and S ⊆ A deciding if S is prudently conflict-free is coNP–
complete even if S contains only two arguments.
Proof. Immediate from the result of Lapaugh and Papadimitriou [35] which shows
deciding the existence of a simple even length path between two specified arguments
in a directed graph to be NP–complete. The extension to odd length simple path is
trivial, so the lemma follows by observing that a prudently conflict-free set is one in
which no simple odd length path is present between two arguments.
Noting the definitions of admissible set, preferred and stable extensions from [22]
and the fact that conflict-freeness is an integral part of these, the result of Fact 1
immediately allows us to deduce that under the prudent semantics (so that “conflict-
free set” becomes “prudently conflict-free”) the respective verification problems are
all coNP–complete. Complexity of credulous and sceptical acceptance under the
prudent semantics has yet to be studied in depth. The intractability status of key
decision problems in this semantics is predicated on the interpretation of “indirect
attack” given by (a), i.e., as a simple path. These do not hold if repeated attacks
– Fig 5.2(c) – are used: here polynomial time methods are available. The status of
allowing repeated arguments – Fig 5.2(b) – is, to the authors’ knowledge still open.
The ideal semantics were originally proposed with respect to ABFs, but have a
natural formulation in AFs: S ⊆ A is an ideal set within A,R if S is both admis-
sible and a subset of every set in EPR(A,R); S is an ideal extension if it a maximal
ideal set. Detailed studies of the complexity of ideal semantics in AFs are presented
in [26, 27]. The treatment of complexity issues presented in these papers exploit a
number of more advanced techniques, however, the hardness proofs continue to be
built on the standard translation of CNF formulae to AFs. In terms of the canoni-
cal problems in Table 5.1, the verification problem (for ideal sets) is shown to be
coNP-complete, placing this decision problem at the same level of complexity as the
verification problem for preferred extensions. Arguably the most radical technique
exploited – although widely applied in a number of earlier complexity-theoretic
analyses – is the use of randomized reductions coupled with structural complexity
results from [8, 9], as opposed to standard many-one reducibility (≤p
m) which fea-
tures in all of the results discussed earlier.10 Combining these elements, the verifica-
tion problem (for ideal extensions) and credulous acceptance problems are shown to
be complete for the (conjectured to be) subclass of PNP in which oracle queries are
non-adaptive (denoted PNP
|| ). We note that the upper bounds from [26, 27] do not use
randomized elements. This complexity class also arises in the known lower bounds
for both credulous and sceptical acceptance in semi-stable semantics presented in
[31]. These lower bounds again apply the structural characterizations of [8] (using
≤p
m reducibility, i.e., not randomized). Upper bounds, however, are Σ p
2 and Π p
2 , i.e.,
exact classifications of reasoning problems in semi-stable semantics is open.
10 Relevant background is outlined in [27] and described in full in [26]. The actual “randomized”
element is not explicit but arises from results of [43] for the satisfiability variant used.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-113-320.jpg)
![100 Paul E. Dunne and Michael Wooldridge
5.2 Properties of restricted frameworks
The complexity lower bounds discussed above describe worst-case scenarios, i.e.,
the fact that, for example CAPR is NP–hard, does not imply that every algorithm
on every instance will entail unrealistic computational overheads. As will be seen
in Chapter 6, if the AF is acyclic, then all of the canonical decision problems of
Table 5.1 have polynomial time solutions. In consequence a natural question to con-
sider is whether other graph-theoretic restrictions also result in frameworks with
efficient decision processes. Examining this question leads to two classes of results:
positive outcomes of the form “decision problem L has a polynomial time algorithm
in AFs satisfying some property P”; and negative classifications of the form “deci-
sion problem L in frameworks satisfying property P are no easier than the general
case”. Results of the first type extend (beyond acyclic frameworks) the range of
AFs for which tractable solutions exist. Recent work has added to the class of such
frameworks: symmetric AFs – those for which x,y ∈ R ⇔ y,x ∈ R – in work
of Coste-Marquis et al. [13]; bipartite AFs (those for which A may be partitioned
into two conflict-free sets) [25]. Using the notion of “treewidth decomposition”, see
e.g., [4] a select number of problems whose instances are single AFs such as EXST ,
NEPR admit linear time algorithms given a treewidth decomposition of width k as
part of the instance: the construction of these algorithms rely on a deep result (Cour-
celle’s Theorem [14, 15]) demonstrating how efficient algorithms for testing graph-
theoretic properties may be obtained given an appropriate logical description of the
property (the so-called Monadic Second Order Logic) and a bounded treewidth de-
composition of the graph. For more details on this approach and its application in
AF settings we refer the reader to [2] and [25].
There are, however, a number of natural properties that fail to yield any reduction
in complexity. Typically the approach adopted in proving such results is to demon-
strate that frameworks with the property of interest are general enough to effectively
“simulate” any framework, e.g., if S is admissible in A,R then S∪T is admissible
in A ∪ B,R′ where the latter AF has a particular property. Using such methods,
[25] shows that no reduction in complexity arises in: k-partite AFs (k ≥ 3); planar
systems; and those in which no argument attacks or is attacked by more than two
other arguments. This remains the case when all restrictions hold simultaneously.
We conclude this overview of recent work by returning to the issue of complexity
in VAFs. In contrast to the class of positive cases that have been identified with AFs,
the situation with VAFs turns outs out to be far more negative. The most extreme
indication of this status is the following result of [25].
Fact 2
a. SBA is NP–complete and OBA is coNP–complete even if the underlying graph is
a binary tree and every v ∈ V is associated with at most three arguments.
b. For every ε 0 SBA is NP–complete and OBA is coNP–complete even if the un-
derlying graph is a binary tree and |V| ≤ |A|ε.
Both constructions use reductions from variants of 3-SAT/3-UNSAT, however, these
differ in a number of ways from the standard translation (which is clearly is not](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-114-320.jpg)
![5 Complexity of Abstract Argumentation 101
a binary tree). The situation highlighted by results such as Fact 2 provides further
indications that the nature of SBA/OBA in VAFs, while superficially similar to, is in
fact radically different from that of CA/SA in AFs.
6 Conclusions and Further Research
In this final section we outline some areas of research which offer a variety of chal-
lenging directions through which the algorithmic and complexity foundations of
abstract argumentation may be further advanced. We stress that our aim is to focus
on general areas rather than particular open questions as such: the reader who has
followed the earlier exposition will have noted that a number of specific open issues
have already been raised in the text.
6.1 Average case properties
As discussed in Section 5.2, the lower bounds on problem complexity are worst-
case, so leaving open the possibility that feasible algorithms may be available in
suitable contexts. In addition to the use of restrictions on the form of instances one
other approach that has been widely considered in the theory of algorithms is the
study of average-case complexity. Underpinning this approach one considers a prob-
ability distribution, µ, on instances of a decision problem – often, but not invariably
so, µ is the uniform distribution whereby each instance is equally likely, proceeding
to define the average-case run time of an algorithm P on instances of size n of L as
∑x∈I(n) µ(x)ρ(P,x) where ρ(P,x) is the run-time of P on instance x. Formal defini-
tions of average-case complexity classes may be found in [36]. To date surprisingly
little work has been carried out concerning the application of average-case methods
to decision problems in AFs either in terms of algorithmic development or in consid-
ering the limitations of such approaches. It remains open to what extent techniques
such as those applied to other intractable problems, e.g., [1] for the NP–complete
Hamiltonian cycle problem, or [46] for CNF satisfiability could be replicated in AF
settings. Of some relevance to such approaches are so-called “phase-transition” ef-
fects, which received much attention in the mid-late 1990s as potential indicators
of factors separating tractable and intractable classes of problem instances, e.g., the
studies of random CNF-SAT from [37, 40]. Analytic studies of such effects appears
to indicate connections between suitable witnessing structures, e.g., satisfying as-
signment, being present “almost certainly” and the performance of algorithms to
identify such structures. Of some interest in the context of AF semantics are the
results of [41, 17] which give conditions ensuring that a random AF “almost cer-
tainly” has a stable extension. There has as yet, however, been no detailed study
of the implications of these results for fast on average methods for identifying or
enumerating stable extensions. In the same way that the analyses of [41, 17] relate](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-115-320.jpg)
![102 Paul E. Dunne and Michael Wooldridge
to the existence of stable extensions in AFs, it would be of some interest to exam-
ine to consider existence properties of other solution structures in random AFs and
algorithmic consequences.
6.2 Approaches to dynamic updates
An important feature of the argumentation forms discussed so far is that, in practice,
these are not static systems: typically an AF, A,R, represents only a “snapshot”
of the environment, and, as further facts, information and opinions emerge the form
of the initial view may change significantly in order to accommodate these. For
example, additional arguments may have to be considered so changing A; existing
attacks may cease to apply and new attacks (arising from changes to A) come into
force. It is clear that accounting for such dynamic aspects raises a number of issues
in terms of assessing the acceptability status of individual arguments. As with the
study of average-case properties, the treatment of algorithms and complexity issues
relating to determining argument status in dynamically changing environments has
been somewhat neglected. Thus, given A,R and S ⊆ A for which S ∈ Es(A,R)
according to some semantics s, natural decision questions are: does x ∈ S continue
to be credulously accepted (w.r.t. to semantics s) in the AF B,S where B results
by removing some arguments from A and replacing these; similarly T modifies the
attack relation R.
Summary
Complexity issues provide an important foundational element of the formal com-
putational theory of abstract argumentation. Our review of the preceding pages is
intended to give a flavour of the class of questions of interest and an appreciation
of the techniques that have been brought to bear in addressing these. While some
notable progress has been achieved since the appearance of [22] – particularly in
understanding of decision properties of the standard semantics and the canonical
problems of Table 5.1, nevertheless a significant number of areas and potential ana-
lytic tools originating from complexity-theoretic studies, remain unexplored.
References
1. D. Angluin and L. Valiant. Fast probabilistic algorithms for hamiltonian circuits and match-
ings. Jnl. of Comp. and System Sci., 18:82–93, 1979.
2. S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Jnl. of
Algorithms, 12:308–340, 1991.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-116-320.jpg)
![106 Sanjay Modgil and Martin Caminada
f. Give all extensions containing A.
g. Give an extension that attacks A.
h. Give all extensions that attack A.
In this chapter, procedures will be described for answering a selection of the
above questions with respect to finite argumentation frameworks A,R (in which
A is finite). Notice that for some semantics, such as the grounded and preferred
semantics, extensions always exist, so that 1a will be answered in the positive for
any framework. Also, for the grounded semantics, at most one extension exists, so
that questions distinguished by reference to ‘an’ or ‘all’ extensions are equivalent
(e.g., questions 2a and 2b).
Sections 2 and 3 will introduce some key concepts underpinning the approaches
that we will use in the description of proof theories and algorithms. Sections 4 - 6
will then focus on application of these approaches to the core semantics defined by
Dung [13]; namely grounded, preferred and stable.
Broadly speaking, two approaches will be presented. Firstly, Section 2 formally
describes the argument graph labelling approach that was originally proposed by
Pollock [23], and has more recently been the subject of renewed analysis and in-
vestigation [5, 6, 25, 27]. The basic idea is that the status assignment to arguments
defined by the extension-based approach (see Chapter 2), can be directly defined
through assignment of labels to the arguments (nodes) in the framework’s corre-
sponding argument graph. Section 2 provides formal underpinnings for the defini-
tion of argument graph labelling algorithms that are used to address a selection of
the above questions in Sections 4 - 6.
Section 3 then describes a framework for argument game based proof theories
[9, 16, 17, 28]. The inherently dialectical nature of argumentation lends itself to for-
mulation of argument games in which a proponent starts with an initial argument to
be tested, and then an opponent and the proponent successively attack each other’s
arguments. The initial argument provably has a certain status if the proponent has a
winning strategy whereby he can win irrespective of the moves made by the oppo-
nent. In Sections 4 - 6 we describe specific games, emphasising the way in which the
rules of each specific game correspond to the semantics they are meant to capture.
2 Labellings
In this section the labelling approach (based on its formulation in [5, 6]) is briefly
reviewed. Given an argumentation framework AF = A,R, a labelling assigns to
each argument exactly one label, which can be either IN, OUT or UNDEC. The la-
bel IN indicates that the argument is justified, OUT indicates that the argument is
overruled, and UNDEC indicates that the status of the argument is undecided.
Definition 6.1. Let A,R be an argumentation framework.
• A labelling is a total function L : A → {IN,OUT,UNDEC}](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-120-320.jpg)
![108 Sanjay Modgil and Martin Caminada
• L is a grounded labelling iff there there does not exist a complete labelling L′
such that in(L′) ⊂ in(L) 1
• L is a preferred labelling iff there there does not exist a complete labelling L′
such that in(L′) ⊃ in(L)
• L is a stable labelling iff undec(L) = /
0
In [6], the following theorem is shown to hold:
Theorem 6.1. Let AF = A,R be an argumentation framework, and E ⊆ A. For
s ∈ {admissible, complete, grounded, preferred, stable}:
E is an s extension of AF iff there exists an s labelling L with in(L) = E 2
In Sections 4 - 6 we will describe algorithms that compute labellings and so
address a subset of the questions enumerated in Section 1. We conclude this section
with an example:
Example 6.1. Consider the framework in Figure 6.1. There exists three complete
labellings: 1. (/
0, /
0, {a,b,c,d,e}); 2. ({a}, {b}, {c,d,e}); and 3. ({b,d}, {a,c,e},
/
0). 1 is the grounded labelling, 2 and 3 are preferred, and 3 is also stable.
a b c
d
e
Fig. 6.1 An argumentation framework
3 Argument Games
In general, proof theories license the way in which pieces of information can be
articulated in order to prove a fact. They therefore provide a basis for algorithm de-
velopment, and proofs constructed according to these theories provide explanations
as to why a given fact is believed to be true. For example, a proof that argument x is
in an admissible extension, would consist of showing how one can establish the exis-
tence of such an extension, rather than simply identifying the extension. Intuitively,
1 Since every framework has a unique minimal fixed point, one could alternatively define L to be
a grounded labelling iff for each complete labelling L′ it holds that in(L) ⊂ in(L′)
2 Note that for s = admissible there is a 1-1 mapping between s extensions and s labellings. An
admissible extension may have more than one admissible labelling. For example, the admissible
extension {c}, of c → b ,c → a, has two admissible labellings: ({c},{b},{a}) and ({c},{b,a}, /
0).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-122-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 109
one would need to show how to defend x by showing that for every argument y that
is put forward (moved) as an attacker of x, one must move an argument z that attacks
y, and then subsequently show how any such z can be reinstated against attacks (in
the same way that z reinstates x). The arguments moved can thus be organised into
a graph of attacking arguments that constitutes an explanation as to why x is in an
admissible extension.
The process of moving arguments and counter-arguments can be implemented as
an algorithm [27]. In this chapter we follow the approach of [9, 14, 16, 17, 26, 28]
and present the moving of arguments as 2-person dialogue games that provide a
natural way in which to lay out and understand the algorithms that implement them.
To be sure, the actual algorithms themselves, should, except for didactic purposes,
not be implemented as dialogue games, but rather as monological procedures (or
methods in OO-languages) that are called recursively.
A dialogue game is played by two players, PRO (for “proponent”) and OPP (for
“opponent”), each of which are referred to as the other’s ‘counterpart’. A game
begins with PRO moving an initial argument x that it wants to put to the test. OPP
and PRO then take turns in moving arguments that attack their counterpart’s last
move. From hereon:
a sequence of moves in which each player moves against its counterpart’s last
move is referred to as a dispute.
If the last move in a dispute is by player Pl, and Pl’s counterpart cannot respond
to this last move, then Pl is said to win the dispute. If a dispute with initial argument
x is won by PRO, we call the dispute a line of defense for x.
The rules of the game encode restrictions on the legality of moves in a dispute,
and different sets of rules capture the different semantics under which justification of
the initial argument x is to be shown, by effectively establishing when OPP or PRO
run out of legal moves. In general, however, a player can backtrack to a counterpart’s
previous move and initiate a new dispute. Consider the dispute aPRO −bOPP −cPRO −
dOPP −ePRO − fOPP won by OPP (xPl −yPl′ denotes player Pl′ moving argument y
against counterpart Pl’s argument x). PRO must then try and backtrack to move an
argument against either bOPP or dOPP and establish an alternative line of defense
for a. Suppose such a line of defense aPRO −bOPP −gPRO. Then OPP can backtrack
and try an alternative line of attack moving h against a, so that PRO must now try
and win the newly initiated dispute aPRO −hOPP. Thus, the ‘playing field’ of a game
— the data structure on the basis of which argument games are played — can be
represented by an argumentation framework’s induced dispute tree, in which every
branch from root to leaf is a dispute:
Definition 6.5. Let AF = A,R be an argumentation framework, and let a ∈ A. The
dispute tree induced by a in AF is a tree T of arguments, such that T’s root node is
a, and ∀x,y ∈ A: x is a child of y in T iff xRy.
Figure 6.2i) shows an argumentation framework, and part of the tree induced by
a is shown in Figure 6.2ii). Notice that multiple instances of arguments are indi-
viduated by numerical indicies. Any game played by PRO and OPP in which PRO](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-123-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 111
Definition 6.7. Let AF = A,R, T the dispute tree induced by a in AF. Let DT be
the set of all disputes in T. Then φ is a legal move function such that φ : DT → 2A
.
Given a dispute tree T induced by a, the legal move function φ for a semantics
s, prunes T to obtain the sub-tree T′ of T that we call the φ tree induced by a. T′ is
the playing field of the game for semantics s. Thus, we define a φ-winning strategy
for a [9, 17] as a sub-tree of the φ dispute tree induced by a, in the same way as
Definition 6.6, except that we replace ‘for any x such that xRy’ in condition 2, with
‘for any x that OPP can φ legally move against y’. Intuitively, φ is defined such
that a is in an admissible extension that conforms to the semantics s iff there is a
φ-winning strategy for a in the φ tree induced by a, where the arguments moved by
PRO in the φ-winning strategy are conflict free (recall that an admissible extension
must contain no arguments that attack each other).
For example, consider games whose legal move function φ prohibits OPP from
repeating arguments in the same dispute. Figure 6.2iii) shows the φ-dispute tree that
is a sub-tree of the dispute tree induced by a (Figure 6.2ii)). After PRO plays a6,
OPP cannot backtrack and extend the dispute d = a1 −b2 −a3 by moving b against
a3, since b has already been moved by OPP in d. Similarly, OPP cannot backtrack
to move c against a8 in order to extend d′ = a1 −c7 −a8. Note also that both DT1 =
{d1 = a1 −b2 −a3 −c5 −a6} and DT2 = {d2 = a1 −c7 −a8 −b10 −a11} are winning
strategies. In the former case, consider the sub-dispute d′
1 = a1 −b2 −a3 of d1. OPP
can legally move c against a3, but there is a dispute in DT1 that extends d′
1 (d1 itself)
in which PRO moves against OPP’s move of c.
To summarise, suppose PRO wishes to show that x is a member of an extension
E under the semantics s. The associated legal move function φ for s defines some
φ-dispute tree T that is a sub-tree of the dispute tree induced by x, and defines all
possible disputes the players can play in a game. The φ-dispute tree T should be
such that: x ∈ E iff there is a φ-winning strategy T′ in T, such that the arguments
moved by PRO in T′ do not attack each other (are conflict free). A φ-winning strat-
egy is a set of disputes won by PRO in which PRO has fulfilled its burden of proof
by countering all possible φ-legal moves of OPP.
4 Grounded Semantics
For any argumentation framework, there is guaranteed to be exactly one grounded
extension. Hence, questions 1b, 1c and 2a - 2h can all be addressed by construc-
tion of a framework’s grounded extension. In Section 4.1 we present an algorithm
that generates the grounded labelling of an argumentation framework. Section 4.2
then describes an argument game for deciding whether a given argument is in the
grounded extension, thus providing an alternative way for addressing the questions
2a and 2b.
The grounded semantics places the highest burden of proof on membership of
the extension that it defines. This equates with Chapter 2’s definition of the exten-
sion as the least fixed point of a framework AF’s characteristic function FAF (i.e.,
the smallest admissible E that contains exactly those arguments that are acceptable](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-125-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 113
a counter attackers-out that represents the number of attackers that are la-
belled OUT. Since all arguments are initially unlabelled, this counter is set to zero
before the actual labelling begins. Every time that an argument is labelled OUT,
it sends a message to each of the arguments that it attacks to increase its variable
attackers-out. Evidently, if this variable equals the number of attackers, the
attacked argument can be labelled IN.
4.2 Argument games for the Grounded Semantics
We have discussed how, in defending an argument x’s membership of the grounded
extension, one must not loop back to x itself, and how the same restriction applies
to any argument moved in x’s line of defence. Intuitively, this is captured by a le-
gal move function φG1
that prohibits PRO from repeating arguments it has already
moved in a dispute.
Definition 6.8. Given A,R, a dispute d such that x is the last argument in d, and
PRO(d) the arguments moved by PRO in d, then φG1
is a legal move function such
that:
• If d is of odd length (next move is by OPP) then φG1
(d) = {y | yRx }
• If d is of even length (next move is by PRO) then:
φG1
(d) = {y |
1. yRx
2. y /
∈ PRO(d)
}
Theorem 6.2. Let AF = A,R be a finite argumentation framework. Then, there
exists a φG1
-winning strategy T for x such that the set PRO(T) of arguments moved
by PRO in T is conflict free, iff x is in the grounded extension of AF.
One can give an intuitive proof of Theorem 6.2 by appealing to the correspon-
dence between the grounded extension and grounded labelling of an argumentation
framework (see Theorem 6.1). That is to say, by showing that:
1. Let T be a φG1
-winning strategy for x such that PRO(T) is conflict free.
Then there is a grounded labelling L with L(x) = IN.
2. Let L be a labelling with L(x) = IN. Then there exists a φG1
-winning
strategy for x such that PRO(T) is conflict free.
Proof of the above correspondences can be found in [21].
Consider the example framework in Figure 6.3i). Part of the dispute tree induced
by a is shown in Figure 6.3ii), and the φG1
dispute tree induced by a is shown in
Figure 6.3iii). Observe that {(a1 −b2 −c3 −d4 −e6)} is a φG1
-winning strategy for
a (a is in the grounded extension {a,c,e}). Finally, consider the example framework
in Figure 6.3iv). In this case the φG1
-winning strategy for a consists of two disputes:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-127-320.jpg)
![114 Sanjay Modgil and Martin Caminada
d
c
e
i)
a1
c3
d4
e6
c5
d7
d8
c10
d11
PRO
OPP
PRO
OPP
PRO
ii) iii)
b
a
b2
OPP
e9
a1
c3
d4
e6
d8
c10
b2
e9
iv)
a
b
d c e
Fig. 6.3 i) shows an argumentation framework and ii) shows the dispute tree induced in a. iii)
shows the φG1 -dispute tree induced by a and the φG1 winning strategy encircled. The φG1 winning
strategy for a, in the framework in iv), consists of two disputes.
{(aPRO −bOPP −dPRO),(aPRO −cOPP −ePRO)}.
Some gain in efficiency can be obtained by a legal move function φG2
that ad-
ditionally prohibits PRO from moving a y that is itself attacked by the x that PRO
moves against (i.e., augmenting 1 and 2 in Definition 6.8 with ¬(xRy)). This is be-
cause if PRO moves such a y against x, then OPP can simply repeat x and move
against y, and then PRO will be prevented from repeating y. The φG2
game is in-
stantiated by Prakken and Sartor for their argument-based system of prioritized ex-
tended logic programming [24]. Amgoud and Cayrol [1] do the same for their argu-
ment based system for inconsistency handling in propositional logic. The following
soundness and completeness result can be proved as a straightforward generalisa-
tion of proofs for the specific systems in [24, 1]. Such a generalised proof can be
found in [4].
Theorem 6.3. Let AF = A,R be a finite argumentation framework. Then, there
exists a φG2
-winning strategy T for x such that the set PRO(T) of arguments moved
by PRO in T is conflict free, iff x is in the grounded extension of AF](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-128-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 115
Since the arguments moved by PRO in a winning strategy are required to be con-
flict free, it is obvious to see that shorter proofs may also be obtained by preventing
PRO from moving arguments in a dispute d that attack themselves or attack or are
attacked by arguments that PRO has already moved in d.
Definition 6.9. Let POSS(d) = {y | ¬(yRy) and ∀z ∈ PRO(d), ¬(zRy) and ¬(yRz)}.
One can then further restrict PRO’s moves in Definition 6.8, by adding the condition
that y ∈ POSS(d), thus obtaining the legal move function φG3
.
Finally, further gains in efficiency can be obtained by noticing that if T is a
φG1
, φG2
or φG3
winning strategy, then PRO(T) is conflict free. Thus, one need
not instigate the conflict free check on winning strategies suggested by the above
soundness and completeness results. To see why, notice that φG1
, φG2
and φG3
make
no restrictions on moves by OPP, and one can show that the following theorem
holds:
Theorem 6.4. Let T be a φ winning strategy such that φ makes no restrictions on
moves by OPP. Then PRO(T) is conflict free.
We refer the reader to [21] for a proof of the above theorem.
5 Preferred Semantics
For any argumentation framework, existence of a preferred extension is guaranteed,
and there can be more than one preferred extension. Hence, the decision questions
2a (credulous membership) and 2b (sceptical membership) are distinct. In Section
5.2 we describe argument games for addressing the credulous membership ques-
tion. Section 5.3 then describes argument games for addressing the more difficult
sceptical membership question. Solution-orientated questions 2c - 2h require proce-
dures for identifying one or all preferred extensions. Such questions become rele-
vant when end-users would like to be informed about the reasons as to how and why
an argument is justified or overruled, and can be addressed by labelling algorithms
that compute one or all preferred labellings. We describe labelling algorithms in the
following section.
5.1 A Labelling Algorithm for the Preferred Semantics
In this Section we review Caminada’s work on labelling algorithms [6]. Theorem
6.1 in Section 2 establishes an equivalence between an argumentation framework’s
preferred extensions and the framework’s preferred labellings. In [5] it is shown
that:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-129-320.jpg)
![116 Sanjay Modgil and Martin Caminada
L is a preferred labelling iff L is an admissible labelling such that for no ad-
missible labelling L′ is it the case that in(L′) ⊃ in(L). (R1)
Hence, a framework’s preferred extensions can be identified by algorithms that
compute admissible labellings that maximise the number of arguments that are
legally IN. In [6], admissible labellings are generated by starting with a labelling
that labels all arguments IN and then iteratively, selects arguments that are illegally
IN and applies a transition step to obtain a new labelling, until a labelling is reached
in which no argument is illegally IN.
Definition 6.10. Let L be a labelling for A,R and x an argument that is illegally
IN in L. A transition step on x in L consists of the following:
1. the label of x is changed from IN to OUT
2. for every y ∈ {x} ∪ {z|xRz}, if y is illegally OUT, then the label of y is changed
from OUT to UNDEC (i.e., any argument made illegally OUT by 1 is changed to
UNDEC)
In what follows, we assume a function transition step that takes as input x and L,
and applies the above operations to yield a labelling L′. We then define a transition
sequence as follows:
A transition sequence is a list [L0, x1, L1, x2, ..., xn, Ln] (n ≥ 0), where for i =
1...n, xi is illegally IN in Li−1, and Li = transition step(Li−1, xi).
A transition sequence is said to be terminated iff Ln does not contain any argument
that is illegally IN.
Let us examine a transition sequence that starts with the initial labelling L0 in
which all arguments are labelled IN (from hereon any such labelling is referred
to as an ‘all-in’ labelling and we assume that any initial labelling L0 is an all-in
labelling). Any labelling containing an argument x that is illegally IN cannot be a
candidate admissible labelling (since not all of x’s attackers are OUT and so x is not
reinstated against all attackers), and so must be relabelled OUT. One might expect
that the second part of the transition step relabels to IN, those arguments that are
made illegally OUT by the first step. However this may not only result in a loop,
but would also ‘overcommit’ arguments to membership of an admissible labelling
(and so extension); just because an argument may be acceptable w.r.t. an admissible
extension E does not mean that it must be in E.
For finite frameworks it can be shown that:
For any terminated transition sequence [L0, x1, L1, x2, ..., xn, Ln], it holds that
Ln is an admissible labelling. (R2)
To see why, observe that L0 contains no arguments that are illegally OUT, and it
is straightforward to show that a transition step preserves the absence of arguments
that are illegally OUT. Hence, since the terminated sequence contains no arguments
that are illegally IN, then by Definition 6.3, Ln is admissible.
In [6], it is also shown that:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-130-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 117
For any preferred labelling L, it holds that there exists a terminated transition
sequence [L0, x1, L1, x2, ..., xn, Ln], where Ln = L. (R3)
The above results R1, R2 and R3, imply that terminated transition sequences whose
final labellings maximise the arguments labelled IN are exactly the preferred la-
bellings. Before presenting the algorithm for generating such sequences, let us con-
sider how admissible labellings are generated for the argumentation framework in
Figure 6.4i). Starting with the initial all-in labelling L0 = ({a,b,c}, /
0, /
0), then se-
lecting a on which to perform a transition step obtains L1 = ({b,c},{a}, /
0). Now
only c is illegally IN, and relabelling it to OUT results in a being illegally OUT, and
so L2 = ({b},{c},{a}). Now b is illegally IN, and relabelling b to OUT results in
both b and c being illegally OUT, so that they are both labelled UNDEC. Thus, the
transition sequence terminates with the labelling L3 = (/
0, /
0,{a,b,c}) in which all
arguments are UNDEC. It is easy to verify that irrespective of whether a, b or c is
selected on the first transition step, every terminated transition sequence will result
in L3.
Consider now the framework in Figure 6.4ii). Starting with the initial all-in la-
belling L0 = ({a,b,c}, /
0, /
0), we observe that B and C are illegally IN:
1. Selecting b for the first transition step obtains the terminated sequence [L0, b,
L1 = ({a,c},{b}, /
0)]. L1 is an admissible and complete labelling, yielding the
admissible and complete extension {a,c}.
2. Selecting c for the first transition step obtains the terminated sequence [L0, c, L1
= ({a,b},{c}, /
0), b, L2 = ({a},{b},{c})], yielding the admissible extension {a}
a b
ii)
c
a
b
c
i)
Fig. 6.4 Two argumentation frameworks
Notice that in the second sequence, the label of c is changed from OUT to UNDEC
since c is made illegally OUT by the second transition step’s assignment of OUT
to the illegally IN b. L2 is an admissible but not complete labelling, since c is
illegally UNDEC. To help avoid non-complete labellings, one can guide the choice
of arguments on which to perform transition steps: choose an argument that is super-
illegally IN, if such an argument is available.
Definition 6.11. An argument x in L that is illegally IN, is also super-illegally IN
iff it is attacked by a y that is legally IN in L, or UNDEC in L.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-131-320.jpg)
![118 Sanjay Modgil and Martin Caminada
Thus, b would preferentially be selected according to the above strategy, since b
and not c is super-illegally IN in ({a,b,c}, /
0, /
0). As shown in [6], both the results
R2 and R3 are preserved under such a strategy.
Algorithm 6.2 Algorithm for Preferred Labellings
1: candidate-labellings := /
0;
2: find labellings(all-in);
3: print candidate-labellings;
4: end.
5: .
6: .
7: procedure find labellings(L)
8: .
9: # if L is worse than an existing candidate labelling then prune the search tree
10: # and backtrack to select another argument for performing a transition step
11: if ∃L′ ∈ candidate-labellings: in(L) ⊂ in(L′) then return;
12: .
13: # if the transition sequence has terminated
14: if L does not have an argument that is illegally IN then
15: for each L′ ∈ candidate-labellings do
16: # if L′’s IN arguments are a strict subset of L’s IN arguments
17: # then remove L′
18: if in(L′) ⊂ in(L) then
19: candidate-labellings :=
20: candidate-labellings − {L′};
21: end if
22: end for
23: # add L as a new candidate
24: candidate-labellings := candidate-labellings ∪ {L};
25: return; # we are done, so try the next possibility
26: else
27: if L has an argument that is super-illegally IN then
28: x := some argument that is super-illegally IN in L;
29: find labellings(transition step(L, x));
30: else
31: for each x that is illegally IN in L do
32: find labellings(transition step(L, x))
33: end for
34: end if
35: end if
36: endproc
We now describe the above listed algorithm for generating preferred labellings.
The main procedure find labellings starts with the all-in labelling, and then
iteratively applies transitions steps in an attempt to generate terminated transition
sequences that update the global variable candidate-labellings. The algo-
rithm preferentially selects from amongst super-illegal arguments for performing
transition steps, if such arguments are available. If at any stage in the generation of
a transition sequence, the arguments that are IN in the labelling Li thus far obtained](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-132-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 119
are a strict subset of in(L′) for some L′ ∈ candidate-labellings, then no
further transition steps on Li can result in a preferred labelling (that maximises the
arguments that are IN). This follows from the result that during the course of a tran-
sition sequence, the set of IN labelled arguments monotonically decreases (as shown
in [6]). Thus, any further transition steps on Li will only reduce the arguments that
are IN. In such cases, the algorithm backtracks to Li−1 and, if possible, selects an-
other argument on which to perform a transition step. In the case that a transition
sequence terminates, the obtained labelling L is compared with all labellings L′
in candidate-labellings. If for any L′, in(L′) is a strict subset of in(L),
then L′ is removed from candidate-labellings. Thus, given a finite argu-
mentation framework A,R, the algorithm calculates the preferred labellings and
so preferred extensions.
5.2 Argument Games for the Credulous Preferred Semantics
Since the admissible extensions of a framework form a complete partial order with
respect to set inclusion (and so every admissible extension is a subset of a preferred
extension), then for argument games addressing the credulous membership question,
it suffices to show an admissible extension containing the argument in question. In
contrast with the grounded semantics, x’s membership of an admissible extension
E can now ‘appeal to’ x itself, in the sense that in defending x’s membership of E,
and membership of all subsequent defenders, one can loop back to x itself. This then
means, that to prevent infinite disputes, it is now OPP, rather than PRO, that should
not be allowed to repeat an argument y it has already moved in a dispute, since y can
then be attacked by PRO repeating the argument it moved against OPP’s first move
of y.
Consider the framework in Figure 6.5i), and the dispute tree induced in a in Fig-
ure 6.5ii). In both disputes (branches) PRO is allowed to repeat its arguments (c5
and d9). OPP repeats its arguments, and the disputes continue with PRO repeatedly
fulfilling its burden of proof w.r.t. c (d). It is of course sufficient that PRO fulfill
its burden of proof only once. Hence, as well as preventing PRO from introduc-
ing a conflict into a dispute, the following legal move function prohibits OPP from
repeating arguments.
Definition 6.12. Given A,R, a dispute d such that x is the last argument in d, and
OPP(d) the arguments moved by OPP in d, then φPC1
is a legal move function such
that:
• If d is of odd length (next move is by OPP) then:
φPC1
(d) = {y |
1. yRx
2. y /
∈ OPP(d)
}](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-133-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 121
2. Let L be an admissible labelling with L(x) = IN. Then there exists a φPC1
-
winning strategy for x such that PRO(T) is conflict free.
Proof of the above correspondences can be found in [21].
Observe that the spectrum of outcomes would not be changed by a function φPC2
that augments φPC1
by prohibiting OPP from moving any argument y (and not just
a y already moved by OPP) that is attacked by an argument x in PRO(d). This is
because PRO can then simply move x against y, and if yRx, prohibiting repetition
by OPP will mean that y cannot be moved against this second move of x by PRO.
Notice that if this prohibition on OPP is in place, then one cannot have a dispute
of the form (...yPRO ...xOPP − yPRO ...) in which PRO repeats an argument, since
the prohibition on OPP would prevent the move xOPP. Hence, shorter proofs can be
obtained by a function φPC2
that augments φPC1
by prohibiting OPP from moving
any argument attacked by an argument in PRO(d), and prohibiting repetition by
PRO. Indeed, [9] prove that the following theorem holds:
Theorem 6.6. Let AF = A,R be a finite argumentation framework. Then, there
exists a φPC2
-winning strategy for x such that the set PRO(T) of arguments moved
by PRO in T is conflict free, iff a is in a preferred extension of AF.
Figure 6.5iv) shows the φPC2
dispute tree induced by a for the framework in
Figure 6.5i), where both disputes in the tree are φPC2
winning strategies. However,
notice that neither dispute fully fulfills the remit of a proof to explain the credulous
membership of a, since neither demonstrates the reinstatement of c, respectively
d, against its attacker d, respectively c, and so provides an explanation for the ad-
missibility of {a,c}, respectively {a,d}. This illustrates a more general point that
efficiency gains often come at the expense of explanatory power.
Finally, note that unlike games for the grounded semantics, checking that the
arguments moved by PRO in a φPC1
(or φPC2
) winning strategy are conflict free, is
required. This is because φPC1
and φPC2
games place restrictions on moves by OPP
(and hence the result concluding Section 4.2 does not hold). For example, consider
that a is not in an admissible, and hence preferred, extension of the framework
in Figure 6.6. Now, {(aPRO − bOPP − cPRO − dOPP − gPRO),(aPRO − eOPP − fPRO −
gOPP − dPRO)} is a φPC1
winning strategy since OPP cannot legally extend either
dispute. However, the arguments moved by PRO are not conflict free (PRO has
moved g and d).
c b a
d
e
f
g
Fig. 6.6 Argument a is not in an admissible and so preferred extension of the above framework.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-135-320.jpg)
![122 Sanjay Modgil and Martin Caminada
5.3 Argument Games for the Sceptically Preferred Semantics
The question of whether an argument is sceptically preferred is much harder to an-
swer than the credulously preferred membership problem. To understand why, it
may first help to realise that the credulous membership problem only requires us to
point at one extension, while the sceptical membership problem requires us to prove
something about all possible extensions. Thus, the credulously preferred member-
ship problem is an existence problem while the sceptically preferred membership
problem is a verification problem. To understand better why verification is hard in
this case, we recall the definition of sceptically preferred membership: an argument
a is sceptically preferred iff it is a member of all preferred extensions. The crux of
the problem is that we have to verify whether there exists preferred extensions that
do not contain a. In so doing, it is not immediately clear where to begin to search
for such extensions.
The following result establishes a connection between a and preferred extensions
that might possibly exclude a (we refer the reader to [21] for a proof of this result). It
basically ensures that the search space for the sceptical decision problem is confined
to elements that are indirectly connected to defense sets of a.
Theorem 6.7 (Complement lemma). An argument a is sceptically preferred if and
only if for every admissible extension B, there is an admissible extension A, contain-
ing a, that is consistent with B.
Thus, conversely, a is not sceptically preferred if there exists an admissible exten-
sion B that conflicts with all admissible extensions around a. Because such an exten-
sion B blocks sceptically preferred membership, such an extension is called a block.
With the help of Theorem 6.7 we may now formulate an abstract and inefficient, but
conceptually correct proof procedure to determine sceptical membership. This pro-
cedure works by falsification, as follows. Try to construct a block B. If this attempt
fails, we may, with the help of Theorem 6.7 conclude that a is sceptically preferred.
The procedure to block a can be described as an argument game that we infor-
mally describe here. The difference with the games described earlier, is that the play-
ers exchange entire admissible extensions rather than single arguments. The game
works as follows. Suppose PRO’s goal is to show that a is sceptically preferred. To
this end, PRO starts by constructing an admissible extension, A{1} around a. Since
A{1} is the only admissible extension known at this stage, it follows that at this
stage a is sceptically preferred. To invalidate this temporary conclusion, the bur-
den of proof shifts to OPP who must show that a is not sceptically preferred. By
virtue of Theorem 6.7 it suffices for OPP to show that there exists an admissible
extension that conflicts with A{1}. If OPP does not manage to construct such an
extension, the procedure ends and OPP has lost. Suppose OPP manages to produce
A{1,1} as a response to A{1}. Thus, A{1,1} is an admissible extension that con-
flicts with A{1}. Once A{1,1} is advanced, a is no longer sceptically preferred, be-
cause A{1,1} conflicts with every admissible extension around a constructed thus
far, viz. A{1}. To invalidate this temporary conclusion, the burden of proof shifts
back to PRO who must now show that there exists another admissible extension](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-136-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 123
around a that does not conflict with A{1,1}. If PRO fails to do so (and PRO’s search
was adequate and exhaustive), it follows that A{1,1} conflicts with all admissible
extensions around A, so that a is not sceptically preferred. Suppose otherwise, i.e,
suppose that PRO is able to construct an admissible extension, A{1,1,1}, that does
not conflict with A{1,1}. OPP must now either extend A{1,1} such that it also con-
flicts with A{1,1,1} or else drop A{1,1} to start all over to attack another member
of A{1}. Continuing this way (including backtracking), OPP is busy with extending
an admissible extension until either PRO is unable to produce another admissible
extension around a, or else until OPP’s admissible extension cannot be further ex-
tended (on pain of becoming inconsistent).
More generally, we may suppose that A{1},...,A{n} are possible begin moves of
PRO, and A{i1,...,ik,m}, k ≥ 1 is the mth possible response of either PRO or OPP
to A{i1,...,ik}. Naturally, all the A{ī} are admissible extensions. The following
constraints hold:
1. Every extension advanced by PRO must contain the main argument, a.
2. Every response of PRO must be consistent with the extension that is previously
advanced by OPP.
3. Every response of OPP must attack PRO’s immediately preceding extension.
4. Within one branch, every extension advanced by OPP must be an extension of
OPP’s previous extension in the same branch.
5. Both parties may backtrack and construct alternative replies.
6. OPP has won if it is able to move last; else PRO has won.
If OPP has won this means that OPP was able to create a block B = A{i1,...,i2k},
where k ≥ 1 (note that we have ‘2k’ since all moves by OPP have an even number of
indices). With B, OPP is able to move last in the particular branch where that block
was created and all sub-branches emanating from the main branch. It must be noted
that all this only works in finitary argument systems, i.e., argument systems where
all arguments have a finite number of attackers. Algorithms for non-finitary argu-
ment systems require additional constraints such as fairness which must guarantee
that every possibility is enumerated eventually.
The above ideas are taken from earlier work on the sceptically preferred mem-
bership problem, notably that of Doutre et al. [12] and Dung et al. [15]. In [12],
the procedure to find a possible block is presented as a so-called meta-acceptance
dialogue. As above, moves in this dialogue are extensions (hence the meta), and a
dialogue is won by OPP if it is able to move last in at least one branch. In Dung et
al. [15] the procedure to construct a “fan” of admissible extensions around A that
together represent all preferred extensions is called generating a complete base for
a. A base for a is a set of admissible extensions, B, such that every preferred exten-
sion around a includes at least one element of B. A complete base for a, then, is a
set of admissible extensions, B, such that every preferred extension includes at least
one element of B. In line with Theorem 6.7, Dung et al. proceed to show that a base
B is incomplete if and only if there exists a preferred extension that attacks every
element of B. Their proof procedure is a combination of a so-called BG-derivation
(base generation derivation) followed by a CB-verification (complete base verifica-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-137-320.jpg)
![124 Sanjay Modgil and Martin Caminada
tion). With BG a base for a is generated, such that every preferred extension around
a contains an element of B. Such a base always exists, but not every base may serve
as a representant of sceptical membership. To check whether B indeed represents
sceptical membership, it is checked for completeness, which effectively means that
it must hold out against every candidate block that might undermine B. Again, all
decision procedures only work in finitary argument systems.
6 Stable and Semi-Stable Semantics
Stable semantics are, what one might call ‘xenophobic’, since every argument out-
side of a stable extension is attacked by an argument in the stable extension. Unlike
the preferred semantics, existence of a stable extension is not guaranteed; consider
that a framework consisting of a single argument that attacks itself has no stable ex-
tension. However, as in the case of the preferred semantics, there may be more than
one extension, and so decision questions 2a (credulous membership) and 2b (scep-
tical membership) are distinct. These questions, questions 1b, 1c, and the solution-
orientated questions 2a - 2h can be addressed by an algorithm (taken from [6])
that generates all stable extensions of a framework. Since a stable labelling makes
all arguments either OUT or IN, one can straightforwardly adapt the algorithm for
preferred labellings in Section 5.1, so as to only yield labellings without UNDEC
labelled arguments. Thus, line 11 in the algorithm is replaced by:
if undec(L) = /
0 then return;
Furthermore, we do not have to compare the arguments made IN by other candidate
labellings, and so we can remove lines 15 to 22. The result is an algorithm that cal-
culates all stable extensions of a finite framework.
Argument games for stable semantics have only recently been studied. In [28],
the authors study coherent argumentation frameworks, in which every preferred ex-
tension is also stable (meaning that the preferred and stable extensions coincide,
since each stable extension is by definition also a preferred extension). Thus, for
coherent argumentation frameworks, one can simply apply existing games for the
preferred semantics to decide membership under stable semantics.
For the general case, where one is not restricted to coherent argumentation frame-
works, the situation is more complex, but can still be expressed in terms of the cred-
ulous games defined in Section 5.2. Given a framework A,R, and letting PRO(T),
respectively OPP(T), denote the arguments moved by PRO, respectively OPP, in a
dispute tree T, then an argument x is in a stable extension iff there exists a set S of
φPC1
winning strategies such that:
1. at least one winning strategy in S is for x.
2.
{PRO(T)|T ∈ S} is conflict free.
3.
{PRO(T)∪OPP(T)|T ∈ S} = A](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-138-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 125
This can be seen as follows. First of all, each φPC1
winning strategy corresponds
to an admissible labelling. A set of winning strategies that do not attack each other
(point 2) again corresponds to an admissible labelling. If this resulting admissible
labelling spans the entire argumentation framework (each argument is either IN or
OUT) then this labelling is also stable (point 3). Then, if x is IN in this labelling,
then x is labelled IN in at least one stable labelling (point 1).
It is also possible to define a single dispute game that determines credulous ac-
ceptance w.r.t. stable semantics. Such a game has recently been stated by Caminada
and Wu [7]. One particular feature of their approach, which builds on the work of
Vreeswijk and Prakken [28], is that they do not use the concept of a winning strat-
egy. Instead, for an argument x to be in a stable extension, it suffices to have at least
one game for x that is won by PRO. Caminada and Wu are able to do this by first
defining a game for credulous preferred in which PRO may repeat its own moves,
but not the moves of OPP, and in which OPP may repeat PRO’s moves but not its
own moves. Moreover, PRO has to react to the directly preceding move of OPP,
whereas OPP is free to react either to the directly preceding move of PRO, or to
a previous PRO move. A dispute is won by PRO iff OPP cannot move. A dispute
is won by OPP iff PRO cannot move, or if OPP managed to repeat one of PRO’s
moves.
Basically, the game can be understood in terms of PRO and OPP building an
admissible labelling in which PRO makes IN moves, and OPP makes OUT moves.
This game can be altered to implement stable semantics by introducing a third kind
of move, which is called QUESTION. By uttering QUESTION x, OPP asks PRO for
an explicit opinion on argument x. PRO is then obliged to reply with either IN x or
with IN y, where y is an attacker of x. Caminada and Wu show that this game indeed
models credulous acceptance under the stable semantics.
Once a procedure for credulous acceptance w.r.t. stable semantics has been de-
fined, the issue of sceptical acceptance w.r.t. stable semantics becomes relatively
straightforward: an argument x is in all stable extensions iff one fails to establish
credulous membership of any attacker of x. For the left to right half, observe that
if x is in all stable extensions, then all attackers of x are attacked by all such ex-
tensions (an argument y is attacked by an extension if it is attacked by an argument
in that extension), and so no attacker of x can be in any such extension, since each
such extension is conflict free. For the right to left half, observe that if any attacker
of x does not belong to any stable extension, then it is attacked by all such exten-
sions. Thus every extension contains an argument that reinstates x, and so contains x.
Caminada has recently proposed semi-stable semantics [5, 6], that unlike the sta-
ble semantics, guarantees that every (finite) framework has at least one semi-stable
extension. In the case that there exists at least one stable extension for a frame-
work, semi-stable semantics yield the same extensions as stable semantics. From
the perspective of argument labellings, semi-stable semantics select those labellings
in which the set of UNDEC arguments is minimal. Referring to Definition 6.4, this
can be expressed as follows:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-139-320.jpg)
![126 Sanjay Modgil and Martin Caminada
Let L be a complete labelling. Then L is a semi-stable labelling iff there
does not exist a complete labelling L′ such that undec(L′) ⊂ undec(L)
For example, consider the framework in Figure 6.4ii) augmented by an additional
argument d that attacks itself. The augmented framework has no stable extension,
but {a,c} is the single semi-stable extension equating with the semi-stable labelling
({a,c},{b},{d}). Notice that although {a,c} is also the single preferred extension,
in general not every preferred extension is a semi-stable extension since not every
preferred extension minimises UNDEC. However, every semi-stable extension is a
preferred extension, which suggests that we can adapt Section 5.1’s algorithm for
preferred labellings in order to compute semi-stable labellings.
In [6] it is also shown that:
L is a semi-stable labelling iff L is an admissible labelling such that for no
admissible labelling L′ is it the case that undec(L′) ⊂ undec(L).
(R1′)
Since every semi-stable extension is a preferred extension then R3 in Section
5.1 also holds for semi-stable labellings L. This result, together with R1′ and R2 in
Section 5.1, implies that terminated transition sequences whose final labellings min-
imise the arguments labelled UNDEC are exactly the semi-stable labellings. Hence,
one can adapt Section 5.1’s algorithm by replacing line 11 by:
if ∃L′ ∈ candidate-labellings: undec(L′) ⊂ undec(L) then return;
In other words, if at any stage in the generation of a transition sequence, the
UNDEC arguments of the labelling Li thus far obtained, are a strict superset of
undec(L′) for some L′ ∈ candidate-labellings, then no further transition
steps on Li can result in a semi-stable labelling, and so one can backtrack to per-
form a transition step on another choice of argument. This follows from the result
that during the course of a transition sequence, the set of UNDEC labelled arguments
monotonically increases (as shown in [6]). Finally, we replace line 18 with:
if undec(L) ⊂ undec(L′);
and we are done. We have an algorithm that calculates the semi-stable labellings of
a finite argumentation framework.
7 Conclusions
In this chapter we have described labelling algorithms and argument game proof
theories for various argumentation semantics. Labellings and argument games can
be seen as alternatives to the extension-based approach to specifying argumentation](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-140-320.jpg)
![6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 127
semantics described in Chapter 2. We conclude with some further reflections on
these different ways of specifying argumentation semantics.
One of the original motivations for developing the labelling approach was to pro-
vide an easy and intuitive account of formal argumentation. After all, principles like
“In order to accept an argument, one has to be able to reject all its counterargu-
ments” and “In order to reject an argument, one has to be able to accept at least one
counterargument” are easy to explain and have therefore been used as the basis of
the labelling approach. Also, our teaching experiences indicate that students who
are new to argumentation tend to find it easier to understand the labelling approach
rather than the extension-based approach to argumentation. In fact, it is often easier
for them to understand the extension-based approach after having been introduced
to the labelling approach.
Another advantage of the labelling approach is that it allows one to specify a
number of relatively small and simple properties, each of which can be individu-
ally satisfied or not, and that collectively define the argumentation semantics. This
modular approach can be of assistance when constructing formal proofs. Also, by
explicitly distinguishing between IN, OUT and UNDEC (instead of merely speci-
fying the set of IN-labelled arguments as in the extension-based approach) one is
provided with more detailed information. For instance, Section 5.1’s algorithm for
generating all preferred extensions, would be much more difficult to specify using
the extension-based approach.
Finally, we note that the labelling approach essentially identifies a graph or ‘net-
work’ labelling problem, suggesting that the approach more readily lends itself to
extensions of argument frameworks that accommodate: different types of relation
between arguments (e.g. support [10] and collective attack [22]); attacks on attacks
[20]; multi-valued and quantitative valuations of arguments [2, 11], and so on. In
essence, these extensions of Dung’s abstract argumentation framework can be un-
derstood as instantiating a more general network reasoning model in which the val-
uations of nodes (arguments) is determined by propagating the valuations of the
connected nodes, as mediated by the semantics of the connecting arcs. Algorithms
for determining these valuations will thus generalise the three value labelling algo-
rithms described in this chapter.
With regard to the argument game approach, we recall that Dung’s abstract ar-
gumentation semantics can be understood as a semantics for a number of non-
monotonic and defeasible logics [3, 13], in the sense that:
α is an inference from a theory Δ in a logic L, iff α is the conclusion of a
justified argument of the argumentation framework A,R defined by Δ and
L.
The argument game approach places an emphasis on the dialectical nature of
argumentation, in the sense that the approach appeals more directly to an inter-
subjective notion of truth: truth becomes that which can be defended in a rational
exchange and evaluation of interacting arguments. Thus, what accounts for the cor-
rectness of an inference is that it can be shown to rationally prevail in the face of](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-141-320.jpg)
![128 Sanjay Modgil and Martin Caminada
arguments for opposing inferences, where it is application of the reinstatement prin-
ciple that encodes logic neutral, rational means for establishing such standards of
correctness. This account of argumentation as a semantics, contrasts with model-
based semantics for formal entailment that appeal to an objective notion of truth:
true is that which holds in every possible model. Notice that dialectical semantics
are not unique to formal argumentation. For instance, Lorenzen and Lorenz [18, 19]
have proposed dialectical devices as a method of demonstration in formal logic.
An advantage of dialectical semantics is that they are able to relate formal en-
tailment to something most people are familiar with in everyday life: debates and
discussions. Argument games of the type described in this chapter are therefore use-
ful not only for providing guidelines and principles for the design of algorithms, but
also for bridging the gap between formal and informal reasoning.
Finally, we note that the dialectical view also accords with our understanding of
reasoning as an incremental process. Rather than have all the arguments and their
attacks defined from the outset, we incrementally acquire knowledge in order to con-
struct arguments required to counter-argue existing arguments. At any stage in this
incremental process we can evaluate the status of arguments, which in turn motivates
acquisition of further knowledge for construction and submission of arguments. Ar-
gument games allow one to model such processes. Provided that there is a well
understood notion of what constitutes an attack between any two arguments, one
can then formalise the games described in this chapter, without reference to a pre-
existing framework. This also allows one to acknowledge that reasoning agents are
resource bounded, and suggests that bounds on reasoning resources may be charac-
terised by bounds on the breadth and depth of the dispute trees constructed in order
to prove the claim of the argument under test.
Acknowledgements The authors would like to thank Gerard Vreeswijk for his con-
tributions to the contents of this chapter. Thanks also to Nir Oren for commenting
on a draft of the chapter.
References
1. L. Amgoud and C. Cayrol. A Reasoning Model Based on the Production of Acceptable Argu-
ments. Annals of Mathematics and Artificial Intelligence, 34(1–3),197–215, 2002.
2. H. Barringer, D. M. Gabbay and J. Woods. Temporal Dynamics of Support and Attack Net-
works: From Argumentation to Zoology. Mechanizing Mathematical Reasoning, 59–98, 2005.
3. A. Bondarenko and P.M. Dung and R.A. Kowalski and F. Toni. An abstract, argumentation-
theoretic approach to default reasoning. Artificial Intelligence, 93:63–101, 1997.
4. M. Caminada. For the sake of the Argument. Explorations into argument-based reasoning.
Doctoral dissertation Free University Amsterdam, 2004.
5. M. Caminada. On the Issue of Reinstatement in Argumentation. In European Conference on
Logic in Artificial Intelligence (JELIA), 111–123, 2006.
6. M. Caminada. An Algorithm for Computing Semi-stable Semantics. In European Conference
on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU), 222–
234, 2007.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-142-320.jpg)
![134 Philippe Besnard and Anthony Hunter
only consider deductive arguments, hence the method of inference for each and
every argument is always entailment according to classical logic.
A counterargument for an argument Φ,α is an argument Ψ,β where the
claim β contradicts the support Φ. Furthermore, we identify a particular kind of
counterargument called a canonical undercut Ψ,β where β is equivalent to ¬(φ1 ∧
.. ∧ φn) and {φ1,...,φn} is the support of the argument being undercut. This is a
valuable form of undercut since it subsumes many other kinds of undercut, and
hence focusing on only canonical undercuts renders the presentation and evaluation
of counterarguments as a more manageable process.
Each undercut to an argument is itself an argument, and so may be undercut, and
hence by recursion each undercut needs to be considered for its undercuts. Exploring
systematically the universe of arguments in order to present an exhaustive synthesis
of the relevant chains of undercuts for a given argument is the basic principle of our
approach.
Following on from the idea that we can capture undercuts, and by recursion un-
dercuts to undercuts, our notion of an argument tree is that it is a synthesis of all the
arguments that challenge the argument at the root of the tree, and it also contains
all counterarguments that challenge these arguments and so on recursively. In each
instance, the only counterarguments we consider are the canonical undercuts.
In the rest of this chapter, we formalize and illustrate arguments and counterar-
guments (including canonical undercuts), and show how these can be collected into
argument trees. We conclude the chapter with a comparison with other approaches
to formalising argumentation. Since the aim of this chapter is to just introduce some
of the basic ideas to argumentation based on classical logic, the interested reader is
requested to refer to [3, 6] for more details including formal results.
2 Preliminaries
We assume the reader has some knowledge of classical logic. We will represent
atoms by lower case roman letters (a,b,c,d,...), formulae by greek letters (α,β,γ,
....), and use ∧,∨,→, and ¬ to denote the logical connectives conjunction, disjunc-
tion, negation, and implication (respectively). We use ⊢ to denote the classical con-
sequence relation, and so if Δ is a knowledgebase, and α is a formula, then Δ ⊢ α
denotes that Δ entails α (or equivalently α is a consequence of Δ). We also use ⊥ to
denote a contradiction, and so Δ ⊢ ⊥ denotes that Δ is contradictory (or equivalently
inconsistent).
For the knowledgebase, we first assume a fixed Δ (a finite set of formulae) and
use this Δ throughout. So when we consider arguments and counterarguments, they
will be formed from this Δ. For examples, we will explicitly give the elements of
the knowledgebase.
We further assume that every subset of Δ is given an enumeration α1,...,αn of
its elements, which we call its canonical enumeration. This really is not a demanding
constraint: In particular, the constraint is satisfied whenever we impose an arbitrary](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-148-320.jpg)
![138 Philippe Besnard and Anthony Hunter
Example 7.6. Consider the following atoms.
p The number is divisible by 10
q The number is divisible by 2
r The number is an even number
We use these for the following set of formulae.
Δ = {p, p → q,q → r}
Hence, the following is an argument with the claim “The number is divisible by 2”.
{p, p → q},q
Similarly, the following is argument with claim “The number is divisible by 2 and
the number is an even number”.
{p, p → q,q → r},r ∧q
However, the first argument {p, p → q},q is more conservative than the second
argument {p, p → q,q → r},r ∧q which can be retrieved from it:
{p, p → q},q
{q,q → r} |= r ∧q
⇒ {p, p → q,q → r},r ∧q
We will use the notion of “more conservative” to help us identify the most useful
counterarguments amongst the potentially large number of counterarguments.
4 Counterarguments
Informally, an argument that disagrees with another argument is described as a coun-
terargument. So counterarguments are an important part of the argumentation pro-
cess. They highlight points of contention.
In logic-based approaches to argumentation, an intuitive notion of counterargu-
ment is captured with the idea of defeaters, which are arguments whose claim refutes
the support of another argument [23, 14, 19, 25, 24, 22]. This gives us a general way
for an argument to challenge another.
Definition 7.3. A defeater for an argument Φ,α is an argument Ψ,β such that
β ⊢ ¬(φ1 ∧...∧φn) for some {φ1,...,φn} ⊆ Φ.
Example 7.7. Let Δ = {¬a,a ∨ b,a ↔ b,c → a}. Then, {a ∨ b,a ↔ b},a ∧ b is a
defeater for {¬a,c → a},¬c. A more conservative defeater for {¬a,c → a},¬c
is {a∨b,a ↔ b},a∨c.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-152-320.jpg)
![148 Philippe Besnard and Anthony Hunter
Dairyman: – Egg was first
Farmer: – Chicken was first
Dairyman: – Egg was first
Farmer: – Chicken was first
. . . – . . .
Here are the formulae again:
p Egg was first
q Chicken was first
r The chicken comes from the egg
s The egg comes from the chicken
¬(p ↔ q) That the egg was first and that the chicken was first are not equivalent
r → ¬q The chicken comes from the egg implies that the chicken was not first
s → ¬p The egg comes from the chicken implies that the egg was not first
So, Δ = {¬(p ↔ q),r → ¬q,s → ¬p,r,s}. The argument tree with the dairyman’s
argument as its root is
{r → ¬q,r,¬(q ↔ p)}, p
↑
{s → ¬p,s,¬(q ↔ p)},✸
but it does not mean that the farmer has the last word nor that the farmer wins
the dispute! The argument tree is merely a representation of the argumentation (in
which the dairyman provides the initial argument). Although the argument tree is
finite, the argumentation here is infinite and unresolved.
We now consider a widely used criterion in argumentation theory for determining
whether the argument at the root of the argument tree is warranted (e.g. [21, 15]).
For this, each node is marked as either U for undefeated or D for defeated.
Definition 7.9. The judge function, denoted Judge, assigns either Warranted or
Unwarranted to each argument tree T such that Judge(T) = Warranted iff Mark(Ar)
= U where Ar is the root node of T. For all nodes Ai in T, if there is child Aj of Ai
such that Mark(Aj) = U, then Mark(Ai) = D, otherwise Mark(Ai) = U.
As a direct consequence of the above definition, the root is undefeated iff all its
children are defeated.
Example 7.27. Returning to Example 7.22, we see that the root of the tree T is un-
defeated, and hence Judge(T) = Warranted.
Example 7.28. Returning to Example 7.26, we see that the root of the tree T is de-
feated, and hence Judge(T) = Unwarranted.
In general, a complete argument tree (i.e. an argument tree with all the canonical
undercuts for each node as children of that node) provides an efficient representation
of the arguments and counterarguments. Furthermore, if Δ is finite, there is a finite
number of argument trees with the root being an argument with claim α that can be
formed from Δ, and each of these trees has finite branching and a finite depth.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-162-320.jpg)
![7 Argumentation Based on Classical Logic 149
7 Discussion
In this chapter, we have reviewed a framework for argumentation based on classical
logic. The key features of this framework are the clarification of the nature of ar-
guments and counterarguments, the identification of canonical undercuts which we
argue are the only undercuts that we need to take into account, and the representa-
tion of argument trees which provide a way of exhaustively collating arguments and
counterarguments. This chapter is based on a particular proposal for logic-based ar-
gumentation for the propositional case [3, 6]. In comparison with the previous pro-
posals based on classical logic (e.g. [1, 20]), our proposal provides a much more
detailed analysis of counterarguments, and ours is the first proposal to consider
canonical undercuts. We believe that without the notion of canonical undercuts, the
number of undercuts available is unmanageable, and so restricting to canonical un-
dercuts reduces the number of undercuts to consider, and renders it manageable.
It is interesting to compare our approach with other logic-based approaches to
argumentation that use logics other than classical logic for the definition of deduc-
tion. The most common alternative is a form of defeasible logic such as defeasi-
ble logic programming [15], defeasible argumentation with specificity-based pref-
erences [23], and argument-based extended logic programming [21]. For a general
coverage of defeasible logics in argumentation see [10, 22, 6].
Whilst there are various positive features of these proposals based on defeasible
logic, including computational viability, and modeling of intuitive features of de-
feasible reasoning, there are complications with these proposals with respect to the
interplay between strict rules and defeasible rules, and the interplay between these
rules and the use of priorities over rules. This can render the deductive process to be
less than transparent. In comparison, we believe that using classical logic provides
a simpler and clear notion of deduction, of argument, and of counterargument.
Furthermore, various questions of rationality have been raised concerning the use
of defeasible logic for deduction in argumentation [8]. One of the proposals made
for rationality is that contrapositive reasoning needs to be supported. Introducing
contrapositive reasoning is controversial in defeasible logic [9], but it is an intrinsic
feature of classical logic. In other words, classical logic offers a nice solution to the
problems raised in [8].
A more general approach to logic-based argumentation is to leave the logic for
deduction as a parameter. This was proposed in abstract argumentation systems [25],
and developed in assumption-based argumentation (ABA) [11]. A substantial part
of the development of the theory and implement of ABA is focused on defeasible
logic. However, ABA is a general framework allowing for the use classical logic for
deduction, and thereby we could instantiate ABA with our proposal.
It is also interesting to compare our approach with abstract argument systems. Su-
perficially, an argument tree could be viewed as an argument framework in Dung’s
system. An argument in an argument tree could be viewed as an argument in a Dung
argument framework, and each arc in an argument tree could be viewed as an attack
relation. However, the way sets of arguments are compared is different.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-163-320.jpg)
![150 Philippe Besnard and Anthony Hunter
Some differences between Dung’s approach and our approach can be seen in the
following examples.
Example 7.29. Consider a set of arguments {A1,A2,A3,A4} with the attack relation
R s.t. A2RA1, A3RA2, A4RA3, and A1RA4. Here there is an admissible set {A1,A3}.
We can try to construct an argument tree with A1 at the root. As a counterpart to the
attack relation, we regard that A1 is undercut by A2, A2 is undercut by A3, and so on.
However, the corresponding sequence of nodes A1,A2,A3,A4,A1 is not an argument
tree because A1 occurs twice in the branch (violating Condition 2 of Definition 7.8).
So, the form of the argument tree for A1 fails to represent the fact that A1 attacks A4.
Example 7.30. Let Δ = {b,b → a,d ∧¬b,¬d ∧¬b}, giving the following argument
tree for a.
{b,b → a},a
ր տ
{d ∧¬b},✸ {¬d ∧¬b},✸
↑ ↑
{¬d ∧¬b},✸ {d ∧¬b},✸
For this let A1 be {b,b → a},a, A2 be {d ∧ ¬b},✸ and A3 be {¬d ∧ ¬b},✸.
Disregarding the difference between the occurrences of ✸, this argument tree
rewrites as A2RA1, A3RA1, A3RA2, and A2RA3 where A1 denotes the root node
{b,b → a},a. In this argument tree, each defeater of the root node is defeated. Yet
no admissible set of arguments contains A1.
Our proposal has been extended in a number of ways. As covered in [6], we have
proposed techniques that take into account intrinsic aspects of the arguments and
counterargument such as the degree of conflict between them, and the degree of
similarity between them, and extrinsic aspects such as the impact on the audience
and the empathy or antipathy the audience may have for individual arguments.
Further developments include formalization of enthymemes in our logic-based
framework [16, 7], a generalization of the framework to also consider the proponent
of each argument, and thereby argue about whether a proponent is an appropriate
proponent for that argument [17], and a refinement of the proposal to reason with
temporal knowledge [18].
We have also generalised the proposal reviewed in this chapter to handle first-
order logic [4]. Indeed, this is straightforward since all we need to do is to use a
first-order language and to use the classical first-order consequence relation instead
of the classical propositional logic. The key finiteness results still hold: So for a finite
number of formulae in the knowledgebase, there is a finite number of argument trees
for a claim, and each of these trees contains a finite number of nodes.
We have developed algorithms for generating arguments [12, 13], formaliza-
tions of decision problems concerning arguments and counterarguments in quan-
tified Boolean formaulas [2], and compilation techniques with the aim of improving
the viability of argumentation based on classical logic [5].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-164-320.jpg)
![Chapter 8
Argument-based Logic Programming
Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
1 Introduction
In this chapter we describe several formalisms for integrating Logic Programming
and Argumentation. Research on the relation between logic programming and ar-
gumentation has been and still is fruitful in both directions: Some argumentation
formalisms were used to define semantics for logic programming and also logic
programming was used for providing an underlying representational language for
non-abstract argumentation formalisms.
One of the first attempts dates back to 1987, when Donald Nute [19] introduced
a formalism called LDR (Logic for Defeasible Reasoning) with a simple represen-
tational language consisting of three types of rules: Strict, defeasible and defeaters.
Although LDR is not a defeasible argumentation formalism in itself, its implemen-
tation, d-Prolog, defined as an extension of PROLOG, was the first language that
introduced defeasible reasoning programming with specificity as a comparison cri-
terion between rules.
Another important step was taken in the nineties, when Phan M. Dung [10] em-
phasized that “there are extremely interesting relations between argumentation and
logic programming”. Dung showed that argumentation can be viewed as a special
form of logic programming with negation as failure. He introduced a general logic
Alejandro J. Garcı́a
Consejo Nacional de Investigaciones Cientı́ficas y Técnicas (CONICET)
Department of Computer Science and Engineering, Universidad Nacional del Sur
e-mail: ajg@cs.uns.edu.ar
Guillermo R. Simari
Department of Computer Science and Engineering, Universidad Nacional del Sur
e-mail: grs@cs.uns.edu.ar
Jürgen Dix
Department of Informatics Clausthal University of Technology
e-mail: dix@tu-clausthal.de
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 153
DOI 10.1007/978-0-387-98197-0 8, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-167-320.jpg)
![154 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
programming based method for generating meta-interpreters for argumentation sys-
tems.
A few years later, in 1997, inspired by legal reasoning, Prakken and Sartor [24]
introduced an argument-based formalism for extended logic programming with de-
feasible priorities. In their formalism, arguments are expressed in a logic program-
ming language with both strong and default negation. Conflicts between arguments
are decided with the help of priorities on the rules. These priorities can be defeasibly
derived as conclusions within the system. The semantics of the system is given by
a fixed point definition, while its proof theory is stated in dialectical style. A proof
takes the form of a dialogue between a proponent and an opponent: An argument
is shown to be justified if the proponent can make the opponent run out of moves in
whatever way the opponent attacks.
Another formalism for combining logic programming and argumentation, Defea-
sible Logic Programming (DeLP) has been introduced by Garcı́a and Simari in [15].
The representational language of DeLP is defined as an extension of a logic pro-
gramming language that considers two types of rules: Strict and defeasible, and
allows for both strong and default negation. A DeLP-query succeeds, i.e., it is war-
ranted from a DeLP-program, if it is possible to build an argument that supports
the query and this argument is found to be undefeated by a warrant procedure. This
process implements an exhaustive dialectical analysis that involves the construction
and evaluation of arguments that either support or interfere with the given query. The
DeLP dialectical analysis is based on previous works on defeasible argumentation
conducted by Simari and Loui [28], and Simari, Chesñevar, and Garcı́a [27].
In [26], Schweimeier and Schroeder formulate a variety of notions of attack for
extended logic programs from combinations of undercut and rebuttal. As shown
in Section 4, their language corresponds to that used by Prakken and Sartor [24]
without strict rules and any priorities. They also define a general hierarchy of argu-
mentation semantics parameterized by the notions of attack chosen by the proponent
and the opponent. They prove the equivalence and subset relationships between the
semantics and examine some essential properties concerning consistency and the
coherence principle, which relates default negation and explicit negation.
The rest of the chapter is organized as follows. Section 2 describes Defeasible
Logic Programming. Section 3 introduces Prakken’s approach of argument-based
extended logic programming with defeasible priorities. Section 4 gives an overview
of argumentation semantics for extended logic programming by Schweimeier and
Schroeder. Section 5 introduces Dung’s approach and Section 6 illustrates Nute’s
framework. Finally, in Section 7 we discuss more recent developments before con-
cluding with Section 8.
2 Defeasible Logic Programming
Defeasible Logic Programming (DeLP), as introduced in [15], is a formalism that
combines techniques of both logic programming and defeasible argumentation. As](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-168-320.jpg)
![8 Argument-based Logic Programming 155
in logic programming, in DeLP knowledge is represented using facts and rules;
however, DeLP also provides the possibility of representing information in the form
of weak rules in a declarative manner. These weak rules are the key element for in-
troducing defeasibility [21] and they are used to represent a relation between pieces
of knowledge that could be defeated when all things have been considered. DeLP
uses a defeasible argumentation inference mechanism for warranting the entailed
conclusions.
A defeasible logic program (DeLP-program for short), is a set of facts, strict rules
and defeasible rules, defined as follows.
• Facts are ground literals representing atomic information or the negation of
atomic information using strong negation “∼”, (e.g., chicken(little) and
∼scared(little)).
• Strict rules, denoted by L0 ← L1,...,Ln, represent non-defeasible information.
The head of the rule, L0, is a ground literal and the body {Li}i0 is a non-empty
set of ground literals, (e.g., bird ← chicken and ∼innocent ← guilty).
• Defeasible rules, denoted by L0 —
L1,...,Ln, represent tentative information.
The head L0 is a ground literal and the body {Li}i0 is a non-empty set of ground
literals (e.g., ∼flies —
chicken or flies —
chicken,scared).
Syntactically, the symbol “—
” is all that distinguishes a defeasible rule from a
strict one. Pragmatically, a defeasible rule is used to represent tentative information
that may be used if nothing could be posed against it. A defeasible rule “H —
B”
is understood as expressing that “reasons to believe in the antecedent B provide
reasons to believe in the consequent H” [28]. For instance, “lights on —
switch on”
expresses that reasons to believe that the switch of a room is on, provide reasons
to believe that lights will be on. Note that this rule represents tentative information
because it may happen that the switch is on and the lights are not on, for example,
if light bulbs are broken, or there is no electricity (as represented by the following
rule: “∼lights on —
∼electricity”).
The information that represents a strict rule is not tentative. For instance, given
“∼innocent ← guilty” and the fact guilty, then it will not be possible to infer
innocent. That is, if guilty has a strict derivation then ∼innocent will also have a
strict derivation and innocent cannot be concluded. However, as it will be shown
below, if guilty is a tentative conclusion, then ∼innocent will also be a tentative
conclusion, and innocent may be concluded if there is information that supports it.
When required, a DeLP-program is denoted by the pair (Π,Δ) distinguishing the
subset Π of facts and strict rules (that represents non-defeasible knowledge), and
the subset Δ of defeasible rules. Observe that strict and defeasible rules are ground.
However, following the usual convention [17], some examples will use “schematic
rules” with variables. To distinguish variables, they are written with an uppercase
letter. In examples, a period may be used as a delimiter between rules or facts. As in
logic programming, queries can be posed to a program. A DeLP-query is a ground
literal that DeLP will try to warrant. For example, dark(a) or illuminated(b) are
DeLP-queries.
Example 8.1. Consider the DeLP-program (Π8.1,Δ8.1) where:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-169-320.jpg)
![158 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
is referred to as the counter-argument point and C,P as the disagreement sub-
argument.
Two contradictory literals trivially disagree (e.g., day and ∼day), however,
two literals that are non-contradictory can also disagree. For instance, in Ex-
ample 8.1, the literals dark(b) and illuminated(b) disagree because from Π8.1∪
{illuminated(b), dark(b)} there exists a strict derivation for ∼dark(b). Hence,
following the example above, A2,illuminated(b) is a counter-argument for
A1,dark(b) (and vice versa). Observe that the argument A3,∼lights on(b) is a
counter-argument for A2,illuminated(b) at (the inner point) lights on(b) and the
disagreement sub-argument in A2 is {lights on(b) —
switch on(b)},lights on(b).
Given an argument A,H, there could be several counter-arguments attacking
different points in A, or different counter-arguments attacking the same point in A.
In DeLP, in order to verify whether an argument is non-defeated, all of its associ-
ated counter-arguments B1, B2, ..., Bk have to be examined, each of them being a
potential reason for rejecting A. If any Bi is (somehow) “better” than, or unrelated
to, A, then Bi is a candidate for defeating A. However, if for some Bi, the argument
A is “better” than Bi, then Bi will not be a defeater for A. To compare arguments
and counter-arguments a preference relation among arguments is needed.
In DeLP the argument comparison criterion is modular, and thus the most ap-
propriate criterion for the domain that is being represented can be introduced. In the
literature of DeLP different criteria have been defined. For example in [15] a crite-
rion that uses rule priorities was introduced. Recently, in [11] a comparison criterion
based on priorities among selected literals of the program was defined. In the follow-
ing examples we will use a syntactic criterion called generalized specificity [15, 29].
This criterion favors two aspects in an argument: It prefers (1) a more precise argu-
ment or (2) a more concise argument. For instance, A2,illuminated(b), is more
specific (more precise) than A1,dark(b). For the following definitions we will
abstract away from the comparison criterion, assuming there exists one (denoted
“≻”).
Definition 8.3 (Defeaters: Proper and Blocking). Let B,S be a counter-argument
for A,H at point P, and C,P the disagreement sub-argument. If B,S ≻ C,P
(i.e., B,S is “better” than C,P) then B,S is a proper defeater for A,H. If
B,S is unrelated by the preference relation to C,P, (i.e., B,S ≻ C,P, and
C,P ≻ B,S) then B,S is a blocking defeater for A,H. Finally, B,S is a
defeater for A,H, if B,S is either a proper or blocking defeater for A,H.
In a previous example, we have shown that A2,illuminated(b) is a counter-
argument for A1,dark(b) (and vice versa). Considering generalized specificity as
the comparison criterion, then A2 ≻ A1. Therefore, A2,illuminated(b) is a proper
defeater for A1,dark(b). The argument A3,∼lights on(b) is a blocking defeater
for A2,illuminated(b) because A3 and the disagreement sub-argument D1 ={
lights on(b) —
switch on(b)} of A2 are unrelated by this comparison criterion.
If an argument A1,H1 is defeated by A2,H2, then A2,H2 represents a rea-
son for rejecting A1,H1. Nevertheless, a defeater A3,H3 for A2,H2 may also](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-172-320.jpg)
![8 Argument-based Logic Programming 159
exist, rejecting A2,H2 and reinstating A1,H1. Note that the argument A3,H3
may be in turn defeated, reinstating A2,H2, and so on. In this manner, a sequence
of arguments (called argumentation line) can arise, where each element is a defeater
of its predecessor.
Definition 8.4 (Argumentation Line). An argumentation line for A1,H1 is a se-
quence of argument structures, Λ = [A1,H1, A2,H2, A3,H3, . . . ], where each
element of the sequence Ai,Hi, i 1, is a defeater of its predecessor Ai−1,Hi−1.
In an argumentation line Λ = [A1,H1, A2,H2, A3,H3, . . . ], the first ele-
ment, A1,H1, becomes a supporting argument for H1, A2,H2 an interfering ar-
gument, A3,H3 a supporting argument, A4,H4 an interfering one, and so on.
Thus, an argumentation line can be split into two disjoint sets: ΛS = {A1,H1,
A3,H3, A5,H5, ...} of supporting arguments, and ΛI = {A2,H2, A4,H4,
...} of interfering arguments. Considering the arguments generated above from
(Π8.1,Δ8.1) of Example 8.1, Λ1 = [A1,dark(b), A2,illuminated(b),
A3,∼lights on(b)] is an argumentation line, with two supporting arguments and
an interfering one.
Note that an infinite argumentation line may arise if an argument structure is rein-
troduced in the sequence. For instance, Λi = [A3,∼lights on(b), D1,lights on(b),
A3,∼lights on(b), D1,lights on(b), ... ]. This is an example of circular argu-
mentation where an argument structure is reintroduced again in the argumentation
line to defend itself. Clearly, this situation is undesirable as it leads to the construc-
tion of an infinite sequence of arguments. Circular argumentation and other forms of
fallacious argumentation were studied in detail in [27, 15]. In DeLP, argumentation
lines have to be acceptable, that is, they have to be finite, an argument cannot appear
twice, and supporting (resp. interfering) arguments have to be non-contradictory.
Definition 8.5 (Acceptable Argumentation Line).
An argumentation line Λ = [A1,H1,...An,Hn] is acceptable iff:
1. Λ is a finite sequence.
2. The set ΛS of supporting arguments (resp. ΛI), is concordant (a set {Ai,Hi}n
i=1
is concordant iff Π ∪
n
i=1 Ai is non-contradictory.).
3. No argument Ak,Hk in Λ is a disagreement sub-argument of an argument
Ai,Hi appearing earlier in Λ (i k.)
4. For all i, such that Ai,Hi is a blocking defeater for Ai−1,Hi−1, if Ai+1,Hi+1
exists, then Ai+1,Hi+1 is a proper defeater for Ai,Hi.
A single acceptable argumentation line for A1,H1 may not be enough to es-
tablish whether A1,H1 is an undefeated argument. The reason is that there can
be several defeaters (B1,Q1, B2,Q2, ..., Bk,Qk) for A1,H1, and therefore
k argumentation lines for A1,H1 need to be considered. Since for each defeater
Bi,Qi there can be in turn several defeaters, then a tree structure (called dialec-
tical tree) is defined. In this tree, the root is labeled with A1,H1 and every node
(except the root) represents a defeater (proper or blocking) of its parent. Each path
from the root to a leaf corresponds to a different acceptable argumentation line. The
definition follows.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-173-320.jpg)
![160 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
Definition 8.6 (Dialectical Tree). A dialectical tree for A1,H1, TA1,H1, is de-
fined as follows:
1. The root of the tree is labeled with A1,H1.
2. Let N be a non-root node labeled An,Hn, and [A1,H1,...,An,Hn] be the
sequence of labels of the path from the root to N. Let {B1,Q1, B2,Q2, ...,
Bk,Qk} be the set of all the defeaters for An,Hn. For each defeater Bi,Qi
(1≤i≤k), such that the argumentation line Λ′
= [A1,H1,...,An,Hn,Bi,Qi]
is acceptable, the node N has a child Ni labeled Bi,Qi. If there is no defeater
for An,Hn or there is no Bi,Qi such that Λ′
is acceptable, then N is a leaf.
In Example 8.1, C1,working at(c) has two defeaters: C2,∼lights on(c),
and C3,∼working at(c), where: C1= {(working at(c) —
illuminated(c)),
(illuminated(c) —
lights on(c),∼day), (lights on(c) —
switch on(c))}; C2=
{∼lights on(c) —
∼electricity(c)} and C3={∼working at(c)—
sunday}. Since C2
and C3 have defeaters, then the following acceptable argumentation lines arise: [C1,
C2, C4] where C4={ working at(c) —
sunday,deadline}, and [C1, C3, C5] where
C5={lights on(c)—
∼electricity(c), emergency lights(c)}. These two lines are con-
sidered in the dialectical tree T
1
∗ of Figure 8.1.
A dialectical tree provides a structure integrating all the possible acceptable ar-
gumentation lines that can be generated for deciding whether an argument is unde-
feated. In order to compute whether the status of the root is defeated or undefeated,
a recursive marking process is introduced next. Let T be a dialectical tree, a marked
dialectical tree, T∗, can be obtained marking every node in T as follows: (1) Each
leaf in T is marked as U in T∗. (2) An inner node N of T is marked as D in T∗ iff N
has at least a child marked as U, otherwise N is marked as U in T∗. Thus, the root
A,H of a dialectical tree is marked with U if all its children are marked as D (i.e.,
all the defeaters for A,H are defeated).
Figure 8.1 shows three different marked dialectical trees. Nodes are depicted as
triangles with marks (D or U) on their right. D-nodes are also grey colored. The
marked dialectical tree T
1
∗ has also the argument names of the example introduced
above as labels inside the triangles.
A ground literal H is considered to be warranted if an argument A for H exists,
and all the defeaters for A,H are defeated. Therefore, this marking procedure
provides an effective way of determining if a DeLP-query H is warranted. Note that
given a DeLP-query H, there can be several arguments that support H. Therefore,
H is warranted if there is at least one argument A for H such that the root of a
dialectical tree for A,H is marked as U.
Definition 8.7 (Warranted Literals). Let P be a DeLP-program, H a ground literal,
A,H an argument from P and T∗ a marked dialectical tree for A,H. Literal H is
warranted from P iff the root of T∗ is marked as U.
Example 8.2. Consider again the DeLP-program (Π8.1,Δ8.1) of Example 8.1. The
set ω includes some literals that are warranted from the program (Π8.1,Δ8.1)
ω ={working at(c), illuminated(c), illuminated(a), dark(b), ∼dark(a)}, whereas
the set ω′ includes literals that are not warranted from (Π8.1,Δ8.1) ω′ ={dark(a),](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-174-320.jpg)
![8 Argument-based Logic Programming 161
U
D U
D D
U
D D D D
C2 C3
U U
D D
U U
U
U
C4
C1
U
C5
U U
U
T1
* T2
* T3
*
Fig. 8.1 Marked Dialectical Trees
illuminated(b), ∼lights on(b), ∼empty(a),illuminated(r)}. Observe that the last
two literals of ω′ have no supporting argument at all.
DeLP Implementation and Extensions
An interpreter for DeLP is available (http://lidia.cs.uns.edu.ar/DeLP). This
interpreter takes a DeLP-program P, and a DeLP-query Q as input. It then returns
one of the following four possible answers: YES, if Q is warranted from P; NO, if the
complement of Q is warranted from P; UNDECIDED, if neither Q nor its complement
are warranted from P; or UNKNOWN, if Q is not in the language of the program P.
For instance, considering the program (Π8.1,Δ8.1) of Example 8.1 the answer for
illuminated(c), is YES, the answer for dark(a) is NO, the answer for ∼lights on(b)
is UNDECIDED, and the answer for ∼empty(a) is UNKNOWN. Observe that for all
the literals of the set ω in Example 8.2 the answer is YES. However, as shown above,
literals of ω′ may have different answers.
The reader may have noticed in Figure 8.1 that in some dialectical trees it is not
necessary to explore the whole tree in order to mark the root. Thus, some pruning is
possible. For instance, in the tree T
2
∗ of Figure 8.1 as one of the children of the root
is marked U, then the mark of the other two children does not affect the mark of the
root. As an optimization, the DeLP interpreter implements a marking procedure with
pruning that is similar to the classic α-β pruning algorithm (see [15] for details). It
is interesting to note that changes in the definition of acceptable argumentation line
may produce a different behavior of the formalism. Therefore, Definition 8.5 could
be modified by providing a way of tuning the system in order to have a different
behavior.
In [15] an extension of DeLP that allows default negation in the body of defeasi-
ble rules was defined (e.g., a —
not b,c,not ∼d). In DeLP ‘not F’ is assumed when
the literal F is not warranted. The definition of argument was adapted for this ex-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-175-320.jpg)
![162 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
tension and a new form of defeat (attack to an assumption) was introduced (see [15]
for details). Thus, default negated literals become new points of attack, and when
the dialectical analysis is carried out, default negated literals can be defeated by
arguments.
In [14] dialectical explanations for DeLP-queries (called δ-Explanations) were
introduced. δ-Explanations allow to visualize the reasoning carried out by the sys-
tem, and the support for the answer. It is clear that without this information at hand it
is very difficult to understand the returned answer. δ-Explanations were also defined
for schematic queries, i.e., a DeLP-query with at least one variable that represents
the set of DeLP-queries that unify with the schematic one (e.g., illuminated(X).)
Possibilistic Defeasible Logic Programming (P-DeLP) represents an important
extension of DeLP in which the elements of the language have the form (ϕ,α),
where ϕ is of the form L0 ← L1,...,Ln or just L0, where the Li,(0 ≤ i ≤ n) are
ground literals as in DeLP, and α is a certainty weight associated with ϕ.
A P-DeLP-program Γ is a set of clauses. To define the possibilistic entailment we
only need the triviality axiom of Possibilistic Gödel Logic (PGL) and the general-
ized modus ponens rule (GMP). The triviality axiom says that we can add (ϕ,0) for
any formula ϕ in the language and the GMP says that from (L0 ← L1,...,Ln,γ)
and (L1,βn),...,(L1,βn) we can obtain (L0,min(γ,β1,...,βn)). A derivation of
a weighted literal (Q,α) from Γ , denoted Γ ⊢ (Q,α), is a sequence of clauses
C1,...,Cm such that Cm = (Q,α), and for each 1 ≤ i ≤ m it holds that Ci ∈ Γ , or Ci
is an instance of the triviality axiom, Ci is obtained by using GMP over its preceding
clauses in the sequence.
The set of clauses Γ of a P-DeLP program can be split, like in DeLP, as (Π,Δ)
distinguishing the subset Π of facts and strict rules with certainty weight α = 1 (rep-
resenting non-defeasible knowledge), and the subset Δ of of clauses with certainty
weight α 1 (representing defeasible knowledge). An argument for a literal (Q,α)
built from a P-DeLP program (Π,Δ) is a subset A ⊆ Δ such that: Π ∪A ⊢ (Q,α),
Π ∪A is non-contradictory, and A is minimal with respect to set inclusion satisfying
the two previous conditions; and it is denoted A,Q,α.
From these extended notions, a complete argumentation system has been devel-
oped which incorporates possibilistic reasoning. Complete details of P-DeLP can be
found in [1, 2].
3 Argument-based extended logic programming with defeasible
priorities
Inspired by legal reasoning, Prakken and Sartor have introduced in [24] an argument-
based formalism for extended logic programming with defeasible priorities. In their
formalism, arguments are expressed in a logic programming language with both
strong and default negation. Conflicts between arguments are decided with the help
of priorities on the rules. These priorities can be defeasibly derived as conclusions
within the system. Its semantics is given with a fixed point definition, while its proof](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-176-320.jpg)
![8 Argument-based Logic Programming 163
theory is stated in dialectical style, where a proof takes the form of a dialogue be-
tween a proponent and an opponent: an argument is shown to be justified if the
proponent can make the opponent run out of moves in whatever way the opponent
attacks.
We give a brief description of this formalism and refer the interested reader
to [24, 22, 25] for details. Their proposed system assumes input information in the
form of an ordered theory (S,D) where S and D are sets of, respectively, strict and
defeasible rules. In their language, ∼ represents default negation and ¬ strong nega-
tion, and rules are denoted by
r : L0 ∧...∧Lj ∧∼Lk ∧...∧∼Lm ⇒ Ln
where r, a term, is the name of the rule, each Li (0 ≤ i ≤ n) is a strong literal, and
each ∼Lk is a weak literal. Only defeasible rules can contain weak literals.
An argument is a finite sequence A = [r0,...,rn] of ground instances of rules
such that
1. for every i (0 ≤ i ≤ n), for every strong literal Lj in the antecedent of ri there is a
k i such that Lj is the consequent of rk; and
2. no two distinct rules in the sequence have the same consequent.
Each consequent of a rule in A is a conclusion of A, and every literal L is an assump-
tion of A iff ∼L occurs in some rule in A. An argument is strict iff it does not contain
a defeasible rule; it is defeasible otherwise. Note that arguments are not assumed to
be either minimal or consistent.
The presence of assumptions in a rule gives rise to two kinds of conflicts between
arguments, conclusion-to-conclusion attack and conclusion-to-assumption attack.
An argument A1 attacks A2 iff there are sequences of strict rules S1 and S2, such that
the concatenation A1 + S1 is an argument with conclusion L and (1) A2 + S2 is an
argument with conclusion L, or (2) L is an assumption of A2. In order to compare
conflicting arguments, priorities among rules (that are relevant to the conflict) are
used. For any two sets R and R′ of defeasible rules, R R′ iff for some r ∈ R and all
r′ ∈ R′ it holds that r r′.
Defeat among arguments is built up from two other notions, rebutting and un-
dercutting an argument. An argument A defeats an argument B iff (1) A=[ ] and B
attacks itself, or else if (2a) A undercuts B or (2b) A rebuts B and B does not undercut
A. As explained in [25], an argument A rebuts an argument B iff A conclusion-to-
conclusion attacks B and either A is strict and B is defeasible, or A’s rules involved in
the conflict have no lower priority than B’s rules involved in the conflict (see [24, 22]
for details). An argument A undercuts an argument B precisely in case of the second
kind of conflict (attack on an assumption). Note that it is not necessary that the rules
responsible for the attack on the assumption have no lower priority than the one
containing the assumption. An argument A strictly defeats B iff A defeats B and B
does not attack A.
As stated above, priorities between rules can be defeasibly derived as conclu-
sions within the system. Hence, the language is extended with a special predicate
≺, where r1 ≺ r2 means that rule r2 has priority over r1. This new predicate symbol](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-177-320.jpg)
![164 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
denotes a strict partial order, and therefore the set S must contain the axioms of a
strict partial order. Thus, the rebut and defeat relations must be made relative to an
ordering relation that might vary during the reasoning process.
Since priorities are not fixed, in order to determine if an argument A is acceptable
with respect to a set T of arguments, the relevant defeat relations are verified relative
to the priority conclusions drawn by the arguments in T. Therefore, given a set of
arguments T, priorities between rules are defined as T ={r r′ | r ≺ r′ is a conclu-
sion of some A ∈ T}. Hence, an argument A (strictly) T-defeats B iff, assuming the
ordering T , A (strictly) defeats B.
An argument A is acceptable with respect to a set T of arguments iff all argu-
ments T-defeating A are strictly T-defeated by some argument in T. Thus, accept-
ability of an argument with respect to a set T depends on the priority conclusions of
the arguments in T. Let Γ be an ordered theory (S,D), and T be any set of arguments
from Γ ; the characteristic function of Γ is FΓ (T)={A | A is acceptable with respect
to T}. Then, the status of an argument A is defined as follows: A is justified iff A is
in the least fixed point of FΓ , A is overruled iff A is attacked by a justified argument,
and A is defensible iff A is neither justified nor overruled.
For example, consider a theory (S,D) where D has the following rules:
r0 : ⇒ a r2 : ∼b ⇒ c r4 : ⇒ r0 ≺ r3 r6 : ⇒ r5 ≺ r4
r1 : a ⇒ b r3 : ⇒ ¬a r5 : ⇒ r3 ≺ r0
The set of justified arguments is constructed as follows. The only /
0-undefeated
argument is [r6], then, FΓ (/
0) = F1 = {[r6]} and F1 = {r5 r4}. Now, the con-
flict between [r4] and [r5] can be solved because [r4] strictly F1-defeats [r5].
Then, FΓ (F1) = F2 = {[r6], [r4]} and F2 = {r5 r4,r0 r3}. Hence, the conflict
between [r0] and [r3] can be solved because [r3] strictly F2-defeats [r0]. Thus,
FΓ (F2) = F3 = {[r6], [r4], [r3]} and F3 = F2 . Finally, FΓ (F3) = F4 = {[r6], [r4],
[r3], [r2]} and FΓ (F4) = F4.
The proof theory of this formalisms is stated in dialectical style, where a proof
takes the form of a dialogue between a proponent (P) and an opponent (O). An
argument is shown to be justified if P can make O run out of moves.
A priority dialogue is a finite sequence of moves (Playeri,Ai), where:
1. Playeri = P iff i is odd, and Playeri = O iff i is even;
2. If Playeri = Playerj = P and i = j, then Ai = Aj;
3. If Playeri = P(i 1) then Ai is a minimal argument such that Ai strictly {Ai}-
defeats Ai−1; or Ai−1 does not {Ai}-defeat Ai−2;
4. If Playeri = O(i 1) then Ai strictly /
0-defeats Ai−1;
A priority dialogue tree is a finite tree of moves such that (1) each branch is a
dialogue and (2) if Playeri = P then the children of movei are all /
0-defeat Ai. A
player wins a dialogue if the other player cannot move, and a player wins a dialogue
tree iff it wins all branches of the tree. Finally, an argument A is provably justified
iff there is a dialogue tree with A as its root, and won by the proponent.
The semantics of this formalism was very much inspired by Dung’s work. If it
is restricted to static priorities (given as input instead of derived within the system)
then it instantiates Dung’s grounded semantics [10]. However, the semantics slightly](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-178-320.jpg)
![8 Argument-based Logic Programming 165
differs from Dung’s when dynamic priorities are considered. In [23] some detailed
legal examples are given, and a third way of attacking arguments, namely by arguing
that a rule is not applicable, is introduced.
4 Argumentation Semantics for Extended Logic Programming
In [26], Schweimeier and Schroeder formulate a variety of notions of attack for ex-
tended logic programs from combinations of undercutting and rebuttal. A general
hierarchy of argumentation semantics is defined, parameterized by the notions of
attack chosen by proponent and opponent, and the equivalence and subset relation-
ships between the semantics is shown. The placement of existing semantics in this
hierarchy is determined.
The language corresponds to [24] without strict rules, and either without priori-
ties, or, equivalently, if all rules have the same priority. An extended logic program
(ELP) P is a (possibly infinite) set of rules of the form L0 ← L1, ..., Lm, not Lm+1,
...,not Lm+n (m,n ≥ 0) where each Li (0 ≤ i ≤ m+n) is an objective literal, i.e., an
atom M or its explicit negation ¬M.
An argument associated with P is a finite sequence A = [r0,...,rn] of ground
instances of rules ri ∈ P such that for every 0 ≤ i ≤ n, for every objective literal
Lj in the body of ri there is a k i such that Lj=head(rk). The head of a rule in
A is called a conclusion of A, and a default literal not L in a rule of A is called an
assumption of A. An argument A with a conclusion L is a minimal argument for L if
there is no subargument of A with conclusion L. Given an ELP P, the set of minimal
arguments associated with P is denoted by ArgsP.
From the two basic notions of attack (undercut u, and rebut r), they introduce
further notions: Attacks (a), defeats (d), strongly attacks (sa) and strongly undercuts
(su). Here are the precise definitions. Let A1 and A2 be arguments.
• A1 undercuts A2 if there is an objective literal L such that L is a conclusion of A1
and not L is an assumption of A2.
• A1 rebuts A2 if there is an objective literal L such that L is a conclusion of A1 and
¬L is a conclusion of A2.
• A1 attacks A2 if A1 undercuts or rebuts A2.
• A1 defeats A2 if A1 undercuts A2, or A1 rebuts A2 and A2 does not undercut A1.
• A1 strongly attacks A2 if A1 attacks A2 and A2 does not undercut A1.
• A1 strongly undercuts A2 if A1 undercuts A2 and A2 does not undercut A1.
A notion of attack is defined as a function x that assigns to each ELP P a bi-
nary relation xP ⊆ ArgsP × ArgsP. Different notions of attack are partially ordered
by defining: x ⊆ y iff ∀P : xP ⊆ yP. The inverse of a notion of attack x, denoted
by x−1, is defined as: x−1
P = {(B,A)|(A,B) ∈ xP}. Thus, a = u ∪ r, d = u ∪ (r-u−1),
sa = (u ∪ r)-u−1, and su = u-u−1. Since their framework does not consider priorities,
undercuts play the prime role and notions of attack which are based on rebuttals,
such as r or r-u−1, are not considered.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-179-320.jpg)
![166 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
Their definition of acceptability is parameterized on the notions of attack allowed
for the proponent and the opponent. Basically, an argument is acceptable if it can be
defended against any attack.
Let x and y be notions of attack, A an argument, and S a set of arguments. Then A
is x/y-acceptable with respect to S if for every argument B such that (B,A) ∈ x there
exists an argument C ∈ S such that (C,B) ∈ y. A particular semantics can be obtained
by choosing one notion of attack for the opponent, and another notion of attack as
defense for the proponent. The uniformity of the definition makes it a convenient
framework for comparing different argumentation semantics.
Based on the notion of acceptability, they define a fixed point semantics for ar-
guments: FP,x/y(S) = {A|A is x/y-acceptable with respect to S} and its least fixed
point is denoted by JP,x/y. Thus, an argument A is x/y-justified if A ∈ JP,x/y, A is x/y-
overruled if it is attacked by an x/y-justified argument, and A is x/y-defensible if it
is neither x/y-justified nor x/y-overruled. For example, Dung’s grounded semantics
is Ja/u, and Prakken and Sartor’s semantics without priorities or strict rules is Jd/su
(see [26] for other examples).
Consider, for instance, P1 = {(p ← not q), (q ← not p), (¬q ← r), (r ← not s),
(¬s ← not s), (s)}. Here, all arguments (except [s]) are undercut by another argu-
ment, and [¬s ← not s] rebuts (but does not defeat) [s]. Thus, [s] is identified as
a justified argument in all semantics, except if a is allowed as an attack. We state
several results (for other semantics see [26]).
JP1,a/x= /
0 JP1,u/sa={[s],[¬q ← r], [p ← not q] }
JP1,d/x={[s]} JP1,sa/su={[s],[p ← not q]}
JP1,u/u={[s],[¬q ← r] } JP1,su/x={[s],[¬q ← r],[q ← not p ],[p ← not q]}
Schweimeier and Schroeder proved a series of theorems, showing that some of
the argumentation semantics defined above are subsumed by others. In fact some are
actually equivalent. Thus, they established a hierarchy of argumentation semantics,
where WFS and WFSXp also occur. One of their results states that WFS=u/u and
WFSXp=u/a. They also showed which semantics satisfy the coherence principle
and which ones generate a conflict free set of justified arguments. Finally, based on
the dialectical proof theory of [24] they introduce a proof theory for x/y-justified
arguments.
5 Dung’s approach
In [10], Phan Minh Dung states that “there are extremely interesting relations be-
tween argumentation and logic programming”. For example, he shows that logic
programming can be shown to be a particular form of argumentation. He considers a
logic program P as a finite set of clauses of the form B0 ← B1,...,Bm,¬Bm+1,...,
¬Bm+n (where the Bi’s are atoms). To capture the semantics of negation as finite
failure (naff), he proposes that P can be transformed in an argumentation frame-
work AFnaff = AR, attacks as follows: AR = {(K,k)| there exists a ground clause
of P with head k and body K} ∪ {({¬k},k)|k is a ground atom}, and (K,h) at-
tacks (K′,h′) iff the complement of h belongs to K′. That is, each ground instance](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-180-320.jpg)
![8 Argument-based Logic Programming 167
of a clause of P constitutes an argument for its head. Then, Dung gives a series
of theorems that relate semantics of logic programs and semantics of argumenta-
tion frameworks. Similarly, an argumentation framework AFnapif that captures the
semantics of negation of possibly infinite failure (general loop checking, as in the
stable semantics) is introduced.
In [10], Dung also shows that argumentation can be “viewed” as logic program-
ming by introducing a general method for generating an interpreter for argumenta-
tion. This method consists of a very simple logic program consisting of the following
two clauses:
C1 = acceptable(A) ← not defeated(A)
C2 = defeated(A) ← attacks(B,A),acceptable(B)
Consider an argumentation framework AF = (AR,attacks) where AR is a set
of arguments and attack a relation over AR × AR. Let PAF be the logic pro-
gram, PAF = {C1,C2} ∪ {attacks(B,A)|(B,A) ∈ attacks}. For each extension E
of AF, let m(E) = {attacks(B,A)|(B,A) ∈ attacks} ∪ {acceptable(A)|A ∈ E}
∪{defeated(B)|B is attacked by some A ∈ E}. Then, Dung shows that
1. E is a stable extension of AF iff m(E) is stable model of PAF .
2. E is grounded extension of AF iff m(E) ∪ {not defeated(A)|A ∈ E} is the well-
founded model of PAF .
6 Nute’s Defeasible Logic
In [19, 20], Donald Nute introduced Logic for Defeasible Reasoning (LDR), a for-
malism that provides defeasible reasoning with a simple representational language.
Although LDR is not a defeasible argumentation formalism in itself, its implemen-
tation, d-Prolog, defined as an extension of PROLOG, was the first language that
introduced defeasible reasoning programming with specificity as a comparison cri-
terion between rules.
In LDR there are three types of rules: strict rules (e.g., emus → birds that rep-
resents “emus are birds”), defeasible rules (e.g., birds ⇒ fly “birds usually fly”)
and defeater rules (e.g., heavy ❀ ¬ fly “heavy animals may not fly”). The purpose
of a defeater rule is to account for the exceptions to defeasible rules. Therefore, in
contrast with the other two types of rules, defeater rules cannot be used to derive
formulas. They can only be used to block an application of a defeasible rule. As
shown below, the derivation of contradictory literals is prevented by the definition
of a defeasible derivation.
The original definitions of strict and defeasible derivation, introduced in [19],
have evolved in successive articles [20, 4, 3]. For strict derivations only facts and
strict rules can be used. However, for defeasible derivations a more sophisticated
analysis is performed. A literal p has a defeasible derivation if p is the consequent
of a rule (strict or defeasible) with antecedent A, such that (1) for every a ∈ A, a](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-181-320.jpg)
![168 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
has a defeasible derivation, (2) the complement of p (p) has no strict derivation, and
(3) for every rule R1 (strict, defeasible or defeater) with consequent p and antecedent
B, it holds that (3a) there is some b ∈ B that has no defeasible derivation or (3b) there
exists a (strict or defeasible) rule R2 with consequent p and antecedent C such that
every c ∈ C has a defeasible derivation and R2 R1. Observe that only rules with
complementary consequents can be compared.
Thus, if a literal has a defeasible derivation, then no further analysis is performed.
Therefore, a defeasible derivation cannot be considered as a single argument. A
defeasible derivation is related to a tree of arguments because it encodes the analysis
of all possible attacks and, by condition (3b), also counter-attacks. A comparison of
d-Prolog with defeasible argumentation formalisms can be found in [25] and [15].
7 Recent Developments
In this section, we will briefly comment some recent developments that extend the
formalisms mentioned above and provide new proposals. We also refer the inter-
ested reader to [5], where an extensive review of current defeasible reasoning im-
plementations is given.
In [13], an argumentative reasoning service for multi-agent systems called
DeLP-server is proposed. A DeLP-server is a stand-alone program that can interact
with multiple client agents. A common (or public) DeLP-program can be stored in
a server, and client agents (that can be distributed in remote hosts) may send queries
to the server and receive the corresponding answer together with the explanation for
that answer [14]. A DeLP-server can be consulted by several agents, and one par-
ticular agent can consult several DeLP-servers, each of them providing a different
shared knowledge base.
To answer queries, a DeLP-server will use the common knowledge stored in
it, together with individual knowledge that clients can send attached to a query,
creating a particular context for that query (see [13] for details). This context is
private knowledge that the server will use for answering the query and will not
affect other future queries. That is, a client agent cannot make permanent changes
to the public DeLP-program stored in a server. The temporal scope of the context
sent in a query [Context, Q] is limited and disappears once the query Q has been
answered.
For instance, consider a DeLP-server that has the set Δ8.1 of Example 8.1 stored.
An agent may send the contextual query [{∼electricity(d), emergency lights(d)}
, lights on(d)] where the context {∼electricity(d), emergency lights(d)} will pro-
vide to the server information about a particular room d. The server will use this
context and Δ8.1 in order to return the answer for the query lights on(d).
Since contextual information can be in contradiction with the information stored
in the server, different types of contextual queries were defined [13]: prioritized,
non-prioritized and restrictive. For instance, a prioritized contextual assigns more](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-182-320.jpg)
![8 Argument-based Logic Programming 169
importance to the information sent by the agent and overrides the information stored
in the server that produces the contradiction.
Agents are not restricted to consult a unique DeLP-server, they may perform the
same contextual query to different servers, and they may share different knowledge
with other agents through different servers. Thus, several configurations of agents
and servers can be established (statically or dynamically). For example, special-
purpose DeLP-servers can be used, each of them representing particular shared
knowledge of a specific domain.
Argue tuProlog (AtuP) [6] is a prototype implementation of an argumentation
engine based on Vreeswijk’s Argumentation System (AS) that can be used to im-
plement a non-monotonic reasoning component in the Internet or agent-based ap-
plications. The following description is taken from [5]. AtuP accepts formulae in
an extended first-order language and returns answers to queries of acceptability on
the basis of the semantics of credulously preferred sets. The language can be con-
sidered to be a conservative extension of the basic language of PROLOG, enriched
with numbers that quantify rule strength and degree of belief. Strict and defeasible
rules can be entered into the system and typically exactly one query is supplied. The
core prover of the implementation attempts to find an argument with a conclusion
for the query. On the macro level arguments are constructed as nodes in a digraph,
and AtuP tries to build an admissible set around an argument for the main claim.
AtuP is currently implemented in Java.
Finally, the issue of studying which properties are satisfied by the set of conse-
quences of an argument-based reasoning system has been gaining attention in the
community [8, 7]. A similar effort was conducted in the general area of nonmono-
tonic reasoning formalisms since very early [12, 16, 18]. In [7] three postulates are
advanced: Closure, Direct Consistency, and Indirect Consistency. The properties
are studied in the context of the possible extensions that could be obtained from
the argumentation system via different semantics such as the ones proposed in [10].
Closure is a form of completeness, i.e., all possible consequences of the knowledge
base are obtained. Consistency (direct and indirect) requires that the set of conclu-
sions be non contradictory in a logical sense.
The idea of introducing postulates such as these is to guide the design of the
reasoners in a manner that their consequences would satisfy the intuitions of the
users of these systems. Even though the importance of this issue cannot be over-
stated, the discussion of the problems of properly characterizing the behavior of an
argument-based consequence operator is out of the scope of this chapter.
8 Conclusions
In this chapter we have described several formalisms that integrate Logic Program-
ming and Argumentation. In particular, we have described Defeasible Logic Pro-
gramming (DeLP), which provides an extension of a logic programming language
with a defeasible argumentation formalism. In DeLP a query succeeds if it is war-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-183-320.jpg)
![170 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari
ranted, i.e., if it is possible to build an argument that supports the query and this
argument is found to be undefeated by a warrant procedure. We have described an
argument-based extended logic programming formalism with defeasible priorities
developed by Prakken and Sartor. In their formalism, arguments are expressed in
a logic programming language with both strong and default negation. Conflicts be-
tween arguments are decided with the help of priorities on the rules. These priorities
can be defeasibly derived as conclusions within the system. An overview of the ar-
gumentation semantics for extended logic programming developed by Schweimeier
and Schroeder was given. Their proposal introduces a variety of notions of attack
for extended logic programs from combinations of undercutting and rebuttal.
As Phan Minh Dung emphasized in [10] “there are extremely interesting re-
lations between argumentation and logic programming”. In this chapter we have
covered some of them.
Acknowledgements This research was funded by CONICET Argentina Project PIP
5050, and SGCyT, Universidad Nacional del Sur, Argentina.
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4. G. Antoniou, D. Billington, and M. J. Maher. Normal forms for defeasible logic. In Pro-
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160–174. MIT Press, 1998.
5. D. Bryant and P. J. Krause. A review of current defeasible reasoning implementations. The
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7. M. Caminada and L. Amgoud. On the evaluation of argumentation formalisms. Artificial
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8. C. Chesñevar and G. Simari. Modelling inference in argumentation through labelled deduc-
tion: Formalization and logical properties. Logica Universalis, 1(1):93–124, 2007.
9. J. Dix, C. Chesñevar, F. Stolzenburg, and G. Simari. Relating Defeasible and Normal Logic
Programming through Transformation Properties. Theoretical Computer Science, 290(1):499–
529, 2002.
10. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic
reasoning and logic programming and n-person games. Artificial Intelligence, 77:321–357,
1995.
11. E. Ferretti, M. Errecalde, A. Garcı́a, and G. Simari. Decision rules and arguments in de-
feasible decision making. In P. Besnard, S. Doutre, and A. Hunter, editors, Proc. 2nd Int.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-184-320.jpg)
![178 John L. Pollock
Fig. 9.3 An inference graph
are simply “given” as premises. Initial nodes are unsupported by arguments, but are
taken to be undefeated. Ultimately, we want to use this machinery to model ratio-
nal cognition. In that case, all that can be regarded as “given” is perceptual input
(construed broadly to include such modes of perception as proprioception, intro-
spection, etc.), in which case it may be inaccurate to take the initial nodes to encode
propositions. It is probably better to regard them as encoding percepts.1
The node-basis of a node is the set of roots of its support-link (if it has one),
i.e., the set of nodes from which the node is inferred in a single step. If a node has
no support-link (i.e., it is initial) then the node-basis is empty. The node-defeaters
are the roots of the defeat-links having the node as their target. Given an inference-
graph, a semantics must determine which nodes encode (the conclusions of) argu-
ments that ought to be accepted, i.e., that are not defeated. This is the defeat-status
computation, and nodes are marked “defeated” or “undefeated”. The defeat-status
computation is made more complex by the fact that some arguments support their
conclusions more strongly than other arguments. For instance, if Jones tells me it
is raining, and Smith denies it, and I regard them as equally reliable, then I have
equally strong arguments both for believing that it is raining and for believing that
it is not raining. In that case, I should withhold belief, not accepting either conclu-
sion. On the other hand, if I regard Jones as much more reliable than Smith, then I
have a stronger argument for believing that it is raining, and if the difference is great
enough, that is the conclusion I should draw. So argument-strengths make a differ-
ence. However, most semantics for defeasible reasoning ignore argument strengths,
pretending that all initial nodes are equally well justified and all inference schemes
equally strong. I will make this same simplifying assumption in this chapter. What
can we say about the semantics in this simplified case?
Let us define:
A node of the inference-graph is initial iff its node-basis and list of
node-defeaters are empty.
It is initially tempting to try to characterize defeat-statuses recursively using the
following two rules:
(D1) Initial nodes are undefeated.
(D2) A non-initial node is undefeated iff all the members of its node-basis
1 See [11] and [12] for a fuller discussion of this.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-192-320.jpg)
![9 A Recursive Semantics for Defeasible Reasoning 179
are undefeated and all node-defeaters are defeated.
However, this recursion turns out to be ungrounded because we can have nodes of
an inference-graph that defeat each other, as in inference-graph (4), where dashed ar-
rows indicate defeasible inferences and heavy arrows indicate defeat-links. In com-
puting defeat-statuses in inference-graph (4), we cannot proceed recursively using
rules (D1) and (D2), because that would require us to know the defeat-status of Q
before computing that of ∼Q, and also to know the defeat-status of ∼Q before com-
puting that of Q. The general problem is that a node Q can have an inference/defeat-
descendant that is a defeater of Q, where an inference/defeat-descendant of a node
is any node that can be reached from the first node by following support-links
and defeat-links. I will say that a node is Q-dependent iff it is an inference/defeat-
descendant of a node Q. So the recursion is blocked in inference-graph (4) by there
being Q-dependent defeaters of Q and ∼Q-dependent defeaters of ∼Q.
Inference-graph (4) is a case of “collective defeat”. For example, let P be “Jones
says that it is raining”, R be “Smith says that it is not raining”, and Q be “It is rain-
ing”. Given P and Q, and supposing you regard Smith and Jones as equally reliable,
what should you believe about the weather? It seems clear that you should withhold
belief, accepting neither Q nor ∼Q. In other words, both Q and ∼Q should be de-
feated. This constitutes a counter-example to rule (D2). So not only do rules (D1)
and (D2) not provide a recursive characterization of defeat-statuses – they are not
even true. The failure of these rules to provide a recursive characterization of defeat-
statuses suggests that no such characterization is possible, and that in turn suggested
to me (see [9, 10]) that rules (D1) and (D2) might be used to characterize defeat-
statuses in another way. Reiter’s default logic [13] proceeded in terms of multiple
“extensions”, and “skeptical default logic” characterizes a conclusion as following
nonmonotonically from a set of premises and defeasible inference-schemes iff it is
true in every extension. There are simple examples showing that this semantics is
inadequate for the general defeasible reasoning of epistemic agents (see below), but
the idea of having multiple extensions suggested to me that rules (D1) and (D2)
might be used to characterize multiple “status assignments”. On this approach, a
partial status assignment is an assignment of defeat-statuses to a subset of the nodes
of the inference-graph in accordance with (D1) and (D2):](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-193-320.jpg)
![182 John L. Pollock
Inference-graphs (5) and (6) constitute intuitive counterexamples to default
logic [13] and the stable model semantics [2] because there are no extensions. Hence
on those semantics, P has the same status as Q, R, and P⊗Q. It is perhaps more ob-
vious that this is a problem for those semantics if we imagine this self-defeating
argument being embedded in a larger inference-graph containing a number of oth-
erwise perfectly ordinary arguments. On these semantics, all of the nodes in all of
the arguments would have to have the same status, because there would still be no
extensions. But surely the presence of the self-defeating argument should not have
the effect of defeating all other (unrelated) arguments.
4 A Problem Case
The multiple-assignment semantics produces the intuitively correct answer for many
complicated inference-graphs. For a number of years, I thought that, given the sim-
plifying assumption that all arguments are equally strong, this semantics was cor-
rect. But I no longer think so. Here is the problem. Contrast inference-graph (4) with
inference-graph (7). Inference-graph (7) involves “odd-length defeat cycles”. For an
example of inference-graph (7), let A = “Jones says that Smith is unreliable”, B =
“Smith is unreliable”, C = “Smith says that Robinson is unreliable”, D = “Robinson
is unreliable”, E = “Robinson says that Jones is unreliable”, F = “Jones is unreli-
able”. Intuitively, this should be another case of collective defeat, with A, C, and
E being undefeated and B, D, and F being defeated. The multiple-assignment se-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-196-320.jpg)
![186 John L. Pollock
I believe that the critical-link semantics gets everything right that the multiple-
assignment semantics got right. Consider a complex example. Inference-graph (11)
illustrates the so called “lottery paradox” [3]. Here P reports a description (e.g., a
newspaper report) of a fair lottery with one million tickets. P constitutes a defeasible
reason for R, which is the description. That is, the newspaper report gives us a
defeasible reason for believing the lottery is fair and has a million tickets. In such
a lottery, each ticket has a probability of one in a million of being drawn, so for
each i, the statistical syllogism gives us a reason for believing ∼Ti (“ticket i will
not be drawn”). The supposed paradox is that although we thusly have a reason for
believing of each ticket that it will not be drawn, we can also infer on the basis of
R that some ticket will be drawn. Of course, this is not really a paradox, because
the inferences are defeasible and this is a case of collective defeat. This results from
the fact that for each i, we can infer Ti from (i) the description R (which entails that
some ticket will be drawn) and (ii) the conclusions that none of the other tickets
will be drawn. This gives us a defeating argument for the defeasible argument to
the conclusion that ∼Ti, as diagrammed in inference-graph (11). The result is that
for each i, there is a status assignment on which ∼Ti is assigned “defeated” and
the other ∼T j’s are all assigned “undefeated”, and hence none of them are assigned
“undefeated” in every status assignment.
I believe that all (skeptical) semantics for defeasible reasoning get the lottery
paradox right. A more interesting example is the “lottery paradox paradox”, dia-
grammed in inference-graph (12). This results from the observation that because
R entails that some ticket will be drawn, from the collection of conclusions of the
form ∼Ti we can infer ∼R, and that is a defeater for the defeasible inference from
∼P to ∼R. This is a self-defeating argument. Clearly, the inferences in the lottery
paradox should not lead us to disbelieve the newspaper’s description of the lottery,
so R should be undefeated. Circumscription [5], in its simple non-prioritized form,
gets this example wrong, because one way of minimizing abnormalities would be to
block the inference from P to R. My own early analysis [8] also gets this wrong. This
was the example that led me to the multiple-assignment semantics. The multiple-
assignment semantics gets this right. We still have the same status assignments as in](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-200-320.jpg)
![9 A Recursive Semantics for Defeasible Reasoning 189
unchanged. B and D are defeated in inference-graph (8), so it follows that P is de-
feated in inference-graph (8). Then because there are no Q-dependent defeat-links
in inference-graph (8), Q is undefeated.
The computation of defeat-statuses in inference-graph (9) works in exactly the
same way, via inference-graph (9*), again producing the result that Q is undefeated.
So on the critical-link semantics, we do not get a divergence between inference-
graphs (8) and (9).
Still, we can ask whether the answer we get for inference-graphs (8) and (9) is
the correct answer. There is some intuitive reason for thinking so. In inference-graph
(8), B and D are defeated, so they should not have the power to defeat P, and hence
P should defeat Q. Similarly, in inference-graph (9), all three of B, D, and F are
defeated, and so again, D should not have the power to defeat P, and hence P should
defeat Q. However, not everyone agrees that this intuitive reasoning is correct. This
issue is closely connected with a question that has puzzled theorists since the earliest
work on the semantics of defeasible reasoning. The multiple-assignment semantics,
as well as default logic, the stable model semantics, circumscription, and almost
every familiar semantics for defeasible reasoning and nonmonotonic logic, supports
what I have called [8] “presumptive defeat”.2 For example, consider inference-graph
(14). On the multiple-assignment semantics, a defeated conclusion like Q that is as-
signed “defeated” in some status assignment and “undefeated” in another retains the
ability to defeat. That is because, in the assignment in which it is undefeated, the
defeatee is defeated, and hence not undefeated in all status-assignments. In the case
of inference-graph (14) this has the consequence that S is assigned “defeated” in
those status-assignments in which Q is assigned “defeated”, but S is assigned “un-
defeated” and ∼S is assigned “defeated” in those status-assignments in which Q is
assigned “undefeated”. Touretzky, Horty, and Thomason [14] called this “ambigu-
ity propagation”, and Makinson and Schlechta [4] called such arguments “Zombie
arguments” (they are dead, but they can still get you). However, the critical-link se-
mantics precludes presumptive defeat. It entails that Q, ∼Q, and hence ∼S, are all
defeated, and S is undefeated. Is this the right answer?
2 The only semantics I know about that does not support presumptive defeat are certain versions
of Nute’s [6] defeasible logic. See also Covington, Nute, and Vellino [1], and Nute [7].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-203-320.jpg)
![Chapter 10
Assumption-Based Argumentation
Phan Minh Dung, Robert A. Kowalski and Francesca Toni
1 Introduction
Assumption-Based Argumentation (ABA) [4, 3, 27, 11, 12, 20, 22] was developed,
starting in the 90s, as a computational framework to reconcile and generalise most
existing approaches to default reasoning [24, 25, 4, 3, 27, 26]. ABA was inspired by
Dung’s preferred extension semantics for logic programming [9, 7], with its dialec-
tical interpretation of the acceptability of negation-as-failure assumptions based on
the notion of “no-evidence-to-the-contrary” [9, 7], by the Kakas, Kowalski and Toni
interpretation of the preferred extension semantics in argumentation-theoretic terms
[24, 25], and by Dung’s abstract argumentation (AA) [6, 8].
Because ABA is an instance of AA, all semantic notions for determining the
“acceptability” of arguments in AA also apply to arguments in ABA. Moreover,
like AA, ABA is a general-purpose argumentation framework that can be instan-
tiated to support various applications and specialised frameworks, including: most
default reasoning frameworks [4, 3, 27, 26] and problems in legal reasoning [27, 13],
game-theory [8], practical reasoning and decision-theory [33, 29, 15, 28, 14]. How-
ever, whereas in AA arguments and attacks between arguments are abstract and
primitive, in ABA arguments are deductions (using inference rules in an underly-
ing logic) supported by assumptions. An attack by one argument against another is
a deduction by the first argument of the contrary of an assumption supporting the
second argument.
Differently from a number of existing approaches to non-abstract argumentation
(e.g. argumentation based on classical logic [2] and DeLP [23]) ABA does not have
explicit rebuttals and does not impose the restriction that arguments have consistent
and minimal supports. However, to a large extent, rebuttals can be obtained “for
Phan Minh Dung
Asian Institute of Technology, Thailand, e-mail: dung@cs.ait.ac.th
Robert A. Kowalski, Francesca Toni
Imperial College London, UK, e-mail: {rak,ft}@doc.ic.ac.uk
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 199
DOI 10.1007/978-0-387-98197-0 10, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-213-320.jpg)
![200 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
free” [27, 11, 33]. Moreover, ABA arguments are guaranteed to be “relevant” and
largely consistent [34].
ABA is equipped with a computational machinery (in the form of dispute deriva-
tions [11, 12, 19, 20, 22]) to determine the acceptability of claims by building and
exploring a dialectical structure of a proponent’s argument for a claim, an oppo-
nent’s counterarguments attacking the argument, the proponent’s arguments attack-
ing all the opponents’ counterarguments, and so on. This computation style, which
has its roots in logic programming, has several advantages over other computational
mechanisms for argumentation. The advantages are due mainly to the fine level of
granularity afforded by interleaving the construction of arguments and determining
their “acceptability”.
The chapter is organised as follows. In Sections 2 and 3 we define the ABA
notions of argument and attack (respectively). In Section 4 we define “acceptability”
of sets of arguments, focusing on admissible and grounded sets of arguments. In
Section 5 we present the computational machinery for ABA. In Section 6 we outline
some applications of ABA. In Section 7 we conclude.
2 Arguments in ABA
ABA frameworks [3, 11, 12] can be defined for any logic specified by means of
inference rules, by identifying sentences in the underlying language that can be
treated as assumptions (see Section 3 for a formal definition of ABA frameworks).
Intuitively, arguments are “deductions” of a conclusion (or claim) supported by a
set of assumptions.
The inference rules may be domain-specific or domain-independent, and may
represent, for example, causal information, argument schemes, or laws and regu-
lations. Assumptions are sentences in the language that are open to challenge, for
example uncertain beliefs (“it will rain”), unsupported beliefs (“I believe X”), or
decisions (“perform action A”). Typically, assumptions can occur as premises of in-
ference rules, but not as conclusions. ABA frameworks, such as logic programming
and default logic, that have this feature are said to be flat [3]. We will focus solely
on flat ABA frameworks. Examples of non-flat frameworks can be found in [3].
As an example, consider the following simplification of the argument scheme
from expert opinion [38]:
Major premise: Source E is an expert about A.
Minor premise: E asserts that A is true.
Conclusion: A may plausibly be taken as true.
This can be represented in ABA by a (domain-independent) inference rule: 1
1 In this chapter, we use inference rule schemata, with variables starting with capital letters, to
stand for the set of all instances obtained by instantiating the variables so that the resulting premises
and conclusions are sentences of the underlying language. For simplicity, we omit the formal defi-
nition of the language underlying our examples.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-214-320.jpg)
![10 Assumption-Based Argumentation 201
h(A) ← e(E,A),a(E,A),arguably(A)
with conclusion h(A) (“A holds”) and premises e(E,A) (“E is an expert about A”),
a(E,A) (“E asserts A”), and an assumption arguably(A). This assumption can be
read in several ways, as “there is no reason to doubt that A holds” or “ the com-
plement of A cannot be shown to hold” or “the defeasible rule – that a conclusion
A holds if a person E who is an expert in A asserts that A is the case – should
not apply”. The inference rule can be understood as the representation of this de-
feasible rule as a strict (unchallangable) rule with an extra, defeasible condition –
arguably(A) – that is open to challenge. This transformation of defeasible rules into
strict rules with defeasible conditions is borrowed from Theorist [31]. Within ABA,
defeasible conditions are always assumptions. Different representations of assump-
tions correspond to different frameworks for defeasible reasoning. For example, in
logic programming arguably(A) could be replaced by not ¬h(A) (here not stands for
negation as failure), and in default logic it could become M h(A). Note that Verheij
[37] also uses assumptions to represent defeasibility of rules. However, his approach
amounts to treating every defeasible rule as an assumption.
In ABA, attacks are always directed at the assumptions in inference rules. The
transformation of defeasible rules into strict rules with defeasible conditions is also
used to reduce rebuttal attacks to undercutting attacks, as we will see in Section 3.
Note that, here and in all the examples given in this chapter, we represent condi-
tionals as inference rules. However, as discussed in [11], this is equivalent to repre-
senting them as object language implications together with modus ponens and and-
introduction as more general inference rules. Representing conditionals as inference
rules is useful for default reasoning because it inhibits the automatic application of
modus tollens to object language implications. However, the ABA approach applies
to any logic specified by means of inference rules, and is not restricted in the way
illustrated in our examples in this chapter.
Suppose we wish to apply the inference rule above to the concrete situation in
which a professor of computer science (cs), say jo, advises that a software product
sw meets a customer’s requirements (reqs) for speed (s) and usability (u). Suppose,
moreover, that professors are normally regarded as experts within (w) their field.
This situation can be represented by the additional inference rules:
reqs(sw) ← h(ok(sw,s)),h(ok(sw,u));
a(jo,ok(sw,s)) ←; a(jo,ok(sw,u)) ←; prof(jo,cs) ←;
w(cs,ok(sw,s)) ←; w(cs,ok(sw,u)) ←;
e(X,A) ← prof(X,S),w(S,A),c prof(X,S)
Note that all these inference rules except the last are problem-dependent. Note also
that, in general, inference rules may have empty premises.
The potential assumptions in the language underlying all these inference rules are
(instances of) the formulae arguably(A) and c prof(X,S) (“X is a credible profes-
sor in S”). Given these inference rules and pool of assumptions, there is an argument
with assumptions {arguably(ok(sw,s)), arguably(ok(sw,u)), c prof(jo,cs)} sup-
porting the conclusion (claim) reqs(sw).
In the remainder, for simplicity we drop the assumptions arguably(A) and re-
place the inference rule representing the scheme from expert opinion simply by](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-215-320.jpg)
![202 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
h(A) ← e(E,A),a(E,A). With this simplification, there is an argument for reqs(sw)
supported by {c prof(jo,cs)}.
Informally, an argument is a deduction of a conclusion (claim) c from a set of as-
sumptions S represented as a tree, with c at the root and S at the leaves. Nodes in this
tree are connected by the inference rules, with sentences matching the conclusion of
an inference rule connected as parent nodes to sentences matching the premises of
the inference rule as children nodes. The leaves are either assumptions or the special
extra-logical symbol τ, standing for an empty set of premises. Formally:
Definition 10.1. Given a deductive system (L,R), with language L and set of infer-
ence rules R, and a set of assumptions A ⊆ L, an argument for c ∈ L (the conclusion
or claim) supported by S ⊆ A is a tree with nodes labelled by sentences in L or by
the symbol τ, such that
• the root is labelled by c
• for every node N
– if N is a leaf then N is labelled either by an assumption or by τ;
– if N is not a leaf and lN is the label of N, then there is an inference rule
lN ← b1,...,bm (m ≥ 0) and
either m = 0 and the child of N is τ
or m 0 and N has m children, labelled by b1,...,bm (respectively)
• S is the set of all assumptions labelling the leaves.
Throughout this chapter, we will often use the following notation
• an argument for (claim) c supported by (set of assumptions) S is denoted by S ⊢ c
in situations where we focus only on the claim c and support S of an argument. Note
that our definition of argument allows for one-node arguments. These arguments
consist solely of a single assumption, say α, and are denoted by {α} ⊢ α.
A portion of the argument {c prof(jo,cs)} ⊢ reqs(sw) is given in Fig. 10.1.
Here, for simplicity, we omit the right-most sub-tree with root h(ok(sw,u)), as this
is a copy of the left-most sub-tree with root h(ok(sw,s)) but with s replaced by u
throughout.
Arguments, represented as trees, display the structural relationships between
claims and assumptions, justified by the inference rules. The generation of argu-
ments can be performed by means of a proof procedure, which searches the space
of applications of inference rules. This search can be performed in the forward direc-
tion, from assumptions to conclusions, in the backward direction, from conclusions
to assumptions, or even “middle-out”. Our definition of tight arguments in [11] cor-
responds to the backward generation of arguments represented as trees. Backward
generation of arguments is an important feature of dispute derivations, presented in
Section 5.2.
Unlike several other authors, e.g. those of [2] (see also Chapter 7) and [23] (see
also Chapter 8), we do not impose the restriction that the support of an argument be
minimal. For example, consider the ABA representation of the scheme from expert](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-216-320.jpg)
![10 Assumption-Based Argumentation 203
opinion, and suppose that our professor of computer science, jo, is also an engineer
(eng). Suppose, moreover, that engineers are normally regarded as experts in com-
puter science. These additional “suppositions” can be represented by the inference
rules
eng(jo) ←; e(X,A) ← eng(X),w(cs,A),c eng(X)
with (instances of) the formula c eng(X) (“X is a credible engineer”) as additional
assumptions. There are now three, different arguments for the claim reqs(sw):
{c prof(jo,cs)} ⊢ reqs(sw), {c eng(jo)} ⊢ reqs(sw), and
{c prof(jo,cs),c eng(jo)} ⊢ reqs(sw).
Only the first two arguments have minimal support. However, all three arguments,
including the third, “non-minimal” argument, are relevant, in the sense that their
assumptions contribute to deducing the conclusion. Minimality is one way to ensure
relevance, but comes at a computational cost. ABA arguments are guaranteed to
be relevant without insisting on minimality. Note that the arguments of Chapter 9,
defined as inference graphs, are also constructed to ensure relevance.
Some authors (e.g. again [2] and [23]) impose the restriction that arguments have
consistent support 2. We will see later, in Sections 3 and 4, that the problems arising
for logics including a notion of (in)consistency can be dealt in ABA by reducing
(in)consistency to the notion of attack and by employing a semantics that insists
that sets of acceptable arguments do not attack themselves.
reqs(sw)
❅
❅
❅
❅
h(ok(sw,s)) h(ok(sw,u))
❅
❅
❅
❅
............
e(jo,ok(sw,s)) a(jo,ok(sw,s))
❅
❅
❅
❅
❅
❅
❅
❅
prof( jo,cs)) w(cs,ok(sw,s)) c prof(jo,cs)) τ
τ τ
Fig. 10.1 An example argument represented as a tree
2 Note that these authors define arguments with respect to an underlying logic with an explicit
negation and hence a notion of consistency, such that inconsistency implies every sentence in the
language. The logic underlying an ABA framework need not have an explicit negation and notion
of inconsistency.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-217-320.jpg)
![204 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
3 Attacks in ABA
In ABA, the notion of attack between arguments is defined in terms of the contrary
of assumptions: one argument S1 ⊢ c1 attacks another (or the same) argument S2 ⊢ c2
if and only if c1 is the contrary of an assumption in S2.
In general, the contrary of an assumption is a sentence representing a challenge
against the assumption. For example, the contrary of the assumption “it will rain”
might be “the sky is clear”. The contrary of the assumption “perform action A”
might be “perform action B” (where the actions A and B are mutually exclusive).
The contrary of the assumption “I believe X” might be “there is evidence against
X”. The contrary of an assumption can also represent critical questions addressed
to an argument scheme. For example, the argument scheme from expert opinion in
Section 2 can be challenged by such critical questions as [38]:
CQ1: How credible is E as an expert source?
CQ2: Is E an expert in the field that A is in?
CQ3: Does E’s testimony imply A?
CQ4: Is E reliable?
CQ5: Is A consistent with the testimony of other experts?
CQ6: Is A supported by evidence?
For simplicity, we focus here solely on CQ1, because modelling the other ques-
tions would require introducing additional assumptions to our earlier representation
of the scheme. 3 Providing negative answers to CQ1 can be understood as prov-
ing the contrary ¬c prof(jo,cs), ¬c eng(jo,cs) of the assumptions c prof(jo,cs),
c eng(jo,cs) respectively.
Contraries may be other assumptions or may be defined by inference rules, e.g.
¬c eng(E,cs) ← ¬prog(E); ¬prog(E) ← theo(E)
where prog stands for “programmer” and theo stands for “theoretician”. The first
rule can be used to challenge the assumption that an engineer is a credible expert in
computer science by arguing that the engineer is not a programmer. The second rule
can be used to show that an engineer is not a programmer by assuming that he/she is
a theoretician (here theo(E) is an additional assumption). Given this representation,
the argument {c eng(jo,cs)} ⊢ reqs(sw) is attacked by the argument {theo(jo)} ⊢
¬c eng(jo,cs).
Definition 10.2. Given a notion of contrary of assumptions 4,
• an argument S1 ⊢ c1 attacks an argument S2 ⊢ c2 if and only if the conclusion c1
of the first argument is the contrary of one of the assumptions in the support S2
of the second argument;
• a set of arguments Arg1 attacks a set of arguments Arg2 if an argument in Arg1
attacks an argument in Arg2.
3 For example, providing negative answers to CQ5 and CQ6 for A can understood as proving
the contrary of the assumption arguably(A) introduced at the beginning of Section 2 but ignored
afterwards.
4 See definition 10.3 for the formal notion of contrary.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-218-320.jpg)
![10 Assumption-Based Argumentation 205
This notion of attack between arguments depends only on attacking (“undercutting”)
assumptions. In many other approaches, however, such as those of Pollock [30]
and Prakken and Sartor [32], an argument can attack (“rebut”) another argument
by deducing the negation of its conclusion. We reduce such “rebuttal” attacks to
“undercutting” attacks, as described in [27, 11, 33, 34]. For example, consider the
inference rules
prog(X) ← works for(X,micro),nor(X); works for(jo,micro) ←
where micro is the name of some company, nor(X) stands for “X is normal”, and the
first inference rule represents the defeasible rule that “normally individuals working
at micro are programmers”. From these and the earlier inference rule for ¬prog, we
can construct both an argument for prog(jo) supported by {nor(jo)} and an argu-
ment for ¬prog(jo) supported by {theo(jo)}. These arguments “rebut” one another
but neither one undercuts the other. However, let us set the contrary of assumption
theo(X) to prog(X) and the contrary of assumption nor(X) to ¬prog(X). Then, the
effect of the rebuttals is obtained by undercutting the assumptions (supporting the
arguments for prog( jo) and ¬prog(jo)).
Note that an alternative approach to accommodate rebuttals could be to introduce
an explicit additional notion of rebuttal attack as done in [10] for logic programming
with two kinds of negation.
To complete our earlier definition of argument and attack we need a definition of
ABA framework:
Definition 10.3. An ABA framework is a tuple L, R, A, where
• (L,R) is a deductive system, with a language L and a set of inference rules R,
• A ⊆ L is a (non-empty) set, whose elements are referred to as assumptions,
• is a total mapping from A into L, where α is the contrary of α.
4 Acceptability of arguments in ABA
ABA can be used to determine whether a given claim is to be “accepted” by a ra-
tional agent. The claim could be, for example, a potential belief to be justified, or a
goal to be achieved, represented as a sentence in L. In order to determine the “ac-
ceptability” of the claim, the agent needs to find an argument for the claim that can
be defended against attacks from other arguments. To defend an argument, other ar-
guments may need to be found and they may need to be defended in turn. As in AA,
this informal definition of “acceptability” can be formalised in many ways, using
the notion of attack between arguments. In this chapter we focus on the following
notions of “acceptable” sets of arguments:
• a set of arguments is admissible if and only if it does not attack itself and it
attacks every argument that attacks it;
• an admissible set of arguments is complete if it contains all arguments that it
defends, where a set of arguments Arg defends an argument arg if Arg attacks all
arguments that attack {arg};](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-219-320.jpg)
![206 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
• the least (with respect to set inclusion) complete set of arguments is grounded.
As for AA (see [8] and Chapter 2), in ABA, given a proposed conclusion c, there
always exists a grounded set of arguments, and this can be constructed bottom-up
[3, 12].
Consider again our formulation of the scheme for expert opinion. The set con-
sisting of the two arguments
arg1={c eng(jo,cs)} ⊢ reqs(sw)
arg2={nor(jo)} ⊢ prog(jo)
is admissible, and as a consequence so is the claim reqs(sw). Indeed, this set does
not attack itself and it attacks the argument
arg3={theo(jo)} ⊢ ¬prog(jo).
However, the set {arg1,arg2} is not (a subset of) the grounded set of arguments. But
the set {arg4} is grounded, where
arg4={c prof(jo,cs)} ⊢ reqs(sw).
The notion of admissibility is credulous, in that there can be alternative, conflicting
admissible sets. In the example above, {arg3} is also admissible, but in conflict with
the admissible {arg1,arg2}.
In some applications, it is more appropriate to adopt a sceptical notion of “ac-
ceptability”. The notion of grounded set of arguments is sceptical in the sense that
no argument in the grounded set is attacked by an admissible set of arguments. Other
notions of credulous and sceptical “acceptable” set of arguments can be employed
within ABA [3, 27, 12].
Note that the notions of “acceptable” sets of arguments given here are more struc-
tured than the corresponding notions of “acceptable” sets of assumptions given in
[3, 27, 11, 12]. The correspondence between “acceptability” of arguments and “ac-
ceptability” of assumptions, given in [12], is as follows:
• If a set of assumptions S is admissible/grounded then the union of all arguments
supported by any subset of S is admissible/grounded;
• If a set of arguments S is admissible/grounded then the union of all sets of as-
sumptions supporting the arguments in S is admissible/grounded.
Note that, if the underlying logic has explicit negation and inconsistency, and we ap-
ply the transformation outlined in Section 3 (to reduce rebuttals to our undercutting
attacks), then an argument has an inconsistent support if and only if it attacks itself.
Thus, in such a case, no argument belonging to an “acceptable” set may possibly
contain an argument with an inconsistent support [34].
5 Computation of “acceptability”
The notion of “acceptability” of sets of arguments provides a non-constructive spec-
ification. In this section we show how to turn the specification into a constructive
proof procedure. As argued in [6, 8], at a conceptual level, a proof procedure for](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-220-320.jpg)
![10 Assumption-Based Argumentation 207
argumentation consists of two tasks, one for generating arguments and one for de-
termining the “acceptability” of the generated arguments. We have already briefly
discussed the computation of arguments in Section 2. Below, we first demonstrate
how to determine the “acceptability” of arguments that are already constructed, in
the spirit of AA, by means of dispute trees [11, 12]. Then, we discuss how dispute
derivations [11, 12, 20, 22], which interleave constructing arguments and determin-
ing their “acceptability”, can be viewed as generating “approximations” to “accept-
able” dispute trees.
Dispute derivations are inspired by SLDNF, the “EK” procedure of [17, 9, 7],
and the “KT” procedure of [36, 26] for logic programming with negation as failure.
Like SLDNF and EK, dispute derivations interleave two kinds of derivations (one
for “proving” and one for “disproving”). Like EK, they accumulate defence assump-
tions and use them for filtering. Like KT, they accumulate culprit assumptions and
use them for filtering.
5.1 Dispute trees
Dispute trees can be seen as a way of representing a winning strategy for a propo-
nent to win a dispute against an opponent. The proponent starts by putting forward
an initial argument (supporting a claim whose “acceptability” is under dispute), and
then the proponent and the opponent alternate in attacking each other’s previously
presented arguments. The proponent wins if he/she has a counter-attack against ev-
ery attacking argument by the opponent.
Definition 10.4. A dispute tree for an initial argument a is a (possibly infinite) tree
T such that
1. Every node of T is labelled by an argument and is assigned the status of propo-
nent node or opponent node, but not both.
2. The root is a proponent node labelled by a.
3. For every proponent node N labelled by an argument b, and for every argument
c attacking b, there exists a child of N, which is an opponent node labelled by c.
4. For every opponent node N labelled by an argument b, there exists exactly one
child of N which is a proponent node labelled by an argument attacking b.
5. There are no other nodes in T except those given by 1-4 above.
The set of all arguments belonging to the proponent nodes in T is called the defence
set of T.
Note that a branch in a dispute tree may be finite or infinite. A finite branch repre-
sents a winning sequence of arguments (within the overall dispute) that ends with an
argument by the proponent that the opponent is unable to attack. An infinite branch
represents a winning sequence of arguments in which the proponent counter-attacks
every attack of the opponent, ad infinitum. Note that our notion of dispute tree intu-
itively corresponds to the notion of winning strategy in Chapter 6.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-221-320.jpg)
![10 Assumption-Based Argumentation 209
• admissible if and only if no argument labels both a proponent and an opponent
node. 5
• grounded if and only if it is finite.
Note that, by theorem 3.1 in [12], any grounded dispute tree is admissible. The re-
lationship between admissible/grounded dispute tree and admissible/grounded sets
of arguments is as follows:
1. the defence set of an admissible dispute tree is admissible;
2. the defence set of a grounded dispute tree is a subset of the grounded set of
arguments;
3. if an argument a belongs to an admissible set of arguments A then there exists
an admissible dispute tree for a with defence set A′ such that A′ ⊆ A and A′ is
admissible;
4. if an argument a belongs to the grounded set of arguments A (and the set of all
arguments supported by assumptions for the given ABA framework is finite) then
there exists a grounded dispute tree for a with defence set A′ such that A′ ⊆ A and
A′ is admissible.
Results 1. and 3. are proven in [12] (theorem 3.2). Results 2. and 4. follow directly
from theorem 3.7 in [26].
Note that the dispute tree in Fig. 10.2 is admissible but not grounded (since it has
an infinite branch). However, the tree with root {c prof(jo,cs)} ⊢ reqs(sw) (arg4
in Section 4) and the two right-most subtrees in Fig. 10.2 is grounded.
We can obtain finite trees from infinite admissible dispute trees by using “filter-
ing” to avoid re-defending assumptions that are in the process of being “defended”
or that have already successfully been “defended”. For example, for the tree in
Fig. 10.2, only the (proponent) child of argument {theo(jo)} ⊢ ¬prog(jo) needs
to be constructed. Indeed, since this argument is already being “defended”, the re-
mainder of the (infinite) branch can be ignored.
R: reqs(sw) ← h(ok(sw,s)),h(ok(sw,u)); h(A) ← e(E,A),a(E,A);
e(X,A) ← eng(X),w(cs,A),c eng(X); e(X,A) ← prof(X,S),w(S,A),c prof(X,S);
a( jo,ok(sw,s)) ←; a( jo,ok(sw,u)); eng(jo) ←; prof(jo,cs) ←;
w(cs,ok(sw,s)) ←; w(cs,ok(sw,u)) ←;
¬c eng(E,cs) ← ¬prog(E); ¬prog(X) ← theo(X);
prog(X) ← works for(X,micro),nor(X); works for(bob,micro) ←;
¬c prof(X,S) ← ret(X),inact(X); ret(jo) ←;
¬c prof(X,S) ← admin(X),inact(X); admin( jo) ←;
act(X) ← pub(X); pub(jo) ←
A: c eng(X); c prof(X,S); theo(X); nor(X); inact(X)
: c eng(X) = ¬c eng(X); c prof(X,S) = ¬c prof(X,S);
theo(X) = prog(X); nor(X) = ¬prog(X); inact(X) = act(X)
Fig. 10.3 ABA framework for the running example.
5 Note that admissible dispute trees are similar to the complete argument trees of [2]. We use the
term “argument tree” for arguments.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-223-320.jpg)
![10 Assumption-Based Argumentation 211
Fig. 10.1 without the dots is an example of a potential argument for reqs(sw),
supported by assumption c prof(jo,cs) and non-assumption h(ok(sw,u)). Fig. 10.4
shows another example of a potential argument. In general there may be one, many
or no actual arguments that can be obtained from a potential argument. For example,
the potential argument in Fig. 10.1 may give rise to two actual arguments (supported
by {c prof(jo,cs)} and {c prof(jo,cs), c eng(jo)} respectively), whereas the po-
tential argument in Fig. 10.4 gives rise to exactly one actual argument (supported
by {inact(jo)}). However, if no inference rules were given for ret, then no actual
argument could be generated from the potential argument in Fig. 10.4.
The benefits of interleaving mentioned earlier can be illustrated as follows:
• A proponent’s potential argument is abandoned if it is supported by assumptions
that would make the proponent’s defence of the claim unacceptable. For example,
in Fig. 10.1, if the assumption c prof(jo,cs) is “defeated” before h(ok(sw,u)) is
expanded, then the entire potential argument can be abandoned.
• An opponent’s potential argument does not need to be expanded into an actual
argument if a culprit can be selected and defeated already in this potential ar-
gument. For example, by choosing as culprit and “defeating” the assumption
inact(jo) in the potential argument in Fig. 10.4, the proponent “defeats” any
argument that can be obtained by expanding the non-assumption ret(jo).
However, when a potential argument cannot be expanded into an actual argument,
defeating a culprit in the potential argument is wasteful. Nonetheless, dispute deriva-
tions employ a selection function which, given a potential or actual argument, se-
lects an assumption to attack or a non-assumption to expand. As a special case,
the selection function can be patient [11, 20], always selecting non-assumptions in
preference to assumptions, in which case arguments will be fully constructed before
they are attacked. Even in such a case, dispute derivations still benefit from filtering.
Informally, a dispute derivation is a sequence of transitions steps from one state
of a dispute to another. In each such state, the proponent maintains a set P of (sen-
tences supporting) potential arguments, representing a single way to defend the ini-
tial claim, and the opponent maintains a set O of potential arguments, representing
all ways to attack the assumptions in P. In addition, the state of the dispute contains
the set D of all defence assumptions and the set C of all culprits already encoun-
tered in the dispute. The sets D and C are used to filter potential arguments, as we
discussed earlier.
A step in a dispute derivation represents either a move by the proponent or a
move by the opponent.
❅
❅
❅
❅
¬c prof(jo,cs)
ret( jo) inact( jo)
Fig. 10.4 A potential argument for ¬c prof(jo,cs), supported by assumption inact(jo) and non-
assumption ret( jo).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-225-320.jpg)
![212 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
A move by the proponent either expands one of his/her potential arguments in P,
or it selects an assumption in one of the opponent’s potential arguments in O and
decides whether or not to attack it. In the first case, it expands the potential argument
in only one way, and adds any new assumptions resulting from the expansion to D,
checking that they are distinct from any assumptions in the culprit set C (filtering
defence assumptions by culprits). In the second case, either the proponent ignores
the assumption, as a non-culprit, or he/she adds the assumption to C (bearing in
mind that, in order to defeat the opponent, he/she needs to counter-attack only one
assumption in each of the opponent’s attacking arguments). In this latter case, he/she
checks that the assumption is distinct from any assumptions in D (filtering culprits
by defence assumptions) and checks that it is distinct from any culprit already in
C (filtering culprits by culprits). If the selected culprit is not already in C, the pro-
ponent adds the contrary of the assumption as the conclusion of a new, one-node
potential argument to P (to construct a counter-attack).
A move by the opponent, similarly, either expands one of his/her potential argu-
ments in O, or it selects an assumption to attack in one of the proponent’s potential
arguments in P. In the first case, it expands a non-assumption premise of the selected
potential argument in all possible ways, replacing the selected potential argument in
O by all the new potential arguments. In the second case, the opponent does not have
the proponent’s dilemma of deciding whether or not to attack the assumption, be-
cause the opponent needs to generate all attacks against the proponent. Thus, he/she
adds the contrary of the assumption as the conclusion of a new, one-node potential
argument to O. 6
A successful dispute derivation represents a single way for the proponent to sup-
port and defend a claim, but all the ways that the opponent can try to attack the
proponent’s arguments. Thus, although the proponent and opponent can attack one
another before their arguments are fully completed, for a dispute derivation to be
successful, all of proponent’s arguments must be actual arguments. In contrast, the
opponent’s defeated arguments may be only potential. In this sense, dispute deriva-
tions compute only “approximations” of dispute trees. However, for every success-
ful dispute derivation, there exists a dispute tree that can be obtained by expanding
the opponent’s potential arguments and dropping the potential arguments that cannot
be expanded, as well as any of the proponent’s unnecessary counter-attacks [22].
We give an informal dispute derivation for the running example:
P: I want to determine the “acceptability” of claim reqs(sw) (D = {} and C = {}
initially).
P: I generate a potential argument for reqs(sw) supported by {h(ok(sw,s)),
h(ok(sw,u))}, and then expand it (through several steps) to one supported by
{c prof(jo,cs), h(ok(sw,u))} (c prof(jo,cs) is added to D).
O: I attack the assumption c prof(jo,cs) in this potential argument by looking
for arguments for its contrary ¬c prof(jo,cs).
6 If computing admissibility, however, the opponent would not attack assumptions that already
belong to D (filtering defence assumptions by defence assumptions).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-226-320.jpg)
![10 Assumption-Based Argumentation 213
O: I generate two potential arguments for ¬c prof(jo,cs), supported by {ret(jo),
inact(jo)} and {admin(jo), inact(jo)} respectively.
P: I choose inact(jo) as culprit in the first potential argument by O (inact(jo) is
added to C), and generate (through several steps) an argument for its contrary
act(jo), supported by the empty set.
O: There is no point for me to expand this potential argument then. But I still
have the attacking argument for ¬c prof(jo,cs), supported by {admin(jo),
inact(jo)}.
P: I again choose inact(jo) as culprit, which I have already defeated (inact(jo) ∈
C).
P: There is no attacking argument that I still need to deal with: let me go back to
expand the argument for reqs(sw).
P: I need to expand h(ok(sw,u)), I can do so and generate (through several steps)
an argument for reqs(sw) supported by {c prof(jo,cs),c eng(jo)} (c eng(jo)
is added to D).
O: I can attack this argument by generating (through several steps) an argument
for the contrary ¬c eng(jo) of c eng(jo): this argument is supported by assump-
tion theo(jo).
P: I can attack this argument by generating (through several steps) a potential
argument for the contrary of theo(jo).
. . .
This dispute corresponds to the top-down and right-to-left construction of (an ap-
proximation of) the dispute tree in Fig. 10.2. The dispute ends successfully for com-
puting admissible sets of arguments, but does not terminate for computing grounded
sets of arguments. Note that dispute derivations are defined in terms of several pa-
rameters: the selection function, the choice of “player” at any specific step in the
derivation, the choice of potential arguments for the proponent/opponent to expand
etc (see [20, 22]). Concrete choices for some of these parameters (e.g. the choice
of the proponent’s arguments) correspond to concrete search strategies for finding
dispute derivations and computing dispute trees. Concrete choices for other param-
eters (e.g. the choice of “player”) determine how the dispute tree is constructed in a
linear manner.
Several notions of dispute derivations have been proposed, differing in the notion
of “acceptability” and in the presentation of the computed set of “acceptable” argu-
ments. More specifically, the dispute derivations of [11, 20, 22] compute admissible
sets of arguments whereas the dispute derivations of [12] compute grounded and
ideal (another notion of sceptical “acceptability”) sets of arguments. Moreover, the
dispute derivations of [11, 12] compute the union of all sets of assumptions sup-
porting the “acceptable” sets of arguments for the given claim, whereas the dispute
derivations of [20, 22] also return explicitly the computed set of “acceptable” argu-
ments and the attacking (potential) arguments, as well as an indication of the attack
relationships amongst these arguments.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-227-320.jpg)
![214 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
6 Applications
In this section, we illustrate recent applications of ABA for dispute resolution (Sec-
tion 6.1, adapted from [20]) and decision-making (Section 6.2, adapted from [15]).
For simplicity, the description of these applications is kept short here. For more
detail see [13] (for dispute resolution applied to contracts) and [33, 16, 29] (for
decision-making in service-oriented architectures). We also show how ABA can be
used to model the stable marriage problem (Section 6.3, building upon [8]).
6.1 ABA for dispute resolution
Consider the following situation, inspired by a real-life court case on contract dis-
pute. A judge is tasked with resolving a disagreement between a software house
and a customer refusing to pay for a product developed by the software house. Sup-
pose that this product is the software sw in the running example of Fig. 10.3. The
judge uses information agreed upon by both parties, and evaluates the claim by the
software house that payment should be made to them.
All parties agree that payment should be made if the product is delivered on
time (del) and is a good product (goodProd). They also agree that a product is
not good (badProd) if it is late (lateProd) or does not meet its requirements. As
before, we assume that these requirements are speed and usability. There is also the
indisputable fact that the software was indeed delivered (del(sw)). This situation
can be modelled by extending the framework of Fig. 10.3 with inference rules
payment(sw) ← del(sw),goodProd(sw); badProd(sw) ← lateProd(sw);
badProd(sw) ← ¬reqs(sw); del(sw) ←
and assumptions goodProd(sw), ¬reqs(sw), with contraries:
goodProduct(sw) = badProduct(sw); ¬reqs(sw) = reqs(sw)
Given the expert opinions of jo (see Section 2), the claim that payment should be
made is grounded (and thus admissible).
6.2 ABA for decision-making
We use a concrete “home-buying” example, in which a buyer is looking for a prop-
erty and has a number of goals including “structural” features of the property, such
as its location and the number of rooms, and “contractual” features, such as the
price, the completion date for the sale, etc. The buyer needs to decide both on a
property, taking into account the features of the properties (Ri below), and general
“norms” about structural properties (Rn below). The buyer also needs to decide and
agree on a contract, taking into account norms about contractual issues (Rc below).
A simple example of buyer is given by the ABA framework L, R, A, with:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-228-320.jpg)
![10 Assumption-Based Argumentation 215
• R = Ri ∪Rn ∪Rc and
Ri : number of rooms = 5 ← house1;
number of rooms = 4 ← house2; price = £400K ← house2
Rn : safe ← council approval,a1; ¬safe ← weak foundations,a2;
council approval ← completion certi ficate,a3
Rc : seller moves abroad ← house2;
quick completion ← seller moves abroad
• A = Ad ∪Au ∪Ac and
Ad = {house1,house2}; Ac = {a1,a2,a3}; Au = {¬council approval }
• house1 = house2, house2 = house1,
a1 = ¬safe, a2 = safe, a3 = ¬council approval,
¬council approval = council approval.
Here, there are two properties for sale, house1 and house2. The first has 5 rooms, the
second has 4 rooms and costs £400K (Ri). The buyer believes that a property ap-
proved by the council is normally safe, a completion certificate normally indicates
council approval, and a property with weak foundations is normally unsafe (Rn).
The buyer also believes that the seller of the second property is moving overseas,
and this means that the seller aims at a quick completion of the sale (Rc). Some
of the assumptions in the example have a defeasible nature (Ac), others amount to
mutually exclusive decisions (Ad) and finally others correspond to genuine uncer-
tainties of the buyer (Au).
This example combines default reasoning, epistemic reasoning and practical rea-
soning. An example of “pure” practical reasoning in ABA applied to a problem
described in [1] can be found in [35].
6.3 ABA for the stable marriage problem
Given two sets M, W of n men and n women respectively, the stable marriage prob-
lem (SMP) is the problem of pairing the men and women in M and W in such a way
that no two people of opposite sex prefer to be with each other rather than with the
person they are paired with. The SMP can be viewed as a problem of finding stable
extensions in an abstract argumentation framework [8].
Here we show that the problem can be naturally represented in an ABA frame-
work L, R, A, with:
• A = {pair(A,B) | A ∈ M,B ∈ W}
• pair(A,B) = contrary pair(A,B)
• R consists of inference rules
contrary pair(A,B) ← prefers(A,D,B), pair(A,D);
contrary pair(A,B) ← prefers(B,E,A), pair(E,B)
together with a set P of inference rules of the form prefer(X,Y,Z) ← such
that for each person A and for each two different people of the opposite sex](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-229-320.jpg)
![216 Phan Minh Dung, Robert A. Kowalski and Francesca Toni
B and C, either prefer(A,B,C) ← or prefer(A,C,B) ← belongs to P, where
prefer(X,Y,Z) stands for X prefers Y to Z.
The standard formulation of the stable marriage problem combines the rules, as-
sumptions and contraries of this framework with the notion that a set of argu-
ments/assumptions is “acceptable” if and only if it is stable, where
• A set of arguments/assumptions is stable if and only if it does not attack itself,
but attacks all arguments/assumptions not in the set.
In SMP, the semantics of stable sets forces solutions to be total, pairing all men and
women in the sets M and W. The semantics of admissible sets is more flexible. It
does not impose totality, and it can be used when the sets M and W have different
cardinalities. Consider for example the situation in which two men, a and b, and two
women, c and d have the following preferences:
prefer(a,d,c) ←; prefer(b,c,d) ←; prefer(c,a,b) ←; prefer(d,b,a) ←
Although there is no stable solution in which all people are paired with their highest
preference, there exist two alternative stable solutions: {pair(a,c), pair(b,d)} and
{pair(a,d), pair(b,c)}. However, suppose a third woman, e, enters the scene, turns
the heads of a and b, and expresses a preference for a over b, in effect adding to P:
prefer(a,e,d) ←; prefer(a,e,c) ←;
prefer(b,e,c) ←; prefer(b,e,d) ←; prefer(e,a,b) ←
This destroys both stable solutions of the earlier SMP, but has a single maximally
admissible solution: {pair(a,e), pair(b,c)}.
Thus, by using admissibility we can drop the requirement that the number of men
and women is the same. Similarly, we can drop the requirement that preferences are
total, namely all preferences between pairs of the opposite sex are given.
7 Conclusions
In this chapter, we have reviewed assumption-based argumentation (ABA), focusing
on relationships with other approaches to argumentation, computation, and applica-
tions. In contrast with a number of other approaches, ABA makes use of under-
cutting as the only way in which one argument can attack another. The effect of
rebuttal attacks is obtained in ABA by adding appropriate assumptions to rules and
by attacking those assumptions instead. The extent to which such undercutting is
an adequate replacement for rebuttals has been explored elsewhere [34], but merits
further investigation. Also, again in contrast with some other approaches, we do not
insist that the support of an argument in ABA be minimal and consistent. Instead of
insisting on minimality, we guarantee that the support of an argument is relevant, as
a side-effect of representing arguments as deduction trees. Instead of insisting that
the support of an argument is consistent, we obtain a similar effect by imposing the
restriction that “acceptable” sets of arguments do not attack themselves.
ABA is an instance of abstract argumentation (AA), and consequently it inherits
its various notions of “acceptable” sets of arguments. ABA also shares with AA the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-230-320.jpg)
![10 Assumption-Based Argumentation 217
computational machinery of dispute trees, in which a proponent and an opponent
alternate in attacking each other’s arguments. However, ABA also admits the com-
putation of dispute derivations, in which the proponent and opponent can attack and
defeat each other’s potential arguments before they are completed. We believe that
this feature of dispute derivations is both computationally attractive and psycholog-
ically plausible when viewed as a model of human argumentation.
The computational complexity of ABA has been investigated for several of its in-
stances [5] (see also Chapter 5). The computational machinery of dispute derivations
and dispute trees is the basis of the CaSAPI argumentation system 7 [19, 20, 22].
Although ABA was originally developed for default reasoning, it has recently
been used for several other applications, including dispute resolution and decision-
making. ABA is currently being used in the ARGUGRID project 8 to support ser-
vice selection and composition of services in the Grid and Service-Oriented Ar-
chitectures. ABA is also being used for several applications in multi-agent systems
[21, 18] and e-procurement [29].
Acknowledgement This work was partially funded by the Sixth Framework IST programme of
the EC, under the 035200 ARGUGRID project.
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argumentation for default reasoning. Artificial Intelligence, 141:57–78, 2002.
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reasoning and logic programming. In Proc. IJCAI’93, pages 852–859. Morgan Kaufmann,
1993.
7. P. M. Dung. An argumentation theoretic foundation of logic programming. Journal of Logic
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reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995.
9. P. M. Dung. Negations as hypotheses: An abductive foundation for logic programming. In
Proc. ICLP, pages 3–17. MIT Press, 1991.
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admissible argumentation. Artificial Intelligence, 170:114–159, 2006.
12. P. M. Dung, P. Mancarella, and F. Toni. Computing ideal sceptical argumentation. Artificial
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7 http://www.doc.ic.ac.uk/˜dg00/casapi.html
8 www.argugrid.eu](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-231-320.jpg)
![Chapter 11
The Toulmin Argument Model in Artificial
Intelligence
Or: how semi-formal, defeasible argumentation
schemes creep into logic
Bart Verheij
1 Toulmin’s ‘The Uses of Argument’
In 1958, Toulmin published The Uses of Argument. Although this anti-formalistic
monograph initially received mixed reviews (see section 2 of [20] for Toulmin’s own
recounting of the reception of his book), it has become a classical text on argumen-
tation, and the number of references to the book (when writing these words1 — by a
nice numerological coincidence — 1958) continues to grow (see [7] and the special
issue of Argumentation 2005; Vol. 19, No. 3). Also the field of Artificial Intelligence
has discovered Toulmin’s work. Especially four of Toulmin’s themes have found
follow-up in Artificial Intelligence. First, argument analysis involves half a dozen
distinct elements, not just two. Second, many, if not most, arguments are substan-
tial, even defeasible. Third, standards of good reasoning and argument assessment
are non-universal. Fourth, logic is to be regarded as generalised jurisprudence. Us-
ing these central themes as a starting point, this chapter provides an introduction to
Toulmin’s argument model and its connections with Artificial Intelligence research.
No attempt is made to give a comprehensive history of the reception of Toulmin’s
ideas in Artificial Intelligence; instead a personal choice is made of representative
steps in AI-oriented argumentation research.
When Toulmin wrote his book, he was worried. He saw the influence of the
successes of formal logic on the philosophical academia of the time, and was afraid
that as a consequence seeing formal logic’s limitations would be inhibited. He wrote
The Uses of Argument to fight the — in his opinion mistaken — idea of formal logic
as a universal science of good reasoning. In the updated edition of The Uses of
Argument [19], he describes his original aim as follows:
to criticize the assumption, made by most Anglo-American academic philosophers, that any
significant argument can be put in formal terms: not just as a syllogism, since for Aristotle
himself any inference can be called a ‘syllogism’ or ‘linking of statements’, but a rigidly
demonstrative deduction of the kind to be found in Euclidean geometry. ([19], vii)
Bart Verheij
Artificial Intelligence, University of Groningen
1 Source: Google Scholar citation count, April 1, 2008.
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 219
DOI 10.1007/978-0-387-98197-0 11, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-233-320.jpg)
![220 Bart Verheij
Fig. 11.1 Toulmin’s layout of arguments with an example ([18], 104–5)
In short: Toulmin wanted to argue that there are other arguments than formal ones.
It is also clear from this quote that Toulmin’s goals were first and foremost aimed at
his fellow philosophers. In the Preface to the 2003 edition, Toulmin says it thus:
In no way had I set out to expound a theory of rhetoric or argumentation: my concern was
with twentieth-century epistemology, not informal logic. ([19], vii)
Let us look closer at some of Toulmin’s points.
1.1 Argument analysis involves half a dozen distinct elements, not
just two
Toulmin is perhaps most often read because of his argument diagram (Figure 1).
Whereas a formal logical analysis uses the dichotomy of premises and conclusions
when analyzing arguments, Toulmin distinguishes six different kinds of elements:
Data, Claim, Qualifier, Warrant, Backing and Rebuttal. Before explaining the roles
of these elements, let us look at Toulmin’s famous example of Harry, who may or
may not be a British subject.
When someone claims that Harry is a British subject, it is natural to ask, so says
Toulmin: What have you got to go on? An answer to that question can provide the
data on which the claim rests, here: Harry was born in Bermuda. But having datum
and claim is not enough. A further important question needs to be answered. Toul-
min phrases it thus: How do you get there? In other words, why do you think that
the datum gives support for your claim? An answer to this question must take the
form of a rule-like general statement, the warrant underlying the step from datum
to claim. In the example, the warrant is that a man born in Bermuda will generally
be a British subject. As the example shows, warrants need not express universal
generalizations. Here the warrant is not that each man born in Bermuda is a British
subject, but merely that a man born in Bermuda will generally be a British sub-
ject. As a result, on the basis of datum and warrant a claim needs to be qualified.
Here the claim becomes that presumably Harry is a British subject. When datum,
qualified claim and warrant have been made explicit, a further question needs to be
asked: Why do you think that the warrant holds? An answer will be provided by the
backing of the warrant. In the example, Toulmin refers to the existence of statutes
and other legal provisions (without specifying them) that can provide the backing](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-234-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 221
for the warrant that who is born in Bermuda will generally be British subjects. The
sixth and final kind of element to be distinguished is that of conditions of exception
or rebuttal (101/93)2. Conditions of rebuttal indicate ‘circumstances in which the
general authority of the warrant would have to be set aside’ or ‘exceptional circum-
stances which might be capable of defeating or rebutting the warranted conclusion’
(101/94). In the example, Harry’s parents could be aliens or he could have become
a naturalized American. Toulmin refers to Hart and Ross as predecessors for his
discussion of rebuttal. Hart coined the term ‘defeasibility’ (see also [9]) and used
it in legal and philosophical settings (contract, free will, responsibility), while Ross
emphasized that moral rules must have exceptions (142/131–2).
Here is Toulmin’s defence of the difference between a datum and the negation
of a rebuttal, which predates discussions about the relation between rule conditions
and exceptions:
[T]he fact that Harry was born in Bermuda and the fact that his parents were not aliens are
both of them directly relevant to the question of his present nationality; but they are relevant
in different ways. The one fact is a datum, which by itself establishes a presumption of
British nationality; the other fact, by setting aside one possible rebuttal, tends to confirm the
presumption thereby created. (102/95)
Summarizing, Toulmin distinguishes six kinds of elements in arguments:
Claim: The Claim is the original assertion that we are committed to and must justify when
challenged (97/90). It is the starting point of the argument.
Datum: The Datum provides the basis of the claim in response to the question: What have
you got to go on? (97–8/90)
Warrant: The Warrant provides the connection between datum and claim. A warrant ex-
presses that ‘[d]ata such as D entitle[s] one to draw conclusions, or make claims, such as
C’. Warrants are ‘general, hypothetical statements, which can act as bridges, and autho-
rise the sort of step to which our particular argument commits’. They are ‘rules, principles,
inference-licences or what you will, instead of additional items of information’. (98/91)
Qualifier: The Qualifier indicates the strength of the step from datum to claim, as conferred
by the warrant (101/94)
Backing: The Backing shows why a warrant holds. Backing occurs when not a particular
claim is challenged, but the range of arguments legitimized by a warrant (103–4/95–6).
Rebuttal: A Rebuttal can indicate ‘circumstances in which the general authority of the war-
rant would have to be set aside’ or ‘exceptional circumstances which might be capable of
defeating or rebutting the warranted conclusion’ (101/94).
1.2 Many, if not most, arguments are substantial, even defeasible
Let us consider another of Toulmin’s example arguments:
(1) Anne is one of Jack’s sisters;
All Jack’s sisters have red hair;
So, Anne has red hair. (123/115)
This example is a variant of the paradigmatic example of a syllogism: ‘Socrates
is a man. All men are mortal. So, Socrates is mortal’. Anyone accustomed to the
2 Page numbers before the slash refer to the original 1958 edition of The Uses of Argument [18],
those after the slash to the updated 2003 edition [19]](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-235-320.jpg)
![224 Bart Verheij
solution to a mathematical problem] may be seducingly elegant, but it could hardly
be less representative’ (127/118).
1.3 Standards of good reasoning and argument assessment are not
universal, but context-dependent
According to Toulmin, our standards for the assessment of real arguments are not
universal, but depend on a context. In a section, where he discusses this issue, he
uses the term ‘possibility’ as an illustration (36/34): whereas in mathematics ‘pos-
sibility’ has to do with the absence of demonstrable contradiction, in most cases
‘possibility’ is based on a stronger standard. His example statement is ‘Dwight
D. Eisenhower will be selected to represent the U.S.A. in the Davis Cup match
against Australia’. This statement involves no contradiction, while still (now former,
then actual) President Eisenhower will not be considered a possible team member.
In other words, ‘possibility’ is judged using different standards, some more for-
mal (‘absence of contradiction’), others more substantial (‘being a top-level tennis
player’).
The example is however an example of different standards for the possible, not
of different standards for the assessment of arguments, his ultimate aim. Here is a
succinct phrasing of his position:
It is unnecessary, we argued, to freeze statements into timeless propositions before admit-
ting them into logic: utterances are made at particular times and in particular situations, and
they have to be understood and assessed with one eye on this context. The same, we can
now argue, is true of the relations holding between statements, at any rate in the majority of
practical arguments. The exercise of the rational judgement is itself an activity carried out in
a particular context and essentially dependent on it: the arguments we encounter are set out
at a given time and in a given situation, and when we come to assess them they have to be
judged against this background. So the practical critic of arguments, as of morals, is in no
position to adopt the mathematician’s Olympian posture. (182–3/168–9; the quote appears
in a section entitled “Logic as a System of Eternal Truths”)
According to Toulmin, the differences between standards of reasoning are reflected
in the backings that are accepted to establish warrants. For instance, he considers
the following three warrants (103–4/96):
A whale will be a mammal.
A Bermudan will be a Briton.
A Saudi Arabian will be a Muslim.
Each of these warrants gives in a similar way the inferential connection between
certain types of data and certain kinds of claims. The first allows inferring that a
particular whale is a mammal, the second that a particular Bermudan is a Briton,
the third that a particular Saudi Arabian is a Muslim (all these inferences, of course,
subject to qualification and rebuttal). The different standards become visible when
information about the corresponding backings is inserted:
A whale will be (i.e. is classifiable as) a mammal
A Bermudan will be (in the eyes of the law) a Briton
A Saudi Arabian will be (found to be) a Muslim](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-238-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 225
Toulmin explains (104/96):
One warrant is defended by relating it to a system of taxonomical classification, another by
appealing to the statutes governing the nationality of people born in the British colonies, the
third by referring to the statistics which record how religious beliefs are distributed among
people of different nationalities.
For Toulmin, the establishment of standards of argument assessment, hence of good
reasoning, is an empirical question, cf. the following excerpt:
Accepting the need to begin by collecting for study the actual forms of argument current
in any field, our starting-point will be confessedly empirical: we shall study ray-tracing
techniques because they are used to make optical inferences, presumptive conclusions and
‘defeasibility’ as an essential feature of many legal arguments, axiomatic systems because
they reflect the pattern of our arguments in geometry, dynamics and elsewhere. (257/237)4
Toulmin goes one step further. Our standards of good reasoning are not only to be
established empirically, they are also to be considered historically: they change over
time and can be improved upon:
To think up new and better methods of arguing in any field is to make a major advance,
not just in logic, but in the substantive field itself: great logical innovations are part and
parcel of great scientific, moral, political or legal innovations. [...] We must study the ways
of arguing which have established themselves in any sphere, accepting them as historical
facts; knowing that they may be superseded, but only as the result of a revolutionary advance
in our methods of thought. (257/237)
Because of his views that standards of good reasoning and argument assessment are
non-universal and depend on field, even context, Toulmin has been said to revive
Aristotle’s Topics (Toulmin 2003, viii).
1.4 Logic is generalised jurisprudence
Toulmin discusses the relation of logic with a number of research areas (3–8/3–8).
When logic is regarded as psychology, it deals with the laws of thought, distinguish-
ing between what is normal and abnormal, thereby perhaps even allowing a kind of
“psychopathology of cognition” (5/5). In logic as psychology, the goal is at heart
descriptive: to formulate generalisations about thinkers thinking. But logic can also
be seen as a kind of sociology. Then it is not individual thinkers that are at issue,
but the focus is on general habits and practices. Here Toulmin refers to Dewey, who
explains the passage from the customary to the mandatory: inferential habits can
turn into inferential norms. Logic can also be regarded as a kind of technology, i.e.,
as providing a set of recipes for rationality or the rules of a craft. Here he speaks
of logic as an art, like medicine. In this analogy, logic aims at the formulation of
maxims, ‘tips’, that remind thinkers how they should think. And then there is logic
as mathematics. There the goal of logic becomes to find truths about logical rela-
tions. There is no connection with thinking and logic becomes an objective science.
4 Toulmin here considers the study of defeasibility an empirical question, to be performed by
looking at the law! Toulmin has predicted history, by foreseeing what actually has happened and
still is happening in the field of AI law.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-239-320.jpg)
![226 Bart Verheij
Finally Toulmin comes to the metaphor that he prefers and uses as the basis for his
work: to view logic as jurisprudence:
Logic is concerned with the soundness of the claims we make-with the solidity of the
grounds we produce to support them, the firmness of the backing we provide for them-or,
to change the metaphor, with the sort of case we present in defence of our claims. (7/7)
The jurisprudence metaphor emphasises the critical, procedural function of logic,
thereby fundamentally changing the perspective on logic. It helps to change logic
from an ‘idealised logic’ to a ‘working logic’ (cf. the title of the fourth essay in The
Uses of Argument). At the end of his book he says that jurisprudence should not be
seen as merely an analogy, but, more strongly, as providing an example to follow, as
being a kind of ‘best practice’:
Jurisprudence is one subject which has always embraced a part of logic within its scope,
and what we called to begin with ‘the jurisprudential analogy’ can be seen in retrospect to
amount to something more than a mere analogy. If the same as has long been done for legal
arguments were done for arguments of other types, logic would make great strides forward.
(255/235)
2 The reception and refinement of Toulmin’s ideas in AI
The reception of Toulmin’s ideas is marked by historical happenstance. It was al-
ready mentioned that his original audience, primarily the positivist, logic-oriented
philosophers of knowledge of the time, was on the whole critical. For Toulmin’s
main messages to be appreciated a fresh crowd was needed. It was found in a radi-
cal movement in academic research and education refocusing on the analysis and as-
sessment of real-life argument. This movement, referred to by names such as speech
communication, informal logic and argumentation theory, started to blossom from
the 1970s, continuing so to the present day (see [22]). One thing that Toulmin and
this movement shared was the relativising, at times antagonistic, attitude towards
logic as a formal science. The swing had swung back to exploring the possibilities
of more formal approaches in the 1990s, when Toulmin’s project of treating logic as
a generalised jurisprudence was almost literally taken up in the field of Artificial In-
telligence and Law (see Feteris’ [4] for a related development in argumentation the-
ory). Successful attempts were made to formalize styles of legal reasoning in a way
that respected actual legal reasoning. The approach taken in this field was rooted
in an independent development in Artificial Intelligence, where so-called nonmono-
tonic logics were studied from the 1980s. In that line of research, formal logical
systems were studied that allowed for the retraction of conclusions when new in-
formation, indicating exceptional or contradictory circumstances, became available.
Also in the 1990s, the study of nonmonotonic logics evolved towards what might
be called argumentation logics. More generally, attention was reallocated to imple-
mented systems and an agent-oriented perspective.
The following does not give a fully representative, historical account of AI work
taking up Toulmin’s ideas. A personal choice of relevant research has been made in](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-240-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 227
order to highlight how Toulmin’s points of view have been adopted and refined in
Artificial Intelligence.
2.1 Reiter’s default rules
An early strand of research in Artificial Intelligence, in which a number of Toulmin’s
key positions are visible, is Reiter’s work on the logic of default reasoning [15]. Re-
iter’s formalism is built around the concept of a default: an expression α : Mβ1, ...,
Mβn / γ, in which α, β1, ..., βn, and γ are sentences of first-order logic. Defaults
are a kind of generalized rules of inference. The sentence α is the default’s prereq-
uisite, playing the role of what Toulmin refers to as the datum. The sentence γ is the
default’s consequent, comparable to Toulmin’s notion of a claim. The sentences βi
are called the default’s justifications. A default expresses that its consequent follows
given its prerequisite, but only when its justifications can consistently be assumed.
Reiter does not refer to Toulmin in his highly influential 1980 paper, nor in his
other work. Being thoroughly embedded in the fertile logic-based AI community
of the time, Reiter does not refer to less formal work. Still, in Reiter’s work two
important ideas defended by Toulmin recur in a formal version. The first is the
idea of defeasibility. As said, Reiter’s defaults are a kind of generalized rules of
inference, but of a defeasible kind. For instance, the default p : M¬e / q expresses
that q follows from p unless ¬e cannot be assumed consistently. Reiter’s formal
definitions are such that, given only p, it follows that q, while if both p and e are
given q does not follow. Reiter’s justifications are hence closely related to Toulmin’s
rebuttals, but as opposites: in our example the opposite e of the default’s justification
¬e is a kind of rebuttal in Toulmin’s sense. This holds more generally: opposites of
justifications can be thought of as formal versions of Toulmin’s rebuttals.
There is a second way in which Reiter’s work formally explicates one of Toul-
min’s prime concerns: defaults are contingent rules of inference, in the sense that
they are not fixed in the logical system, as is the case for the natural deduction rules
of first-order logic. Concretely, in Reiter’s approach, defaults are part of the the-
ory from which consequences can be drawn, side by side with the other, factual,
information. One can therefore say that Toulmin’s creed that standards of reason-
ing are field-dependent has found a place in Reiter’s work. There is one important
limitation however. Although Reiter’s defaults can be used to construct arguments
— in what Toulmin refers to as warrant-using arguments —, they cannot be argued
about. In other words, there is no counterpart of warrant-establishing arguments. A
default can for instance not have a default as its conclusion. Since Reiter’s defaults
are givens, it is not possible to give reasons for why they hold. Whereas in Toulmin’s
model warrants do not stand by themselves, but can be given support by backings,
this has no counterpart for Reiter’s defaults. (See section 2.7 for an approach to
warrant-establishing arguments.)
How did Reiter extend or refine Toulmin? The first way is obvious: Reiter has
given a precise explication of a part of Toulmin’s notions, which is a direct conse-
quence of the fact that Reiter’s approach is formally specified, whereas Toulmin’s
only exists in the form of an informal philosophical essay. Reiter has shown that it
is possible to give a formal elaboration of rebuttals and of warrants.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-241-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 229
2.2 Pollock’s undercutting and rebutting defeaters
Pollock’s work on the philosophy and AI of argumentation has rightly achieved
recognition in today’s argumentation research. He can be regarded as being the first
who combined theoretical, computational and practical considerations in his design
of an ‘artificial person’, OSCAR (see, e.g., [12]). In this high ambition, he has had
no followers. Pollock’s work started with roots close to Toulmin’s original audi-
ence, namely philosophers of knowledge. Gradually he began using methods from
the field of Artificial Intelligence, where his ideas have gained most attention. Pol-
lock does not seem to have been directly influenced by Toulmin. In his [11], where
Pollock connects philosophical approaches to defeasibility with AI approaches, he
cites work on defeasibility by Chisholm (going back to 1957, hence a year before
Toulmin’s The Uses of Argument) and himself (going back to 1967).
Here are some of Pollock’s definitions [11]:
P is a prima facie reason for S to believe Q if and only if P is a reason for S to believe Q
and there is an R such that R is logically consistent with P but (P R) is not a reason for S
to believe Q. R is a defeater for P as a prima facie reason for Q if and only if P is a reason
for S to believe Q and R is logically consistent with P but (P R) is not a reason for S to
believe Q.
So prima facie reasons are reasons that sometimes lead to their conclusion, but not
always, namely not when there is a defeater. There is a close connection with non-
monotonic consequence relations: when P is a prima facie reason for Q, and R is a
defeater for P as a reason, then Q follows from P, but not from P R. Pollock goes
on to distinguish between two kinds of defeaters:
R is a rebutting defeater for P as a prima facie reason for Q if and only if R is a defeater and
R is a reason for believing ∼Q. R is an undercutting defeater for P as a prima facie reason
for S to believe Q if and only if R is a defeater and R is a reason for denying that P wouldn’t
be true unless Q were true.
Undercutting defeaters only attack the inferential connection between reason and
conclusion, whereas a defeater is rebutting if it is also a reason for the opposite
of the conclusion. Pollock remarks that ‘P wouldn’t be true unless Q were true’ is
a kind of conditional, different from the material conditional of logic, but having
learnt from an initial analysis, which he no longer finds convincing, he maintains
that it is otherwise not clear how to analyze this conditional ([11], 485).5
Pollock’s finding that there are different kinds of defeaters has been recognized
as an important contribution both for the theory and for the practical analysis of
arguments. Nothing of the sort can be found in Toulmin’s The Uses of Argument.6
(See section 2.7 for more on different conceptions of a rebuttal.)
5 As far as I know, Pollock’s later work (e.g., his [12]) does not contain a new analysis of this
conditional. See section 2.7 for an approach addressing this.
6 Pollock’s work contributes significantly to several other aspects of argumentation (e.g., argu-
ment evaluation, semi-formal rules of inference and software implementation). See also Verheij’s
discussion [26], 104–110.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-243-320.jpg)
![230 Bart Verheij
2.3 Prakken, Sartor Hage on reasoning with legal rules
Toulmin’s idea that logic should be regarded as a generalised form of jurisprudence
(section 1.4), was taken up seriously in the 1990s in the field of Artificial Intel-
ligence and Law. The work by Prakken, Sartor and Hage on reasoning with legal
rules [13, 6] is representative.7
Influenced by logic-based knowledge representation (see, e.g., chapter 10 of
[16]), Prakken Sartor and Hage use an adapted first-order language as the basis
of their formalism. For instance, here is a formal version of the rule that someone
has legal capacity unless he can be shown to be a minor ([13], 340):
r1: ∼x is a minor ⇒ x has legal capacity
Here r1 is the name of the rule, which can be used to refer to it, and ‘x is a minor’
and ‘x has legal capacity’ are unary predicates. The tilde represents so-called weak
negation, which here means that the rule’s antecedent is fulfilled when it cannot
be shown that x is a minor. If ordinary negation were used, the fulfilment of the
antecedent would require something stronger, namely that it can be shown that x is
not a minor.
In the system of Prakken Sartor, arguments are built by applying Modus po-
nens to rules. There are two ways in which arguments can attack each other. First, an
argument can attack a weakly negated assumption in the antecedent of a rule used
in the attacked argument. Second, two arguments can have opposite conclusions.
Information about rule priorities (expressed using the rules names) is then used to
compare the arguments. Argument evaluation is defined in terms of winning strate-
gies in dialogue games: an argument is called justified when it can be successfully
defended against an opponent’s counterarguments.
Hage’s approach [6], in several ways similar to Prakken Sartor’s, is more am-
bitious and philosophically radical.8 For Hage, rules are first-and-foremost to be
thought of as things with properties. As a result, a rule is formalized as a structured
term. A rule’s properties are then formalized using predicates. For instance, the fact
that the rule that thieves are punishable, is valid is formalized as
Valid(rule(theftl, thief(x), punishable(x))).
Here ‘theft1’ is the name of the rule, ‘thief(x)’ the rule’s antecedent and ‘punish-
able(x)’ its consequent. Hage’s work takes the possibilities of a knowledge repre-
sentation approach to the modelling of legal reasoning to its limits. For instance,
there are dedicated predicates to express reasons, rule validity, rule applicability
and the weighing of reasons.
How does the work by Prakken, Sartor Hage relate to Toulmin’s views? First,
they have provided an operationalisation of Toulmin’s idea of law-inspired logic, by
formalizing aspects of legal reasoning. Second, they have refined Toulmin’s treat-
ment of argument. Notably, Prakken Sartor have modelled specific kinds of re-
buttal, namely by the attack of weakly negated assumptions and on the basis of
7 Some other important AI Law work concerning argumentation is for instance [1, 2, 5, 10].
8 Hage’s philosophical and formal theory of rules and reasons Reason-Based Logic was initiated
by Hage and further developed in cooperation with Verheij.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-244-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 231
rule priorities, and embedded them in an argumentative dialogue. Hage has added a
further kind of rebuttal, namely by the weighing of reasons. Also, Hage has distin-
guished the validity of a rule from its applicability. The former can be regarded as an
expression of a warrant in Toulmin’s sense, and since in Hage’s system rule validity
can depend on other information, it is natural to model Toulmin’s backings as rea-
sons for the validity of a rule. And perhaps most importantly: Prakken Sartor and
Hage (and other AI law researchers) have worked on the embedding of defeasible
argumentation in a genuine procedural, dialogical setting (see also section 2.5). A
further refinement of Toulmin’s view is given by Verheij and colleagues [28], who
show how two kinds of warrants (viz. legal rules and legal principles) with apparent
logical differences, can be seen as extremes of a spectrum.
2.4 Dung’s admissible sets
Dung’s paper [3] has supplied an abstract mathematical foundation for formal
work on argumentation. Following earlier mathematically flavoured work (e.g.,
[17, 29, 30]), his abstraction of only looking at the attack relation between argu-
ments has helped organize the field, e.g., by showing how several formal systems
of nonmonotonic reasoning can be viewed from the perspective of argument attack.
A set of (unstructured) arguments with an attack relation is called an argumentation
framework.
Dung has studied the mathematics of three types of subsets of the set of argu-
ments of an argumentation framework: stable, preferred and grounded extensions.
A set of arguments is a stable extension if it attacks all arguments not in the set. A
set of arguments is a preferred extension if it is a maximal set of arguments without
internal conflicts and attacking all arguments attacking the set. The grounded exten-
sion (there is only one) is the result of an inductive process: starting from the empty
set, consecutively arguments are added that are only attacked by arguments already
defended against.
Stable extensions can be regarded as an ‘ideal’ interpretation of an argumentation
framework. When an extension is stable, all conflicts between arguments can be
regarded as solved. It turns out that sometimes there are distinct ways of resolving
the conflicts (e.g., when two arguments attack each other, each argument by itself is
a stable extension) and that sometimes there is no way (e.g., when an argument is
self-attacking). Preferred extensions are a generalization of stable extensions, as all
stable extensions are also preferred. However, an argumentation framework always
has a preferred extension (perhaps several). Preferred extensions can be regarded
as showing how as many conflicts as possible can be resolved by counterattack.
The grounded extension exists always and is a subset of all preferred and stable
extensions.
Dung’s work shows that the mathematics of argument attack is non-trivial and
interesting. Thereby he has significantly extended our understanding of Toulmin’s
concept of rebuttal.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-245-320.jpg)
![232 Bart Verheij
2.5 Walton’s argumentation schemes
Toulmin’s proposal that the maxims provided by a standard formal logical system
(such as first order predicate logic) are not the only criteria for good reasoning
and argument assessment, posed a new problem: if there are other, more field- and
context-dependent standards of reasoning, what are they? For, though many recog-
nized the shortcomings of formal logic for practical argument assessment, few were
happy with the possible relativistic implication that anything goes. A good way to
avoid the trap of uncontrolled relativism is to provide a systematic specification of
standards of good reasoning.
One approach in this direction, which is especially close to Toulmin’s concep-
tion of warrants, can be found in Walton’s work on argumentation schemes (e.g.,
[31]). Argumentation schemes can be thought of as a semi-formal generalization of
the rules of inference found in formal logic. Argument from expert opinion is an
example ([31], 65):
E is an expert in domain D.
E asserts that A is known to be true.
A is within D.
Therefore, A may (plausibly) be taken to be true.
As Walton’s argumentation schemes are context-dependent, not universal; defeasi-
ble, not strict; and concrete, not abstract, there is a strong analogy with Toulmin’s
warrants. There are two important differences though. First, Walton’s argumentation
schemes are structured, whereas Toulmin’s warrants are not. Walton’s argumenta-
tion schemes have premises, consisting of one or more sentences (often with in-
formal variables), and a conclusion;9 Toulmin’s warrants are expressed as rule-like
statements, such as ‘A man born in Bermuda will generally be a British subject’.10
By giving generic inference licenses more structure, as in Walton’s work, the ques-
tion arises whether they become formal enough to give rise to a kind of ‘concrete
logic’. Verheij [24] argues that it is a matter of choice, perhaps: taste, whether one
draws the border between form and content on either side of argumentation schemes.
To indicate the somewhat ambiguous status of argumentation schemes, the term
‘semi-formal’ may be most appropriate.
Second, Walton’s argumentation schemes have associated critical questions. Crit-
ical questions help evaluating applications of an argumentation scheme. As a result,
they play an important role in the evaluation of practical arguments. For instance,
Walton lists the following critical questions for the scheme ‘Argument from expert
opinion’ ([31], 65):
1. Is E a genuine expert in D?
2. Did E really assert A?
9 Sometimes Walton’s schemes take another form, e.g., small chains of argument steps or small
dialogues; see Verheij’s [24] for a format for the systematic specification of argumentation schemes
inspired by knowledge engineering technology.
10 Occasionally, a bit more structure is made explicit. For instance, when Toulmin phrases a warrant
in an ‘if ... then ...’ form (e.g., ‘If anything is red, it will not also be black’, 98/91), thereby making
an antecedent and consequent recognizable.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-246-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 233
3. Is A relevant to domain D?
4. Is A consistent with what other experts in D say?
5. Is A consistent with known evidence in D?
Critical questions are related to argument attack, as they point to circumstances
in which application of the scheme is problematic (e.g., [24]). For instance, the
question ‘Is E a genuine expert in D?’ questions whether the element ‘E is an expert
in domain D’ in the premises of the scheme really holds. Some critical questions
are like Toulmin’s notion of rebuttal. For instance, the question ‘Is A consistent with
what other experts in D say?’ points to a rebuttal ‘A is not consistent with what other
experts in D say’, which, if accepted, can raise doubt whether the conclusion can
justifiably be drawn. In general, four types of critical questions can be distinguished
[24]:
1. Critical questions concerning the conclusion of an argumentation scheme. Are there other
reasons, based on other argumentation schemes for or against the scheme’s conclusion?
2. Critical questions concerning the elements of the premises of an argumentation scheme.
Is E an expert in domain D? Did E assert that A is known to be true? Is A within D?
3. Critical questions based on the exceptions of an argumentation scheme. Is A consistent
with what other experts in D say? Is A consistent with known evidence in D?
4. Critical questions based on the conditions of use of an argumentation scheme. Do experts
with respect to facts like A provide reliable information concerning the truth of A?
The critical questions associated with an argumentation scheme point to the dia-
logical setting of argumentation. Toulmin mentions the dialogical and procedural
setting of argumentation (as, e.g., when discussing the jurisprudence metaphor for
logic), but the discussion is not elaborate. Much work on the relation between argu-
mentation and dialogue has been done. There is for instance the pragma-dialectical
school (e.g., [21]), but also Walton’s conception of argumentation is embedded in a
procedural, dialogical setting. For instance, Walton [32] expresses a view on how to
determine the relevance of an argument in a dialogue. There are six issues to take
into account: the dialogue type11, the stage the dialogue is in, the dialogue’s goal,
the type of argument, which is determined by the argumentation scheme underlying
the argument, the prior sequence of argumentation, and the institutional and social
setting.
In conclusion, Walton’s work has played a significant role in two developments
in AI with respect to Toulmin’s main themes. First, the study of argumentation
schemes by Walton and others has made a start with the systematic specification
of context-dependent, defeasible, concrete standards of argument assessment, as
sought for by Toulmin. And, second, the idea of considering argumentation from
a procedural, dialogue perspective has been elaborated upon.
2.6 Reed Rowe’s argument analysis software
Further steps towards the realization of Toulmin’s goals have been made by the
recent advent of software-support of argumentative tasks, often using argument dia-
grams [8, 26]. In this connection, Reed Rowe’s work on the Araucaria tool [14] is
11 See also Walton Krabbe’s [33], a treatment of dialogue types that is especially influential in
research in AI and multi-agent systems.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-247-320.jpg)
![234 Bart Verheij
especially relevant for the achievement of Toulmin’s goals, as they have presented
Araucaria specifically as a software tool for argument analysis. Araucaria uses an
argument diagramming format, in which the recursive tree-structure of reasons sup-
porting conclusions is depicted. It is also possible to indicate statements that are in
conflict. Araucaria’s standard diagramming format12 is different from Toulmin’s in
several ways, but especially by not graphically distinguishing warrants from data.
In an interestingly different way, however, Araucaria’s standard format does include
the idea of context-dependent types of reasoning as argued for by Toulmin, namely
by its incorporation of Walton-style argumentation schemes (cf. section 2.5). Ar-
gumentation schemes can be used in Araucaria to label argumentative steps. For
instance, a concrete argument ‘There is smoke. Therefore, there is fire’ could be la-
belled as an instance of the scheme ‘Argument from sign’, thereby giving access two
critical questions, such as ‘Are there other events that would more reliably account
for the sign?’. By this possibility, Reed Rowe’s Araucaria is a useful step towards
software-supported argument assessment. The tool provides a significant extension
of Toulmin’s aim to change logic from an ‘idealised logic’ to a ‘working logic’.
2.7 Verheij’s formal reconstruction of Toulmin’s scheme
Already the examples in this chapter show a wide variety of approaches to — what
might be called — semi-formal defeasible argumentation; and this is just the tip of
the iceberg. By this embarrassment of riches, the question arises whether there are
fundamental differences, e.g., between explicitly Toulmin-oriented approaches and
other; or is the similarity of subject matter strong enough to allow for a synthesis of
approaches? Looking for answers, I have attempted to reconstruct Toulmin’s scheme
using modern formal tools [25]. I used the abstract argumentation logic DefLog
[23]. DefLog uses two connectives × and ❀: the first for expressing the defeat of
a prima facie justified statement (‘negation-as-defeat’, the semantics of which falls
outside the scope of this chapter), the second for expressing a conditional relation
between statements (‘primitive implication’, validating Modus Ponens, but lacking
a so-called introduction rule).13 Toulmin’s notion of a qualifier has been left out of
the reconstruction.
The key to the translation of Toulmin’s scheme into DefLog is to explicitly ex-
press that a datum leads to a claim; in DefLog: D ❀ C. DefLog’s primitive im-
plication can be thought of as expressing a specific inference license. It is an ex-
plicit expression of what Toulmin refers to as a ‘logical gulf’ (9/9) that seems to
exist between a reason and the state of affairs it supports. In this way, the licens-
ing of concrete argument steps is removed from the logic, i.e., the fixed formalized
background specifying general argument validity, and shifted to the contingent in-
formation. In this way, it becomes possible to express substantial arguments about
concrete inferential bridges.
12 In later versions, two alternative formats are provided: Toulmin’s and Wigmore’s.
13 DefLog is formally an extension of Dung’s abstract argumentation framework (section 2.4), as
Dung’s attack between two arguments A and B can be expressed as A ❀ ×B. DefLog analogues
of Dung’s stable and preferred semantics are defined and proven to coincide with Dung’s when
DefLog’s language is restricted to Dung’s.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-248-320.jpg)
![11 The Toulmin Argument Model in Artificial Intelligence 235
In particular, the role of a warrant can now be expressed as a reason for such a
conditional statement: W ❀ (D ❀ C). To formally show that a datum and claim are
specific, whereas a warrant is to be thought of as a generic inference license, we can
use variables and their instances: W ❀ (D(t) ❀ C(t)). This clarifies the distinctions
between the following three:
(1) A man born in Bermuda will generally be a British subject.
(2) If Person was born in Bermuda, then generally Person is a British subject.
(3) If Harry was born in Bermuda, then generally he is a British subject.
The first is the ordinary language expression of a warrant as a rule-like statement
(formally: W). The third is a conditional sentence (formally: D(t) ❀ C(t)), instan-
tiating the second, which is a scheme of conditional sentences (formally: D(x) ❀
C(x), where x is a variable, that can be instantiated by the concrete term t). Note that
only (1) and (3) can occur in actual texts, whereas (2) is — by its use of a variable
Person — an abstraction. One could say however that (1) and (2) imply each other: a
warrant corresponds to a scheme of argumentative steps from datum to claim. Given
this analysis of warrants and their relation to datum and claim, backings are simply
reasons for warrants: B ❀ W.
Rebuttals are an ambiguous concept in Toulmin’s treatment. He associates rebut-
tals with ‘circumstances in which the general authority of the warrant would have to
be set aside’ (101/94), ‘exceptional circumstances which might be capable of defeat-
ing or rebutting the warranted conclusion’ (101/94) and with the (non )applicability
of a warrant (102/95). It turns out that these three can be distinguished, and, given
the present analysis of the warrant-datum-claim part of Toulmin’s scheme, even ex-
tended to five kinds of rebuttals, as there are five different statements that can be
argued against: the datum D, the claim C, the warrant W, the conditional D(t) ❀
C(t), expressing the inferential bridge from datum to claim, and the conditional W
❀ (D(t) ❀ C(t)), which expresses the application of the warrant in the concrete situ-
ation (see [25] for a more extensive explanation). A rebuttal of the latter conditional
coincides conceptually with an undercutting defeater in the sense of Pollock’s. Note
that the analysis suggests an answer to Pollock’s open issue of how to analyze the
conditional ‘P wouldn’t be true unless Q were true’ (section 2.2). If U undercuts P
as a reason for Q, we would write U ❀ ×(P ❀ Q).
In this analysis, there is a natural extension of Toulmin’s concept of warrant: just
as it is necessary to specify which data imply claims (by Toulmin’s warrants), it is
necessary to specify which rebuttals block the application of warrants. Informally: it
is a matter of substance, not logic, which statements are rebuttals. It must be shown
by argument whether some statement is a rebuttal. In the law, for instance, not only
legal rules (a kind of warrants) find backing in statutes, but also exceptions to rules.
In the present analysis, dealing with ‘rebuttal warrants’ is a matter of course, since
there is an explicit expression that R is a rebuttal.14
A side effect of the reconstruction is that arguments modelled according to Toul-
min’s scheme can be formally evaluated. For instance, assuming that datum and
14 When R is a rebuttal in the sense of a Pollockian undercutter, this requires a sentence of the form
R ❀ ×(W ❀ (D ❀ C)), expressing that, if R holds, the warrant W is not applicable.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-249-320.jpg)
![236 Bart Verheij
Fig. 11.2 An entangled dialectical argument
warrant hold, but not a rebuttal, the claim follows; when also a rebuttal is assumed,
the claim does not follow.15 A rebuttal of a rebuttal can be shown to reinstate a claim.
Verheij [24] extends the approach to include Walton’s argumentation schemes.
The result of the formal reconstruction of Toulmin’s scheme showed some ex-
tensions, while retaining the original flavour. Figure 2 (using a diagramming format
used in [27]) illustrates the basic relations between statements as distinguished here:
Claims can have reasons for and against them (Jim’s testimony supporting the as-
sault by Jack, and Paul’s attacking it), and the inferential bridges (the conditionals
connecting reasons with their conclusions, here drawn as arrows) can be argued
about just like other statements. The resulting argument structures are dialectical,
by their incorporation of pros and cons, and entangled by their allowing the support
and attack of inferential bridges.
3 Concluding remarks
It has been shown that central points of view argued for by Toulmin (1958), in par-
ticular the defeasibility of argumentation, the substantial, instead of formal, nature
of standards of argument assessment, and the richer set of building blocks for argu-
ment analysis, are very much alive. Also Toulmin’s ‘research program’ of treating
logic as generalised jurisprudence has been taken up (with or without reference to
him) and proven to be fertile.
There have also been refinements and extensions. It is now known that defeasible
argumentation has interesting (and intricate) formal properties. There exist formal
systems for the evaluation of defeasible arguments. Toulmin’s argument diagram
and its associated set of building blocks for argument analysis have been made pre-
cise and become refined, e.g., by distinguishing between kinds of argument attack.
Not only is there now a wealth of studies of domain-bound, concrete forms of ar-
15 Formally: using W ❀ (D ❀ C), R ❀ ×(W ❀ (D ❀ C)), two sentences expressing that W is a
warrant and R a rebuttal blocking its application, respectively, as background, and then assuming
W, and D, one finds a unique dialectical interpretation, in which C holds, whereas adding R to the
assumptions leads to a unique dialectical interpretation in which C does not hold.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-250-320.jpg)
![Chapter 12
Proof Burdens and Standards
Thomas F. Gordon and Douglas Walton
1 Introduction
This chapter explains the role of proof burdens and standards in argumentation, il-
lustrates them using legal procedures, and surveys the history of research on compu-
tational models of these concepts. It also presents an original computational model
which aims to integrate the features of these prior systems.
The ‘mainstream’ conception of argumentation in the field of artificial intelli-
gence is monological [6] and relational [14]. Argumentation is viewed as taking
place against the background of an inconsistent knowledge base, where the knowl-
edge base is a set of propositions represented in some formal logic. Argumentation
in this conception is a method for deducing warranted propositions from an incon-
sistent knowledge base. Which statements are warranted depends on attack relations
among the arguments [10] which can be constructed from the knowledge base.
The notions of proof standards and burden of proof become relevant only when
argumentation is viewed as a dialogical process for making justified decisions. The
input to the process is an initial claim or issue. The goal of the process is to clarify
and decide the issues, and produce a justification of the decision which can with-
stand a critical evaluation by a particular audience. The role of the audience could
be played by the respondent or a neutral-third party, depending on the type of di-
alogue. The output of this process consists of: 1) a set of claims, 2) the decision
to accept or reject each claim, 3) a theory of the generalizations of the domain and
the facts of the particular case, and 4) a proof justifying the decision of each issue,
showing how the decision is supported by the theory.
Notice that a theory or knowledge-base is part of the output of argumentation
dialogues, not, as in the relational conception, its input. This is because, as has been
Thomas F. Gordon
Fraunhofer FOKUS, Berlin, Germany, e-mail: thomas.gordon@fokus.fraunhofer.de
Douglas Walton
University of Windsor, Windsor, Canada, e-mail: dwalton@uwindsor.ca
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 239
DOI 10.1007/978-0-387-98197-0 12, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-253-320.jpg)
![240 Thomas F. Gordon and Douglas Walton
repeatedly recognized [31, 33, 21], the generalizations (rules) of some domain and
the particular facts of a problem or case are dependent on one another and need to
be constructed together, in an iterative process. For example, one of the founders of
the field of computer science and law, Jon Bing, wrote in 1982:
Legal reasoning is not primarily deductive, but rather a modeling process of shaping an
understanding of the facts, based on evidence, and an interpretation of the legal sources, to
a construct a theory for some legal conclusion. [7]
The concept of proof in argumentation is weaker than it is in mathematics. The
proof need not demonstrate that a proposition is necessarily true, given a set of ax-
ioms assumed to be true. Rather, as in law, a proof in argumentation is a structure
which demonstrates to a particular audience that a proposition statisfies its applica-
ble proof standard. Since expressive logics are undecidable or intractable, the theory
constructed during the dialogue cannot usually serve as a proof. A burden of proof
is not discharged if the audience must solve a hard problem to construct the proof
for themselves from the theory.
There are several kinds of proof burdens. The distinctions between them can only
be understood with a deeper analysis of particular argumentation processes. There
are many kinds of argumentation processes, each regulated by its own procedural
rules, usually called ‘protocols’ in AI. Walton has developed a typology of dialogue
types, classifying persuasion dialogues, negotiation, and deliberation, among other
types [37].
For our purpose of illustrating different kinds of proof burdens, it is sufficient to
use a simplified description of civil procedure, roughly based on the law of Califor-
nia [34]. A civil case begins by the plaintiff filing a complaint, stating a claim against
the defendant. The complaint is the first step in the pleadings phase of the case. It
contains, in addition to the claim, assertions about the facts of the case which the
plaintiff contends are sufficient, if true, to prove the defendant has breached some
obligation legally entitling the plaintiff to some remedy or compensation. The de-
fendant then has several options for responding to the complaint. For the sake of
brevity we will mention just one, filing an answer in which the factual allegations
are each denied or conceded and asserting additional facts, called an affirmative de-
fense, which may be useful for defeating or undercutting later arguments put forward
by the plaintiff. The final step in the pleadings phase gives the plaintiff an opportu-
nity to file a reply in which he concedes or denies the additional facts alleged by the
defendant in his answer. The next phase of the process provides the parties various
methods to discover evidence, for example by interviewing witnesses under oath,
called taking depositions. At the trial, this evidence is presented to the judge, and
possibly a jury, and further evidence is produced by examining and cross-examining
witnesses during the trial. At the end of the trial, the evidence is passed on to the
trier-of-fact, either the judge or the jury, if there is one. If there is a jury, the judge
first instructs the jury about the relevant law, since the jury is only responsible for
finding the facts. After the jury has completed its deliberations, it reports its verdict
to the judge, who then enters his judgment upon the verdict. The judgment may be
appealed by the losing party, but we will end our exposition of legal procedure here.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-254-320.jpg)
![12 Proof Burdens and Standards 241
Our account of legal burdens of proof below is based in part on [30]. The first
kind of burden of proof is called the burden of claiming. A person who feels he has
a right to some legal remedy has the burden of initiating the proceeding by filing
a complaint, which must allege facts sufficient to prove the operative facts of legal
rules entitling him to some remedy. The second type of burden of proof is called
the burden of questioning or contesting. During pleading, any allegations of fact by
either party are implicitly conceded unless they are denied. The third type of burden
is called the burden of production. It is the burden to discover and bring forward
evidence supporting the contested factual allegations in the pleadings. The fourth
type of burden of proof is the burden of persuasion. In a civil proceeding, this burden
becomes operative only at the end of the trial, when the evidence and arguments are
put to the jury to decided the factual issues. In a civil proceeding, the plaintiff has
the burden of persuasion for all operative facts of his complaint and the defendant
has the burden of persuasion for all affirmative defenses, i.e. exceptions. In criminal
cases this is different. The prosecution has the burden of persuasion for all facts of
the case, whether or not they are the operative facts of the elements of the alleged
crime, or defenses, such as self-defense in a murder case. The fifth type of burden is
called the tactical burden of proof. During the trial, arguments are put forward by
both parties, pro and con the various claims at issue. At a finer level of granularity,
the argumentation phase can be broken down conceptually into a sequence of stages,
where each stage consists of all the arguments which have been put forward by both
parties so far in the proceeding. The parties take turns putting forward arguments,
by introducing new evidence. The next stage is constructed by adding the arguments
put forward during this turn to all the previous arguments. The tactical burden arises
from considering whether the arguments of a stage would be sufficient to meet the
burden of persuasion with regard to some issue, if hypothetically the trial were to
end at the stage and the issues where immediately put to the jury. The tactical burden
of proof is the only burden of proof which, strictly speaking, can shift back and forth
between the parties during the proceeding.
How does the burden of persuasion operate? Essentially the jury has the task of
weighing the arguments pro and con each proposition at issue. If the pro arguments
are not deemed to sufficiently outweigh the con arguments, then the jury must reject
the alleged fact by deciding that the alleged fact is not true. Because of the way the
burden of persuasion is allocated, this amounts to accepting the default truth value
of the proposition at issue.
When do pro arguments ‘sufficiently’ outweigh con arguments to meet the bur-
den of persuasion? This leads us to our final topic, proof standards. The question
is how to aggregate or ‘accrue’ [24] arguments pro and con some claim. In the le-
gal domain, four proof standards for factual issues exist, at least in common law
jurisdictions. The scintilla of evidence proof standard is met if there is “any evi-
dence at all in a case, even a scintilla, tending to support a material issue . . . ” [8,
p. 1207] The preponderance of evidence proof standard is met by “evidence which
as a whole shows that the fact sought to be proved . . . is more credible and convinc-
ing to the mind.” [8, p. 1064]. The clear and convincing evidence proof standard
is the “measure or degree of proof which will produce in mind of trier of facts a](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-255-320.jpg)
![242 Thomas F. Gordon and Douglas Walton
firm belief or conviction as to allegations sought to be established; it is intermedi-
ate, being more than preponderance, but not to extent of such certainty as is required
beyond reasonable doubt . . . ” [8, p. 227]. Finally, the beyond reasonable doubt stan-
dard is the strongest legal proof standard, applicable in criminal cases. It requires
evidence which leaves the trier of fact “fully satisfied, entirely convinced, . . . to a
moral certainty” [8, p. 147].
In our view proof standards cannot and should not be interpreted probabilisti-
cally. The first and most important reason is that probability theory is applicable
only if statistical knowledge about prior and conditional probabilities is available.
Presuming the existence of such statistical information would defeat the whole pur-
pose of argumentation about factual issues, which is to provide methods for making
justified decisions when knowledge of the domain is lacking. Another argument
against interpreting proof standards probabilistically is more technical. Arguments
for and against some proposition are rarely independent. What is needed is some
way to accrue arguments which does not depend on the assumption that the argu-
ments or evidence are independent.
Prakken has identified three principles any formal account of accrual must satisfy
[24]: 1) Combining several arguments pro or con some proposition can not only
strengthen one’s position, but also weaken it. 2) Once several arguments have been
accrued, the individual arguments, considered separately, should have no impact on
the acceptability of the proposition at issue, and 3) Finally, any argument which is
‘flawed’ may not take part in the aggregation process. The models of proof standards
presented in the section are designed to respect these principals.
2 Formal Model
Our goal in this section is to define an abstract formal model of argumentation as
a theory and proof construction process for making justified decisions. Inspired by
Dung’s model of abstract argumentation frameworks, the model shall be as abstract
and simple as possible while being sufficient for capturing the distinctions between
the various types of proof burdens and proof standards identified in the introduction
and meeting other known requirements. It is not intended to be a comprehensive
formal model of argumentation. We will also take care to abstract from the details
of the legal domain and, in particular, the law of civil procedure.
We begin with the concept of an argument. Unlike Dung, we cannot leave this
concept fully abstract, since our aim is to model burden of proof and proof stan-
dards. The proponent of an argument has the burden of production for its ordinary
premises; while the respondent has the burden of production for any exceptions.
Moreover, since the task of proof standards is to aggregate arguments pro and con
some proposition at issue, the model must represent not only the premises of argu-
ments, and distinctions between types of premises, but also their conclusions. These
considerations lead us to the following definition of argument.
Definition 12.1 (argument). Let L be a propositional language. An argument is a
tuple P,E,c where P ⊂ L are its premises, E ⊂ L are its exceptions and c ∈ L](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-256-320.jpg)
![12 Proof Burdens and Standards 243
is its conclusion. For simplicity, c and all members of P and E must be literals, i.e.
either an atomic proposition or a negated atomic proposition. Let p be a literal. If p
is c, then the argument is an argument pro p. If p is the complement of c, then the
argument is an argument con p.
Since all conclusions of arguments are literals according to this definition, the
axioms of the theory constructed during argumentation consist only of literals. Other
propositions of the theory can be derived from these axioms using the inference rules
of classical logic and the argumentation schemes of the domain.
To model the distinctions between the various kinds of burden of proof, we must
model argumentation as a process, consisting of several phases. It is sufficient to
distinguish three phases, the opening, argumentation and closing phases of the pro-
cess. Since typically argumentation takes place in dialogues, we will use the term
‘dialogue’ as the generic name for argumentation processes.
Definition 12.2 (dialogue). A dialogue is a tuple O,A,C, where O, A and C, the
opening, argumentation, and closing phases of the dialogue, respectively, are each
sequences of stages. A stage is a tuple arguments,status, where arguments is a
set of arguments and status is a function mapping the conclusions of the arguments
in arguments to their dialectical status in the stage, where the status is a member
of {claimed,questioned}. In every chain of arguments, a1,...an, constructable from
arguments by linking the conclusion of an argument to a premise of another argu-
ment, a conclusion of an argument ai may not be a premise of an argument aj, if
j i. A set of arguments which violates this condition is said to contain a cycle and
a set of arguments which complies with this condition is called cycle-free.
Notice that the cycles defined here are not the same as cycles in a Dung argu-
mentation framework. Whereas the links (arcs) between arguments in the directed
graph induced by a Dung argumentation framework model the attack relation, the
links in the directed graph induced by arguments in our system model the premise
and conclusion relations. Notice that, in our system, arguments both pro and con
some proposition can be included in a set of arguments without causing a cycle.
Constraining the arguments of a stage to be cycle-free is intended to simplify
the evaluation of arguments. The set of arguments is intended to model the current
state of the proof being constructed by the parties in the dialogue, not a ‘pool of
information’ for constructing proofs. Intuitively, proofs should not contain cycles.
Next we need a structure for evaluating arguments, to assess the acceptability of
propositions at issue. As in value-based argumentation frameworks [4, 5] arguments
are evaluated with respect to an audience, such as the trier-of-fact (judge or jury) in
legal trials.
Definition 12.3 (audience). An audience is a structure assumptions,weight, where
assumptions ⊂ L is a consistent set of literals assumed to be acceptable by the au-
dience and weight is a partial function mapping arguments to real numbers in the
range 0.guatda.com/cmx.p0...1.0, representing the relative weights assigned by the audience to the
arguments.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-257-320.jpg)
![12 Proof Burdens and Standards 245
Now we are ready to define the proof standards, beginning with scintilla of the
evidence. A proposition satisfies the scintilla standard, in our model, if it is sup-
ported by at least one applicable pro argument.
Definition 12.7 (scintilla of evidence). Let stage,audience,standard be an argu-
ment evaluation structure and let p be a literal in L. scintilla(p,stage,audience) =
true if and only if there is at least one applicable argument pro p in stage.
Scintilla is the weakest of the proof standards we will define and is the only
one which can be met by complementary literals in the same argument evaluation
structure. That is, if p is an atomic proposition, both p and ¬p can be acceptable
in an argument evaluation structure using the scintilla of evidence standard. This
would be the case if p has an applicable con argument, as well as an applicable pro
argument.
Let us now turn our attention to the three most important legal proof standards:
preponderance of the evidence, clear and convincing evidence and beyond reason-
able doubt. Intuitively, preponderance is satisfied if the pro arguments outweigh the
con arguments, by however much. The issue we have to face when formalizing pre-
ponderance is how to aggregate the weights of a set of arguments for the purpose of
this comparison. The clear and convincing evidence standard requires more proof
than the preponderance standard: not only must the pro arguments outweigh the
con arguments, the weight of the pro arguments and the difference in weight of the
pro and con arguments both must exceed some thresholds. Finally, the beyond a
reasonable doubt standard goes further. Not only must the arguments be clear and
convincing, but, as the name of the standard suggests, the weight of the con argu-
ments must be below the threshold of ‘reasonable doubt’.
Definition 12.8 (preponderance of the evidence). Let stage,audience,standard
be an argument evaluation structure and let p be a literal in L. preponderance
(p,stage,audience) = true if and only if
• there is at least one applicable argument pro p in stage and
• the maximum weight assigned by the audience to the applicable arguments pro
p is greater than the maximum weight of the applicable arguments con p.
The preponderance of the evidence standard was called the best argument stan-
dard in [18].
Definition 12.9 (clear and convincing evidence). Let stage,audience,standard
be an argument evaluation structure and let p be a literal in L. clear-and-convincing
(p,stage,audience) = true if and only if
• the preponderance of the evidence standard is met,
• the maximum weight of the applicable pro arguments exceeds some threshold α,
and
• the difference between the maximum weight of the applicable pro arguments and
the maximum weight of the applicable con arguments exceeds some threshold β.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-259-320.jpg)
![246 Thomas F. Gordon and Douglas Walton
Definition 12.10 (beyond reasonable doubt). Let stage,audience,standard be an
argument evaluation structure stage,audience,standard and let p be a literal in L.
beyond-reasonable-doubt(p,stage,audience) = true if and only if
• the clear and convincing evidence standard is met and
• the maximum weight of the applicable con arguments is less than some threshold
γ.
We assume the α, β and γ thresholds used by the clear and convincing evidence
and beyond a reasonable doubt standards are set by the applicable protocol of the
dialogue.
At first glance, using maximum weights to aggregate pro and con arguments
might seem unintuitive. One might be inclined to compare the sums of the weights
of the applicable pro and con arguments. However, since arguments cannot be as-
sumed to be independent, summing weights would risk taking the same information
or reasons into account multiple times. When several weak arguments can be com-
bined to make a stronger argument, this can be achieved by joining their premises
together into a single argument, as discussed further next. We leave it up to the au-
dience to judge the effect of any possible interdependencies among the premises
on the weight of the argument. Both alternatives, summing weights or taking their
maximum weight, have the property of taking all arguments into account.
Assuming arguments are stated in their strongest form, aggregating arguments
using their maximum weight satisfies all three of Prakken’s principles of accrual
[24]: 1) Aggregated arguments can be evaluated by the audience to be stronger or
weaker than the arguments considered separately; 2) Once several arguments have
been accrued, the individual arguments, considered separately, have no effect on the
acceptability of the proposition at issue; and 3) Any argument which is ‘flawed’
does not take part in the aggregation process.
By ‘stating arguments in their strongest form’ we mean the following. Let p and q
be two propositions which, when they are true, are evidence pro a third proposition,
r. This can be expressed as either two convergent arguments or as a linked argument
[38]. The convergent arguments would be:
a1: r since p.
a2: r since q.
The linked form of this argument ‘accrues’ p and q into a single argument:
a3: r since p and q.
If the linked argument, a3, is more persuasive, that is if the party putting forward
this argument estimates it would be given more weight by the audience than either
a1 or a2, then we assume a3 will be put forward.
To illustrate this more concretely, let’s return to Prakken’s example about jog-
ging when it is both hot and rainy. The strongest arguments con jogging are the
convergent arguments:
a4: not jogging since hot.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-260-320.jpg)
![12 Proof Burdens and Standards 247
a5: not jogging since rainy.
The strongest argument pro jogging is the following linked argument:
a6: jogging since hot and rainy.
Returning to Prakken’s three principles of accrual: 1) The audience can decide
whether to give a6 greater or lesser weight than each of the arguments a4 and a5;
2) If the accrued argument, a6, is given greater weight, then a6 will defeat both
a4 and a5, rendering them ineffective, using any of the proofs standards which ag-
gregate arguments by weight; and 3) Inapplicable arguments are not be taken into
consideration using any proof standard.
In [18] one further proof standard was defined, called dialectical validity. For the
sake of completeness we include its definitions here.
Definition 12.11 (dialectical validity). Let stage,audience,standard be an argu-
ment evaluation structure stage,audience,standard and let p be a literal in L.
dialectical-validity(p,stage,audience,) = true if and only if there is at least one ap-
plicable argument pro p in stage and no argument con p in stage is applicable.
The dialectical validity standard is suitable for aggregating arguments from gen-
eral rules and exceptions, where any applicable exception is enough to override the
general rule.
One of the requirements identified in the introduction is that checking proofs
should be an easy task. In more computational terms, using our formal model, the is-
sue is whether the acceptability of a proposition in an argument evaluation structure
is tractably decidable. We conjecture that this is the case, but will not try to prove
this formally here. The reasons for this conjecture are many. The argument eval-
uation structure has been designed in part to achieve this goal, by making several
restrictions: 1) The language is propositional, not first-order; 2) premises, excep-
tions and conclusions of arguments must be literals; and 3) the set of arguments of
a stage is finite and, by definition, acyclic. Of course the computational complexity
of acceptability also depends on the complexity of the proof standards applied.
Now we are ready to turn to modeling the various kinds of burden of proof. The
burdens of claiming and questioning must be met during the opening phase of the
dialogue. The burden of production and the tactical burden of proof are relevant only
during the argumentation phase. Finally, the burden of persuasion comes into play
in the closing phase, but is also used hypothetically to estimate the tactical burden
of proof during the argumentation phase.
The purpose of the opening stage of a dialogue is to frame the issues. Arguments
put forward in the argumentation stage must be relevant to the issues raised in the
opening stage. Depending on the protocol of the dialogue, a proposition claimed in
the opening stage may be deemed conceded unless it is questioned, requiring the
audience to assume it is true, following the principal of “silence implies consent.”
Definition 12.12 (burdens of claiming and questioning). Let s1,...,sn be the
stages of the opening phase of a dialogue. Let argumentsn,statusn be the last stage,](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-261-320.jpg)
![248 Thomas F. Gordon and Douglas Walton
sn, of the opening phase. A party has met the burden of claiming a proposition p
if and only if statusn(p) ∈ {claimed,questioned}, that is, if and only if statusn(p)
is defined. The burden of questioning a proposition p has been met if and only if
statusn(p) = questioned.
Notice that a questioned proposition satisfies the burden of claiming, since it
is assumed that only propositions which have been claimed in an earlier stage are
questioned.
This simple model defines only minimal requirements for raising issues in the
opening phase of a dialogue. The argumentation protocol of a dialogue may state
additional requirements. For example, according to the law of civil procedure in Cal-
ifornia, the plaintiff must state a cause of action: the facts claimed must be sufficient
to give the plaintiff a right to judicial relief, as a matter of law.
The burden of production is relevant only during the argumentation phase of a
dialogue. The burden of production for some proposition is satisfied if it is accept-
able at the end of the argumentation phase using the the weakest proof standard,
scintilla of the evidence. The party who puts forward an argument has the burden of
production for its premises. Similarly, the respondent to an argument has the burden
of production for each exception.
The audience used to assess the burden of production depends on the protocol of
the particular dialogue. In civil proceedings in California, the judge is the audience
during the argumentation phase, i.e. the trial.
Definition 12.13 (burden of production). Let s1,...,sn be the stages of the argu-
mentation phase of a dialogue. Let argumentsn,statusn be the last stage, sn, of
the argumentation phase. Let audience be the relevant audience for assessing the
burden of production, depending on the protocol of the dialogue. Let AES be the
argument evaluation structure sn,audience,standard, where standard is a function
mapping every proposition to the scintilla of evidence proof standard. The burden
of production for a proposition p has been met if and only if p is acceptable in
AES.
Even though the weakest proof standard, scintillia of the evidence, is used to
test whether the burden of production has been met, an arbitrary, or silly, argument
would not be sufficient, since only applicable arguments are taken into consider-
ation by all proof standards. A silly argument can be defeated by questioning or
attacking its premises. Arguments can be undercut in our system by first revealing,
if necessary, an implicit premise about the applicability of the warrant underlying
the argument to this case and then attacking this premise [18].
Since arguments put forward to met the burden of production can be defeated
by further arguments, the burden of production may be met at some stage, si, of a
dialogue, but not met at some later stage, sj, where j i. If the burden of production
is not met at the end of the argumentation phase, the audience in the closing phase
may be required, depending on the dialogue type, to assume that the proposition is
false. In this case, the burden of persuasion for this proposition becomes irrelevant.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-262-320.jpg)
![250 Thomas F. Gordon and Douglas Walton
respondent is the defendant and the audience is the judge or jury. In such cases it
will be necessary to make assumptions about the audience.
Definition 12.15 (tactical burden of proof). Let s1,...,sn be the stages of the ar-
gumentation phase of a dialogue. Assume audience is the audience which will as-
sess the burden of persuasion in the closing phase. Assume standard is the function
which will be used in the closing phase to assign a proof standard to each propo-
sition. For each stage si in s1,...,sn, let AESi be the argument evaluation structure
si,audience,standard. The tactical burden of proof for a proposition p is met at
stage si if and only if p is acceptable in AESi.
Both sides in a dialogue can have a tactical burden. Intuitively, a party has a
tactical burden of proof for a proposition p at some stage si only if p is not acceptable
in si and the party has an interest in proving p, either because proving p is relevant
for proving some claim of the party or disproving some claim of the other party,
given the arguments which have been put forward. A fuller, more complete account
of the tactical burden of proof would require the parties, their claims and the concept
of relevance to be modeled.
This completes our formal model of proof standards and burden of proof. Again,
we do not claim this is a comprehensive dialogical model of argumentation. Many
important elements of argumentation have been abstracted out for the sake of
brevity, such as the parties, argumentation schemes and their critical questions, di-
alogue types, argumentation protocols and commitment stores. Our aim here was
to define the simplest, most abstract possible model which is sufficient for distin-
guishing the various kinds of proof standards and burdens of proof which have been
discussed in the computational models of argument literature. Of course, we cannot
prove that we have achieved this goal. We leave it up to others in the field to try to
develop a simpler model with this scope.
In the introduction we formulated our view of argumentation as a kind of pro-
cess for making justified decisions, where the input to the process is an initial claim
or issue and the output consists of a set of claims, the decision to accept or reject
each claim, a theory and a proof. Unlike assumption-based instantiations of Dung’s
model of abstract argumentation frameworks [9], in which a theory or knowledge
base is presumed as part of the input, in our model a theory and a proof are con-
structed together during argumentation and are part of the output. The theory con-
structed is the deductive closure, in classical propositional logic, of the set of all
propositions which have been assumed by the audience or, if neither the proposi-
tion or its complement has been assumed, are acceptable in the final stage of the
closing phase of the dialogue. The proof constructed is represented by the argument
evaluation structure of the final stage.
3 Survey of Prior Research
Early work in the field of Artificial Intelligence and Law recognized the utility of
defeasible rules, subject to exceptions, as a tool for allocating the burden of proof](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-264-320.jpg)
![12 Proof Burdens and Standards 251
and developed nonmonotonic logics for reasoning with such rules [13, 35]. But
different proof standards or kinds of proof burdens were not yet distinguished in
these models.
In the Pleadings Game [14], inspired by legal proceedings, argumentation was
modeled as dialogical process consisting of several phases, in which theories and
arguments are constructed dynamically by the parties during the process. The Plead-
ings Game modeled the burdens of claiming, questioning and production, but it did
not explicitly use this terminology. Proof standards and the burden of persuasion,
were not modeled, as they do not play a role in the opening phase.
To our knowledge, the first effort to develop a computational model of proof
standards was by Freeman and Farley in 1996 [12]. They modeled argumentation
as a dialectical process during which an acyclic argument graph is constructed by
putting forward pro and con arguments constructed (‘invented’) from a propositional
rule-base. The model of proof standards comes into play when evaluating the argu-
ments in the graph. An argument is defendable if each premise satisfies its proof
standard. Five proof standards were defined (scintilla of evidence, preponderance,
dialectical validity, beyond a reasonable doubt and beyond a doubt). The relative
weights of arguments were not assigned by an audience, but rather computed from
certainty factors assigned to premises and the kind of argumentation scheme used to
construct the argument, using the weakest premise principle [32]. The model of ar-
gumentation developed in this paper has borrowed much from Freeman and Farley,
but there are some important differences. By restricting the arguments which can be
put forward to those which can be constructed from a static rule-base and model of
the facts, provided as input to the dialectical process, Freeman and Farley’s model
is more of a relational model of argumentation than a theory construction model.
The relative strength of arguments is determined by the rules and facts, rather than
by the intersubjective judgment of an audience. Only the burden of persuasion has
been modeled by this work. Finally, no attempt was made to model proof burdens
other than the burden of persuasion in this work.
The Zeno system [17] was inspired by Freeman and Farley’s work. Zeno’s model
of the structure of argument graphs was based on Kunz and Rittel’s [22] concept of
issue-based information systems (IBIS). Zeno supported arguments about both fac-
tual issues and issues about which action to take to solve some problem or achieve
some goal (practical reasoning). For factual issues, three proof standards were mod-
eled (scintilla of evidence, preponderance of the evidence and beyond reasonable
doubt). For issues about actions, two further proof standards were provided (no bet-
ter alternative and best choice). The relative strength of arguments in Zeno was
computed from qualitative constraints (equations and inequalities) over proposi-
tions. The qualitative constraints were issues which could be argued about. Only
the burden of persuasion was modeled in Zeno. As in Freeman and Farley’s work,
Zeno did not explicitly model an audience. On the contrary, in Zeno it was assumed
the parties would argue about the evaluation of their own arguments, by putting
forward and arguing about constraints.
Prakken formulated the three principles of argument accrual [24] explained in the
introduction, which our model has been designed to satisfy. He compared automatic](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-265-320.jpg)
![252 Thomas F. Gordon and Douglas Walton
and manual approaches to accrual. In the manual approach, arguments are accrued
by changing their representation in the model. In the automatic approach, argu-
ments are accrued by the argument evaluation process, without having to modify
the representation of the arguments. Prakken illustrated this approach with a novel
system, using a rule-based instantiation of Dung’s model of an abstract argumenta-
tion framework, in which the conclusions of rules are labeled with the set of their
premises. The attack relation of the argumentation framework is defined so that an
argument A does not attack an argument B if the set of premises of the conclu-
sion of the argument A is a subset of the (accrued) set of premises of the argument
B. This leads to a bottom-up inference process which, Prakken notes, is similar to
Reason-Based Logic [20]: first all arguments pro and con some leaf proposition
are combined into two competing sets of accruals; next the conflict between these
accruals is resolved; and finally the process iterates moving up the tree, with only
the winning defeasible conclusion being used. The way arguments are evaluated in
our model has much in common with the bottom-up inference approach taken by
both Reason-Based Logic and Prakken’s system. Nonetheless, our approach to the
accrual problem is manual, not automatic, as responsibility for accruing arguments
is allocated to the parties who put them forward in dialogues. But unlike Bayesian
Networks, which require a complete conditional probabilities table to be provided
as input to the process, our approach does not require all possible arguments to be
formulated. Accruing arguments when possible to strengthen one’s case is part of
the burden of proof allocated to the parties. Prakken points out that there is a trade-
off between automatic and manual approaches to accrual: in the former unwanted
inferences must be expressly blocked while in the latter accruals must be expressly
formulated. So neither approach is clearly superior.
In 2006, Prakken and Sartor published the first formal account of the distinctions
between the burden of production, the burden of persuasion and the tactical burden
of proof [28]. Their model is based on an interpretation of presumptions as default
rules, formalized using an extended version of their Inference System (IS) defeasi-
ble logic [27], called the Litigation Inference System (LIS) [23], which includes as
part of the input, together with the defeasible rules, an assignment of the burden of
persuasion for literals to either the plaintiff or defendant in the proceeding. Prakken
and Sartor’s model of the distinctions between these three proof burdens is more
concrete than our model, as it is limited to arguments constructed from defeasible
rules. Our model abstracts away the process of constructing arguments and can be
instantiated with models of various argumentation schemes, for example for argu-
ments from cases as well as defeasible rules.1 Prakken and Sartor did not attempt
to model the burdens of claiming or questioning or support the use of various proof
standards.
Continuing work began with Reed and Walton in 2005 on dialogues about the
burden of proof [26], Prakken and Sartor in 2007 presented a model of arguments
about the allocation of the burden of persuasion [29]. They note that ‘the argumen-
1 We have done some research recently on constructing arguments automatically from formal mod-
els of ontologies, rules, and cases, in a way which is well integrated with the model of argument
presented here [16].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-266-320.jpg)
![12 Proof Burdens and Standards 253
tation games we define in this paper are not intended as a model of actual legal
dialogue but as a proof theory for a nonmonotonic logic. . . . All we claim is that
our games draw the correct defeasible inferences from a given body of information
and an associated allocation of the burden of persuasion. It remains to be seen how
the present logical model can be integrated with dialogical and procedural mod-
els of legal argument”. Clearly, their work is a relational model of argumentation,
not a theory construction model. In their conclusion, Prakken and Sartor express
concern that their system “lacks an extension-based semantics in the style of [10]”
and note, citing [23], that this “raises questions about the adequacy of ‘mainstream’
nonmonotonic logics for representing legal reasoning”.2
The idea of using audiences in our model was inspired by work by Bench-Capon,
Doutre and Dunne [5]. Bench-Capon’s focus is on modeling practical reasoning, i.e.
the process of making decisions about which action to take in order to achieve goals
which promote an agent’s values. Since different agents can have different values,
as well as different priority orderings on their set of values, arguments about action
can only be evaluated against the values of a particular agent, or ‘audience’. Bench-
Capon contrasted practical reasoning with argumentation about “what is true in a sit-
uation”, i.e. the facts of a case, and appears to consider audiences to be relevant only
for the former. As illustrated by legal trials, however, audiences can be important in
argumentation dialogues about factual issues as well, since different persons will
judge the probative weight of evidence, such as witness testimony, differently. Our
model of an audience is more abstract, since it orders arguments by their strength,
regardless of the kind of argumentation scheme which has been applied to construct
the argument. Formally, Bench-Capon’s model is an extension of Dung’s concept
of an abstract argumentation framework. Although a formal dialogue game is de-
fined, the game serves as a proof theory for a relational model of argument. The
input to the system is a Value-Based Argumentation Framework (VAF) consisting
of a set of arguments, an attack relation over these arguments, a set of values, and
a mapping from arguments to values. An audience is defined to be a binary rela-
tion over these values, modeling the preference ordering over these values of the
audience. The output of the dialogue game is the set of arguments which are objec-
tively or subjectively acceptable. Arguments which are objectively acceptable are,
or should be, acceptable by all (rational) audiences. Conversely, arguments which
are subjectively acceptable are acceptable to a particular, given audience. Bench-
Capon compares the distinction between objective and subjective acceptability with
the concept of credulous and skeptical inference familiar from nonmonotonic logic
and Dung argumentation frameworks. The distinction between objective and sub-
jective acceptability, like the distinction between credulous and skeptical inference,
possibly can be viewed as a simple two-level model of proof standards, but it is
questionable that this distinction is of much use outside of the procedural context of
dialogues in which the allocation of the burden of proof matters, and such dialogues
were outside the scope of this work.
2 But see [36] for a defense of nonmonotonic logic for legal reasoning, even when the burden of
persuasion is divided among the parties.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-267-320.jpg)
![254 Thomas F. Gordon and Douglas Walton
Atkinson and Bench-Capon recently modeled a variety of proof standards us-
ing Dung-style argumentation frameworks [2]. This paper also recognizes the need
for audiences in dialogues about factual issues, as well as teleological issues about
values promoted by alternative courses of action. The basic idea of this paper elab-
orates on the idea discussed above, of using the distinction between various kinds
of acceptability in Dung argumentation frameworks to define proof standards. The
scintilla of evidence standard is modeled as membership in some preferred exten-
sion (credulous acceptability). Preponderance of the evidence corresponds to mem-
bership in all preferred extensions (skeptical acceptability). And, finally, beyond
reasonable doubt corresponds to membership in the grounded extension, if there
is one. Ways of modeling proof standards between preponderance of the evidence
and beyond reasonable doubt, such as the clear and convincing evidence standard,
are proposed which make use of an assignment of ‘probabilities’, i.e. weights, to
arguments. However this idea is discarded with the argument that this information
is not usually available. In our model this information is provided by the intersub-
jective judgment of the audience. Later in their paper, Atkinson and Bench-Capon
suggest the use of audiences to derive preferences over arguments, starting with
Bench-Capon’s value-based argumentation frameworks for arguing about teleolog-
ical issues. This idea is extended to arguments about factual issues by ordering evi-
dence, for example by ordering witness testimony using information about witness
credibility.
An advantage of Bench-Capon and Atkinson’s line of research is that it elabo-
rates rationality constraints on an audience’s assignment of strengths to arguments,
using for example its value preferences or its assessment of the relative credibil-
ity of witnesses. Although our model leaves the weights assigned by audiences to
arguments completely unconstrained, it has the advantage of being more general,
applying to arguments from any argumentation scheme. Perhaps it is possible to
combine the benefits of these two approaches. This may be an interesting topic for
future research.
A disadvantage of modeling proof standards using Dung argumentation frame-
works is the computational complexity of testing whether the proof standard has
been met. We have argued that, intuitively, to satisfy a burden of proof, the party
with the burden must present the proof in a form which is easy to check. The other
party shouldn’t have to solve an undecidable or intractable problem to check the
proof. As discussed previously, in Section 2, this criterion could be satisfied for the
proposed scintilla of evidence proof standard, modeled as credulous acceptability,
by requiring the proponent to produce an admissible set of arguments. The admissi-
ble set could serve as a representation of the proof, since checking whether the set
is admissible and whether an argument is a member of this set are both tractable
problems. But it is not clear what structures could serve as proofs for the models
proposed by Atkinson and Bench-Capon for the stricter proof standards. A related
issue is whether or not such proofs could be presented in a form which is compre-
hensible to human users, using for example some kind of argument mapping or other
visualization technique. Existing argument mapping methods have been developed
for presenting proofs (argument graphs) of the kind we have presented in this paper](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-268-320.jpg)
![12 Proof Burdens and Standards 255
[3, 11, 1, 15]. While Dung argumentation frameworks are often displayed as graphs,
these graphs do not represent proofs of the acceptability of any of the arguments in
the graph.
Prakken recently published a formal model which explicates the role of judges
in deciding issues regarding the admissibility of evidence and the allocation of the
burdens of production and persuasion [25]. Interestingly, dialogues are divided into
two, rather than tree phases, in the model, called the pleadings phase and the de-
cision phase. The pleadings phases encompasses the opening and argumentation
phases of our model. The decision phase corresponds to our closing phase. The
judge plays a role in the decision phase comparable to the audience in the closing
phase of our model, but arguments are evaluated by having the judge put forward
further arguments, which are then evaluated using Prakken’s LIS nonmonotonic
logic [23], discussed above. Arguments are not weighed and proof standards are
not part of the model.
Prakken and Sartor’s chapter on a “Logical Analysis of Burdens of Proof” in
[30] highly influenced our model of the distinctions among the various kinds of
proof burdens. The main difference between our systems is that they use Dung’s
abstract argumentation framework to evaluate the arguments at each stage of the
dialogue. We have already expressed our reservations about this approach, which
does not take into consideration that a burden of proof entails an obligation to put
forward the proof in a form which can be tractably checked. In their conclusion they
point out that their use of a nonmonotonic logic for evaluating arguments in a stage
could be replaced by any formalism “which accepts as input a description of an ev-
idential problem and produces as output a fallible assessment whether a claim has
been proven”. This is one way looking at we have done here, by replacing their non-
monotonic logic with a structure designed to make argument evaluation tractable for
a variety of common proof standards. Proof standards are not given much attention
in their chapter, except to suggest that they could be handled using some mecha-
nism for ordering arguments by their strength. Finally, although they recognize the
role of the finder-of-fact, they did not explictly model audiences or their impact on
assessing proof burdens. For example, the evalution of the tactical burden of proof
in their model does not require a party to estimate an audience’s assessment of the
weight of arguments. Rather, this information appears to be assumed as input to the
dialogue, available as common knowledge to both parties.
The model of argumentation developed in this paper is most closely related to
the Carneades system, which consists of both a mathematical model of argumen-
tation and software tools for supporting argumentation tasks based on this model.3
Carneades is work in progress and thus prior publications about the system differ in
their details. For example, the version of the model in [19] gave the term ‘presump-
tion’ a technical meaning which is confusing for those familar with the concept of a
presumption in the legal domain. This was pointed out by Prakken and Sartor [28]
and corrected when Prakken joined us for the next version of Carneades [18]. Simi-
larly, the use of weights to order arguments in [19] was replaced by a partial order in
3 http://carneades.berlios.de](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-269-320.jpg)
![256 Thomas F. Gordon and Douglas Walton
[18], but now we have come full circle, by again using numeric weights. The initial
models of the preponderance of the evidence and beyond reasonable doubt proof
standards of [19] were removed from the [18] version, since we had our doubts
about their adequacy as models of these legal standards. We feel confident enough
about the models of these legal standards in this paper to want to publish them
in order to obtain critical feedback. Aside from improved models of various proof
standards, the main contribution of the new version presented here is that it more
clearly and explicitly models argumentation as a theory and argument construction
process, consisting of a sequence of stages divided into opening, argumentation and
closing phases. Although we had already suggested in [18] how to use Carneades to
model the distinction between the burden of production and the burden of persua-
sion, the version in this article models these distinctions more explicitly and extends
the model to cover the burdens of claiming and questioning in the opening phase
and the tactical burden of proof. This version also introduces an explicit model of
audiences. Previous versions had used the concept of an argumentation context to
model argument strengths or priorities, with no reference to an audience.
4 Concluding Remarks
Viewing argumentation as dialogical process for making justified decisions raises
a number of issues which have no place in relational, monological accounts of ar-
gumentation, proof burdens and standards among them. Thus it should come as no
surprise that the concept of proof has thus far received little attention in mainstream
accounts of argumentation in artificial intelligence. An argument may be acceptable
in a Dung-style argumentation framework, or a proposition may be warranted by a
default theory in some nonmonotonic logic, but what mathematical structures are
adequate as models of proofs of these or other inference relations? We have argued
that checking a proof should be an easy, tractable problem. Argumentation frame-
works and default theories do not, in general, meet this requirement. And other,
less computational requirements could also be formulated, such as transparency and
comprehensibility for the intended class of audiences.
Another distinction between relational and dialogical conceptions of argumenta-
tion concerns their input/out relations. Whereas in relational accounts an argumen-
tation framework or default theory is provided as input and the task is to derive ac-
ceptable arguments or warranted propositions, argumentation dialogues begin with
a claim or issue and construct, as part of their output, theories and proofs. Argumen-
tation dialogues includes synthetic as well as analytic tasks.
Abstract argumentation frameworks, which focus on attack relations among ar-
guments, are not well-suited to modeling proof standards, at least not the familiar
proof standards from the legal domain. The intuitive idea of legal proof standards
since Roman times, symbolized by the scales of the goddess Justitia, involves the
weighing of arguments or evidence pro and con some claim. This simple idea can-
not be directly represented using abstract argumentation frameworks. Attempts to](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-270-320.jpg)
![Chapter 13
Dialogue Games for Agent Argumentation
Peter McBurney and Simon Parsons
1 Introduction
The rise of the Internet and the growth of distributed computing have led to a ma-
jor paradigm shift in software engineering and computer science. Until recently,
the notion of computation has been variously construed as numerical calculation,
as information processing, or as intelligent symbol analysis, but increasingly, it is
now viewed as distributed cognition and interaction between intelligent entities [60].
This new view has major implications for the conceptualization, design, engineer-
ing and control of software systems, most profoundly expressed in the concept of
systems of intelligent software agents, or multi-agent systems [99]. Agents are soft-
ware entities with control over their own execution; the design of such agents, and
of multi-agent systems of them, presents major research and software engineering
challenges to computer scientists.
One key challenge is the design of means of communication between intelli-
gent agents. Considerable research effort has been expended on the design of ar-
tificial languages for agent communications, such as DARPA’s Knowledge Query
and Manipulation Language (KQML) [33] and the Foundation for Intelligent Phys-
ical Agents’ (now IEEE FIPA) Agent Communications Language (FIPA ACL) [35].
These languages, and languages like them, have been designed to be widely ap-
plicable. As well as being a strength, this feature can also be a weakness: agents
participating in conversations have too many choices of what to utter at each turn,
and thus agent dialogues may endure a state-space explosion.
Allowing sufficient flexibility of expression while avoiding state-space explosion
had led agent communications researchers to the study of formal dialogue games;
these are rule-governed interactions between two or more players (or agents), where
Department of Computer Science, University of Liverpool, UK
e-mail: mcburney@liverpool.ac.uk
Department of Computer and Information Science, Brooklyn College, New York, USA
e-mail: parsons@sci.brooklyn.cuny.edu
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 261
DOI 10.1007/978-0-387-98197-0 13, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-275-320.jpg)
![262 McBurney and Parsons
each player “moves” by making utterances, according to a defined set of rules. Al-
though their study dates to at least the time of Aristotle [5], dialogue games have
found recent application in philosophy, computational linguistics and Artificial In-
telligence (AI). In philosophy, dialogue games have been used to study fallacious
reasoning [41, 62] and to develop a game-theoretic semantics for various logics, e.g.,
intuitionistic and for classical logics [59]. In linguistics, they have been used to ex-
plain sequences of human utterances [57], with subsequent application to machine-
based natural language processing and generation [49], and to human-computer in-
teraction [9]. Within computer science and AI, they have been applied to modeling
complex human reasoning, for example in legal domains [81], and to requirements
specification for complex software systems [34]. Dialogue games differ from the
games of economic game theory in that payoffs for winning or losing a game are
not considered, and, indeed, the notions of winning and losing are not always ap-
plicable to dialogue games. They also differ from the abstract games recently used
as a semantics for interactive computation [1], since these latter games do not share
the rich rule structure of dialogue games, nor are these latter intended to themselves
have a semantic interpretation involving the co-ordination of actions among a group
of agents.
This chapter considers the application of formal dialogue games for agent com-
munication and interaction using argumentation. We begin, in the next subsection,
with a brief overview of an influential typology of human dialogues, which have
proven useful in classifying agent interactions. Because the design of artificial lan-
guages for communication between software agents shares much with the study of
natural human languages, we structure this chapter according to the standard divi-
sion within linguistic theory between syntax, semantics and pragmatics; we do this
despite this division being imprecise and contested within linguistics (e.g., [58]).
Very broadly (following [58]), we may view: syntax as being concerned with the sur-
face form and combinatorial properties of utterances, words and their components;
semantics as being concerned with the truth or falsity of utterances; and pragmatics
as being concerned with those aspects of the meaning of utterances other than their
truth or falsity.1 Section 2 thus presents a model of a formal dialogue game protocol,
focusing primarily on the syntax of such dialogues. We follow this in Section 3 with
a discussion of the semantics and the pragmatics of agent dialogues. Section 4 then
presents an illustrative example, taken from [68], while Section 5 considers protocol
design and assessment. The chapter ends with a brief conclusion in Section 6.
1.1 Types of dialogues
An influential model of human dialogues is the typology of primary dialogue types
of argumentation theorists Douglas Walton and Erik Krabbe [96]. This categoriza-
tion is based upon the information the participants have at the commencement of a
1 Note that the word semantics is used differently here than in the study of argumentation frame-
works, as in Chapter 2 of this volume.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-276-320.jpg)
![13 Dialogue Games 263
dialogue (of relevance to the topic of discussion), their individual goals for the di-
alogue, and the goals they share. Information-Seeking Dialogues are those where
one participant seeks the answer to some question(s) from another participant, who
is believed by the first to know the answer(s). In Inquiry Dialogues the participants
collaborate to answer some question or questions whose answers are not known
to any one participant. Persuasion Dialogues involve one participant seeking to
persuade another to accept a proposition he or she does not currently endorse. In
Negotiation Dialogues, the participants bargain over the division of some scarce
resource. If a negotiation dialogue terminates with an agreement, then the resource
has been divided in a manner acceptable to all participants. Participants of Delib-
eration Dialogues collaborate to decide what action or course of action should be
adopted in some situation. Here, participants share a responsibility to decide the
course of action, or, at least, they share a willingness to discuss whether they have
such a shared responsibility. Participants may have only partial or conflicting in-
formation, and conflicting preferences. As with negotiation dialogues, if a deliber-
ation dialogue terminates with an agreement, then the participants have decided on
a mutually-acceptable course of action. In Eristic Dialogues, participants quarrel
verbally as a substitute for physical fighting, aiming to vent perceived grievances.
Several comments are important to make here. The first is that although Walton
and Krabbe talk about the goal of a dialogue and the goal of a dialogue type,2 only
participants can have goals since only they are sentient. Participants may believe
that a dialogue interaction they enter has an ostensible purpose, but their own goals
or the goals of the other participants may not be consistent with this purpose. For ex-
ample, participants may enter a negotiation dialogue in order to reach an agreement
(a deal) over the allocation of some resource; or they may enter it to prevent any
such agreement being reached, or to delay agreement [24], or to prove that no such
agreement is possible, or to gather information from the other participants, or even
to signal something to some third party, not in the dialogue. Participants in dialogues
may also seek to hide their true goals from the other participants [25, 64]. Instead
of dialogue goals it makes sense only to speak of participant goals and dialogue
outcomes [74].
Secondly, most actual dialogue occurrences — both human and agent — involve
mixtures of these dialogue types. A purchase transaction, for example, may com-
mence with a request from a potential buyer for information from a seller, proceed
to a persuasion dialogue, where the seller seeks to persuade the potential buyer of
the importance of some feature of the product, and then transition to a negotia-
tion, where each party offers to give up something he or she desires in return for
something else. The two parties may or may not be aware of the different nature of
their discussions at each phase, or of the transitions between phases. Instances of
individual dialogue types contained entirely within other dialogue types are said to
be embedded [96]. Several formalisms have been suggested for computational rep-
resentation of combinations of dialogue: first, the Dialogue Frames of Reed [84],
which enable iterated, sequential and embedded dialogues to be represented; sec-
2 as do others, e.g., [80].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-277-320.jpg)
![264 McBurney and Parsons
ond, the Agent Dialogue Frameworks of McBurney and Parsons, based on PDL
[66], which permit iterated, sequential, parallel and embedded dialogues to be rep-
resented; and third, the more abstract RASA frameworks of Miller and McBurney
[73], which permit iterated, sequential, parallel and embedded combinations of any
types of agent interaction protocols. All these formalisms are neutral with regard to
the modeling of the primary dialogue types themselves, allowing the primary types
to be represented in any convenient form, and allowing for types other than the six
of the Walton and Krabbe typology to be included. Walton and Krabbe do not claim
their typology is comprehensive, and some recent research has explored other types
and sub-types, e.g., [14].
Researchers in multi-agent systems and in argumentation have articulated dia-
logue game protocols for many of the types in the Walton and Krabbe typology.
For example, the two-party protocol of Amgoud, Maudet and Parsons [3], which is
based on MacKenzie’s philosophical dialogue game DC [62], supports persuasion,
inquiry and information-seeking dialogues; a subsequent extension of this protocol
with additional locutions supports negotiation dialogues [4]. Information-seeking
dialogues have been considered by Hulstijn [49], and analyzed by Cogan, Parsons
and McBurney [14]; indeed this latter work, which examines the pre- and post-
conditions of dialogues over beliefs in fine detail, identifies several new types of
dialogues not explicitly included in the Walton and Krabbe typology. A study of dif-
ferent persuasion protocols can be found in the review paper by Prakken [80]; other
protocols for persuasion dialogues include the PADUA protocol for arguments from
experience by Wardeh, Bench-Capon and Coenen [97] and a protocol for arguments
over access to information by Doutre and colleagues [22, 23, 78].
Protocols for multi-agent inquiry dialogues have been proposed and studied by
McBurney and Parsons [65], who consider the circumstances under which an in-
quiry dialogue may converge to the truth, and by Black and Hunter [11], whose
agent reasoning architecture enables generative inquiry dialogues, i.e., those where
new proposals may emerge for consideration and possible endorsement by the
agents participating. For dialogues over beliefs (information-seeking, inquiry and
persuasion dialogues), Parsons and Sklar consider the question of convergence of
beliefs of agents engaged in repeated dialogues with one another [77]. In addition
to [4] cited above, protocols for negotiation dialogues include those of Sadri, Toni
and Torroni [87], McBurney, van Eijk, Parsons and Amgoud [63], and Karunatil-
lake [54]. Regarding dialogues over action which are not negotiations: McBurney,
Hitchcock and Parsons [64] and Tang and Parsons [92] have presented protocols for
deliberation dialogues; Atkinson, Bench-Capon and McBurney have given a repre-
sentation for proposals for actions and a dialogue game protocol to discuss these
proposals [6]; and Atkinson, Girle, McBurney and Parsons have presented a dia-
logue game protocol for dialogues over commands [7]. Finally, the dialogue-game
protocols presented in the work of Dignum, Dunin-Kȩplicz and Verbrugge [20, 21]
are intended to enable agents to form teams and to agree joint intentions, respec-
tively.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-278-320.jpg)
![13 Dialogue Games 265
2 Syntax
The syntax of a language concerns the surface form of words and phrases, and how
these may be combined. Accordingly, defining the syntax of an agent dialogue game
protocol usually involves the specification of the possible utterances which agents
can make (the locutions) and the rules which govern the order in which utterances
can be made. Since the work of Hamblin [41], it has become standard to talk of
speakers in a dialogue incurring commitments: a speaker who asserts a statement as
being true, for example, may be committed to justifying this assertion when chal-
lenged by another participant, or else allowed (or even forced) to retract the asser-
tion. Although such dialogical commitments may be viewed as aspects of the se-
mantics (the meaning) of utterances, the rules regarding commitments are typically
included in the specification of dialogue syntax because these rules often influence
the order of utterances. The various commitments of the participants are usually
tracked in a publicly-readable database, called a commitment store.
Within the agents communications community, it has become standard to view
utterances as composed of two layers: an inner layer comprising the topics of dis-
cussion, and an outer (or wrapper) layer, comprising the locutions. An utterance can
thus be seen as an instantiated locution, with one variable of instantiation being the
topic. This structure, adopted for both KQML and FIPA ACL, provides great flexi-
bility, since agents encoded appropriately may use the same wrappers to undertake
dialogues over different topics.
We now present a generic framework for specification of a dialogue game pro-
tocol in terms of its key components, adapted from [66].3 We first assume that the
topics of discussion between the agents (the inner layer) can be represented in some
logical language, whose well-formed formulae are denoted by the lower-case Ro-
man letters, p, q, r, etc. A dialogue game specification then comprises the following
elements, each of which concern the wrapper layer of communications:
Commencement Rules: Rules which define the circumstances under which the
dialogue commences.
Locutions: Rules which indicate what utterances are permitted. Typically, legal
locutions permit participants to assert propositions, permit others to question or
contest prior assertions, and permit those asserting propositions which are sub-
sequently questioned or contested to justify their assertions. Justifications may
involve the presentation of a proof of the proposition or an argument for it. The
dialogue game rules may also permit participants to utter propositions to which
they assign differing degrees of commitment, for example: one may merely pro-
pose a proposition, a speech act which entails less commitment than would an
assertion of the same proposition.
Rules for Combination of Locutions: Rules which define the dialogical contexts
under which particular locutions are permitted or not, or obligatory or not. For
3 We are also informed by [80]; note, however, that work defines a mathematical model for analyz-
ing multi-party dialogues, rather than defining a framework for specification of dialogue protocols
for agent communications.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-279-320.jpg)
![266 McBurney and Parsons
instance, it may not be permitted for a participant to assert a proposition p and
subsequently the proposition ¬p in the same dialogue, without in the interim
having retracted the former assertion.
Commitments: Rules which define the circumstances under which participants
incur dialogical commitments by their utterances, and thus alter the contents of
the participants’ associated commitment stores. For example, a question posed
by one agent to another may impose a commitment on the second to provide a
response; until provided, this commitment remains undischarged.
Rules for Combination of Commitments: Rules which define how commitments
are combined or manipulated when utterances incurring conflicting or comple-
mentary commitments are made. For example, the rules may allow a speaker to
assert the truth of a proposition and then to assert its negation, with the commit-
ment store holding only the most recent asserted proposition, or the store may
hold the earlier proposition until explicitly retracted. These rules become partic-
ularly important when multiple dialogues are involved, as when one dialogue is
embedded within another; in such a case, the commitments incurred in the inner
dialogue may take priority over those of the outer dialogue, or vice versa [66].
Rules for Speaker Order: Rules which define the order in which speakers may
make utterances. It may be that any speaker may speak at any time, as in FIPA
ACL, or that there are rules regarding turn-taking.
Termination Rules: Rules that define the circumstances under which the dialogue
ends.
It is worth noting here that more than one notion of commitment is present in the
literature on dialogue games. For example, Hamblin treats commitments in a purely
dialogical sense: “A speaker who is obliged to maintain consistency needs to keep a
store of statements representing his previous commitments, and require of each new
statement he makes that it may be added without inconsistency to this store. The
store represents a kind of persona of beliefs; it need not correspond with his real
beliefs . . .” [41, p. 257]. In contrast, Walton and Krabbe [96, Chapter 1] treat com-
mitments as obligations to (execute, incur or maintain) a course of action, which
they term action commitments. These actions may be utterances in a dialogue, as
when a speaker is forced to defend a proposition he has asserted against attack from
others; so Walton and Krabbe also consider propositional commitment as a special
case of action commitment [96, p. 23]. As with Hamblin’s treatment, such dialogi-
cal commitments to propositions may not necessarily represent a participant’s true
beliefs. In contrast, Singh’s social semantics [90], requires participants in an interac-
tion to express publicly their beliefs and intentions, and these expressions are called
social commitments. These include both expressions of belief in some propositions
and expressions of intent to execute or incur some future actions.4 Our primary mo-
tivation is the use of dialogue games as the basis for interaction protocols between
autonomous agents. Because such agents will typically enter into these interactions
4 It is worth noting that all these notions of commitment differ from that commonly used in discus-
sion of agent’s internal states, namely the idea of the persistence of a belief or an intention [99, p.
205].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-280-320.jpg)
![268 McBurney and Parsons
Different types of semantics may serve these various purposes to varying degrees,
and so it may be useful to develop more than one semantics for a communications
language or protocol. 5 In addition, an articulation of semantics could be undertaken
at one or more different levels: for each individual utterance, or speech act; for
specified short sequences of utterances,6 such as a question-and-answer sequence;
for complete sequences of utterances, or dialogues; and for dialogue protocols. Most
current published work on agent dialogue protocols presents a semantics defined in
terms of individual utterances. In the terms defined below, these semantics are most
often axiomatic or operational, and are much less often denotational.
3.2 Types of Semantics
We have inherited two conflicting notions of semantics, one deriving from linguis-
tics and the other from mathematical logic. As linguists normally understand these
terms, the syntax of a language is “the formal relation of signs to one another”
and the semantics of the language “the relations of signs to the objects to which
the signs are applicable” (Morris [75], cited in [58, p. 1]). Thus, it makes sense
to speak of the truth of a sign (or of an utterance in a language using such signs),
since this indicates that the sign has a relationship to external objects in the world.
Within mathematics and mathematical logic, a different understanding of semantics
has arisen, beginning with Pieri [79] and Hilbert [43] and first articulated formally
by Tarski [94]. In this tradition, a semantics for a formal language is a relationship
between that language and a space M of mathematical structures, called models.
A statement S in the language specifies a subset M(S) of M. Such a statement is
said to be true in a particular model M0 if M0 ∈ M(S). A statement is said to be
logically true if it is true in every model, i.e., if M(S) = M.7 These two notions
of semantics — one linguistic, one mathematical — collide.8 In particular, in the
mathematical framework, benefit may be gained from defining different semantic
mathematical structures for the one language; Tarski himself, for instance, defined
topological [93] and discrete lattice [71] semantics for propositional logic. The ben-
efits of this are that different semantic frameworks may enable different properties
of the language to be studied and may provide different insights. Insight may also
be gained by comparing the structures with each other, a subject known as model
theory or metamathematics [48]. But, defining and comparing alternative structures
5 Traditional mathematical communications theory, due to Shannon and Weaver [89], explicitly
ignores the semantics of messages, and so provides little guidance to designers, developers or
users of agent communications languages and protocols.
6 These are known as conversations in the agent communications literature, e.g., [39].
7 Note that Tarski only applied his framework to formal, or mathematical, languages, and was
skeptical about its applicability to natural language [94, pp. 163–165].
8 Their first skirmish was the argument between Hilbert and Frege over the meaning of Hilbert’s ax-
ioms for geometry: Frege took what we are calling a linguistic approach, Hilbert a model-theoretic
approach; see [95, pp. 408–412] and [46, pp. 7–10].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-282-320.jpg)
![13 Dialogue Games 269
in this way makes no sense in the linguistic understanding of semantics: how could
a language admit more than one set of relationships to the truth?
Agent communications languages and dialogue protocols straddle this divide.
Because they are formal languages, insight into their properties can be gained by
defining semantic relationships to mathematical structures, and studying these struc-
tures. However, because they are also intended as media for communication, just as
natural language is, each agent using a particular communications language or pro-
tocol will wish to ensure that all users share a common understanding of utterances.9
To verify that agents have the same understanding — the same semantics — for a
communications language ultimately requires some form of inspection of their in-
ternal states or, equivalently, their program code. This is a challenging, and perhaps
conceptually impossible, undertaking since a sufficiently-clever agent can always
simulate insincerely any required internal state.10 Rather, in this chapter, our use
of semantic frameworks differs from that in linguistics: first, as in model theory,
semantic structures are a means to understand the properties of a formal agent com-
munications language, and, second, because our focus is on computer systems, these
structures are a means to support the engineering of multi-agent systems software
and to aid uniformity of implementation when software engineering is undertaken
by different development teams.
It is therefore helpful to consider several different types of semantic frameworks
for formal languages. In doing so, we draw on the summary of the literature on pro-
gramming language semantics presented by van Eijk in [29, Section 1.2.2]; however,
we make no claims that the typology is comprehensive. One type of semantics de-
fines each locution of a communications language in terms of the pre-conditions
which must exist before the locution can be uttered, and possibly also the post-
conditions which apply following its utterance, in a STRIPS-like fashion [32]. This
is called an axiomatic semantics [29, 72]. For agent communications languages and
dialogue protocols we distinguish between public and private axiomatic approaches.
In public axiomatic approaches, the pre-conditions and post-conditions all describe
states or conditions of the dialogue which can be observed by all participants. In
private axiomatic approaches, at least some of the pre- or post-conditions describe
states or conditions which are internal to one or more of the participants, and thus
not directly observable by the others. For example, the semantic language, SL, for
the locutions of the Agent Communications Language, FIPA ACL, of the Founda-
tion for Intelligent Physical Agents (FIPA), is a private axiomatic semantics of the
speech acts of the language, defined in terms of the beliefs, desires and intentions
of the participating agents [35]. For example, the inform locution in the FIPA ACL
language, allows one agent, say agent A, to tell another agent, say B, some propo-
sition p. The FIPA ACL semantics of inform only permits agent A to do this if [35,
p. 10]: (a) agent A believes p to be true, (b) agent A intends that agent B believes p
to be true, and (c) agent A believes that agent B does not already have a belief about
9 Friedrich Dürrenmatt’s novel, Die Panne, shows what tragic consequences may follow when
participants assign very different meanings to the same conversation [28].
10 For more on this, see [98].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-283-320.jpg)
![270 McBurney and Parsons
the truth of p.11 Similarly, the semantics defined for many dialogue game protocols
for agent interaction, e.g., [3], are also private axiomatic semantics. In contrast, the
semantics provided for dialogue games used for modeling legal reasoning in [10] is
a public axiomatic semantics.
A second type of semantics, an operational semantics, considers the dialogue
locutions as computational instructions which operate successively on the states
of some abstract machine. Under this approach, the participating agents and their
shared dialogue are viewed conceptually as parts of a large abstract or virtual com-
puter, whose overall state may be altered by the utterance of valid locutions or by
internal decision processes of the participants; it is as if these locutions or decisions
were commands in some computer programme language acting on the virtual ma-
chine.12 The utterances and agent decision-mechanisms are thus seen as state tran-
sition operators, and the operational semantics defines these transitions precisely
[29]. This approach to the semantics of agent communications languages makes ex-
plicit any link between the internal decision mechanisms of the participating agents
and their public utterances to one another. The semantics therefore enables the re-
lationships between the mental states of the participants and the public state of the
dialogue to be seen explicitly, and shows how these relationships change as a re-
sult of utterances and internal agent decisions. Thus, an operational semantics will
typically make assumptions about the internal decision-mechanisms of the agents
participating in the interaction; the actual agents engaged in a communicative in-
teraction may not necessarily use the decision-process or realize the mental states
assumed. Operational semantics have recently been defined for some agent commu-
nications languages, for example, in [30, 44] and for some dialogue protocols, e.g.,
information-passing interactions [19], negotiation dialogue protocols [54, 63], and
a general argumentation protocol [68].
Third, in denotational semantics, each element of the language syntax is as-
signed a relationship to an abstract mathematical entity, its denotation. The possi-
ble worlds, or Kripkean, semantics defined for modal logic syntax is an example of
such a semantics for a logical language [56]. However, two decades before Kripke’s
work, a denotational semantics mapping logical formulae to subsets of a topologi-
cal space was given for the modal logic system S4 [91]. For argumentation systems,
three denotational semantics have been provided for the ICRF’s Logic of Argu-
mentation LA [55]. In the first of these, Ambler [2] articulated a category-theoretic
semantics [61] for LA, by extending to arguments the Curry-Howard isomorphism,
which connects proofs in a deductive logic to the morphisms of a free cartesian
closed category. In this semantics, propositions (i.e., premises or claims) correspond
to objects in a particular enriched category, and arguments linking propositions to
morphisms between the associated objects. A second denotational semantics for LA,
due to Parsons [76], connects argumentation systems to qualitative probabilistic net-
works (QPNs). In this semantics, propositions correspond to nodes in a QPN, and
arguments linking propositions to edges between the associated nodes. Das [17]
11 Note that condition (a) enforces sincerity on the speaker, which is not necessarily desirable.
Also, condition (c) precludes the use of inform in authentication dialogues.
12 This virtual machine is purely a conceptual construct and does not need to exist in reality.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-284-320.jpg)
![13 Dialogue Games 271
articulated a third denotational semantics for logics of argumentation, based on a
Kripkean possible-worlds structure. In this semantics, different arguments are as-
sumed classified according to the degree of support they provide for propositions;
these differential degrees of support are translated into separate hyper-relations over
the accessibility relations of the Kripke structure.13
Perhaps the first example of a denotational semantics for a dialogue protocol was
the possible-worlds semantics for question-response interactions defined by Ham-
blin in 1956 [40]. Although possible-worlds and category-theoretic denotational se-
mantics have a long subsequent history in mathematical linguistics, only recently
have denotational semantics been defined for agent dialogue protocols. In [67],
McBurney and Parsons articulated a category-theoretic semantics, called a Trace
Semantics, for a broad class of deliberation dialogue protocols. In this semantics,
articulation of proposals for action by participants correspond to the creation of ob-
jects in certain categories, while participant preferences between these proposals
correspond to the existence of arrows (morphisms) between the corresponding ob-
jects. Thus, the semantics is constructed jointly and incrementally by the dialogue
participants as the dialogue proceeds, in a manner similar to the natural language se-
mantics of Discourse Representation Theory [53] (which uses possible worlds), or
the argumentation graph of Gordon’s Pleadings Game [38]. A similar denotational
semantics, constructed jointly and incrementally by the participants, is outlined by
Atkinson and colleagues in [6], for a dialogue protocol for arguments over propos-
als for action. In this semantics, the mathematical entities constructed are topoi and
maps between them, rather than simply categories.14
For the denotational semantics approach to be useful, we must be able to derive
the semantic mapping of a compound statement in the language from the seman-
tic mappings of its elements, a property called compositionality. This property is
not always present; for example, it may be absent if the language contains com-
pound statements with infinite combinations of elements or if compound statements
have denotations which differ from the composition of those of their elements, as
in Hintikka’s Independence-Friendly (IF) Logic [47]. In these cases, a specific type
of denotational semantics, game-theoretic semantics, has sometimes proven use-
ful [45]. In this semantics, each well-formed statement in the language is associated
with a conceptual game between two players, a protagonist and an antagonist. A
statement in the language is considered to be true when and only when a winning
strategy exists for the protagonist in the associated game; a winning strategy for
a player is a rule giving that player moves for the game such that executing these
moves guarantees the player can win the game, no matter what moves are made by
the opposing player. Game semantics have been articulated for propositional and
predicate logics [59], linear logic [1], and for probability theory [18], among others.
13 This semantics may be viewed as a form of quantification over possible worlds, of which a
more general formalism is that developed subsequently (and independently) by van Eijk and his
colleagues to compare network topologies [31].
14 Topoi are generalizations of the category of sets, and incorporate a categorial analogue of the
notion of set membership [37].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-285-320.jpg)
![272 McBurney and Parsons
What value do these different types of semantics have? Axiomatic semantics
show the pre- and post-conditions of individual utterances in a communications in-
teraction. They may also be used to show the pre- and post-conditions of sequences
of utterances, or even entire dialogues [14, 74]. Thus, they provide a set of rules
to regulate participation in rule-governed interactions. Operational semantics, by
showing the state transitions effected by utterances, may be used to identify dia-
logue states which are not reachable or from which no legal utterance may be made.
These semantics can be used, therefore, to demonstrate that termination of dialogues
between participants using a particular protocol is or is not possible. Operational
semantics also identify which internal agent decision-mechanisms are needed by
agents in order to issue and comprehend received utterances. Properties of dialogue
protocols may also be demonstrated using denotational semantics. In [67], we used
the Deliberation Trace Semantics to generalize a result of Harsanyi [42] regarding
the pareto-optimality of deals achieved using Zeuthen’s Monotonic Concession Pro-
tocol (MCP) [100]. Game semantics have also been used to study the properties of
formal argumentation systems and dialogue protocols, such as their computational
complexity [26], or the extent of truth-convergence under an inquiry dialogue pro-
tocol [65], and to identify acceptable sets of arguments in argument frameworks
[13, 51].
3.3 Pragmatics
Following Levinson [58], we view the study of language pragmatics as dealing with
those aspects of linguistic meaning not covered by considerations of truth and fal-
sity. Chief among these aspects are the desires and intentions of speakers, and these
are usually communicated by means of speech acts, non-propositional utterances in-
tended to or perceived to change the state of the world. Examples of speech acts are
utterances in which a speaker proposes that some action be undertaken, or promises
to undertake it, or commands another to perform it. Modern speech act theory was
initially due to Austin [8] and Searle [88], who classified spoken utterances by their
intended and actual effects on the world (including the internal mental states of
those hearing the utterances), and developed pre-conditions for those effects to be
realized. Drawing on this theory, Bretier, Cohen, Levesque, Perrault and Sadek
were able to present pre- and post-conditions for agent utterances in terms of the
mental states of the participants [12, 15, 16]. This work formed the basis for the
axiomatic Semantic Language SL of the FIPA Agent Communications Language
ACL mentioned above [35]. One of the criticisms made of FIPA ACL is that the
language does not support argumentation [70]; accordingly, McBurney and Parsons
[68] extended this language by defining five additional locutions to enable agents to
assert, question, challenge, justify and retract statements with one another. A set of
locution-combination rules are given (although any such rules are absent from the
specification of FIPA ACL itself), along with an axiomatic semantics in the style of](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-286-320.jpg)
![13 Dialogue Games 273
SL and an operational semantics. This protocol is discussed in the next Section, as
an example of these ideas.
Long before Austin and Searle, Reinach [85] had noted that speech acts typically
require endorsement, or uptake, from the hearer before changing the state of the
world; a speaker may promise a hearer to perform some action, but the speaker is
only obligated to act once the promise is accepted by the hearer. Speech acts are thus
essentially social activities and cannot, normally, be executed by a lone reasoner:
their natural home is a multi-party dialogue. This observation is particularly true
for those speech acts for which the speaker does not have power of retraction or
revocation. In [69], McBurney and Parsons presented an analysis of the differences
in meaning between, for instance, commands and promises. Once uptaken (i.e.,
once in effect), a command may only be revoked by the original speaker, whereas a
promise may only be revoked by the agent to which the promise was made, not the
speaker.
However, capturing such differences in the syntax of utterances can be difficult.
For example, the syntactical form of the two utterances:
I command you to wash the car.
I promise you to wash the car.
is identical, but the illocutionary force, the effects on the world, the nature of the
obligation incurred, the identity of the agent with revocation powers, and even
the identity of the agent intended by the speaker to perform the action are differ-
ent. Although formal agent communications languages should be less ambiguous
than natural language, an interpretation of the syntax of utterances is required for
elimination of any ambiguity in meaning. McBurney and Parsons [69] dealt with
this problem by modifying the denotational trace semantics of [67] to map action-
utterances to tuples in a partitioned tuple space [36]. The different dialogical powers
that participating agents have of issuance, endorsement and revocation for particular
types of utterance then correspond to different permissions to write, copy and delete
(respectively) tuples from associated sub-spaces of the tuple space.
4 Example
As an example of the ideas in this chapter, we present the Fatio protocol of [68].
This protocol comprises five locutions which may be added to the 22 locutions of
FIPA ACL, in order to enable communicating agents to engage in rational argument:
assert, question, challenge, justify and retract. These five locutions are subject to
six locution-combination rules, which together encode a particular dialectical ar-
gumentation theory. Because these locutions are intended to be complementary to
FIPA ACL, there are no locutions for commencing or terminating a dialogue.15 For
reasons of space, we only give examples of two of the five legal locutions of Fatio:
15 It would be easy to take these from another protocol, such as [63].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-287-320.jpg)
![274 McBurney and Parsons
F1: assert(Pi,φ): A speaker Pi asserts a statement φ ∈ C (a belief, an intention,
a social connection, an external commitment, etc). In doing so, Pi creates a di-
alectical obligation within the dialogue to provide justification for φ if required
subsequently by another participant.
F3: challenge(Pj,Pi,φ): A speaker Pj challenges a prior utterance of assert(Pi,φ)
by another participant Pi, and seeks a justification for φ. In contrast to a ques-
tion, with this locution, Pj also creates a dialectical obligation on himself to pro-
vide a justification for not asserting φ, for example an argument against φ, if
questioned or challenged. Thus, challenge(Pj,Pi,φ) is a stronger utterance than
question(Pj,Pi,φ).
For illustration, we present two of the six Fatio locution-combination rules. Here,
Φ ⊢+ φ indicates that Φ is an argument in support of φ.
CR2: The utterances question(Pj,Pi,φ) and challenge(Pj,Pi,φ) may be made
at any time following an utterance of assert(Pi,φ). Similarly, the utterances
question(Pj,Pi,Φ) and challenge(Pj,Pi,Φ) may be made at any time following
an utterance of justify(Pi,Φ ⊢+ φ).
CR3: Immediately following an utterance of question(Pj,Pi,φ) or challenge(Pj,
Pi,φ), the speaker Pi of assert(Pi,φ) must reply with justify(Pi,Φ ⊢+ φ), for some
Φ ∈ A.
In [68], both an axiomatic and an operational semantics for this protocol are ar-
ticulated. The axiomatic semantics is defined in terms of the beliefs, desires and
intentions of the participating agents consistent with the axiomatic semantics SL of
FIPA ACL. For example, the semantics of the locution assert(.) is defined as fol-
lows, with Biφ indicating that “Agent i believes that φ is true”, and Diφ that “Agent
i desires that φ be true.”
• assert(Pi,φ)
Pre-conditions: A speaker Pi desires that each participant Pj(j = i), believes that
Pi believes the proposition φ ∈ C.
((Pi,φ,+) ∈ DOS(Pi))∧(∀j = i)(DiBjBiφ).
Post-conditions: Each participant Pk(k = i), believes that participant Pi desires
that each participant Pj(j = i), believe that Pi believes φ.
(Pi,φ,+) ∈ DOS(Pi) ∧(∀k = i)(∀j = i)(BkDiBjBiφ).
Dialectical Obligations: (Pi,φ,+) is added to DOS(Pi), the Dialectical Obliga-
tions Store of speaker Pi.
Similarly, the operational semantics for Fatio defined in [68] articulates the state-
transition effected by an assert(.) utterance (here labeled F1) on the mental states
of, firstly, the agent who uttered it and, secondly, on any agent who heard it.
TR2: Pi, D1, utter-assert(φ)
F1
→ Pi, D5, listen
TR3: Pi, D1, utter-assert(φ)
F1
→ Pj, D5, do-mech(D2)
These excerpts from the semantics of Fatio are intended simply for illustration. Full
details are given with the protocol definition [68].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-288-320.jpg)
![13 Dialogue Games 275
5 Protocol design and assessment
The science and software engineering of agent communications and interactions is
still in its infancy. Accordingly, designers of multi-agent dialogue game protocols
still have little guidance for design questions such as: How many locutions should
there be? What types of locutions should be included, e.g., assertions, questions, etc?
What are the appropriate rules for the combination of locutions? When should be-
havior be forbidden, e.g., repeated utterance of one locution? Under what conditions
should dialogues be made to terminate? When are dialogues conducted according to
a particular protocol guaranteed to terminate? What are the properties of a proposed
protocol?
Similarly, the immaturity of the discipline means that software developers and
their agents still lack answers to questions such as: How may different protocols
be compared and differences measured? Which protocols are to be preferred under
which circumstances? In other words, what are their advantages and disadvantages?
How should a system developer (or an agent) choose between two protocols which
both support the same type of dialogue, for example, two negotiation protocols?
When are dialogue game protocols preferable to other forms of agent interaction,
such as auction mechanisms or general agent communications languages, such as
FIPA ACL?
Some work has been undertaken which would assist with such questions. For ex-
ample, McBurney, Parsons and Wooldridge [70], proposed thirteen desirable prop-
erties of agent interaction protocols using dialogue games, and then applied these
properties to assess several dialogue game protocols and FIPA ACL; all were found
wanting, to a greater or lesser extent. From an empirical perspective, Karunatil-
lake [54] undertook an evaluation of various negotiation protocols. This work used
simulation studies to compare performance in negotiation interactions for agents
using protocols with and using protocols without argumentation, in order to iden-
tify the circumstances under which argumentation-based negotiation was beneficial.
From a theoretical perspective, Johnson, McBurney and Parsons [52] defined var-
ious measures of protocol equivalence, both syntactical and semantic, and showed
the relationship of these measures to one another. Knowing that two protocols are
equivalent allows inference about their properties (such as termination), and about
their compliance with a given specification, such as that laid down as the standard
for interacting within some electronic institution [86].
6 Conclusion
In this chapter we have given a brief introduction to the theory of dialogue game
protocols for agent interaction and argument, a subject which has become impor-
tant with the rise of multi-agent systems. We have focused on the syntax and the
semantics of these protocols because these topics are important, not only for anal-
ysis of protocols, but also for the software engineering specification, design and](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-289-320.jpg)
![276 McBurney and Parsons
implementation of agent interaction systems. In only a chapter, there are many top-
ics we do not have space to discuss, for example: the computational complexity
of decision-making involved in making utterances, and in deciding whether or not
these comply with a protocol, e.g., [27]; strategic issues over what utterances an
agent should make and when, under a given protocol [82, 83]; properties of specific
protocols, e.g. [3, 11, 87]; experiences arising from implementation [23]; and al-
lowing agents to choose protocols themselves, even at run-time [50, 74]. As can be
seen, there are many avenues to explore in this rich and exciting subject.
Acknowledgements We are grateful for partial financial support received from the EC Informa-
tion Society Technologies (IST) programme, through project ASPIC (IST-FP6-002307). This work
was also partially supported by the US Army Research Laboratory and the UK Ministry of Defence
under Agreement Number W911NF-06-3-0001. The views and conclusions contained in this doc-
ument are those of the authors and should not be interpreted as representing the official policies,
either expressed or implied, of the US Army Research Laboratory, the US Government, the UK
Ministry of Defense, or the UK Government. The US and UK Governments are authorized to re-
produce and distribute reprints for Government purposes notwithstanding any copyright notation
hereon.
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10. T. J. M. Bench-Capon, T. Geldard, and P. H. Leng. A method for the computational modelling
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11. E. Black and A. Hunter. A generative inquiry dialogue system. In M. Huhns, O. Shehory,
E. H. Durfee, and M. Yokoo, editors, Proceedings of the Sixth International Joint Conference
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2007. IFAAMAS, ACM Press.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-290-320.jpg)
![Chapter 14
Models of Persuasion Dialogue
Henry Prakken
1 Introduction
This chapter1 reviews formal dialogue systems for persuasion. In persuasion dia-
logues two or more participants try to resolve a conflict of opinion, each trying to
persuade the other participants to adopt their point of view. Dialogue systems for
persuasion regulate how such dialogues can be conducted and what their outcome
is. Good dialogue systems ensure that conflicts of view can be resolved in a fair
and effective way [6]. The term ‘persuasion dialogue’ was coined by Walton [13]
as part of his influential classification of dialogues into six types according to their
goal. While persuasion aims to resolve a difference of opinion, negotiation tries to
resolve a conflict of interest by reaching a deal, information seeking aims at trans-
ferring information, deliberation wants to reach a decision on a course of action,
inquiry is aimed at “growth of knowledge and agreement” and quarrel is the ver-
bal substitute of a fight. This classification leaves room for shifts of dialogues of
one type to another. In particular, other types of dialogues can shift to persuasion
when a conflict of opinion arises. For example, in information-seeking a conflict of
opinion could arise on the credibility of a source of information, in deliberation the
participants may disagree about likely effects of plans or actions and in negotiation
they may disagree about the reasons why a proposal is in one’s interest.
The formal study of dialogue systems for persuasion was initiated by Hamblin
[5]. Initially, the topic was studied only within philosophical logic and argumenta-
tion theory [15, 7], but later several fields of computer science also became inter-
ested in this topic. In general AI the embedding of nonmonotonic logic in models
of persuasion dialogue was seen as a way to deal with resource-bounded reasoning
[6, 2], while in AI Law persuasion was seen as an appropriate model of legal
Henry Prakken
Department of Information and Computing Sciences, Utrecht University, and Faculty of Law, Uni-
versity of Groningen, e-mail: henry@cs.uu.nl
1 This chapter is a revised and updated version of [11].
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 281
DOI 10.1007/978-0-387-98197-0 14, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-295-320.jpg)
![282 Henry Prakken
procedures [4]. In intelligent tutoring, systems for teaching argumentation skills
have been founded on models of persuasion dialogue [16]. Finally, in the field of
multi-agent systems dialogue systems have been incorporated into models of ratio-
nal agent interaction [8].
To delineate the scope of this chapter, it is useful to discuss what is the subject
matter of dialogue systems. According to Carlson [3] dialogue systems define the
principles of coherent dialogue, that is, the conditions under which an utterance
is appropriate. The leading principle here is that an utterance is appropriate if it
furthers the goal of the dialogue. For persuasion this means that an utterance should
contribute to the resolution of the conflict of opinion that triggered the persuasion.
Thus according to Carlson the principles governing the use of utterances should not
be defined at the level of individual speech acts but at the level of the dialogue in
which the utterance is made. Carlson therefore proposes a game-theoretic approach
to dialogues, in which speech acts are viewed as moves in a game and rules for their
appropriateness are formulated as rules of the game. Most work on formal dialogue
systems for persuasion follows this approach and therefore this chapter will assume
a game format of dialogue systems. It should be noted that the term dialogue system
as used in this chapter only covers the rules of the game, i.e., which moves are
allowed; it does not cover principles for playing the game well, i.e., strategies and
heuristics for the individual players. The latter are instead aspects of agent models.
Below in Section 2 an example persuasion dialogue will be presented, which will
be used for illustration throughout the paper. Then in Section 3 a formal framework
for specifying dialogue game systems is proposed, which in Section 4 is instantiated
for persuasion dialogues and in Section 5 is used for discussing and comparing three
systems proposed in the literature.
2 An example persuasion dialogue
The following example persuasion dialogue exhibits some typical features of per-
suasion and will be used in this chapter to illustrate different degrees of expressive-
ness and strictness of the various persuasion systems.
Paul: My car is safe. (making a claim)
Olga: Why is your car safe? (asking grounds for a claim)
Paul: Since it has an airbag, (offering grounds for a claim)
Olga: That is true, (conceding a claim) but this does not make your car safe. (stat-
ing a counterclaim)
Paul: Why does that not make my care safe? (asking grounds for a claim)
Olga: Since the newspapers recently reported on airbags expanding without cause.
(stating a counterargument by providing grounds for the counterclaim)
Paul: Yes, that is what the newspapers say (conceding a claim) but that does not
prove anything, since newspaper reports are very unreliable sources of technologi-
cal information. (undercutting a counterargument)](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-296-320.jpg)
![14 Models of Persuasion Dialogue 283
Olga: Still your car is still not safe, since its maximum speed is very high. (alter-
native counterargument)
Paul: OK, I was wrong that my car is safe.
This dialogue illustrates several features of persuasion dialogues.
• Participants in a persuasion dialogue not only exchange arguments and coun-
terarguments but also express various propositional attitudes, such as claiming,
challenging, conceding or retracting a proposition.
• As for arguments and counterarguments it illustrates the following features.
– An argument is sometimes attacked by constructing an argument for the oppo-
site conclusion (as in Olga’s two counterarguments) but sometimes by saying
that in the given circumstances the premises of the argument do not support
its conclusion (as in Paul’s counterargument). This is Pollock’s well-known
distinction between rebutting and undercutting counterarguments [9].
– Counterarguments are sometimes stated at once (as in Paul’s undercutter and
Olga’s last move) and are sometimes introduced by making a counterclaim (as
in Olga’s second and third move).
– Natural-language arguments sometimes leave elements implicit. For example,
Paul’s second move arguably leaves a commonsense generalisation ‘Cars with
airbags usually are safe’ implicit.
• As for the structure of dialogues, the example illustrates the following features.
– The participants may return to earlier choices and move alternative replies: in
her last move Olga states an alternative counterargument after she sees that
Paul had a strong counterattack on her first counterargument. Note that she
could also have moved the alternative counterargument immediately after her
first, to leave Paul with two attacks to counter.
– The participants may postpone their replies, sometimes even indefinitely: with
her second argument why Paul’s car is not safe, Olga postpones her reply to
Paul’s counterattack on her first argument for this claim; if Paul fails to suc-
cessfully attack her second argument, such a reply might become superfluous.
3 Elements of dialogue systems
In this section a formal framework for specifying dialogue systems is proposed. To
summarise, dialogue systems have a dialogue goal and at least two participants,
who can have various roles. Dialogue systems have two languages, a a communica-
tion language wrapped around a topic language. Sometimes, dialogues take place
in a context of fixed and undisputable knowledge, such as the relevant laws in a le-
gal dispute. The heart of a dialogue system is formed by a protocol, specifying the
allowed moves at each point in a dialogue, the effect rules, specifying the effects
of utterances on the participants’ commitments, and the outcome rules, defining the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-297-320.jpg)
![284 Henry Prakken
outcome of a dialogue. Two kinds of protocol rules are sometimes separately de-
fined, viz. turntaking and termination rules.
Let us now specify these elements more formally. The definitions below of dia-
logues, protocols and strategies are based on Chapter 12 of [1] as adapted in [10].
As for notation, the complement ϕ of a formula ϕ is ¬ϕ if ϕ is a positive formula
and ψ if ϕ is a negative formula ¬ψ.
Definition 14.1. (Dialogue systems) A dialogue system consists of the following
elements.
• A topic language Lt, closed under classical negation.
• A communication language Lc, consisting of a set of speech acts with a content.
The set of dialogues, denoted by M≤∞, is the set of all sequences from Lc, and
the set of finite dialogues, denoted by M∞, is the set of all finite sequences from
Lc. For any dialogue d = m1,...,mn,..., the subsequence m1,...,mi is denoted
with di.
• A dialogue purpose.
• A set A of participants (or ‘players’) and a set R of roles, defined as disjoint
subsets of A. A participant a may or may not have a, possibly inconsistent, belief
base Σa ⊆ Pow(Lt), which may or may not change during a dialogue. Further-
more, each participant has a, possibly empty set of commitments Ca ⊆ Lt, which
usually changes during a dialogue.
• A context K ⊆ Lt, containing the knowledge that is presupposed and must be
respected during a dialogue. The context is assumed consistent and remains the
same throughout a dialogue.
• A logic L for Lt, which may or may not be monotonic and which may or may
not be argument-based.
• A set of effect rules C for Lc, specifying for each utterance ϕ ∈ Lc its effects on
the commitments of the participants. These rules are specified as functions
– Ca : M∞ −→ Pow(Lt)
Changes in commitments are completely determined by the last move in a dia-
logue and the commitments just before making that move:
– If d = d′ then Ca(d,m) = Ca(d′,m)
• A protocol Pr for Lc, specifying the allowed (or ‘legal’) moves at each stage of
a dialogue. Formally, A protocol on Lc is a function Pr with domain the context
plus a nonempty subset D of M∞ taking subsets of Lc as values. That is:
– Pr : Pow(Lt)×D −→ Pow(Lc)
such that D ⊆ M∞. The elements of D are called the legal finite dialogues. The
elements of Pr(d) are called the moves allowed after d. If d is a legal dialogue
and Pr(d) = ∅, then d is said to be a terminated dialogue. Pr must satisfy the
following condition: for all finite dialogues d and moves m, d ∈ D and m ∈ Pr(d)
iff d,m ∈ D.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-298-320.jpg)
![286 Henry Prakken
• sa : Da −→ Lc
• ha : D′
a −→ Pow(Lc)
4 Persuasion
Let us now become more precise about persuasion. Walton Krabbe [14] define
persuasion dialogues as dialogues with as goal to resolve a conflict of points of
view between at least two participants. A point of view with respect to a proposition
can be positive (for), negative (against) or doubtful. The participants aim to per-
suade the other participant(s) to accept their point of view. According to Walton
Krabbe a conflict is resolved if all parties share the same point of view on the propo-
sition that is at issue. They distinguish disputes as a subtype of persuasion dialogues
where two parties disagree about a single proposition ϕ, such that at the start of the
dialogue one party has a positive (ϕ) and the other party a negative (¬ϕ) point of
view towards the proposition.
Dialogue systems for persuasion can be formally defined as a particular class of
instantiations of the general framework.
Definition 14.4. (dialogue systems for persuasion) A dialogue system for persua-
sion is a dialogue system with at least the following instantiations of Definition 14.1.
• The dialogue purpose is resolution of a conflict of opinion about one or more
propositions, called the topics T ⊆ Lt. This dialogue purpose gives rise to the
following participant roles and outcome rules.
• The participants can have the following roles. To start with, prop(t) ⊆ A, the
proponents of topic t, is the (nonempty) set of all participants with a positive point
of view towards t. Likewise, opp(t) ⊆ A, the opponents of t, is the (nonempty)
set of all participants with a doubtful point of view toward a topic t. Together,
the proponents and opponents of t are called the adversaries with respect to t.
For any t, the sets prop(t) and opp(t) are disjoint but do not necessarily jointly
exhaust A. The remaining participants, if any, are the third parties with respect
to t, assumed to be neutral towards t.
Note that this allows that a participant is a proponent of both t and ¬t or has
a positive attitude towards t and a doubtful attitude towards a topic t′ that is
logically equivalent to t. Since protocols can deal with such situations in various
ways, this should not be excluded by definition.
• The Outcome rules of systems for persuasion dialogues define for a dialogue d,
context K and topic t the winners and losers of d with respect to topic d. More
precisely, O consists of two partial functions w and l:
– w : D×Pow(Lt)×Lt −→ Pow(A)
– l : D×Pow(Lt)×Lt −→ Pow(A)
such that they are defined at least for all terminated dialogues but only for those
t that are a topic of d. These functions will be written as wK
t (d) and lK
t (d) or,](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-300-320.jpg)
![288 Henry Prakken
If the system has an anytime outcome notion, then another distinction can be
made [6]: a protocol is immediate-response if the turn shifts just in case the speaker
is the ‘current’ winner and if it then shifts to a ‘current’ loser.
As for the communication language and effect rules, some common elements can
be found throughout the literature. Below are the most common speech acts, with
their informal meaning and the various names they have been given in the literature.2
• claim ϕ (assert, statement, ...). The speaker asserts that ϕ is the case.
• why ϕ (challenge, deny, question, ...) The speaker challenges that ϕ is the case
and asks for reasons why it would be the case.
• concede ϕ (accept, admit, ...). The speaker admits that ϕ is the case.
• retract ϕ (withdraw, no commitment, ..) The speaker declares that he is not com-
mitted (any more) to ϕ. Retractions are ‘really’ retractions if the speaker is com-
mitted to the retracted proposition, otherwise it is a mere declaration of non-
commitment (for example, in reply to a question).
• ϕ since S (argue, argument, ...) The speaker provides reasons why ϕ is the case.
Some protocols do not have this move but instead require that reasons be pro-
vided by a claim ϕ or claim S move in reply to a why ψ move (where S is a
set of propositions). Also, in some systems the reasons provided for ϕ can have
structure, for example, of a proof three or a deduction.
• question ϕ The speaker asks the hearers’ opinion on whether ϕ is the case.
Paul and Olga (ct’d): In this communication language our example from Section 2
can be more formally displayed as follows:
P1: claim safe O2: why safe
P3: safe since airbag O4: concede airbag
O5: claim ¬ safe
P6: why ¬ safe O7: ¬ safe since newspaper
P8: concede newspaper
P9: so what since ¬ newspapers reliable O10: ¬ safe since high max. speed
P11: retract safe
Most dialogue systems have a notion of typical replies to certain speech acts,
although usually this is left implicit in the replies that are allowed by the protocol
rules. In most systems these typical replies are as displayed in Table 14.1.
Paul and Olga (ct’d): With this table our running example can be displayed as in
Figure 14.1, where the boxes stand for moves and the links for reply relations.
The reply notion induces another distinction between dialogue protocols.
Definition 14.6. A dialogue protocol is unique-reply if at most one reply to a move
is allowed throughout a dialogue; otherwise it is multiple-reply.
Paul and Olga (ct’d): The protocol governing our running example is multiple-
reply, as illustrated by the various branches in Figure 14.1.
2 To make this chapter more uniform, the present terminology will be used even if the original
publication of a system uses different terms.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-302-320.jpg)
![14 Models of Persuasion Dialogue 289
Table 14.1 Locutions and typical replies
Locutions Replies
claim ϕ why ϕ, claim ϕ, concede ϕ
why ϕ ϕ since S (alternatively: claim S), retract ϕ
concede ϕ
retract ϕ
ϕ since S why ψ (ψ ∈ S), concede ψ (ψ ∈ S), ϕ′ since S′
question ϕ claim ϕ, claim ϕ, retract ϕ
Fig. 14.1 Reply structure of the example dialogue
As for the commitment rules, the following ones are generally accepted in the
literature. (Below pl denotes the speaker of the move; effects on the other parties’
commitments are only specified when a change is effected.)
• If pl(m) = claim(ϕ) then Cpl(d,m) = Cpl(d)∪{ϕ}
• If pl(m) = why(ϕ) then Cpl(d,m) = Cpl(d)
• If pl(m) = concede(ϕ) then Cpl(d,m) = Cpl(d)∪{ϕ}
• If pl(m) = retract(ϕ) then Cpl(d,m) = Cpl(d) −{ϕ}
• If pl(m) = ϕ since S then Cpl(d,m) ⊇ Cpl(d)∪prem(A)
The rule for since uses ⊇ since such a move may commit to more than just the
premises of the moved argument. For instance, in [10] the move also commits to
ϕ, since arguments can also be moved as counterarguments instead of as replies to](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-303-320.jpg)
![290 Henry Prakken
challenges of a claim. And in some systems that allow incomplete arguments, such
as [14], the move also commits the speaker to the material implication S → ϕ.
Paul and Olga (ct’d): According to these rules, the commitment sets of Paul and
Olga at the end of the example dialogue are
- CP(d11) ⊇ {airbag, newspaper, ¬ newspapers reliable}
- CO(d11) ⊇ {¬ safe, airbag, newspaper, high max. speed}
5 Three systems
Now three persuasion protocols from the literature will be discussed. The first is pri-
marily based on commitments, the second defines protocols as finite state machines,
while the third exploits an explicit reply structure on the communication language.
5.1 Walton and Krabbe (1995)
The first system to be discussed is Walton Krabbe’s dialogue system PPD for “per-
missive persuasion dialogues” [14]. In PPD, dialogues have no context. The players
are called White (W) and Black (B). They are assumed to declare zero or more “as-
sertions” and “concessions” in an implicit preparatory phase of a dialogue. Each
participant is proponent of his own and opponent of the other participant’s initial
assertions. B must have declared at least one assertion, and W starts a dialogue. The
communication language consists of challenges, (tree-structured) arguments, con-
cessions, questions, resolution demands (“resolve”), and two retraction locutions,
one for assertion-type and one for concession-type commitments. It has no explicit
reply structure but the protocol reflects the reply structure of Table 14.1 above.
The logical language is that of propositional logic and the logic consists of an
incomplete set of deductively valid inference rules: they are incomplete to reflect
that for natural language no complete logic exists. Although an argument may thus
be incomplete, its mover becomes committed to the material implication premises
→ conclusion, which is then open for discussion.
The commitment rules are standard but Walton Krabbe distinguish between
several kinds of commitments for each participant, viz. assertions, concessions and
dark-side commitments. Initial assertions and premises of arguments are placed in
the assertions while conceded propositions are placed in the concessions. Only as-
sertions can be challenged. Dark-side commitments are hidden or veiled commit-
ments of an agent, of which they are often unaware. This makes them hard to model
computationally, for which reason they will be ignored below.
The protocol is driven by two main factors: the contents of the commitment sets
and the content of the last turn. W starts and in their first turn both W and B either
concede or challenge each initial assertion of the other party. Then each turn must](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-304-320.jpg)
![14 Models of Persuasion Dialogue 293
In sum, two features of PPD prevent a fully natural modelling of our example:
the monotonic nature of the underlying logic and the requirement to reply to each
claim or argument of the other participant.
5.2 Parsons, Wooldridge Amgoud (2003)
In a series of papers Parsons, Wooldridge Amgoud have developed an approach
to specifying dialogue systems for various types of dialogues. Here the persuasion
system of [8] will be discussed.
The system is for dialogues on a single topic between two players called White
(W) and Black (B). Dialogues have no context but each participant has a, possibly
inconsistent, belief base Σ. The communication language consists of claims, chal-
lenges, and concessions; it has no explicit reply structure but the protocol largely
conforms to Table 14.1. Claims can concern both individual propositions and sets
of propositions. The logical language is propositional. Its logic is an argument-
based nonmonotonic logic in which arguments are classical proofs from consistent
premises and counterarguments negate a premise of their target. Conflict relations
between arguments are resolved with a preference relation on the premises such that
arguments are as good as their least preferred premises. Argument acceptability is
defined with grounded semantics. In dialogues, arguments cannot be moved as such
but only implicitly as claim S replies to challenges of another claim ϕ, such that S is
consistent and S ⊢ ϕ. Finally, the commitment rules are standard and commitments
are only used to enlarge the player’s belief base with the other player’s commit-
ments; they do not constrain move legality nor determine the dialogue’s outcome.
An important feature of the system is that the players are assumed to adopt an
assertion and an acceptance attitude, which they must respect throughout the dia-
logue. The attitudes are defined relative to their internal belief base (which remains
constant throughout a dialogue) plus both players’ commitment sets (which may
vary during a dialogue). The following assertion attitudes are distinguished: a con-
fident agent can assert any proposition for which he can construct an argument, a
careful agent can do so only if he can construct such an argument and cannot con-
struct a stronger counterargument, and a thoughtful agent can do so only if he can
construct an acceptable argument for the proposition. The corresponding acceptance
attitudes also exist: a credulous agent accepts a proposition if he can construct an
argument for it, a cautious agent does so only if in addition he cannot construct a
stronger counterargument and a skeptical agent does so only if he can construct an
acceptable argument for the proposition.
It can be debated whether such the requirement to respect these attitudes must
be part of a protocol or of a participant’s heuristics. According to one approach, a
dialogue protocol should only enforce coherence of dialogues [14, 10]; according
to another approach, it should also enforce rationality and trustworthiness of the
agents engaged in a dialogue [8]. The second approach allows protocol rules to
refer to an agent’s internal belief base and therefore such protocols do not have a](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-307-320.jpg)
![296 Henry Prakken
Here the nonrepetition rule makes the dialogue terminate without agreement. Note
that only Paul could develop his arguments. To give Olga a chance to develop her
arguments, let us make her careful and skeptical while the players still reason with
each others commitments. Then:
P1: claim safe O2: claim ¬ safe
In the new dialogue state Paul’s argument for ‘safe’ is not acceptable any more,
since it is not preferred over its attacker ({¬ safe}, ¬ safe). So he must challenge.
P3: why ¬ safe O4: claim {newspaper, newspaper → ¬ safe }
Although Paul can construct an argument for Olga’s first premise, namely,
({¬(newspaper → ¬ safe’}, safe), it is not acceptable since it is not preferred over
its attacker based on Olga’s second premise. So he must challenge.
P5: why newspaper O6: claim {newspaper}
Olga had to reply with a (trivial) argument for her first premise, after which Paul
cannot repeat his challenge, so he has to go to the second premise of O4. Based on
his beliefs and Olga’s commitments he can construct (trivial) arguments both for
and against it and neither of these is acceptable. So he must again challenge.
P7: why newspaper → ¬ safe O8: claim {newspaper → ¬ safe}
Here the nonrepetition rule again makes the dialogue terminate without agreement.
In this dialogue only Olga could develop her arguments (although she could not
state her second counterargument).
In conclusion, the PWA persuasion protocol leaves little room for choice and ex-
ploring alternatives, and induces one-sided dialogues in that at most one side can
develop their arguments for a certain issue. Also, the examples suggest that if a
claim is accepted, it is accepted in the first ‘round’ of moves (but this should be
formally verified). On the other hand, the strictness of the protocol induces short di-
alogues which are guaranteed to terminate, which promotes efficiency. Also, thanks
to the strong assumptions on the logic and the participants’ beliefs and reasoning
behaviour, PWA have been able to prove several interesting properties of their pro-
tocols. Finally, without the requirement to respect the assertion and acceptance atti-
tudes the protocol would be much more liberal while still enforcing some coherence.
5.3 Prakken (2005)
In [10] I proposed a formal framework for systems for two-party persuasion dia-
logues and instantiated it with some example protocols. The participants have pro-
ponent and opponent role, and their beliefs are irrelevant to the protocols, so that
these have a public semantics. Dialogues have no context. The framework abstracts
from the communication language except for an explicit reply structure. It also ab-
stracts from the logical language and the logic, except that the logic is assumed to](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-310-320.jpg)
![14 Models of Persuasion Dialogue 297
be argument-based and to conform to grounded semantics and that arguments are
trees of deductive and/or defeasible inferences, as in e.g. [9].
A main motivation of the framework is to ensure focus of dialogues while yet
allowing for freedom to move alternative replies and to postpone replies. This is
achieved with two main features of the framework. Firstly, Lc has an explicit reply
structure, where each move either attacks or surrenders to its target. An example Lc
of this format is displayed in Table 14.3. Secondly, winning is defined for each dia-
Table 14.3 An example Lc in Prakken’s framework
Acts Attacks Surrenders
claim ϕ why ϕ concede ϕ
ϕ since S why ψ(ψ ∈ S) concede ψ
(ψ ∈ S)
ϕ′ since S′ concede ϕ
(ϕ′ since S′ defeats ϕ since S)
why ϕ ϕ since S retract ϕ
concede ϕ
retract ϕ
logue, whether terminated or not, and it is defined in terms of a notion of dialogical
status of moves. The dialogical status of a move is recursively defined as follows,
exploiting the tree structure of dialogues generated by the reply structure on Lc. A
move is in if it is surrendered or else if all its attacking replies are out. (This implies
that a move without replies is in). And a move is out if it has a reply that is in. Then
a dialogue is (currently) won by the proponent if its initial move is in while it is
(currently) won by the opponent otherwise.
Together, these two features of the framework support a notion of relevance that
ensures focus while yet leaving a degree of freedom: a move is relevant just in case
making its target out would make the speaker the current winner. Termination is
defined as the situation that a player is to move but has no legal moves. Various
results are proven about the relation between being the current winner of a dialogue
and what is defeasibly implied by the arguments exchanged during the dialogue.
As for dialogue structure, the framework allows for all kinds of protocols. The
instantiations of [10] are all multi-move and multi-reply; one of them has the com-
munication language of Table 14.3 and is constrained by the requirement that each
move be relevant. This makes the protocol immediate-response, which implies that
each turn consists of zero or more surrenders followed by one attacker. Within these
limits postponement of replies is allowed, sometimes even indefinitely.
Let us next discuss some examples, assuming that the protocol is further in-
stantiated with Prakken Sartor’s argument-based version of prioritised extended
logic programming [12]. This logic uses grounded semantics and supports argu-
ments about rule priorities. (The examples below should speak for themselves so
no formal definitions about the logic will be given. Note that since the rules are
logic-programming rules, they do not satisfy contraposition or modus tollens. Rule](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-311-320.jpg)
![298 Henry Prakken
connectives are tagged with a rule name, which is needed to express rule priorities
in the object language). Consider two agents with the following belief bases:
ΣP = {p, p ⇒r1 q, q ⇒r2 r, p∧s ⇒r3 r2 r4}
ΣO = {t,t ⇒r4 ¬r}.
Then the following is legal in [10]’s so-called relevant protocol (with each move
its target is indicated between square brackets):
P1[−]: claim r O2[P1]: why r
P3[O2]: r since q,q ⇒ r O4[P3]: why q
P5[O4]: q since p, p ⇒ q O6[P5]: concede p ⇒ q
O7[P5]: why p
(Note that unlike in [8] but like in [14], arguments can be stepwise built in sev-
eral moves.) Here P has several allowed moves, viz. retracting any of his argument
premises or his claim, or giving an argument for p. All these moves are relevant
but if P makes any retraction then an argument for p ceases to be relevant, since it
cannot make P the current winner. Moreover, if P retracts r as a reply to P1 then the
dialogue terminates with a win for O.
O could at all points after P3 have moved her argument against r. For instance:
O7[P3]: ¬r since t,t ⇒ ¬r
P8[O7]: r2 r4 since p,s, p∧s ⇒ r1 r4
P8 is a priority argument which makes P3 strictly defeat O7 (note that the fact that s
is not in P’s own belief base does not make the move illegal). At this point, P1 is in;
O has various allowed moves, viz. challenging or conceding any (further) premise
of P’s arguments, moving a counterargument to P5 or a second counterargument to
P3, and conceding P’s initial claim.
This example shows that the participants have much more freedom in this system
than in the one of [8], since they are not bound by assertion and acceptance attitudes
and the protocol is structurally less strict. The downside of this is that dialogues
can be much longer, that the participants can lie and that they can prevent losing by
simply continuing to attack the other participant.
Another drawback of the present approach is that not all natural-language dia-
logues have an explicit reply structure. For example, often one player tries to extract
seemingly irrelevant concessions from the other player with the aim to lure her into
a contradiction, as in as in the following witness cross-examination dialogue:
Witness: Suspect was at home with me that day.
Prosecutor: Are you a student?
Witness: Yes.
Prosecutor: Was that day during summer holiday?
Witness: Yes.
Prosecutor: Aren’t all students away during summer holiday?
In [14] such dialogues can be modelled with the question locution but at the price
of decreased coherence and focus.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-312-320.jpg)
![Chapter 15
Argumentation for Decision Making
Leila Amgoud
1 Introduction
Decision making, often viewed as a form of reasoning toward action, has raised the
interest of many scholars including economists, psychologists, and computer scien-
tists for a long time. Any decision problem amounts to selecting the “best” or suf-
ficiently “good” action(s) that are feasible among different alternatives, given some
available information about the current state of the world and the consequences of
potential actions. Available information may be incomplete or pervaded with un-
certainty. Besides, the goodness of an action is judged by estimating how much
its possible consequences fit the preferences of the decision maker. This agent is
assumed to behave in a rational way [29], at least in the sense that his decisions
should be as much as possible consistent with his preferences.
Classical decision theory, as developed mainly by economists, has focused on mak-
ing clear what is a rational decision maker. Thus, they have looked for principles
for comparing different alternatives. The inputs of this approach are a set of can-
didate actions, and a function that assesses the value of their consequences when
the actions are performed in a given state, together with complete or partial infor-
mation about the current state of the world. In other words, such an approach dis-
tinguishes between knowledge and preferences, which are respectively encoded in
practice by a distribution function assessing the plausibility of the different states of
the world, and by a utility function encoding preferences by estimating how good a
consequence is. The output is a preference relation between actions encoded by the
associated principle. Note that such an approach aims at rank-ordering a group of
candidate actions rather than focusing on a candidate action individually. Moreover,
the candidate actions are supposed to be feasible. What is worth noticing is that
in such an approach, the principles that are defined for comparing pairs of alterna-
tives are given in terms of analytical expressions that summarize the whole decision
Institut de Recherche en Informatique de Toulouse, IRIT-UPS
118 route de Narbonne, 31062 Toulouse, Cedex, France, e-mail: amgoud@irit.fr
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 301
DOI 10.1007/978-0-387-98197-0 15, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-315-320.jpg)
![302 L. Amgoud
process. It is then hard for a person who is not familiar with the abstract decision
methodology, to understand why a proposed alternative is good, or better than an-
other. It is thus important to have an approach in which one can better understand
the underpinnings of the evaluation. Argumentation is the most appropriate way to
advocate a choice thanks to its explanatory power.
Argumentation has been introduced in decision making analysis by several re-
searchers only in the last few years (e.g. [15, 20, 23]). Indeed, in everyday life, deci-
sion is often based on arguments and counter-arguments. Argumentation can be also
useful for explaining a choice already made. Recently, in [1], a decision model in
which some decision criteria were articulated in terms of a two-steps argumentation
process has been proposed. At the first step, called inference step, the model uses a
Dung style system in which arguments in favor/against each option are built, then
evaluated using a given acceptability semantics. At the second step, called compar-
ison step, pairs of alternatives are compared using a given criterion. This criterion
is generally based on the “accepted” arguments computed at the inference step. The
model returns thus, an ordering on the set of options, which may be either partial
or total depending on the decision criterion that is encoded. This approach presents
a great advantage since not only the best alternative is provided to the user but also
the reasons justifying this recommendation. In what follows, we will develop that
argument-based model for decision making.
2 A general framework for argumentative decision making
Solving a decision problem amounts to defining a pre-ordering, usually a complete
one, on a set D of possible options (or candidate decisions), on the basis of the
different consequences of each decision. Let us illustrate this problem through a
simple example borrowed from [20].
Example 15.1 (Having or not a surgery). The example is about having a surgery
(sg) or not (¬sg), knowing that the patient has colonic polyps. The knowledge base
contains the following information:
• having a surgery has side-effects,
• not having surgery avoids having side-effects,
• when having a cancer, having a surgery avoids loss of life,
• if a patient has cancer and has no surgery, the patient would lose his life,
• the patient has colonic polyps,
• having colonic polyps may lead to cancer.
In addition to the above knowledge, the patient has also some goals like: “no side
effects” and “to not lose his life”. Obviously it is more important for him to not lose
his life than to not have side effects.
Let L denote a logical language. From L, a finite set D = {d1,...,dn} of n op-
tions is identified. An option di may be a conjunction of other options in D. Let us,](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-316-320.jpg)
![15 Argumentation for Decision Making 303
for instance, assume that an agent wants a drink and has to choose between tea, milk
or both. Thus, there are three options: d1 : tea, d2 : milk and d3 : tea and milk. In
Example 15.1, the set D contains two options: d1 : sg and d2 : ¬sg.
Argumentation is used in this chapter for ordering the set D. An argumentation-
based decision process can be decomposed into the following steps:
1. Constructing arguments in favor/against statements (beliefs or decisions)
2. Evaluating the strength of each argument
3. Determining the different conflicts among arguments
4. Evaluating the acceptability of arguments
5. Comparing decisions on the basis of relevant “accepted” arguments
Note that the first four steps globally correspond to an “inference problem” in which
one looks for accepted arguments, and consequently warranted beliefs. At this step,
one only knows what is the quality of arguments in favor/against candidate deci-
sions, but the “best” candidate decision is not determined yet. The last step answers
this question once a decision principle is chosen.
2.1 Types of arguments
As shown in Example 15.1, decisions are made on the basis of available knowledge
and the preferences of the decision maker. Thus, two categories of arguments are
distinguished: i) epistemic arguments justifying beliefs and are themselves based
only on beliefs, and ii) practical arguments justifying options and are built from
both beliefs and preferences/goals. Note that a practical argument may highlight
either a positive feature of a candidate decision, supporting thus that decision, or a
negative one, attacking thus the decision.
Example 15.2 (Example 15.1 cont.). In this example, α = [“the patient has colonic
polyps”, and “having colonic polyps may lead to cancer”] is considered as an ar-
gument for believing that the patient may have cancer. This epistemic argument
involves only beliefs. While δ1 = [“the patient may have a cancer”, “when having
a cancer, having a surgery avoids loss of life”] is an argument for having a surgery.
This is a practical argument since it supports the option “having a surgery”. Note
that such argument involves both beliefs and preferences. Similarly, δ2 = [“not hav-
ing surgery avoids having side-effects”] is a practical argument in favor of “not
having a surgery”. However, the two practical arguments δ3 = [“having a surgery
has side-effects”] and δ4 = [“the patient has colonic polyps”, and “having colonic
polyps may lead to cancer”, “if a patient has cancer and has no surgery, the patient
would lose his life”] are respectively against surgery and no surgery since they point
out negative consequences of the two options.
In what follows, Ae denotes a set of epistemic arguments, and Ap denotes a
set of practical arguments such that Ae ∩ Ap = ∅. Let A = Ae ∪ Ap (i.e. A will](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-317-320.jpg)
![304 L. Amgoud
contain all those arguments). The structure and origin of the arguments are assumed
to be unknown. Epistemic arguments will be denoted by variables α1,α2,..., while
practical arguments will be referred to by variables δ1,δ2,... When no distinction is
necessary between arguments, we will use the variables a,b,c,...
Example 15.3 (Example 15.1 cont.). Ae = {α} while Ap = {δ1,δ2,δ3,δ4}.
Let us now define two functions that relate each option to the arguments support-
ing it and to the arguments against it.
• Fp : D → 2Ap is a function that returns the arguments in favor of a candidate
decision. Such arguments are said pro the option.
• Fc : D → 2Ap is a function that returns the arguments against a candidate deci-
sion. Such arguments are said cons the option.
The two functions satisfy the following requirements:
• ∀d ∈ D, ∄δ ∈ Ap s.t. δ ∈ Fp(d) and δ ∈ Fc(d). This means that an argument is
either in favor of an option or against that option. It cannot be both.
• If δ ∈ Fp(d) and δ ∈ Fp(d′) (resp. if δ ∈ Fc(d) and δ ∈ Fc(d′)), then d = d′.
This means that an argument refers only to one option.
• Let D = {d1,...,dn}. Ap = (
Fp(di)) ∪ (
Fc(di)), with i = 1,...,n. This means
that the available practical arguments concern options of the set D.
When δ ∈ Fx(d) with x ∈ {p,c}, we say that d is the conclusion of δ, and we
write Conc(δ) = d.
Example 15.4 (Example 15.1 cont.). The two options of the set D = {sg,¬sg} are
supported/attacked by the following arguments: Fp(sg) = {δ1}, Fc(sg) = {δ3},
Fp(¬sg) = {δ2}, and Fc(¬sg) = {δ4}.
2.2 Comparing arguments
As pointed out by several researchers (e.g. [14, 18]), arguments may have forces of
various strengths. These forces play two key roles: i) they may be used in order to
refine the notion of acceptability of epistemic or practical arguments, ii) they allow
the comparison of practical arguments in order to rank-order candidate decisions.
Generally, the strength of an epistemic argument reflects the quality, such as the
certainty level, of the pieces of information involved in it. Whereas the strength of
a practical argument reflects both the quality of knowledge used in the argument, as
well as how important it is to fulfill the preferences to which the argument refers.
In our particular application, three preference relations between arguments are
defined. The first one, denoted by ≥e, is a (partial or total) preorder1 on the set Ae.
The second relation, denoted by ≥p, is a (partial or total) preorder on the set Ap.
1 A preorder is a binary relation that is reflexive and transitive](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-318-320.jpg)
![306 L. Amgoud
Each preference relation ≥x (with x ∈ {e, p,m}) is combined with the conflict
relation Rx into a unique relation between arguments, denoted by Defx and called
defeat relation, in the same way as in ([4], Definition 3.3, page 204).
Definition 15.1 (Defeat relation). Let A be a set of arguments, and a, b ∈ A.
(a,b) ∈ Defx iff (a,b) ∈ Rx, and (b,a) /
∈x.
Let Defe, Defp and Defm denote the three defeat relations corresponding to the three
attack relations. In case of Defm, the second bullet of Definition 15.1 is always true
since epistemic arguments are strictly preferred (in the sense of ≥m) to any practical
arguments. Thus, Defm = Rm (i.e. the defeat relation is exactly the attack relation
Rm). The relation Defp is the same as Rp, thus it is empty. However, the relation
Defe coincides with its corresponding attack relation Re in case all the arguments
of the set Ae are incomparable.
2.4 Extensions of arguments
Now that the sets of arguments and the defeat relations are identified, we can define
the decision system.
Definition 15.2 (Decision system). Let D be a set of options. A decision system
for ordering D is a triple AF = (D,A,Def) where A = Ae ∪Ap
2 and Def = Defe ∪
Defp ∪Defm
3.
Note that a Dung style argumentation system is associated to a decision system
AF = (D,A,Def), namely the system (A,Def). This latter can be seen as the union
of two distinct argumentation systems: AFe = (Ae,Defe), called epistemic system,
and AFp = (Ap,Defp), called practical system. The two systems are related to each
other by the defeat relation Defm.
Due to Dung’s acceptability semantics defined in [17] or their extensions defined
in [8], it is possible to identify among all the conflicting arguments, which ones
will be kept for ordering the options. Recall that an acceptability semantics amounts
to define sets of arguments that satisfy a consistency requirement and must defend
all their elements. In what follows, E1,...,Ex denote the different extensions of
the system (A,Def) under a given semantics. Using these extensions, a status is
assigned to each argument of AF as follows.
Definition 15.3 (Argument status). Let AF = (D,A,Def) be a decision system, and
E1,...,Ex its extensions under a given semantics. Let a ∈ A.
• a is skeptically accepted iff a ∈ Ei, ∀Ei=1,...,x, Ei = ∅.
• a is credulously accepted iff ∃Ei such that a ∈ Ei and ∃Ej such that a /
∈ Ej.
2 Recall that options are related to their supporting and attacking arguments by the functions Fp
and Fc respectively.
3 Since the relation Defp is empty, then Def = Defe ∪Defm.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-320-320.jpg)
![15 Argumentation for Decision Making 309
system AFp is needed.
Depending on what sets are considered and how they are handled, one can
roughly distinguish between three categories of principles:
Unipolar principles: are those that only refer to either the arguments pros or the
arguments cons.
Bipolar principles: are those that take into account both types of arguments at the
same time.
Non-polar principles: are those where arguments pros and arguments cons a given
choice are aggregated into a unique meta-argument. It results that the negative
and positive polarities disappear in the aggregation.
Whatever the category is, a relation ' should suitably satisfy the following min-
imal requirements:
1. Transitivity: The relation should be transitive (as usually required in decision
theory).
2. Completeness: Since one looks for the “best” candidate decision, it should then
be possible to compare any pair of choices. Thus, the relation should be complete.
2.5.1 Unipolar principles
In this section we present basic principles for comparing decisions on the basis of
only arguments pros. Similar ideas apply to arguments cons. We start by presenting
those principles that do not involve the strength of arguments, then their respective
refinements when strength is taken into account. A first natural criterion consists of
preferring the decision that has more arguments pros.
Definition 15.4 (Counting arguments pros). Let AF = (D,A,Def) be a decision
system and Acc(AF) its accepted arguments. Let d1, d2 ∈ D.
d1 ' d2 iff |Fp(d1)∩Acc(AF)| ≥ |Fp(d2)∩Acc(AF)|.
Property 15.4. This relation is a complete preorder.
Note that when the decision system has no accepted arguments (i.e. Acc(AF) = ∅),
all the options in D are equally preferred w.r.t. the relation '. It is also worth men-
tioning that with such a principle, one may prefer a decision d, which has three
arguments pointing all to the same positive consequence, to decision d′, which is
supported by two arguments pointing to different consequences.
When the strength of arguments is taken into account in the decision process,
one may think of preferring a choice that has a dominant argument, i.e. an argument
pros that is preferred w.r.t. the relation ≥p⊆ Ap ×Ap to any argument pros the other
choices. This principle is called promotion focus principle in [2].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-323-320.jpg)
![15 Argumentation for Decision Making 313
ture (eg. [3, 9, 26]), and their handling does not make new problems in the general
setting of Section 2, even in the decision process perspective of this chapter. More-
over, they only play a role when the knowledge base is inconsistent. Before pre-
senting the different types of practical arguments, we start first by introducing the
logical language as well as the different bases needed in a decision making problem.
3.1 Logical representation of knowledge and preference
This section introduces the representation setting of knowledge and preference
which are here distinct, as it is in classical decision theory. Moreover, preferences
are supposed to be handled in a bipolar way meaning that what the decision maker
is really looking for may be more restrictive than what it is just willing to avoid. In
what follows, a vocabulary P of propositional variables contains two kinds of vari-
ables: decision variables, denoted by v1,...,vn, and state variables. Decision vari-
able are controllable, that is their value can be fixed by the decision maker. Making
a decision then amounts to fixing the truth value of every decision variable. On the
contrary, state variables are fixed by nature, and their value is a matter of knowl-
edge by the decision maker. He has no control on them (although he may express
preferences about their values).
1. D is a set of formulas built from decision variables. Elements of D represent
the different alternatives, or candidate decisions. Let us consider the example of
an agent who wants to know whether she should take her umbrella, her raincoat
or both. In this case, there are two decision variables: umb (for umbrella) and
rac (for raincoat). Assume that this agent hesitates between the three following
options: i) d1 : umb (i.e. to take only her umbrella), ii) d2 : rac (i.e. to take only
her raincoat), or iii) d3 : umb ∧ rac (i.e. to take both). Thus, D = {d1,d2,d3}.
Note that elements of D are not necessarily mutually exclusive. In the example,
if the agent chooses the option d3 then the two other options are satisfied.
2. G is a set of propositional formulas built from state variables. It gathers the goals
of an agent (the decision maker). A goal represents what the agent wants to
achieve, and has thus a positive flavor. This means that if g ∈ G, the decision
maker wants that the chosen decision leads to a state of affairs where g is true.
This base may be inconsistent. In this case it would be for sure impossible to
satisfy all the goals, which would induce the simultaneous existence of practical
arguments pros and cons. In general G contains several goals. Clearly, an agent
should try to satisfy all goals in its goal base G if possible. This means that G
may be thought as a conjunction. However, the two goal bases G = {g1,g2} and
G′ = {g1 ∧g2} although they are logically equivalent, will not be handled in the
same way in an argumentative perspective, since in the second case there is no
way to consider intermediary objectives such as here satisfying g1, or satisfying
g2 only, in case it turns out that it is impossible to satisfy g1 ∧g2. This means that
our approach is syntax-dependent.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-327-320.jpg)
![314 L. Amgoud
3. The set R contains propositional formulas built from state variables. It gathers
the rejections of an agent. A rejection represents what the agent wants to avoid.
Clearly rejections express negative preferences. The set {¬r|r ∈ R} describing
what is acceptable for the agent is assumed to be consistent, since acceptable
alternatives should satisfy ¬r due to the rejection of r, and at least there should
remain some possible worlds that are not rejected. There are at least two rea-
sons for separately considering a set of goals and a set of rejections. First, since
agents naturally express themselves in terms of what they are looking for (i.e.
their goals), and in terms of what they want to avoid (i.e. their rejections), it is
better to consider goals and rejections separately in order to articulate arguments
referring to them in a way easily understandable for the agents. Moreover, recent
cognitive psychology studies [13] have confirmed the cognitive validity of this
distinction between goals and rejections. Second, if r is a rejection, this does not
necessarily mean that ¬r is a goal, and thus rejections cannot be equivalently
restated as goals. For instance, in case of choosing a medical drug, one may have
as a goal the immediate availability of the drug, and as a rejection its availability
only after at least two days. In such a case, if the candidate decision guarantees
the availability only after one day, this decision will for sure avoid the rejection
without satisfying the goal. Another simple example is the case of an agent who
wants to get a cup of either coffee or tea, and wants to avoid getting no drink. If
the agent obtains a glass of water, again he would avoid its rejection, without be-
ing completely satisfied. We can imagine different forms of consistency between
the goals and the rejections. A minimal requirement is to have G∩R = ∅.
4. The set K represents the background knowledge that is not necessarily assumed
to be consistent. The argumentation framework for inference presented in Section
2 will handle such inconsistency, namely with the epistemic system. Elements of
K are propositional formulas built from the alphabet P, and assumed to be put
in a clausal form. The base K contains basically two kinds of clauses: i) those
not involving any element from D which encode pieces of knowledge or factual
information (possibly involving goals) about how the world is; ii) those involving
one negation of a formula d of the set D, and which states what follows when
decision d is applied.
Thus, the decision problem we consider will be always encoded with the four above
sets of formulas (with the restrictions stated above). Moreover, we may suppose that
each of the three bases K, G, and R are stratified. Having K stratified would mean
that we consider that some pieces of knowledge are fully certain, while others are
less certain (maybe distinguishing between several levels of partial certainty such as
“almost certain”, “rather certain”, etc.). Clearly, formulas that are not certain at all
cannot be in K. Similarly, having G (resp. R) stratified means that some goals (resp.
rejections) are imperative, while some others are less important (one may have more
than two levels of importance). Completely unimportant goals (resp. rejections) do
not appear in any stratum of G (resp. R). It is worth pointing out that we assume that
candidate decisions are all considered as a priori equally potentially suitable, and
thus there is no need to have D stratified.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-328-320.jpg)
![15 Argumentation for Decision Making 317
would like benefit G = {b}. O2 does not want large money loss R = {lml}, and
would like benefit G = {b}. In such case, it is expected that O1 prefers ¬d to d,
since there is a NC argument against d and no argument for ¬d. For O2, there is no
longer any NC argument against d. He might even prefer d to ¬d, if he is optimistic
and he considers that there is a possibility that it does not rain (leading to a potential
PP argument under the hypothesis to have ¬r in K.
Due to the asymmetry in the human mind between what is rejected and what is
desired, the former being usually considered as stronger than the latter, one may
assume that NC arguments are stronger than PC arguments, and conversely PP argu-
ments are stronger than NP arguments.
4 Related work
Some works have been done on arguing for decision. Quite early, in [22] Brewka
and Gordon have outlined a logical approach to decision (for negotiation purposes),
which suggests the use of defeasible consequence relation for handling prioritized
rules, and which also exhibits arguments for each choice. However, arguments are
not formally defined. In the framework proposed by Fox and Parsons in [20], no ex-
plicit distinction is made between knowledge and goals. However, in their examples,
values (belonging to a linearly ordered scale) are assigned to formulas which repre-
sent goals. These values provide an empirical basis for comparing arguments using a
symbolic combination of strengths of beliefs and goals values. This symbolic com-
bination is performed through dictionaries corresponding to different kinds of scales
that may be used. In this work, only one type of arguments is considered in the style
of arguments in favor of beliefs. In [10], Bonet and Geffner have also proposed an
approach to qualitative decision, inspired from Tan and Pearl [27], based on “action
rules” that link a situation and an action with the satisfaction of a positive or a nega-
tive goal. However in contrast with the previous two works and the work presented
in this paper, this approach does not refer to any model in argumentative inference.
Other researchers in AI, working on practical reasoning, starting with the generic
question “what is the right thing to do for an agent in a given situation” [24, 25], have
proposed a two steps process to answer this question. The first step, often called de-
liberation [29], consists of identifying the goals of the agent. In the second step, they
look for ways of achieving those goals, i.e. for plans, and thus for intermediary goals
and sub-plans. Such an approach raises issues such as: how are goals generated ? are
actions feasible ? do actions have undesirable consequences ? are sub-plans compat-
ible ? are there alternative plans for achieving a given goal, etc. In [12], it has been
argued that this can be done by representing the cognitive states, namely agent’s
beliefs, desires and intentions (thus the so-called BDI architecture). This requires a
rich knowledge/preference representation setting, which contrasts with the classical
decision setting that directly uses an uncertainty distribution (a probability distri-
bution in the case of expected utility), and a utility (value) function. Besides, the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-331-320.jpg)
![318 L. Amgoud
deliberation step is merely an inference problem since it amounts to finding a set of
desires that are justified on the basis of the current state of the world and of condi-
tional desires. Checking if a plan is feasible and does not lead to bad consequences
is still a matter of inference. A decision problem only occurs when several plans or
sub-plans are possible, and one of them has to be chosen. This latter issue may be
viewed as a classical decision problem. What is worth noticing in most works on
practical reasoning is the use of argument schemes for providing reasons for choos-
ing or discarding an action (e.g. [19, 21]). For instance, an action may be considered
as potentially useful on the basis of the practical syllogism [28]:
• G is a goal for agent X
• Doing action A is sufficient for agent X to carry out goal G
• Then, agent X ought to do action A
The above syllogism is in essence already an argument in favor of doing action
A. However, this does not mean that the action is warranted, since other argu-
ments (called counter-arguments) may be built or provided against the action. Those
counter-arguments refer to critical questions identified in [28] for the above syllo-
gism. In particular, relevant questions are “Are there alternative ways of realizing
G?”, “Is doing A feasible?”, “Has agent X other goals than G?”, “Are there other
consequences of doing A which should be taken into account?”. Recently in [6, 7],
the above syllogism has been extended to explicitly take into account the reference
to ethical values in arguments.
5 Conclusion
The chapter has proposed an abstract argumentation-based framework for decision
making. The main idea behind this work is how to define a complete preorder on a
set of candidate decisions on the basis of arguments. The framework distinguishes
between two types of arguments: epistemic arguments that support beliefs and prac-
tical arguments that justify candidate decisions. Each practical argument concerns
only one candidate decision, and may be either in favor of that decision or against
it. The framework follows two main steps: i) an inference step in which arguments
are evaluated using acceptability semantics. This step amounts to return among the
practical arguments, those which are warranted in the current state of information,
i.e. the “accepted” arguments. ii) A pure decision step in which candidate decisions
are compared on the basis of accepted practical arguments. For the second step of
the process, we have proposed three families of principles for comparing pairs of
choices. An axiomatic study and a cognitive validation of these principles are worth
developing, in particular in connection with [11, 16]. The proposed approach is very
general and includes as particular cases already studied argumentation-based deci-
sion systems. Moreover it has been shown in [5] that the approach is suitable for
multiple criteria decision making as well as decision making under uncertainty. In
particular, the approach has been shown to fully agree with qualitative decision mak-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-332-320.jpg)
![Chapter 16
Argumentation and Game Theory
Iyad Rahwan and Kate Larson
1 What Game Theory Can Do for Argumentation
In a large class of multi-agent systems, agents are self-interested in the sense that
each agent is interested only in furthering its individual goals, which may or may
not coincide with others’ goals. When such agents engage in argument, they would
be expected to argue strategically in such a way that makes it more likely for their
argumentative goals to be achieved. What we mean by arguing strategically is that
instead of making arbitrary arguments, an agent would carefully choose its argu-
mentative moves in order to further its own objectives.
The mathematical study of strategic interaction is Game Theory, which was pi-
oneered by von Neuman and Morgenstern [13]. A setting of strategic interaction is
modelled as a game, which consists of a set of players, a set of actions available
to them, and a rule that determines the outcome given players’ chosen actions. In
an argumentation scenario, the set of actions are typically the set of argumentative
moves (e.g. asserting a claim or challenging a claim), and the outcome rule is the
criterion by which arguments are evaluated (e.g. a judge’s attitude or a social norm).
Generally, game theory can be used to achieve two goals:
1. undertake precise analysis of interaction in particular strategic settings, with a
view to predicting the outcome;
2. design rules of the game in such a way that self-interested agents behave in some
desirable manner (e.g. tell the truth); this is called mechanism design;
Both these approaches are quite useful for the study of argumentation in multi-
agent systems. On one hand, an agent may use game theory to analyse a given
argumentative situation in order to choose the best strategy. On the other hand, we
Iyad Rahwan
British University in Dubai, UAE University of Edinburgh, UK, e-mail: irahwan@acm.org
Kate Larson
University of Waterloo, Canada e-mail: klarson@cs.uwaterloo.ca
I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 321
DOI 10.1007/978-0-387-98197-0 16, c
Springer Science+Business Media, LLC 2009](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-335-320.jpg)
![322 Iyad Rahwan and Kate Larson
may use mechanism design to design the rules (e.g. argumentation protocol) in such
a way as to promote good argumentative behaviour. In this chapter, we will discuss
some early developments in these directions.
In the next section, we motivate the usefulness of game theory in argumentation
using a novel game. After providing a brief background on game theory in Section
3, we introduce our Argumentation Mechanism Design approach in Section 4 and
present some preliminary results in Section 5. Finally, we discuss related work in
Section 6 and conclude in Section 7
2 The “Argumentative Battle of the Sexes” Game
Consider the following situation involving the couple Alice (A) and Brian (B), who
want to decide on an activity for the day.1 Brian thinks they should go to a soc-
cer match (argument α1) while Alice thinks they should attend the ballet (argument
α2). There is time for only one activity, however (hence α1 and α2 defeat one an-
other). Moreover, while Alice prefers the ballet to the soccer, she would still rather
go to a soccer match than stay at home. Likewise, Brian prefers the soccer match
to the ballet, but also prefers the ballet to staying home. Formally, we can write
uA(ballet) uA(soccer) uA(home) and uB(soccer) uB(ballet) uB(home).
Alice has a strong argument which she may use against going to the soccer,
namely by claiming that she is too sick to be outdoors (argument α3). Brian simply
cannot attack this argument (without compromising his marriage at least). Likewise,
Brian has an irrefutable argument against the ballet; he could claim that his ex-wife
will be there too (argument α4). Alice cannot stand her! Using Dung’s abstract ar-
gumentation model [1], which is described in detail in Chapter 2, the argumentative
structure of this situation can be modelled as shown in Figure 16.1(a).
Alice can choose to say nothing, utter argument α2 or α3 or both. Similarly,
Brian can choose to say nothing, utter argument α1 or α4 or both. For the sake of
the example, we will suppose that Alice and Brian use the grounded semantics as
the argumentative foundation of their marriage! The question we are interested in
here is: What will Alice and Bob say? or at least: What are they likely to say?
The strategic encounter, on the other hand, can be modelled as shown in the table
in Figure 16.1(b). Each cell corresponds to a strategy profile in which Alice and
Brian reveal a particular set of arguments. The numbers in the cells correspond to
the utilities they obtain once the grounded extension is calculated on their revealed
arguments. For example, if Alice utters {α2} while Brian utters {α1,α4}, we end up
with a sub-graph of Figure 16.1(a) in which α3 is missing. The grounded extension
of this argument graph admits arguments {α1,α4}. This corresponds to a situation
where Brian wins and the couple head to the soccer. Thus, he gets the highest utility
of 2, while Alice gets her second-preferred outcome with utility 1. This representa-
tion, shown in Figure 16.1(b), is known as a normal form game.
1 We call this the argumentative battle of the sexes game. It is similar, but not identical to the
well-known “Battle of the Sexes” game.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-336-320.jpg)
![326 Iyad Rahwan and Kate Larson
A number of researchers have proposed using extensive-form games of perfect
information to model argumentation. For example, Procaccia and Rosenschein [10]
proposed a game-based argumentation framework where they extend Dung’s
abstract argumentation framework by mapping argumentation frameworks into
extensive-form games of perfect information. A similar approach has recently been
proposed by Riveret et al [12], giving an extensive-form game characterisation of
Prakken’s dialectical framework [8]. In both cases, the authors show how standard
backward induction techniques can be used to eliminate dominated strategies and
characterise Nash equilibrium strategies.
3 Technical Background
Before we present a precise formal mapping of abstract argumentation into game
theory, in this section, we give a brief background on key game-theoretic concepts.
Readers who lack background in game theory may consult a more comprehensive
introduction to the field, such as [5].
3.1 Game Theory
The field of game theory studies strategic interactions of self-interested agents. We
assume that there is a set of self-interested agents, denoted by I. We let θi ∈ Θi
denote the type of agent i which is drawn from some set of possible types Θi. The
type represents the private information and preferences of the agent. An agent’s
preferences are over outcomes o ∈ O, where O is the set of all possible outcomes. We
assume that an agent’s preferences can be expressed by a utility function ui(o,θi)
which depends on both the outcome, o, and the agent’s type, θi. Agent i prefers
outcome o1 to o2 when ui(o1,θi) ui(o2,θi).
When agents interact, we say that they are playing strategies. A strategy for agent
i, si(θi), is a plan that describes what actions the agent will take for every decision
that the agent might be called upon to make, for each possible piece of information
that the agent may have at each time it is called to act. That is, a strategy can be
thought as a complete contingency plan for an agent. We let Σi denote the set of
all possible strategies for agent i, and thus si(θi) ∈ Σi. When it is clear from the
context, we will drop the θi in order to simplify the notation. We let strategy profile
s = (s1(θ1),...,sI(θI)) denote the outcome that results when each agent i is playing
strategy si(θi). As a notational convenience we define
s−i(θ−i) = (s1(θi),...,si−1(θi−1),si+1(θi+1),...,sI(θI))
and thus s = (si,s−i). We then interpret ui((si,s−i),θi) to be the utility of agent i with
type θi when all agents play strategies specified by strategy profile (si(θi),s−i(θ−i)).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-340-320.jpg)
![328 Iyad Rahwan and Kate Larson
Definition 16.3 (Bayes-Nash Equilibrium). A strategy profile s∗ = (s∗
i ,s∗
−i) is a
Bayes-Nash equilibrium if
Eθ−i [ui((s∗
i (θi),s∗
−i(·)),θi)] ≥ Eθ−i [ui((s′
i(θi),s∗
−i(·)),θi)] ∀θi,∀s′
i.
3.2 Mechanism Design
The problem that mechanism design studies is how to ensure that a desirable system-
wide outcome or decision is made when there is a group of self-interested agents
who have preferences over the outcomes. In particular, we often want the outcome to
depend on the preferences of the agents. This is captured by a social choice function.
Definition 16.4 (Social Choice Function). A social choice function is a rule f :
Θ1 × ... ×ΘI → O, that selects some outcome f(θ) ∈ O, given agent types θ =
(θ1,...,θI).
The challenge, however, is that the types of the agents (the θ′
i s) are private and
known only to the agents themselves. Thus, in order to select an outcome with the
social choice function, one has to rely on the agents to reveal their types. However,
for a given social choice function, an agent may find that it is better off if it does
not reveal its type truthfully, since by lying it may be able to cause the social choice
function to choose an outcome that it prefers. Instead of trusting the agents to be
truthful, we use a mechanism to try to reach the correct outcome.
A mechanism M = (Σ,g(·)) defines the set of allowable strategies that agents
can chose, with Σ = Σ1 × ··· × ΣI where Σi is the strategy set for agent i, and an
outcome function g(s) which specifies an outcome o for each possible strategy pro-
file s = (s1,...,sI) ∈ Σ. This defines a game in which agent i is free to select any
strategy in Σi, and, in particular, will try to select a strategy which will lead to an
outcome that maximizes its own utility. We say that a mechanism implements social
choice function f if the outcome induced by the mechanism is the same outcome
that the social choice function would have returned if the true types of the agents
were known.
Definition 16.5 (Implementation). A mechanism M = (Σ,g(·)) implements social
choice function f if there exists an equilibrium s∗ such that
∀θ ∈ Θ, g(s∗
(θ)) = f(θ).
While the definition of a mechanism puts no restrictions on the strategy spaces of
the agents, an important class of mechanisms are the direct-revelation mechanisms
(or simply direct mechanisms).
Definition 16.6 (Direct-Revelation Mechanism). A direct-revelation mechanism
is a mechanism in which Σi = Θi for all i, and g(θ) = f(θ) for all θ ∈ Θ.
In words, a direct mechanism is one where the strategies of the agents are to an-
nounce a type, θ′
i to the mechanism. While it is not necessary that θ′
i = θi, the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-342-320.jpg)
![16 Argumentation and Game Theory 329
important Revelation Principle (see below for more details) states that if a social
choice function, f(·), can be implemented, then it can be implemented by a direct
mechanism where every agent reveals its true type [5]. In such a situation, we say
that the social choice function is incentive compatible.
Definition 16.7 (Incentive Compatible). The social choice function f(·) is incen-
tive compatible (or truthfully implementable) if the direct mechanism M = (Θ,g(·))
has an equilibrium s∗ such that s∗
i (θi) = θi.
If the equilibrium concept is the dominant-strategy equilibrium, then the social
choice function is strategy-proof. In this chapter we will on occasion call a mecha-
nism incentive-compatible or strategy-proof. This means that the social choice func-
tion that the mechanism implements is incentive-compatible or strategy-proof.
3.3 The Revelation Principle
Determining whether a particular social choice function can be implemented, and in
particular, finding a mechanism which implements a social choice function appears
to be a daunting task. In the definition of a mechanism, the strategy spaces of the
agents are unrestricted, leading to an infinitely large space of possible mechanisms.
However, the Revelation Principle states that we can limit our search to a special
class of mechanisms [5, Ch 14].
Theorem 16.1 (Revelation Principle). If there exists some mechanism that imple-
ments social choice function f in dominant strategies, then there exists a direct
mechanism that implements f in dominant strategies and is truthful.
The intuitive idea behind the Revelation Principle is fairly straightforward. Sup-
pose that you have a, possibly very complex, mechanism, M, which implements
some social choice function, f. That is, given agent types θ = (θ1,...,θI) there ex-
ists an equilibrium s∗(θ) such that g(s∗(θ)) = f(θ). Then, the Revelation Principle
states that it is possible to create a new mechanism, M′, which, when given θ, will
then execute s∗(θ) on behalf of the agents and then select outcome g(s∗(θ)). Thus,
each agent is best off revealing θi, resulting in M′ being a truthful, direct mechanism
for implementing social choice function f.
The Revelation Principle is a powerful tool when it comes to studying imple-
mentation. Instead of searching through the entire space of mechanisms to check
whether one implements a particular social choice function, the Revelation Prin-
ciple states that we can restrict our search to the class of truthful, direct mecha-
nisms. If we can not find a mechanism in this space which implements the social
choice function of interest, then there does not exist any mechanism which will
do so.
It should be noted that while the Revelation Principle is a powerful analysis tool,
it does not imply we should only design direct mechanisms. Some reasons why one
rarely sees direct mechanisms in the “real world” include (among others);](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-343-320.jpg)
![330 Iyad Rahwan and Kate Larson
• they can place a high computational burden on the mechanism since it is required
to execute agents’ strategies,
• agents’ strategies may be computationally difficult to determine, and
• agents may not be willing to reveal their true types because of privacy concerns.
4 Argumentation Mechanism Design
Mechanism design (MD) is a sub-field of game theory concerned with the follow-
ing question: what game rules guarantee a desirable social outcome when each
self-interested agent selects the best strategy for itself? In other words, while game
theory is concerned with a given strategic situation modelled as a game, mechanism
design is concerned with designing the game itself. As such, one might actually call
it reverse game theory.
In this section we define the mechanism design problem for abstract argumenta-
tion. We dub this new approach ‘Argumentation Mechanism Design’ (ArgMD).
Let AF = A,R be an argumentation framework with a set of arguments A and
a binary defeat relation R. We define a mechanism with respect to AF and semantics
S, and we assume that there is a set of I self-interested agents. We define an agent’s
type to be its set of arguments.
Definition 16.8 (Agent Type). Given an argumentation framework A,R, the type
of agent i, Ai ⊆ A, is the set of arguments that the agent is capable of putting for-
ward.
There are two things to note about this definition. Firstly, an agent’s type can
be seen as a reflection of its expertise or domain knowledge. For example, medical
experts may only be able to comment on certain aspects of forensics in a legal case,
while a defendant’s family and friends may be able to comment on his/her character.
Also, such expertise may overlap, so agent types are not necessarily disjoint. For
example, two medical doctors might have some identical argument, and so on.
The second thing to note about the definition is that agent types do not include
the defeat relation. In other words, we implicitly assume that the notion of defeat is
common to all agents. That is, given two arguments, no agent would dispute whether
one attacks another. This is a reasonable assumption in systems where agents use
the same logic to express arguments or at least multiple logics for which the notion
of defeat is accepted by everyone (e.g. conflict between a proposition and its nega-
tion). Disagreement over the defeat relation itself requires a form of hierarchical
(meta) argumentation [7], which is a powerful concept, but is beyond the scope of
the present chapter.
Given the agents’ types (argument sets) a social choice function f maps a type
profile into a subset of arguments;
f : 2A
×...×2A
→ 2A](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-344-320.jpg)
![16 Argumentation and Game Theory 331
While our definition of an argumentation mechanism will allow for generic social
choice functions which map type profiles into subsets of arguments, we will be
particularly interested in argument acceptability social choice functions. We denote
by Acc(A,R,S) ⊆ A the set of acceptable arguments according to semantics S.3
Definition 16.9 (Argument Acceptability Social Choice Functions). Given an ar-
gumentation framework A,R with semantics S, and given an agent type profile
(A1,...,AI), the argument acceptability social choice function f is defined as the
set of acceptable arguments given the semantics S. That is,
f(A1,...,AI) = Acc(A1 ∪...∪AI,R,S).
As is standard in the mechanism design literature, we assume that agents have
preferences over the outcomes o ∈ 2A
, and we represent these preferences using
utility functions where ui(o,Ai) denotes agent i’s utility for outcome o when its
type is argument set Ai.
Agents may not have incentive to reveal their true type because they may be able
to influence the final argument status assignment by lying, and thus obtain higher
utility. There are two ways that an agent can lie in our model. On one hand, an
agent might create new arguments that it does not have in its argument set. In the
rest of the chapter we will assume that there is an external verifier that is capable of
checking whether it is possible for a particular agent to actually make a particular
argument. Informally, this means that presented arguments, while still possibly de-
feasible, must at least be based on some sort of demonstrable ‘plausible evidence.’
If an agent is caught making up arguments then it will be removed from the mech-
anism. For example, in a court of law, any act of perjury by a witness is punished,
at the very least, by completely discrediting all evidence produced by the witness.
Moreover, in a court of law, arguments presented without any plausible evidence
are normally discarded (e.g. “I did not kill him, since I was abducted by aliens at
the time of the crime!”). For all intents and purposes this assumption (also made by
Glazer and Rubinstein [2]) removes the incentive for an agent to make up facts.
A more insidious form of manipulation occurs when an agent decides to hide
some of its arguments. By refusing to reveal certain arguments, an agent might be
able to break defeat chains in the argument framework, thus changing the final set of
acceptable arguments. For example, a witness may hide evidence that implicates the
defendant if the evidence also undermines the witness’s own character. This type of
lie is almost impossible to detect in practice, and it is this form of strategic behaviour
that we will be the most interested in.
As mentioned earlier, a strategy of an agent specifies a complete plan that de-
scribes what action the agent takes for every decision that a player might be called
upon to take, for every piece of information that the player might have at each time
that it is called upon to act. In our model, the actions available to an agent involve
announcing sets of arguments. Thus a strategy si ∈ Σi for agent i would specify for
3 Here, we assume that S specifies both the classical semantics used (e.g. grounded, preferred,
stable) as well as the acceptance attitude used (e.g. sceptical or credulous).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-345-320.jpg)
![336 Iyad Rahwan and Kate Larson
6 Related Work
6.1 Pareto Optimality of Outcomes
A well-known property of the grounded semantics is that it is extremely sceptical,
accepting only undefeated arguments and arguments defended by undefeated argu-
ments. An interesting question, then, is whether it is possible to be more inclusive
(i.e. being more credulous) in order to produce argumentation outcomes that are
more socially desirable. For example, consider the simple argument graph in Figure
16.5 and suppose we have two agents with types A1 = {α1} and A2 = {α2} who
both reveal their arguments. The grounded extension (Figure 16.5(a)) is empty here.
α2 α1
(a)
α2 α1
(b)
α2
α1
(c)
Fig. 16.5 Preferred extensions ‘dominate’ the grounded extension
Suppose the judge chooses one of the preferred extensions instead (Figure
16.5(b) or (c)). Clearly, when compared to outcome (a), each preferred extension
makes one agent better-off without making the other worse-off. Formally, we say
that outcomes (b) and (c) each Pareto dominates outcome (a). An outcome that is
not Pareto dominated is called Pareto optimal.
Recently, Rahwan and Larson [11] presented an extensive analysis of Pareto op-
timality in abstract argumentation, and established correspondence results between
different semantics on one hand and the Pareto optimal outcomes on the other.
6.2 Glazer and Rubinstein’s Model
Another game-theoretic analysis of argumentation was presented by Glazer and Ru-
binstein [2]. The authors explore the mechanism design problem of constructing
rules of debate that maximise the probability that a listener reaches the right con-
clusion given arguments presented by two debaters. They study a very restricted
setting, in which the world state is described by a vector ω = (w1,...,w5), where
each ‘aspect’ wi has two possible values: 1 and 2. If wi = j for j ∈ {1,2}, we say
that aspect wi supports outcome Oj. Presenting an argument amounts to revealing
the value of some wi. The setting is modelled as an extensive-form game and anal-
ysed. In particular, the authors investigate various combinations of procedural rules
(stating in which order and what sorts of arguments each debater is allowed to state)
and persuasion rules (stating how the outcome is chosen by the listener). In terms of
procedural rules, the authors explore: (1) one-speaker debate in whichone debater](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-350-320.jpg)
![16 Argumentation and Game Theory 337
chooses two arguments to reveal; (2) simultaneous debate in which the two debaters
simultaneously reveal one argument each; and (3) sequential debate in which one
debater reveals one argument followed by one argument by the other. Our mecha-
nism is closer to the simultaneous debate, but is much more general as it enables
the simultaneous revelation of an arbitrary number of arguments. Glazer and Rubin-
stein investigate a variety of persuasion rules. For example, in one-speaker debate,
one rule analysed by the authors states that ‘a speaker wins if and only if he presents
two arguments from {a1,a2,a3} or {a4,a5}.’ In a sequential debate, one persuasion
rule states that ‘if debater D1 argues for aspect a3, then debater D2 wins if and only if
he counter-argues with aspect a4.’ These kinds of rules are arbitrary and do not fol-
low an intuitive notion of persuasion (e.g. like scepticism). The sceptical mechanism
presented in this chapter provides a more natural criterion for argument evaluation,
supplemented by a strong solution concept that ensures all agents have incentive to
reveal their arguments, and thus for the listener to reach the correct outcome. More-
over, our framework for argumentation mechanism design is more general in that it
can be used to model a variety of more complex argumentation settings.
6.3 Game Semantics
It is worth contrasting our work with work on so-called game semantics for logic,
which was pioneered by logicians such as Paul Lorenzen [4] and Jaakko Hintikka
[3]. Although many specific instantiations of this notion have been presented in
the literature, the general idea is as follows. Given some specific logic, the truth
value of a formula is determined through a special-purpose, multi-stage dialogue
game between two players, the verifier and falsifier. The formula is considered true
precisely when the verifier has a winning strategy, while it will be false whenever
the falsifier has the winning strategy. Similar ideas have been used to implement
dialectical proof-theories for defeasible reasoning (e.g. by Prakken and Sartor [9]).
In a related development, Matt and Toni recently proposed a game-theoretic ap-
proach to characterise argument strength [6]. Thus, the acceptability of each argu-
ment is rated between 0 and 1 by using a two-person zero-sum game with imperfect
information between a proponent and an opponent.
There is a fundamental difference between the aims of game semantics and our
ArgMD approach. In game semantics, the goal is to interpret (i.e., characterise the
truth value of) a specific formula by appealing to a notion of a winning strategy.
As such, each player is carefully endowed with a specific set of formulae to en-
able the game to characterise semantics correctly (e.g. the verifier may own all the
disjunctions in the formula, while the falsifier is given all the conjunctions).
In contrast, ArgMD is about designing rules for argumentation among self-
interested players who may have incentives to manipulate the outcome given a va-
riety of possible individual preferences (specified in arbitrary instantiations of a
utility function). Our interest is in conditions that guarantee truth-revelation given
different classes of preferences. Game semantics have no similar notion of strategic](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-351-320.jpg)
![338 Iyad Rahwan and Kate Larson
manipulation by hiding information. Moreover, our framework allows an arbitrary
number of players (as opposed to two agents).
6.4 Argumentation and Cooperative Games
Another notable early link between argumentation and cooperative game theory has
been proposed by Dung in his seminal paper [1]. Let A = {a1,...,a|A|} be a set
of agents. A cooperative game is defined by specifying a value to V(C) to each
coalition C ⊆ A of agents. An outcome of the game is a vector u = (ua1 ,...,ua|A|
) ∈
R|A| specifying a vector of utilities, one per agent.
Outcome u dominates outcome u′ if there is a (nonempty) coalition K ⊆ A in
which agents get more utility (as a whole) in u than in u′. An outcome is said to be
stable if no outcome dominates it (i.e. if no subset of agents has incentive to leave
their own coalition and be all individually better off). A solution of the cooperative
game is a set of outcomes S satisfying the following conditions:
1. No s ∈ S is dominated by an s′ ∈ S.
2. Every s /
∈ S is dominated by some s′ ∈ S.
Dung argued that an n-person game can be seen as an argumentation framework
A,R in which the set of arguments A is the set of all possible outcomes of the co-
operative game, and the defeat relation is defined as R = {(u,u′) | u dominates u′}.
Dung shows that with this characterisation, the set of solutions to the cooperative
game corresponds to the set of stable extensions of an abstract argumentation frame-
work. This enabled Dung to describe the well-known stable marriage problem as a
problem of finding a stable extension.
Another important notion in cooperative game theory is that of the core: the
set of (feasible) outcomes which are not dominated by any other outcome. Dung
showed that the core corresponds to F(∅) where F is the characteristic function of
the corresponding argumentation framework.
7 Conclusion
In this chapter, our aim was to demonstrate the importance of game theory as a
tool for analysing strategic argumentation. We showed how normal form games and
extensive-form games can be used to analyse equilibrium strategies in strategic ar-
gumentation. We then introduced Argumentation Mechanism Design (ArgMD) as
a new framework for designing and analysing argument evaluation criteria. With
ArgMD, designing new argument acceptance criteria becomes akin to designing
auction protocols in strategic multi-agent settings. The goal is to design rules that
ensure, under precise conditions, that agents have no incentive to manipulate the](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-352-320.jpg)
![342 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
ing the links between both areas. Based on such insights, we will develop a con-
ceptual view on this topic, which is based on the understanding of argumentation
and belief revision being complementary disciplines for the broad picture sketched
above. Each needs the other’s support if we want to model successful decision mak-
ing in a real world application. We will also discuss how one area may contribute to
the other, enriching the respective framework.
2 Basic facts on argumentation and belief revision
An argument A for α is a set of interrelated pieces of knowledge supporting α
from evidence. Abstract approaches to argumentation such as [13] make use of ar-
gumentation frameworks A,R with is a finite set A of arguments and an attack
relation R among arguments. The main issue of such argumentation frameworks is
the selection of acceptable sets of arguments called extensions on the basis of which
argumentation semantics can be defined. We assume that the reader is familiar with
the basic concepts from argumentation theory such as conflict, attack, non-attack,
defeat, evaluation of arguments, etc.
In classical belief revision frameworks, a fixed finite language L with a complete
set of boolean connectives is usually used. Formulae in L are be denoted by lower
case Greek characters α,β,δ,..., while sets of formulae from L will be denoted by
upper case letters A,B,C,.... The underlying logic contains a consequence operator
Cn (Cn : 2L
⇒ 2L
) and it is assumed to be supraclassical (i.e., include classical
propositional calculus), compact (Cn(A) = Cn(B) for some finite subset B of A) and
to satisfy the deduction theorem (α → β ∈ Cn(A) if and only if β ∈ Cn(A∪{α})).
Sometimes, the relation ⊢ is used as an alternative notation of the consequence op-
erator: A ⊢ α if and only if α ∈ Cn(A).
There are many different frameworks for belief revision with their respective
epistemic models. The epistemic model is the formalism in which beliefs are rep-
resented and in which different kinds of operators can be defined. The basic repre-
sentation of epistemic states is through belief sets (sets of sentences closed under
logical consequence) or belief bases (sets of sentences not necessarily closed). Op-
erators may be presented in two ways: by giving an explicit construction (algorithm)
for the operator, or by giving a set of rationality postulates to be satisfied. Rational-
ity postulates determine constraints that the operators should satisfy. They treat the
operators as black boxes by describing their response behavior to inputs in basic
cases, but not the internal mechanisms used.
The distinction between belief sets and belief bases is similar to the distinction
between the coherence approach and the foundational approach to belief revision.
The coherence approach focuses on logical relations among beliefs rather than on
inferential relations, that is, no belief is more fundamental than others [11]. Beliefs
provide each other with mutual support; therefore, a belief set represents the limit
case of this approach. On the other hand, the foundational approach divides beliefs
into two classes: explicit beliefs and those beliefs justified by the explicit beliefs.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-356-320.jpg)
![17 Belief Revision and Argumentation Theory 343
The explicit beliefs can be seen as “self-justified beliefs” whereas the other beliefs
are considered as derived, justified or supported beliefs. The foundational approach
provides explanation of beliefs by requiring that each belief is supportable by means
of non-circular arguments from explicit or basic beliefs [11]. Since a belief α may
be justified by several independent beliefs, if some of the justifications for α are
removed, α may be retained because it is supported by other beliefs. Belief base
revision provides a good example of the foundational approach.
The AGM paradigm [1] has been extensively studied during the last three decades
and it is the most influential model of belief revision so far, serving as a frame of
reference for improvements, extensions or criticisms of the original proposal. The
AGM model is conceived as an idealistic theory of rational change in which epis-
temic states are represented by belief sets and the epistemic input is represented by
a sentence. AGM theory studies the changes at the knowledge level whereas some
others approaches studies the changes at the symbol level. The distinction about
knowledge and symbol level was proposed by Allen Newell [43]. According to
Newell, the knowledge level lies above the symbol level where all of the knowledge
in the system is represented. Some belief bases with different symbol representa-
tions may represent the same knowledge. Suppose that p and q are logically inde-
pendent propositions. Then, K1 = {p,q} and K2 = {p, p → q} are different belief
bases. Since their closure is the same Cn(K1) = Cn(K2) = Cn({p,q}) they repre-
sent the same beliefs at the knowledge level. Although they are statically equivalent
(they represent the same beliefs), they could be dynamically different: changes of
K1 can be different from changes of K2 because p and q are totally independent in
K1 but interrelated in K2 by the sentence p → q.
Suppose that the set K represents the beliefs of an agent. The AGM theory defines
three basic types of change operations for K:
• Expansions: the result of expanding K by a sentence α is a larger set which
infers α;
• Contractions: the result of contracting K by α is a smaller set which does not
infer α;
• Revisions: the result of revising K by α is a set that neither extends nor is part
of the set K. In general, if K infers ¬α then α is consistently inferred from the
revision of K by α.
In the classical AGM framework and all coherence approaches, the internal structure
of beliefs is ignored and the focus is on belief sets. Given a belief set K1 and a
sentence α, the expansion of K by α is denoted by K+α and is defined as K+α =
Cn(K∪{α}).
Suppose that ÷ is a contraction operator. The contraction of K by α is denoted
as K÷α. The basic postulates for contraction are [1, 20] the following:
• Closure: K÷α = Cn(K÷α).
• Inclusion: K÷α ⊆ K.
1 Belief sets (i.e., logically closed sets of sentences) are typically denoted by Latin letters in bold-
face.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-357-320.jpg)
![344 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
• Success: If α then α ∈K÷α.
• Vacuity: If α ∈K then K÷α = K.
• Recovery: K ⊆ (K÷α)+α.
• Extensionality: If ⊢ α ↔ β then K÷α = K÷β.
Recovery is one of the most controversial postulates of the AGM theory: it says that
a sequence of first contracting by a sentence α, and then expanding by the same
sentence α leaves the belief state unchanged. In other words, so much is retained
in a belief set that everything can be recovered by reinstatement of the contracted
sentence.
Revision operators can be defined through Levi identity [38, 20] from contraction
operators; in order to revise a belief set K with respect to a sentence α, we contract
with respect to ¬α and then expand the new epistemic state with respect to α.
That is, the revision of K by α, noted by K∗α, is defined as K∗α = (K÷¬α)+α.
There is a set of basic postulates for revision that correspond to the above postulates
for contraction. One of the most important (though not indisputable) properties is
success (α ∈ K∗α) which specifies that the new information has primacy over the
beliefs of an agent.
2.1 Changes on belief sets
Partial meet contractions cover exactly the basic AGM-contractions. They are based
on the concept of remainder sets.
Definition 17.1. [2] Let K be a set of sentences and α a sentence. Then K⊥α is
the set of all H such that H ⊆ K, H α and if H ⊂ K′ ⊆ K,H = K′, then K′ ⊢ α.
The set K⊥α is called the remainder set of K with respect to α, and its elements are
called the α-remainders of K.
In order to define a partial meet contraction operator, a selection function is
needed. This function makes a selection among the α-remainders, choosing those
candidate sets for contraction which are preferred.
Definition 17.2. Let K be a set of sentences. A selection function for K is a function
γ defined on {K⊥α|α ∈ L} such that for any sentence α ∈ L, it holds that:
1) If K⊥α = ∅, then γ(K⊥α) is a non-empty subset of K⊥α.
2) If K⊥α = ∅, then γ(K⊥α) = K.
Definition 17.3. Given a set of sentences K, a sentence α and a selection function
γ for K, the partial meet contraction of K by α, denoted by K÷γα, is defined as
K÷γα = ∩γ(K⊥α). That is, K÷γα is equal to the intersection of the α-remainders
of K selected by γ. If γ selects a single element, then the induced contraction is
called maxichoice contraction. If γ selects all elements of the remainder set, then
the induced contraction is called full meet contraction.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-358-320.jpg)
![17 Belief Revision and Argumentation Theory 345
Representation theorems (also called axiomatic characterizations) characterize
an operation in terms of axioms or postulates. They are widely used in belief re-
vision in order to show an interrelation between constructions and postulates. The
following is a representation theorem for partial meet contractions.
Theorem 17.1. [1, 20] Let K be a belief set and ÷ be a contraction operator for
K. Then ÷ is a partial meet contraction for K if and only if ÷ satisfies closure,
inclusion, success, vacuity, recovery and extensionality.
Partial meet revisions are defined from partial meet contractions using Levi iden-
tity. On the other hand, partial meet contractions on belief sets can be defined from
partial meet revisions using the Harper identity [20]: K÷α = K∩(K∗¬α). A rep-
resentation theorem for partial meet revisions on belief sets can be found in [1].
2.2 Changes on belief bases
Following the seminal work of the AGM trio, almost all works on belief revision
employed models in which epistemic states are represented by theories or belief sets,
that is, set of sentences closed under logical consequence. However, it was observed
by Alchourrón and Makinson [2] that “the intuitive processes of contraction and
revision, contrary to casual impressions, are never really applied to theories as a
whole, but rather to more or less clearly identified bases for them”. Perhaps inspired
by this observation, operations on belief bases have been investigated, among other,
by Hansson [26, 29, 30], Fuhrmann [17], and Nayak [42].
Fuhrmann [17] defined a contraction operator − on a belief base K and then
studied the properties of its related contraction operator ÷ on belief sets K = Cn(K)
defined as K÷α = Cn(K−α). That is, the definition of K÷α actually depends on
K and α. Fuhrmann proposed a set of postulates for the operator − on belief bases
based on the basic postulates on belief sets proposed by Gärdenfors, showing that
recovery does not hold on belief bases.
Hansson proposed an alternative approach, studying the changes on belief bases
and not on their associated belief sets. He proposed an alternative contraction, called
kernel contraction. Kernel contractions [31] are based on a selection among the
sentences that imply the information to be retracted and they are a natural non-
relational generalization of safe contractions [3].
Definition 17.4. [31] Let K be a set of sentences and α a sentence. Then K⊥
⊥α is
the set of all X such that X ⊆ K, X ⊢ α, and if K′ ⊂ X, then K′ α. The set K⊥
⊥α
is called the kernel set, and its elements are called the α-kernels of K.
In order to retract some information α from K, some sentences in each α-kernel
must be erased. This is done by incision functions.
Definition 17.5. [31] Let K be a set of sentences. An incision function for K is
a function σ defined on {K⊥
⊥α|α ∈ L} such that for any sentence α ∈ L, the
following statements hold:](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-359-320.jpg)
![346 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
1) σ(K⊥
⊥α) ⊆ ∪(K⊥
⊥α).
2) If X ∈ K⊥
⊥α and X = ∅ then (X ∩σ(K⊥
⊥α)) = ∅.
In the limit case when K⊥
⊥α = ∅ then σ(K⊥
⊥α) = ∅.
So, incision functions cut into each α-kernel, removing at least one sentence.
Since all α-kernels are minimal subsets implying α, from the resulting sets it is no
longer possible to derive α. Hence, incision functions may be used to derive (kernel)
contraction operations.
Definition 17.6. [31] Given a set of sentences K, a sentence α and an incision
function σ for K, the kernel contraction of K by α, denoted by K÷σ α, is defined
as: K÷σ α = K σ(K⊥
⊥α).
That is, K÷σ α can be obtained by erasing from K the sentences cut out by σ.
In the following, we recall a set of postulates for contraction that can be consid-
ered on every arbitrary set of sentences K:
• Inclusion: K÷α ⊆ K.
• Success: If α ∈Cn(∅) then α ∈Cn(K÷α).
• Vacuity: If α ∈Cn(K) then K÷α = K.
• Core-Retainment [27]: If β ∈ K and β ∈K÷α then there is a set K′ such that
K′ ⊆ K such that α ∈Cn(K′) but α ∈ Cn(K′ ∪{β}).
• Uniformity [28]: If it holds for all subsets K′ of K that α ∈ Cn(K′) if and only
if β ∈ Cn(K′) then K÷α = K÷β.
• Relative Closure [27]: K ∩Cn(K÷α) ⊆ K÷α.
The following is a representation theorem for kernel contractions on belief bases.
Theorem 17.2. [31] Let K a belief base and ÷ be a contraction operator for K.
Then ÷ is a kernel contraction for K if and only if ÷ satisfies inclusion, success,
core-retainment and uniformity.
Smooth kernel contractions are kernel contractions that satisfy relative closure.
Hansson showed that smooth kernel contractions and partial meet contractions are
equivalent on belief sets. That is, smooth kernel contractions and partial meet con-
tractions are just two different ways to construct the same class of operations on
belief sets. However, kernel contractions are more general than partial meet con-
tractions on belief bases. More relations about partial meet and kernel contractions
can be found in [14]. The textbook [32] gives a good overview on axiomatic work
in general (belief) base revision.
2.3 Alternative approaches and iterated revision
There are other approaches to belief change realizing AGM change operations, for
instance, Grove’s sphere system [23], epistemic entrenchment [22], and safe con-
traction [3]. Grove’s sphere system is very similar to the sphere semantics for coun-
terfactuals proposed by Lewis [39]. Concentric spheres allow the ordering of pos-
sible worlds according to some notion of similarity or the like. Then, the result of](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-360-320.jpg)
![17 Belief Revision and Argumentation Theory 347
revising by α can be semantically described by the set of α-models that are clos-
est to the K-models. Epistemic entrenchment allows that a revision can be based
on a relation ≤ among sentences that when forced to choose between two beliefs,
an agent will give up the less entrenched one. Gärdenfors and Makinson [22] pro-
posed an epistemic entrenchment relation from which it is possible to define AGM
contractions.
Belief updating [33] is a change operation of a different kind. Whereas in belief
revision, both the old beliefs and the new information refer to the same situation, in
updating the new information is about a current, possibly changed situation. Recent
work relates update to actions [37]. Much current work in belief revision focuses
on iterated revision that deals with changing epistemic relations such as epistemic
entrenchment [7]. For a recent paper that considers iterated belief (base) revision in
a broad epistemic framework, cf. [36].
3 Related work
Before entering into the discussion on the backgrounds of belief revision and argu-
mentation theories, and possible connections in between, we will give an overview
over work addressing this broad topic.
3.1 Truth Maintenance Systems [1979–1986]
Doyle [10] presents initially the truth maintenance system, or TMS, which is a
knowledge representation method for representing both beliefs and their justifica-
tions. The aim of such systems is to restore consistency when a new justification
has been added. A TMS associates a special data structure, called a node, with each
problem solver datum which includes database entries, inference rules, and proce-
dures. It records justifications, i.e., arguments, for potential inferred beliefs, so as
to compute the current set of beliefs by manipulating the status of nodes repre-
senting beliefs and evaluating justifications that represent reasons to believe. The
truth maintenance process starts when a new justification for a node is added, and
runs through basic steps of argumentation, such as evaluating the justifications to
determine the status of nodes and checking justifications and contradictions, while
avoiding circular relations among justifications and nodes.
A different type of truth maintenance systems was proposed by de Kleer’s
assumption-based truth maintenance systems (ATMS)[8]. Instead of evaluating jus-
tifications in a global process, ATMS label each datum with the corresponding sets
of assumptions, representing the contexts under which it holds. These assumption
sets are computed by the ATMS from the justifications that are supplied by the prob-
lem solver. The idea is that the assumptions provide contexts from which beliefs can
be derived and hence may serve as arguments. The assumptions are primitive data,](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-361-320.jpg)
![348 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
and all other data can be derived from them. There is no necessity that the overall
database be consistent; it is easy to refer to contexts, and moving to different points
in the search space requires very little effort.
3.2 Inconsistencies from multiple sources [1995]
Benferhat et al. [4] present an article primarily oriented towards the treatment of in-
consistency caused by the use of multiple sources of information. Knowledge bases
are stratified, namely each formula in the knowledge base is associated with its level
of certainty corresponding to the layer to which it belongs.
The authors investigate two classes of approaches to deal with inconsistency in
knowledge bases: coherence theories and foundation theories. The first insists on
revising the knowledge base and restoring consistency, while the latter accepts in-
consistency and copes with it. Coherence theories propose to give up some formulas
of the knowledge base in order to get one or several consistent subbases, and to apply
classical entailment on these consistent subbases to deduce plausible conclusions of
the knowledge base. Foundation theories proceed differently since they retain all
available information but each plausible conclusion inferred from the knowledge
base is justified by some strong argumentative reason for believing in it.
The claim of the paper is that it does not always make sense to revise an inconsis-
tent knowledge base, in particular, if the information comes from multiple sources.
It is not even necessary to restore consistency in order to make sensible inferences
from an inconsistent knowledge base, since inference based on argumentation can
derive conclusions and reasons to believe them, independently of the consistency of
the knowledge base.
3.3 Belief revision and epistemology [2000]
Pollock and Gillies [46] studied the dynamics of a belief revision system considering
relations among beliefs in a “derivational approach” trying to derive a theory of
belief revision from a more concrete epistemological theory. According to them, one
of the goals of belief revision is to generate a knowledge base in which each piece
of information is justified (by perception) or warranted by arguments containing
previously held beliefs.
The difficulty is that the set of justified beliefs can exhibit all kinds of logical
incoherencies because it represents an intermediate stage in reasoning. Therefore,
the authors propose a theory of belief revision concerned with warrant rather than
justification. The set of warranted propositions already takes account of all possible
inferences, so there is only one way to acquire new warranted propositions: through
perception. However, perception adds a percept to the inference-graph, not a belief,](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-362-320.jpg)
![17 Belief Revision and Argumentation Theory 349
so the effect of perception on warrant cannot be represented as the addition of a
proposition to a belief state.
3.4 Deductive explanations and belief revision [2002]
Falappa et al. [16] present a kind of non-prioritized revision operator based on
the use of explanations. The idea is that an agent, before incorporating informa-
tion which is inconsistent with its knowledge, requests an explanation supporting it.
The authors build on the classical distinction between the explanandum, which is
the final conclusion, and the explanans, represented by a set of sentences supporting
the conclusion. The structure of an explanation is very similar to the structure of
a deductive argument; the main difference is that every belief of an explanation is
undefeasible (in a moment of time) whereas some beliefs of an argument may be
defeasible or tentatively inferred.
In this framework, every explanation contains rules and factual knowledge. If
the sentences in the explanans are better or more plausible than the sentences in
the original belief base, then the explanation is incorporated. Therefore, not beliefs,
but explanations (and hence arguments) supporting a belief are used for the change
process. The authors considered both kernel and partial meet revision by a set of
sentences and gave representation theorems for them. These operators may partially
accept the new information, so they are non-prioritized.
3.5 Data-oriented Belief Revision [2006]
Paglieri and Castelfranchi [45] join argumentation and belief revision in the same
conceptual framework by following Toulmin’s layout of argumentation, which is
intended to be used to analyze the rationality of arguments typically found in court-
rooms. The connection to belief revision is made by considering argumentation as
“persuasion to believe”, hence argumentation is supposed to initiate successful re-
vision processes. The authors propose Data-oriented Belief Revision (DBR) as an
alternative to the AGM approach. Two basic informational categories, data and be-
liefs, are put forward in their approach, to account for the distinction between pieces
of information that are simply gathered and stored by the agent (data), and pieces of
information that the agent considers (possibly up to a certain degree) truthful rep-
resentations of states of the world (beliefs). The beliefs are a subset of the data: an
agent might well be aware of a datum that he does not believe (i.e., he does not con-
sider reliable enough). Data structures are conceived as networks of nodes (data),
linked together by the relations of support, contrast and union. Data are selected (or
rejected) as beliefs on the basis of their properties, described by relevance, credibil-
ity, importance, and likeability.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-363-320.jpg)
![350 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
According to Paglieri and Castelfranchi [45], the Toulmin’s layout schema is li-
able of immediate implementation in DBR, since it uses a specific data structure.
The union of data and warrant supports the claim, and the warrant is in turn sup-
ported by its backing and contrasted by the rebuttal, i.e., supports of the rebuttal
make the warrant less reliable.
3.6 Merging Dung argumentation systems [2007]
Coste-Marquis et al. [6] proposed a general framework for merging Dung’s style
argumentation systems. They presented a framework for deriving reasonable infor-
mation from a collection of Dung argumentation systems. Their approach consists
in merging such systems assuming that all agents do not share the same sets of ar-
guments. No assumption is made concerning the meaning of the attack relations,
so that such relations may differ not only because agents have different points of
view on the way arguments interact but more generally may disagree on what an
interaction is.
Every argumentation system is expanded to a partial argumentation system, and
such partial systems are built over the same set of arguments. Merging is used on
the expanded systems as a way to solve the possible conflicts between them, and a
set of argumentation systems which are as close as possible to the whole profile is
generated. Finally, the last step consists in selecting the acceptable arguments at the
group levels from the set of argumentation systems.
3.7 Prioritized revision by arguments [2008]
Rotstein et al. [47] introduce an abstract theory that captures the dynamics of an
argumentation framework through the application of belief revision concepts. They
define a dynamic abstract argumentation theory including dialectical constraints,
and then present argument revision techniques to describe the fluctuation of the
set of active arguments (the ones considered by the inference process of the theory).
Expansion, contraction, and revision operators are realized in this framework, where
the revision can be expressed in terms of the other two, leading to an identity similar
to the one defined by Isaac Levi [38]. Their abstract theory allows the introduction
of an argument ensuring it can be believed afterwards. The expansion operator is
quite straightforward, but the definition of the contraction operator allows a wide
range of possibilities from affecting unrestrictedly any number of arguments in the
system to keeping this perturbation to a minimum, following the minimal change
principle of the AGM theory. Contraction may also have an indirect impact on the
attack relation among arguments.
In a second paper, Moguillansky et al. [41] proposed an instantiation of these
change operators to Defeasible Logic Programming, DeLP [19]. In particular, a](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-364-320.jpg)
![17 Belief Revision and Argumentation Theory 351
warrant-prioritized argument revision operator (WPA) is defined that implements
successful change. To be more precise, when a program is revised by an argument
A,α (where A is an argument for α), the revised program will be such that A is
an undefeated argument, and α will therefore be warranted. The main issue under-
lying WPA revision lies in the selection of arguments and the incisions that have to
be made over them. An argument selection criterion determines which arguments
should not be present and once this selection is made, incisions (in the form of
deletion of rules) will make those arguments “disappear” following some minimal
change principle.
3.8 Adding arguments to Dung systems [2008]
Cayrol et al. [5] proposed a Dung-style abstract argumentation system that allows
the addition of a new argument which may interact with previous arguments. An
argumentation framework A,R is identified with an associated attack graph G.
The revision process produces a new framework represented by a graph G′ and a
new set of extensions. By considering how the set of extensions is modified under
the revision process, the authors propose a typology of different revisions including
decisive revision, when there is only one acceptable set of arguments in the revised
framework, and expansive revision, when the revision simply adds the new argument
to the existing extensions.
3.9 Relating reinstatement and recovery [2008]
Boella et al. [24] try to show a direct relation between argumentation and belief
revision on the level of abstract properties. Like Paglieri and Castelfranchi [45],
the authors also consider argumentation as persuasion to believe, and persuasion
should be related to belief revision. They establish a link between reinstatement
and recovery. Reinstatement plays an important role in argumentation; it refers to
the situation that an argument that is not acceptable because of the existence of
an attacking argument become acceptable again when an attacker to the attacking
argument exists. Recovery reflects the central idea of minimal change in AGM belief
revision; according to it, expansion by α should recover what was lost when α was
contracted.
More recently, Boella et al. [25] present the application of a belief revision ma-
chinery to argumentation in a multi-agent system. The authors define an argument
base revision that gives priority to the last introduced argument. As they are inter-
ested particularly in the persuasive power of argumentation, they express appealing-
ness of arguments in terms of belief revision.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-365-320.jpg)
![17 Belief Revision and Argumentation Theory 353
query “Suppose α holds, would you believe β?”, whatever α and β are, reasoning
would include a hypothetical change process that the current epistemic state of the
agent must undergo.
From this embedding into a complex reasoning process, the complementary char-
acters of argumentation and belief revision become obvious: while argumentation
can make substantial contributions to the evaluation step, belief revision theory
should be employed in the belief change part. But this is not the end of the story.
Evaluation might include hypothetical change processes, considering what would
happen if the new information were to be believed, and belief change implicitly
relies on logical links between pieces of information which can be represented by
arguments. In the end, both from argumentation processes and from belief revision
processes, plausible beliefs can be obtained, but both areas focus only on parts of
the dynamic reasoning process while at the same time providing general and ver-
satile frameworks. Revision operators can not only be applied to beliefs, but also
to intentions, preferences, theories, ontologies, law codes, etc. Argumentation can
be used for negotiation (when the agents have conflicting interests and they try to
make the best out of a deal for themselves), inquiry (when the agents have general
ignorance in some subject and they try to find a proof or destroy one), deliberation
(when the agents collaborate to decide what course of action to take) or information
seeking (when the agents have personal ignorance on some subject and every agent
seeks the answer to some questions from other agents). This is a much more general
view than considering argumentation as persuasion to believe, as has been proposed
in [45, 25].
These contemplations result in a most complex, highly interrelated view on ar-
gumentation and belief revision. Consequently, discussing links between both fields
must reflect this complexity, taking into regard that proposed frameworks on either
side might only implement some but not all aspects of the corresponding field. In
particular, the famous AGM theory [1] in belief revision is more concerned with
judging the results of change in a very abstract way than with the change process
itself; moreover, argumentative evaluation of justifications to believe the given in-
formation in the sense described above is not at all a topic of AGM. That is why suc-
cess is one of the basic postulates of the AGM theory according to which the agent is
forced to believe the new information. Here, it is implicitly assumed that evaluation
has been done beforehand. Hence, a comparison between AGM and some approach
that makes use of a sophisticated argumentative evaluation of information to select
beliefs, as has been developed e.g. in [45], is very likely to shed a bad light on AGM.
AGM is no more on argumentative reasons for belief than is Dung’s framework [13]
on change processes. Both are highly abstract, declarative approaches to the respec-
tive field, hence (necessarily) over-simplifying in some respects but nevertheless
extremely valuable as reference points.
Therefore, an investigation of the connections between belief revision and ar-
gumentation theory that is to do justice to both areas must go beneath the surface
of abstract frameworks. It must study methods and the rationale underlying these
methods, as well as purposes and intentions guiding the application of techniques.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-367-320.jpg)
![354 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
We will first identify characteristic and prototypic concepts on either side and find
basic correspondences as well as crucial differences. Afterwards, we will investigate
how either domain may enrich the other domain’s framework.
4.2 Comparing belief revision and argumentation
At first glance, the differences between argumentation and belief revision prevail.
To begin with representational issues, the syntactic and semantic foundations of
both areas have not much in common. In standard belief revision, logical formulas
are used for knowledge representation, and the result of change processes are logi-
cal formulas. In more advanced frameworks, epistemic states are changed in which
some qualitative relation allows for ranking worlds or sentences with respect to en-
trenchment, plausibility, and the like. In order to verify the results of the change
processes, classical logical semantics is used. On the contrary, argumentation the-
ory focuses on the interactions of arguments as pieces of information that may attack
one another, and a relation between arguments may give priority to one argument
or another. The arguments themselves, however, are very heterogeneous. In most
approaches, they are quite complex argument structures built up by some kind of
rules, using standard or non-standard derivation to implement reasoning. However,
they might as well be abstract objects as in Dung’s framework [13] without any in-
ternal structure. A special semantics makes precise what good arguments are, and
there are several such semantics possible, implemented by preferred, stable, and
grounded extensions.
As to the common grounds, both disciplines aim at resolving conflicts which are
usually based on logical grounds, i.e., on contradictions, and make use of prefer-
ence relations to achieve this aim. However, belief revision theory provides a highly
declarative framework for that, based on postulates, whereas argumentation theory
is more concerned with practical, justification-based techniques.
For a direct comparison of argumentation and belief revision, it is tempting to
take the standard AGM approach [1], as no other work has influenced the develop-
ment of belief revision theory in a similar way. And the AGM postulates offer a clear
framework that makes fundamental views explicit. However, one of the most basic
assumptions of the standard AGM theory, namely its focus on deductively closed
sets of formulas as representing belief sets, does not fit at all to argumentation the-
ory, as it abstracts from the basic steps of reasoning, blurring the distinction between
what is explicitly given or serves as assumptions, and what is justified belief. Hence,
works on belief base revision (e.g. [32, 14]), or on iterated epistemic change (e.g.
[7, 34, 35, 9]) offer a much richer base for comparison, since they allow one to take
deeper insights into change processes.
Belief revision methods and most argumentation frameworks can be used directly
for reasoning, although this is not their principal concern. The connection between
belief revision and nonmonotonic inference has been made clear by [40] and further
developed for iterative change operations in [36]; this is the BRDI view on belief](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-368-320.jpg)
![17 Belief Revision and Argumentation Theory 355
revision, as Dubois [12] calls it. As to argumentation, those approaches that are
based on rules or use some sort of derivation provide logical chains that establish
links between what is presupposed and what is concluded. Also with respect to that
aspect, belief revision theories remain on an abstract level, describing by axioms
what good inferences are, while argumentation is more concerned with how and
why conclusions are drawn, making reasons for belief apparent.
These considerations make obvious that belief revision and argumentation theory
are basically complementary areas of research. In the following, we will discuss how
they may complement each other.
4.3 Argumentation in Belief Revision
In this subsection we study how argumentation concepts and techniques can be used
in belief revision theory. Among the works relating argumentation and belief revi-
sion, many of them add belief revision capabilities to an argumentation system.
However, there are some papers that propose to use an “argumentative machin-
ery” for a belief revision process, like e.g. [46, 45] (see also section 3). The first
approach to use argumentation in belief revision is justification-based truth mainte-
nance systems (TMS’s) proposed by Doyle [10] (cf. section 3.1). Doyle studies the
interactions between justifications when a new justification has been added, to find
out which conclusions can be justified. He does not consider retraction of justifica-
tions. Assumption-based truth maintenance systems (ATMS) [8] are more concerned
with managing assumptions instead of implementing change processes. As in DBR
[45], we have data and beliefs: a data is believed in some contexts represented by
assumptions sets. As in [4], there is no necessity that the overall database be consis-
tent; it is easy to refer to contexts and move to different points in the search space
representing the proper context. Both types of truth maintenance systems work on
belief bases and are very early approaches to belief revision, dating before AGM
theory.
In [16], we combine the ATMS idea with base revision and propose a system
that uses argumentative structures in the form of explanations for nonprioritized
revisions of belief bases K. Different from standard approaches to belief revision,
the new information not only consists of a proposition α but of reasons to believe
a proposition, i.e., of rules and prerequisites A from which α can be deductively
derived. So, A can be considered an explanans for the explanandum α. In order
to integrate an argumentative evaluation of the new information into the revision
process, we defined partial acceptance revision operators in the following way: i)
the epistemic input is the set of sentences A as explanans for the explanandum α, ii)
A is initially accepted, that is, A is joined to K (possibly producing an inconsistent
intermediate state), iii) all possible inconsistencies of K ∪A are removed, returning a
consistent revised belief base K ◦A. This operator is an operator of external revision.
The name “external” indicates that the revision process takes place outside of the
original set.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-369-320.jpg)
![356 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari
Whether α is accepted or not depends on the evaluation of its explanans A with
respect to the current belief base. The acceptance of the explanans forces the ac-
ceptance of the explanandum in the revised set, since explanation here is based on
classical deduction. However, there is an important remark to be made regarding the
degree of acceptance of the explanans and the explanandum. While the explanans
can be explicitly included in the revised set, the explanandum may be inferred from
it without actually being included. So, the distinction between explicitly given in-
formation (as data in [45]) and inferred beliefs is respected and can be implemented
only when working with belief bases instead of belief sets.
The main difference between the process of revision by a set of sentences and the
process of argumentation is significant: in revision, external beliefs are compared
with internal beliefs and, after a selection process, some sentences are discarded,
other ones are accepted. In argumentation, the process is more procedural: we have
an argument, attack this by counterarguments, defend it by counterarguments to
counterarguments, and so on. Nevertheless, the rationale behind partial acceptance
revision operators matches the intentions leading argumentation in that not the new
information itself, but reasons to believe it, are evaluated.
4.4 Belief Revision in Argumentation
Belief revision theory offers nice methods to implement dynamical features in an
argumentation framework. The papers by Rotstein, Moguillansky et al. [47, 41] and
Boella et al. [25] are among the most comprehensive approaches to address such
a revision theory for argument systems (cf. section 3). Several different ways of
applying belief revision in argumentation may be distinguished:
• Changing by adding or deleting an argument.
• Changing by adding or deleting a set of arguments.
• Changing the attack (and/or defeat) relation among arguments.
• Changing the status of beliefs (as conclusions of arguments).
• Changing the type of an argument (from strict to defeasible, or vice versa).
The distinction between the two first ways is similar to the distinction among sin-
gle change (as in AGM model) and multiple change (as in multiple contraction
[18, 44]). Adding or removing an argument or a set of arguments may trigger a
change in the justified conclusions. The consideration of such changes may lead
to a base revision theory which deals with changes of argumentative bases, and in
which deduction is replaced by an argumentation process. The method of kernel
contraction making use of incision functions to eliminate base elements (cf. section
2) is of particular interest here; first steps towards this direction can be found in [41].
Changing the attack and/or defeat relation among arguments may lead to a com-
pletely different behavior of the system, the understanding and the control of which
constitutes substantially new challenges for belief revision theory. Boella et al. [25]
have proposed an approach that changes the attack relation in a dictatorial way in](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-370-320.jpg)
![17 Belief Revision and Argumentation Theory 357
favor of the new information. More sophisticated change processes are conceivable,
taking ideas from the methods for epistemic belief change (or iterated belief change)
that deal with modifying relations on possible worlds (cf. e.g. [7, 35]). Although ar-
guments are very different from possible worlds, those works might be considered
as approaches to realize minimal change processes for general relations. However,
before applying those techniques from epistemic belief revision to argumentative at-
tack or defeat relations, a careful investigation has to be done concerning conceptual
differences between both frameworks.
Changing the status of beliefs may be a consequence of previous changes to the
argument system. In classical, Dung-style argument systems, the criterion to be a
justified belief is based on extensions. Cayrol et al. [5] have studied how extensions
change when new arguments are added. In argumentation frameworks where ar-
gumentation is based on conclusions (e.g. DeLP [19]), the addition of an argument
may change the status of a claim from unwarranted to warranted and vice versa. This
touches basic concerns of the reasoning aspect of belief revision, and investigations
on the level of belief revision postulates might be very useful. The link between
reinstatement (in argumentation) and recovery (in belief revision) proposed in [24]
is in this line of research.
Finally, changing the type of an argument from strict to defeasible, or vice versa,
addresses novel issues in belief revision. Such modifications are not dealt with prop-
erly by chaining retraction and addition of arguments, as the old and the new ar-
gument are linked by a syntactic or semantic relation which gets lost when these
operations are carried out independently. The basic idea is that inconsistencies that
arise when new information has to be incorporated into the stock of beliefs can be
eliminated not only by removing arguments (resp. beliefs), but by weakening strict
beliefs to defeasible rules (resp. conditionals). The procedure of dynamic classifica-
tion of generic rules has been frequently used in the evolution of humanity’s knowl-
edge. The possibility of changing the status of beliefs from undefeasible to defea-
sible induces revision operations of a new quality, with important consequences for
argumentation, as arguments are formed very often by defeasible beliefs. In [16],
we have introduced a new type of base revision that implements such a dynamic
classification of beliefs and is likewise interesting for argumentation and belief re-
vision theory. We proposed a framework in which defeasible conditionals can be
generated by revising belief structures composed of defeasible rules and undefeasi-
ble knowledge. The approach preserves consistency in the undefeasible knowledge
and it provides a mechanism to dynamically qualify the beliefs as undefeasible or
defeasible, providing a more complete set of epistemic attitudes and extending the
inference power of knowledge based systems. As a particular case, Falappa et al.
[15] propose to extend the application of this non-prioritized revision operator to
Defeasible Logic Programming (DeLP) [19].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-371-320.jpg)
![17 Belief Revision and Argumentation Theory 359
complexity than each of the areas can. Moreover, their diverseness makes it possi-
ble that each of argumentation and belief revision provides enriching aspects for the
respective other discipline beyond any differences. We pointed out various starting
points for work in this direction, and proposed future lines of research along the
borderline between argumentation and revision.
Acknowledgements This research was funded by Consejo Nacional de Investigaciones Cientı́ficas
y Técnicas (CONICET), Agencia Nacional de Promocion Cientı́fica y Tecnológica (ANPCyT),
Universidad Nacional del Sur (UNS), Ministerio de Ciencia y Tecnologı́a (MinCyT) [Argentina]
and Fundação para a Ciência e a Tecnologia (FCT) [Portugal].
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monotonic Logic. Lecture Notes in Computer Science, 465:183–205, 1991.
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Theory Change: Revision Upon Warrant. In Proceedings of The Twenty-Third Conference on
Artificial Intelligence, AAAI 2008, pages 132–137, 2008.
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Revision Theories, pages 359–377. Berlin, Springer, 2006.
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Theory Change: Revision upon Warrant. In Proceedings of The Computational Models of
Argument, COMMA 2008, pages 336–347, 2008.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-374-320.jpg)
![364 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor
out in his spare time? What is the precise nature of the agreements entered into?
Once an interpretation has been selected, the argument must be organised into the
form considered most likely to persuade, both to advocate the client’s position and
to rebut anticipated objections. Some precedents may point to one result and others
to another. In that case, further arguments may be produced to suggest following the
favourable precedent and ignoring the unfavourable one. Or the rhetorical presenta-
tion of the facts may prompt one interpretation rather than the other. Surely all this
requires the skill, experience and judgement of a human being? Granted that this is
true, much effort has been made to design computer programs that will help people
in these tasks, and it is the purpose of this chapter to describe the progress that has
been made in modelling and supporting this kind of sophisticated legal reasoning.
We will review1 systems that can store conflicting interpretations and that can
propose alternative solutions to a case based on these interpretations. We will also
describe systems that can use legal precedents to generate arguments by drawing
analogies to or distinguishing precedents. We will discuss systems that can argue
why a rule should not be applied to a case even though all its conditions are met.
Then there are systems that can act as a mediator between disputing parties by struc-
turing and recording their arguments and responses. Finally we look at systems that
suggest mechanisms and tactics for forming arguments.
Much of the work described here is still research: the implemented systems are
prototypes rather than finished systems, and much work has not yet reached the stage
of a computer program but is stated as a formal theory. Our aim is therefore to give
a flavour (certainly not a complete survey) of the variety of research that is going
on and the applications that might result in the not too distant future. Also for this
reason we will informally paraphrase example inputs and outputs of systems rather
than displaying them in their actual, machine-readable format; moreover, because
of space limitations the examples have to be kept simple.
2 Early systems for legal argumentation
In this section we briefly discuss some early landmark systems for legal argumenta-
tion. All of them concern the construction of arguments and counterarguments.
2.1 Conflicting interpretations
Systems to address conflicting interpretations of legal concepts go back to the very
beginnings of AI and Law. Thorne McCarty (e.g. [25, 27]) took as his key problem
a landmark Supreme Court Case in US tax law which turned on differing interpreta-
tions of the concept of ownership, and set himself the ambitious goal of reproducing,
1 This chapter is a revised and updated version of [6].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-378-320.jpg)
![18 Argumentation in Legal Reasoning 365
in his TAXMAN system, both the majority and the dissenting opinions express-
ing these interpretations. This required highly sophisticated reasoning, constructing
competing theories and reasoning about the deep structure of legal concepts to map
the specific situation onto paradigmatic cases. Although some aspects of the system
were prototyped, the aim was perhaps too ambitious to result in a working sys-
tem, certainly given the then current state of the art. This was not McCarty’s goal,
however: his motivation was to gain insight into legal reasoning through a computa-
tional model. McCarty’s main contribution was the recognition that legal argument
involves theory construction as well as reasoning with established knowledge. Mc-
Carty [26] summarises his position as follows: “The task for a lawyer or a judge in
a “hard case” is to construct a theory of the disputed rules that produces the desired
legal result, and then to persuade the relevant audience that this theory is preferable
to any theories offered by an opponent” (p. 285). Note also the emphasis on persua-
sion, indicating that we should expect to see argumentation rather than proof. Both
the importance of theory construction and the centrality of persuasive argument are
still very much part of current thinking in AI and Law.
Another early system was developed by Anne Gardner [15] in the field of of-
fer and acceptance in American contract law. The task of the system was “to spot
issues”: given an input case, it had to determine which legal questions arising in
the case were easy and which were hard, and to solve the easy ones. The system
was essentially rule based, and this simpler approach offered more possibilities for
practical exploitation than did McCarty’s system. One set of rules was derived from
the Restatement of Contract Law, a set of 385 principles abstracting from thousands
of contract cases. These rules were intended to be coherent, and to yield a single
answer if applicable. This set of rules was supplemented by a set of interpretation
rules derived from case law, common sense and expert opinion, intended to link
these other rules to the facts of the case. Gardner’s main idea was that easy ques-
tions were those where a single answer resulted from applying these two rule sets,
and hard questions, or issues, were either those where no answer could be produced,
because no interpretation rule linked the facts to the substantive rules, or where con-
flicting answers were produced by the facts matching with several rules. Some of
the issues were resolved by the program with a heuristic that gives priority to rules
derived from case law over restatement and commonsense rules. The rationale of
this heuristic is that if a precedent conflicts with a rule from another source, this is
usually because that rule was set aside for some reason by the court. The remaining
issues were left to the user for resolution.
Consider the following example, which is a very much simplified and adapted
version of Gardner’s own main example2. The main restatement rule is
R1: An offer and an acceptance constitute a contract
Suppose further that there are the following commonsense (C) and expert (E) rules
on the interpretation of the concepts of offer and acceptance:
2 We in particular abstract from Gardner’s refined method for representing knowledge about
(speech act) events.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-379-320.jpg)
![366 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor
C1: A statement “Will supply ...” in reply to a request for offer is an offer.
C2: A statement “Will you supply ...” is a request for offer.
C3: A statement “I accept ...” is an acceptance.
E1: A statement “I accept” followed by terms that do not match the terms of
the offer is not an acceptance.
Suppose that Buyer sent a telegram to Seller with “Will you supply carload salt at
$2.40 per cwt?” to which Seller replied with “Will supply carload at $2.40, terms
cash on delivery”, after which Buyer replied with her standard “Purchase Order”
indicating “I accept your offer of 12 July” but which also contained a standard pro-
vision “payment not due until 30 days following delivery”.
Applying the rules to these events, the “offer” antecedent of R1 can be estab-
lished by C1 combined with C2, since there are no conflicting rules on this issue.
However, with respect to the “acceptance” antecedent of R1 two conflicting rules
apply, viz. C3 and E1. Since we have no way of giving precedence to C3 or E1, the
case will be a hard one, as there are two conflicting notions of “acceptance”. If the
case is tried and E1 is held to have precedence, E1 will now be a precedent rule, and
any subsequent case in which this conflict arises will be easy, since, as a precedent
rule, E1 will have priority over C3.
2.2 Reasoning with precedents
The systems described in the last section do recognise the importance of precedent
cases as a source of legal knowledge, but they make use of them by extracting the
rationale of the case and encoding it as a rule. To be applicable to a new case,
however, the rule extracted may need to be analogised or transformed to match
the new facts. Nor is extracting the rationale straightforward: judges often leave
their reasoning implicit and in reconstructing the rationale a judge could have had
in mind there may be several candidate rationales, and they can be expressed at a
variety of levels of abstraction. These problems occur especially in so-called ‘factor-
based’ domains, i.e., domains where problems are solved by considering a variety of
factors that plead for or against a solution. In such domains a rationale of a case often
just expresses the resolution of a particular set of factors in a specific case. A main
source of conflict in such domains is that a new case often will not exactly match a
precedent but will share some features with it, lack some of its other features, and/or
have some additional features. Moreover, cases are more than simple rationales:
matters such as the context and the procedural setting can influence the way the case
should be used. In consequence, some researchers have attempted to avoid using
rules and rationales altogether, instead representing the input, often interpreted as
a set of factors, and the decisions of cases, and defining separate argument moves
for interpreting the relation between the input and decision (e.g. [23, 1], both to be
discussed below). This approach is particularly associated with researchers in US,
where the common law tradition places a greater stress on precedent cases and their
particular features than is the case with the civil law jurisdictions of Europe. None](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-380-320.jpg)
![18 Argumentation in Legal Reasoning 367
the less cases are also used in civil law jurisdictions and the reasoning techniques
are similar.
The most influential system of this sort is HYPO [2], developed by Edwina Riss-
land and Kevin Ashley in the domain of US Trade Secrets Law, which can be seen
as a factor-based domain. In HYPO cases are represented according to a number
of dimensions. A dimension is some aspect of the case relevant to the decision, for
example, the security measures taken by the plaintiff. One end of the dimension
represents the most favourable position for the plaintiff (e.g. specific non-disclosure
agreements), while the other end represents the position most favourable to the de-
fendant (e.g. no security measures at all). Typically a case will lie somewhere be-
tween the two extremes and will be more or less favourable accordingly. HYPO
then uses these dimensions to construct three-ply arguments. First one party (say
the plaintiff) cites a precedent case decided for that side and offers the dimensions
it shares with the current case as a reason to decide the current case for that side. In
the second ply the other party responds either by citing a counter example, a case
decided for the other side which shares a different set of dimensions with the cur-
rent case, or distinguishing the precedent by pointing to features which make the
precedent more, or the current case less, favourable to the original side. In the third
ply the original party attempts to rebut the arguments of the second ply, by distin-
guishing the counter examples, or by citing additional precedents to emphasise the
strengths or discount the weaknesses in the original argument.
Subsequently Ashley went on, with Vincent Aleven, to develop CATO (most
fully reported in [1]), a system designed to help law students to learn to reason with
precedents. CATO simplifies HYPO in some respects but extends it in others. In
CATO the notion of dimensions is simplified to a notion of factors. A factor can be
seen as a specific point of the dimension: it is simply present or absent from a case,
rather than present to some degree, and it always favours either the plaintiff or defen-
dant. A new feature of CATO is that these factors are organised into a hierarchy of
increasingly abstract factors, so that several different factors can be seen as meaning
that the same abstract factor is present. One such abstract factor is that the defendant
used questionable means to obtain the information, and two more specific factors in-
dicating the presence of this factor are that the defendant deceived the plaintiff and
that he bribed an employee of the plaintiff: both these factors favour the plaintiff.
The hierarchy allows for argument moves that interpret the relation between a case’s
input and its decision, such as emphasising or downplaying distinctions. To give an
example of downplaying, if in the precedent defendant used deception while in the
new case instead defendant bribed an employee, then a distinction made by the de-
fendant at this point can be downplayed by saying that in both cases the defendant
used questionable means to obtain the information. To give an example of empha-
sising a distinction, if in the new case defendant bribed an employee of plaintiff
while in the precedent no factor indicating questionable means was present, then
the plaintiff can emphasise the distinction “unlike the precedent, defendant bribed
an employee of plaintiff” by adding “and therefore, unlike the precedent defendant
used questionable means to obtain the information”.](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-381-320.jpg)
![368 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor
Perhaps the most elaborate representation of cases was produced in Karl Brant-
ing’s Grebe system in the domain of industrial injury, where cases were represented
as semantic networks [13]. The program matched portions of the network for the
new case with parts of the networks of precedents, to identify appropriate analogies.
Of all this work, HYPO in particular was highly influential, both in the explicit
stress it put on reasoning with cases as constructing arguments, and in providing
a dialectical structure in which these arguments could be expressed, anticipating
much other work on dialectical procedures.
3 Logical accounts of reasoning under disagreement
The systems discussed in the previous section were (proposals for) implemented
systems, based on informal accounts of some underlying theory of reasoning. Other
AI Law research aims at specifying theories of reasoning in a formal way, in
order to make general reasoning techniques from nonmonotonic logic and formal
argumentation available for implementations.
The first AI Law proposals in this vein, for example, [16, 30], can be regarded
as formal counterparts of Gardner’s ideas on issue spotting. Recall that Gardner
allows for the presence in the knowledge base of conflicting rules governing the
interpretation of legal concepts and that she defines an issue as a problem to which
either no rules apply at all, or conflicting rules apply. Now in logical terms an issue
can be defined as a proposition such that either there is no argument about this
proposition or there are both arguments for the proposition and for its negation.
Some more recent work in this research strand has utilised a very abstract AI
framework for representing systems of arguments and their relations developed by
Dung [14]. For Dung, the notion of argument is entirely abstract: all that can be
said of an argument is which other arguments it attacks, and which it is attacked
by. Given a set of arguments and the attack relations between them, it is possible
to determine which arguments are acceptable: an argument which is not attacked
will be acceptable, but if an argument has attackers it is acceptable only if it can
be defended, against these attackers, by acceptable arguments which in turn attack
those attackers. This framework has proved a fruitful tool for understanding non-
monotonic logics and their computational properties. Dung’s framework has also
been made use of in AI Law. It was first applied to the legal domain by Prakken
Sartor [35], who defined a logic for reasoning with conflicting rules as an instan-
tiation of Dung’s framework. (See below and Chapter 8 of this book) Bench-Capon
[3] has explored the potential of the fully abstract version of the framework to rep-
resent a body of case law. he uses preferred semantics, where arguments can defend
themselves: in case of mutual attack this gives rise to multiple sets of acceptable
arguments, which can explain differences in the application of law in different juris-
dictions, or at different times in terms of social choices. Dung’s framework has also
been extended to include a more formal consideration of social values (see Chapter 3
of this book).](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-382-320.jpg)
![18 Argumentation in Legal Reasoning 369
3.1 Reasoning about conflicting rules
Generally speaking, the proposed systems discussed so far attempt to identify con-
flicting interpretations and arguments, but do not attempt to resolve them, leaving it
to the user to choose which argument will be accepted. As we saw above, Gardner’s
system went somewhat further in that it gave priority to rules derived from case law
over restatement and commonsense rules. Thus her system was able to solve some
of the cases to which conflicting rules apply. This relates to much logical work in
Artificial Intelligence devoted to the resolution of rule conflicts in so-called com-
monsense reasoning. If we have a rule that birds can fly and another that ostriches
cannot fly, we do not want to let the user decide whether Cyril the ostrich can fly or
not: we want the system to say that he cannot, since an ostrich is a specific kind of
bird. Naturally attempts have been made to apply these ideas to law.
One approach was to identify general principles used in legal systems to establish
which of two conflicting rules should be given priority. These principles included
preferring the more specific rule (as in the case of the ostrich above, or where a
law expresses an exception to a general provision), preferring the more recent rule,
or preferring the rule deriving from the higher legislative authority (for instance,
‘federal law precedes state law’). To this end the logics discussed above were ex-
tended with the means to express priority relations between rules in terms of these
principles so that rule conflicts would be resolved. Researchers soon realised, how-
ever, that general priority principles can only solve a minority of cases. Firstly, as
for the specificity principle, whether one rule is more specific than another often
depends on substantive legal issues such as the goals of the legislator, so that the
specificity principle cannot be applied without an intelligent appreciation of the par-
ticular issue. Secondly, general priority principles usually only apply to rules from
regulations and not to, for instance, case rationales or interpretation rules derived
from cases. Accordingly, in many cases the priority of one rule over another can be
a matter of debate, especially when the rules that conflict are unwritten rules put
forward in the context of a case. For these reasons models of legal argument should
allow for arguments about which rule is to be preferred.
As an example of arguments about conflicting case rationales, consider three
cases discussed by, amongst others, [10, 7, 32] and [8] concerning the hunting of
wild animals. In all three cases, the plaintiff (P) was chasing wild animals, and the
defendant (D) interrupted the chase, preventing P from capturing those animals. The
issue to be decided is whether or not P has a legal remedy (a right to be compensated
for the loss of the game) against D. In the first case, Pierson v Post, P was hunting
a fox on open land in the traditional manner using horse and hound, when D killed
and carried off the fox. In this case P was held to have no right to the fox because he
had gained no possession of it. In the second case, Keeble v Hickeringill, P owned
a pond and made his living by luring wild ducks there with decoys, shooting them,
and selling them for food. Out of malice, D used guns to scare the ducks away
from the pond. Here P won. In the third case, Young v Hitchens, both parties were
commercial fisherman. While P was closing his nets, D sped into the gap, spread his](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-383-320.jpg)
![370 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor
own net and caught the fish. In this case D won. The rules we are concerned with
here are the rationales of these cases:
R1 Pierson: If the animal has not been caught, the defendant wins
R2 Keeble: If the plaintiff is pursuing his livelihood, the plaintiff wins
R3 Young: If the defendant is in competition with the plaintiff and the animal
is not caught, the defendant wins.
Note that R1 applies in all cases and R2 in both Keeble and Young. In order to
explain the outcomes of the cases we need to be able to argue that R3 R2 R1.
To start with, note that if, as in HYPO, we only look at the factual similarities and
differences, none of the three precedents can be used to explain the outcome of one
of the other precedents. For instance, if we regard Young as the current case, then
both Pierson and Keeble can be distinguished. A way of arguing for the desired
priorities, first mooted in [10], is to refer to the purpose of the rules, in terms of the
social values promoted by following the rules.
The logic of [35] provides the means to formalise such arguments. Consider an-
other case in which only plaintiff was pursuing his livelihood and in which the ani-
mal was not caught. In the following (imaginary) dispute the parties reinterpret the
precedents in terms of the values promoted by their outcomes, in order to find a con-
trolling precedent (we leave several details implicit for reasons of brevity; a detailed
formalisation method can be found in [32]; see also [8].
Plaintiff: I was pursuing my livelihood, so (by Keeble) I win
Defendant: You had not yet caught the animal, so (by Pierson) I win
Plaintiff: following Keeble promotes economic activity, which is why Keeble takes
precedence over Pierson, so I win.
Defendant: following Pierson protects legal certainty, which is why Keeble does not
take precedence over Pierson, so you do not win.
Plaintiff: but promoting economic activity is more important than protecting legal
certainty since economic development, not legal certainty is the basis of this coun-
try’s prosperity. Therefore, I am right that Keeble takes precedence over Pierson, so
I still win.
This dispute contains priority debates at two levels: first the parties argue about
which case rationale should take precedence (by referring to values advanced by
following the rationale), and then they argue about which of the conflicting pref-
erence rules for the rationales takes precedence (by referring to the relative order
of the values). In general, a priority debate could be taken to any level and will be
highly dependent on the context and jurisdiction. Various logics proposed in the AI
Law literature are able to formalise such priority debates, such as [17, 35, 19] and
[21].](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-384-320.jpg)
![18 Argumentation in Legal Reasoning 371
3.2 Other arguments about rules
Besides priority debates in case of conflicting rules, these logics can also model
debates about certain properties of rules, such as their legal validity or their appli-
cability to a legal case. The most fully developed logical theory about what it takes
to apply a rule is reason-based logic, developed jointly by Jaap Hage and Bart Ver-
heij (e.g. [19, 45]). They claim that applying a legal rule involves much more than
subsuming a case under the rule’s conditions. Their account of rule application can
be briefly summarised as follows. First in three preliminary steps it must be deter-
mined whether the rule’s conditions are satisfied, whether the rule is legally valid,
and whether the rule’s applicability is not excluded in the given case by, for instance,
a statutory exception. If these questions are answered positively (and all three are
open to debate), it must finally be determined that the rule can be applied, i.e., that
no conflicting rules or principles apply. On all four questions reason-based logic al-
lows reasons for and against to be provided and then weighed against each other to
obtain an answer.
Consider by way of illustration a recent Dutch case (HR 7-12-1990, NJ 1991,
593) in which a male nurse aged 37 married a wealthy woman aged 97 whom he
had been nursing for several years, and killed her five weeks after the marriage.
When the woman’s matrimonial estate was divided, the issue arose whether the
nurse could retain his share. According to the relevant statutes on Dutch matrimo-
nial law the nurse was entitled to his share since he had been the woman’s husband.
However, the court refused to apply these statutes, on the grounds that applying it
would be manifestly unjust. Let us assume that this was in turn based on the legal
principle that no one shall profit form his own wrongdoing (the court did not explic-
itly state this). In reason-based logic this case could be formalised as follows (again
the full details are suppressed for reasons of brevity).
Claimant: Statutory rule R is a valid rule of Dutch law since it was enacted ac-
cording to the Dutch constitution and never repealed. All its conditions are satisfied
in my case, and so it should be applied to my case. The rule entitles me to my late
wife’s share in the matrimonial estate. Therefore, I am entitled to my wife’s share in
the matrimonial estate.
Defendant: Applying rule R would allow you to profit from your own wrongdoing:
therefore rule R should not be applied in this case.
Court: The reason against applying this rule is stronger than that for applying the
rule, and so the rule does not apply.
Of course, in the great majority of cases the validity or applicability of a statute
rule is not at issue but instead silently presumed by the parties (recall the differ-
ence between arguments and proofs described in the introduction). The new logical
techniques alluded to above can also deal with such presumptions, and they can be
incorporated in reason-based logic.
Reason-based logic also has a mechanism for ‘accruing’ different reasons for
conclusions into sets and for weighing these sets against similar sets for conflict-](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-385-320.jpg)
![372 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor
ing conclusions. Thus it captures that having more reasons for a conclusions may
strengthen one’s position. Prakken [33] formalises a similar mechanism for accruing
arguments in the context of Dung’s framework. Prakken also proposes three prin-
ciples that any model of accrual should satisfy. In Chapter 12 of this book Gordon
and Walton show that the Carneades logic of [18] has an accrual mechanism that
satisfies these principles.
One way to argue about the priority of arguments is to claim that the argument is
preferred if it is grounded in the better or more coherent legal theory3. While there
has been considerable progress in seeing how theories can be constructed on the
basis of a body of past cases, evaluation of the resulting theories in terms of their
coherence is more problematic, since coherence is a difficult notion to define pre-
cisely4. Bench-Capon Sartor [8] describe some features of a theory which could
be used in evaluation, such as simplicity of a theory or the number of precedent
cases explained by the theory. As an (admittedly somewhat simplistic) example of
the last criterion, consider again the three cases on hunting animals, and imagine
two theories that explain the case decisions in terms of the values of promotion of
economic activity and protection of legal certainty. A theory that gives precedence
to promoting economic activity over protecting legal certainty explains all three
precedents while a theory with the reverse value preference fails to explain Keeble.
The first theory is therefore on this criterion the more coherent one. However, how
several coherence criteria are to be combined is a matter for further resear](https://image.slidesharecdn.com/argumentationinartificialintelligence-230807174532-a57c8ce4/85/Argumentation-in-Artificial-Intelligence-pdf-386-320.jpg)
Argumentation in Artificial Intelligence.pdf
- 3. “This page left intentionally blank.”
- 4. Argumentation in Artificial Intelligence Edited by Iyad Rahwan Guillermo R. Simari 123
- 5. Editors Dr. Iyad Rahwan Faculty of Informatics British University in Dubai P.O.Box 502216, Dubai United Arab Emirates irahwan@acm.org Guillermo R. Simari Department of Computer Science & Engineering Universidad Nacional del Sur Alem 1253 (8000) Bahı́a Blanca - Argentina grs@cs.uns.edu.ar and School of Informatics University of Edinburgh Edinburgh EH8 9AB, UK ISBN 978-0-387-98196-3 e-ISBN 978-0-387-98197-0 DOI 10.1007/978-0-387-98197-0 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009927013 c Springer Science+Business Media, LLC 2009 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
- 6. To Zoe, for her love and support. – I.R. To my family with love. – G.S.
- 7. “This page left intentionally blank.”
- 8. Foreword Argumentation is all around us. Letters to the Editor often make points of consis- tency, and “Why” is one of the most frequent questions in language, asking for rea- sons behind behaviour. And argumentation is more than ‘reasoning’ in the recesses of single minds, since it crucially involves interaction. It cements the coordinated social behaviour that has allowed us, in small bands of not particularly physically impressive primates, to dominate the planet, from the mammoth hunt all the way up to organized science. This volume puts argumentation on the map in the field of Artificial Intelligence. This theme has been coming for a while, and some famous pioneers are chapter authors, but we can now see a broader systematic area emerging in the sum of topics and results. As a logician, I find this intriguing, since I see AI as ‘logic continued by other means’, reminding us of broader views of what my discipline is about. Logic arose originally out of reflection on many-agent practices of disputation, in Greek Antiq- uity, but also in India and China. And logicians like me would like to return to this broader agenda of rational agency and intelligent interaction. Of course, Aristotle also gave us a formal systems methodology that deeply influenced the field, and eventually connected up happily with mathematical proof and foundations. Thus, I see two main paradigms from Antiquity that come together in the modern study of argumentation: Platos’ Dialogues as the paradigm of intelligent interaction, and Euclid’s Elements as the model of rigour. Of course, some people also think that formal mathematical proof is itself the ultimate ideal of reasoning - but you may want to change your mind about reasoning’s ‘peak experiences’ when you see top mathematicians argue interactively at a seminar. But more themes went into the mixture of this Book. Leibniz and those after him, from Boole to Turing or McCarthy, added computation as a major category in understanding reasoning. Now, this is not necessarily congenial to argumentation: Leibniz’ famous ‘Calculemus’ calls for replacing interactive disputation by mechan- ical computing. But modern computation itself is distributed and interactive, so we are in tune again. Also relevant to understanding this Book is the emergence of ‘Argumentation Theory’ in the 20th century, partly in opposition to formal logic. In particular, Toul- min gave us a much richer view of actual inference than just a bleak jump from premises to conclusion, and placed it in a historical tradition of dynamic legal pro- cedure (what he calls the ‘formalities’) rather than just the static mathematical form of statements. Indeed, Mathematics and Law seem two major pillars of our culture, with the latter often under-estimated as an intellectual force. This tandem seems sig- nificant to me, since it fits the Dynamic Turn I have long advocated toward logical studies of cognitive actions, and indeed multi-agent interaction. Strategic responses to others, and ‘logical empathy’ putting yourself in someone else’s place, are keys to rational behaviour. And argumentation is one of the major processes that make this interaction happen. Thus, pace Toulmin, logic and argumentation theory can form happy unions after all, witness the work of colleagues like van Eemeren, Krabbe Walton, Gabbay Woods, etc. vii
- 9. viii Foreword And even beyond these strands, the land of rational agency is populated by other tribes, many equipped with mathematical tools. Game theorists study social mech- anisms, social scientists care about social choice and decisions, and philosophers, too, have long studied rational interaction. Think of Kant’s categorical imperative of treating others as an end like yourself, not just a means. This only makes sense in a society of agents. AI lets all these strands come together: logic, mathematics, computation, and human behaviour. It has long been a sanctuary for free-thinkers about reasoning and other intelligent activities, taking a fresh look at the practice of common sense all around us. Indeed, I see the above perspective as an appropriate extension of the very concept of ‘common sense’, which is not just ‘sense’ about how single agents represent the world and make inferences about it, but equally much ‘common’ about how they sensibly interact with others. And once more, argumentation is a major mechanism for doing so. The content of this rich volume is definitely not exhausted by the above. It contains methods from computer science, mathematics, philosophy, law, and eco- nomics, merging artificial with natural intelligence. Its formal methods range from logic programs to abstract argumentation systems, and from non-monotonic default logics and belief revision to classical proof theory. It also highlights multi-agent di- alogue and decision making, including connections with game theory - where our rich practices of argumentation and debate pose many unsolved challenges. Just try to understand how we successfully conduct meetings, and ‘play’ arguments of var- ious strengths over time! Finally, I would mention an intriguing feature in many studies of argumentation, viz. attention to fallacies and errors. Once I was taken to task by a prominent medical researcher, who claimed that the most interesting in- formation about the human body and mind is found with patients deviating from the norm, and coping with ‘disturbance’ in unexpected creative ways. He did not understand why logicians would wilfully ignore the corresponding rich evidence in the case of reasoning, concentrating just on angelic correctness. I agree, and linking up with empirical psychology and cognitive science seems an attractive next step, given the suggestive material collected here. This volume tries to stake out a new field, and hence: papers, careers, tenure. But something broader is at stake. Original visions of AI tended to emphasize hugely uninspiring, if terrifying, goals like machines emulating humans. A Dutch book with ‘vision statements’ by leading scientists once revealed a disturbing uniformity: all described a technological end goal for their field of which all said they hoped to be dead long before it was achieved. I myself prefer goals that I could live with. Understanding argumentation means understanding a crucial feature of ourselves, perhaps using machines to improve our performance, helping us humans be better at what we are. I am happy that books like this are happening and I congratulate the editors and authors. Amsterdam and Stanford, December 2008 Johan van Benthem
- 10. Preface This book is about the common ground between two fields of inquiry: Argumenta- tion Theory and Artificial Intelligence. On the one hand, formal models of argumen- tation are making significant and increasing contributions to Artificial Intelligence, from defining semantics of logic programs, to implementing persuasive medical diagnostic systems, to studying negotiation dialogues in multi-agent systems. On the other hand, Artificial Intelligence has also made an impact on Argumentation Theory and Practice, for example by providing formal tools for argument analysis, evaluation, and visualisation. The field of Argumentation in Artificial Intelligence has grown significantly in the past few years resulting in a substantial body of work and well-established technical literature. A testimony to this is the appearance of several special is- sues in leading scientific journals in recent years, (e.g., Springer’s Journal of Au- tonomous Agents and Multiagent Systems 2006; Elsevier’s Artificial Intelligence Journal 2007; IEEE Intelligent Systems 2007; Wiley’s International Journal of In- telligent Systems 2007). Another evidence of the maturity of this area is the es- tablishment of a new biannual international conference in 2006 (see www.comma- conf.org). In addition, two series of workshops have been co-located with major AI conferences: the Argumentation in Multi-Agent Systems (ArgMAS) workshop se- ries running annually alongside AAMAS since 2004, and the Computational Models of Natural Argument (CMNA) workshop running series alongside IJCAI and ECAI since 2001. Yet, although valuable survey papers exist, there is no comprehensive presentation of the major achievements in the field. This volume is a response to a growing need for an in-depth presentation of this fast-expanding area. As such it can be seen as a confluence of deep exposition and comprehensive exploration of the underlying themes in the various areas, done by leading researchers. While no single volume on Argumentation and Artificial Intelligence could cover the entire scope of this dynamic area, these selected writings will give the reader an insightful view of a landscape of stimulating ideas that drive forward the fundamental research and the creation of applications. This book is aimed at new and current researchers in Argumentation Theory and in Artificial Intelligence interested in exploring the rich terrain at the intersection between these two fields. In particular, the book presents an overview of key con- cepts in Argumentation Theory and of formal models of Argumentation in AI. After laying a strong foundation by covering the fundamentals of argumentation and for- mal argument modeling, the book expands its focus to more specialised topics, such as algorithmic issues, argumentation in multi-agent systems, and strategic aspects of argumentation. Finally, as a coda, the book presents some practical applications of argumentation in AI and applications of AI in argumentation. Although the book is an edited collection, the chapters’ topics and order was done carefully to produce a highly organised text containing a progressive develop- ment of intuitions, ideas and techniques, starting from philosophical backgrounds, to abstract argument systems, to computing arguments, to the appearance of appli- cations presenting innovative results. Authors had the chance to review each others’ ix
- 11. x Preface work at various stages of writing in order to coordinate content, ensuring unified notation (when possible) and natural progression. Readers of this book will acquire an appreciation of a wide range of topics in Argumentation and Artificial Intelligence covering, for the first time, a breadth of hot topics. Throughout the chapters the authors have provided extensive examples to ensure that readers develop the right intuitions before they move from one topic to another. The primary audience is composed of researchers and graduate students work- ing in Autonomous Agents, AI and Law, Logic in Computer Science, Electronic Governance, Multi-agent Systems, and the growing research represented by the in- terdisciplinary inquiry carried out in many areas such as Decision Support Systems. Given the scope and depth of the chapters of this book, its content provides an ex- cellent foundation for several different graduate courses. The book begins with an “Introduction to Argumentation Theory” by Douglas Walton, who was one of the argumentation theorists who pioneered joint work with AI researchers. The rest of the book’s twenty three chapters have been organised into four parts: “Abstract Argument Systems”, “Arguments with Structure”, “Argu- mentation in Multi-Agent Systems”, and “Applications”. Chapters in this book have been written by researchers that have helped shape the field. As such, we are confi- dent that this book will be an essential resource for graduate students and researchers coming to the area. The value of this book is in the ideas it presents. Thus we gratefully acknowledge efforts by all authors who shared their ideas and deep insights of this fertile area of research in such a clear manner. Furthermore, they also acted as peer reviewers of other chapters and helped to significantly improve the quality and the flow of the book. We would also like to thank all the contributions made by the different organisations that supported the authors of this book as they individually recognise in each chapter. We are grateful to the Springer team, and in particular Melissa Fearon and Va- lerie Schofield, for supporting the creation of this book from early discussions right through to final editorial work. Last but not least, we are always grateful to our families for their endless love and support. Dubai, Edinburgh and Bahia Blanca, Iyad Rahwan December 2008 Guillermo Simari
- 12. Contents 1 Argumentation Theory: A Very Short Introduction . . . . . . . . . . . . . . . 1 Douglas Walton Part I Abstract Argument Systems 2 Semantics of Abstract Argument Systems . . . . . . . . . . . . . . . . . . . . . . . 25 Pietro Baroni and Massimiliano Giacomin 3 Abstract Argumentation and Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Trevor Bench-Capon and Katie Atkinson 4 Bipolar abstract argumentation systems . . . . . . . . . . . . . . . . . . . . . . . . 65 Claudette Cayrol and Marie-Christine Lagasquie-Schiex 5 Complexity of Abstract Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . 85 Paul E. Dunne and Michael Wooldridge 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Sanjay Modgil and Martin Caminada Part II Arguments with Structure 7 Argumentation Based on Classical Logic . . . . . . . . . . . . . . . . . . . . . . . . 133 Philippe Besnard and Anthony Hunter 8 Argument-based Logic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari 9 A Recursive Semantics for Defeasible Reasoning . . . . . . . . . . . . . . . . . 173 John L. Pollock xi
- 13. xii Contents 10 Assumption-Based Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Phan Minh Dung, Robert A. Kowalski and Francesca Toni 11 The Toulmin Argument Model in Artificial Intelligence. . . . . . . . . . . . 219 Bart Verheij 12 Proof Burdens and Standards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 Thomas F. Gordon and Douglas Walton Part III Argumentation in Multi-Agent Systems 13 Dialogue Games for Agent Argumentation . . . . . . . . . . . . . . . . . . . . . . 261 Peter McBurney and Simon Parsons 14 Models of Persuasion Dialogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 Henry Prakken 15 Argumentation for Decision Making . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Leila Amgoud 16 Argumentation and Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Iyad Rahwan and Kate Larson 17 Belief Revision and Argumentation Theory . . . . . . . . . . . . . . . . . . . . . . 341 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari Part IV Applications 18 Argumentation in Legal Reasoning. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor 19 The Argument Interchange Format . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Iyad Rahwan and Chris Reed 20 Empowering Recommendation Technologies Through Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 Carlos Iván Chesñevar, Ana Gabriela Maguitman and Marı́a Paula González 21 Arguing on the Semantic Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Paolo Torroni, Marco Gavanelli and Federico Chesani 22 Towards Probabilistic Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . 443 Ingrid Zukerman 23 Argument-Based Machine Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463 Ivan Bratko, Jure Žabkar and Martin Možina
- 14. Contents xiii A Description Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 B Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
- 15. Chapter 1 Argumentation Theory: A Very Short Introduction Douglas Walton 1 Introduction Since the time of the ancient Greek philosophers and rhetoricians, argumentation theorists have searched for the requirements that make an argument correct, by some appropriate standard of proof, by examining the errors of reasoning we make when we try to use arguments. These errors have long been called fallacies, and the logic textbooks have for over 2000 years tried to help students to identify these fallacies, and to deal with them when they are encountered. The problem was that deduc- tive logic did not seem to be much use for this purpose, and there seemed to be no other obvious formal structure that could usefully be applied to them. The radical approach taken by Hamblin [7] was to refashion the concept of an argument to think of it not just as an arbitrarily designated set of propositions, but as a move one party makes in a dialog to offer premises that may be acceptable to another party who doubts the conclusion of the argument. Just after Hamblin’s time a school of thought called informal logic grew up that wanted to take a new practical approach to teach- ing students skills of critical thinking by going beyond deductive logic to seek other methods for analyzing and evaluating arguments. Around the same time, an interdis- ciplinary group of scholars associated with the term ‘argumentation,’ coming from fields like speech communication, joined with the informal logic group to help build up such practical methods and apply them to real examples of argumentation [9]. The methods that have been developed so far are still in a process of rapid evolu- tion. More recently, improvements in them have been due to some computer scien- tists joining the group, and to collaborative research efforts between argumentation theorists and computer scientists. Another recent development has been the adop- tion of argumentation models and techniques to fields in artificial intelligence, like multi-agent systems and artificial intelligence for legal reasoning. In a short paper, it is not possible to survey all these developments. The best that can be done is to offer Douglas Walton University of Windsor, e-mail: dwalton@uwindsor.ca I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 1 DOI 10.1007/978-0-387-98197-0 1, c Springer Science+Business Media, LLC 2009
- 16. 2 Douglas Walton an introduction to some of the basic concepts and methods of argumentation theory as they have evolved to the present point, and to briefly indicate some problems and limitations in them. 2 Arguments and Argumentation There are four tasks undertaken by argumentation, or informal logic, as it is also often called: identification, analysis, evaluation and invention. The task of identifi- cation is to identify the premises and conclusion of an argument as found in a text of discourse. A part of this task is to determine whether a given argument found in a text fits a known form of argument called an argumentation scheme (more about schemes below). The task of analysis is to find implicit premises or conclusions in an argument that need to be made explicit in order to properly evaluate the argu- ment. Arguments of the kind found in natural language texts of discourse tend to leave some premises, or in some instances the conclusion, implicit. An argument containing such missing assumptions is traditionally called an enthymeme. The task of evaluation is to determine whether an argument is weak or strong by general cri- teria that can be applied to it. The task of invention is to construct new arguments that can be used to prove a specific conclusion. Historically, recent work has mainly been directed to the first three tasks, but there has been a tradition of attempting to address the fourth task from time to time, based on the tradition of Aristotelian topics [1, ch. 8]. There are differences in the literature in argumentation theory on how to define an argument. Some definitions are more minimal while others are more inclusive. We start here with a minimal definition, however, that will fit the introduction to the elements of argumentation presented below. An argument is a set of statements (propositions), made up of three parts, a conclusion, a set of premises, and an infer- ence from the premises to the conclusion. An argument can be supported by other arguments, or it can be attacked by other arguments, and by raising critical questions about it. Argument diagramming is one of the most important tools currently in use to assist with the tasks of analyzing and evaluating arguments. An argument diagram is essentially a box and arrow representation of an argument where the boxes con- tain propositions that are nodes in a graph structure and where arrows are drawn from nodes to other nodes representing inferences. At least this is the most common style of representation. Another style growing in popularity is the diagram where the nodes represent arguments and the boxes represent premises and conclusions of these arguments. The distinction between a linked argument and a convergent ar- gument is important in argumentation theory. A linked argument is one where the premises work together to support the conclusion, whereas in a convergent argument each premise represents a separate reason that supports the conclusion. Arguments fitting the form of an argumentation scheme are linked because all of the premises are needed to adequately support the conclusion. Here is an example of a convergent
- 17. 1 Argumentation Theory: A Very Short Introduction 3 Fig. 1.1 Example of an Argument Diagram argument: gold is malleable; it can be easily made into jewelry, and my metallurgy textbook says it is malleable. In the example shown in Figure 1.1, two linked argu- ments are combined in a chain of reasoning (called a serial argument). The linked argument on the right at the bottom has a colored border and the label Argument from Expert Opinion is shown in a matching color at the top of the con- clusion. This label represents a type of argument called an argumentation scheme. Figure 1.1 was drawn with argument diagramming tool called Araucaria [14]. It assists an argument analyst using a point-and-click interface, which is saved in an Argument Markup Language based on XML [14]. The user inserts the text into Arauacaria, loads each premise or conclusion into a text box, and then inserts ar- rows showing which premises support which conclusions. As illustrated above, she can also can insert implicit premises or conclusions and label them. The output is an argument diagram that appears on the screen that can be added to, exported or printed (http://araucaria.computing.dundee.ac.uk/). The other kind of format for representing arguments using visualization tools is shown in the screen shot in Figure 1.2. According to this way of represent- ing the structure of the argument, the premises and conclusions appear as state- ments in the text boxes, while the nodes represent the arguments. Information about the argumentation scheme, and other information as well, is contained in a node (http://carneades.berlios.de/downloads/). Both arguments pictured in Figure 1.2 are linked. Convergent arguments are rep- resented as separate arguments. Another chapter in this book (see Chapter 12) shows how Carneades represents different proof standards of the kinds indicated on the lower right of the screen shot.
- 18. 4 Douglas Walton Fig. 1.2 Carneades Screen Shot of an Argument Argument diagrams are very helpful to display premises and conclusions in an argument and to show how groups of premises support conclusions that can in turn be used as premises in adjoining arguments. Smaller arguments are chained together into longer sequences, and an argument diagram can be very useful to help an an- alyst keep track of the chain of reasoning in its parts. However, typical argument diagrams are made up only of statements in text boxes joined together by arrows. Such an argument diagram is of limited or less use when it comes to representing critical questions and the relationship of these questions to an argument. The definition of ‘argument’ relied on so far could be called a minimal inferential definition, and the method of argument diagramming shown so far fits this minimal definition. The boxes represent propositions and the arrows represent inferences from some propositions to others. The general approach or methodology of argumentation can be described as dis- tinctively different from the traditional approach based on deductive logic. The tra- ditional approach concentrated on a single inference, where the premises and con- clusion are designated in advance, and applied formal models like propositional calculus and quantification theory determine whether the conclusion conclusively follows from the premises. This approach is often called monological. In contrast, the argumentation approach is called dialogical (or dialectical) in that it looks at two sides of an argument, the pro and the contra. According to this ap- proach, the method of evaluation is to examine how the strongest arguments for and against a particular proposition at issue interact with each other, and in particular how each argument is subject to probing critical questioning that reveals doubts
- 19. 1 Argumentation Theory: A Very Short Introduction 5 about it. By this dialog process of pitting the one argument against the other, the weaknesses in each argument are revealed, and it is shown which of the two argu- ments is the stronger.1 To fill out the minimal definition enough to make it useful for the account of argumentation in the paper, however, some pragmatic elements need to be added, that indicate how arguments are used in a dialog between two (in the simplest case) parties. Argumentation is a chain of arguments, where the conclusion of one infer- ence is a premise in the next one. There can be hypothetical arguments, where the premises are merely assumptions. But generally, the purpose of using an argument in a dialog is to settle some disputed (unsettled) issue between two parties. In the speech act of putting forward an argument, one party in the dialog has the aim of trying to get the other party to accept the conclusion by offering reasons why he should accept it, expressed in the premises. This contrasts with the purpose of using an explanation, where one party has the aim of trying to get the other party to under- stand some proposition that is accepted as true by both parties. The key difference is that in an argument, the proposition at issue (the conclusion) is doubted by the one party, while in an explanation, the proposition to be explained is not in doubt by either party. It is assumed to represent a factual event. 3 Argument Attack and Refutation One way to attack an argument is to ask an appropriate critical question that raises doubt about the acceptability of the argument. When this happens, the argument temporarily defaults until the proponent can respond appropriately to the critical question. Another way to attack an argument is to question one of the premises. A third way to attack an argument is to put forward counter-argument that opposes the original argument, meaning that the conclusion of the opposing argument is the opposite (negation) of the conclusion of the original argument. There are other ways to attack an argument as well [10]. For example, one might argue that the premises are not relevant to the conclusion, or that the argument is not relevant in relation to the issue that is supposedly being discussed. One might also argue that the original argument commits a logical fallacy, like the fallacy of begging the question (argu- ing in a circle by taking for granted as a premise the very proposition that is to be proved). However, the three first ways cited above of attacking an argument are es- pecially important for helping us to understand the notion of argument refutation. A refutation of an argument is an opposed argument that attacks the original argument and defeats it. A simple way to represent a sequence of argumentation in the dialogical style is to let the nodes in a graph represent arguments and the arrows represent attacks on 1 This approach has been neglected for a long time in the history of logic, but it is not new. Ci- cero, based on the work of his Greek predecessors in the later Platonic Academy, Arcesilaus and Carneades, adopted the method of dialectical inquiry that, by arguing for and against competing views, reveals the one that is the more plausible [16, p. 4].
- 20. 6 Douglas Walton arguments [5]. In this kind of argument representation, one argument is shown as attacking another. In this example, argument A1 attacks both A2 and A3. A2 attacks A6. A6 attacks A7, and A7 attacks A6. A3 attacks A4, and so forth. Fig. 1.3 Example of a Dung-style Argument Representation Notice that arguments can attack each other. A6 attacks A7 and A7 also attacks A6. An example [4, p. 23] is the following pair of arguments. Richard is a Quaker and Quakers are pacifists, so he is a pacifist. Richard is a Republican and Republicans are not pacifists, so he is a not a pacifist. In Dung’s system, the notions of argument attack are undefined primitives, but the system can be used to model criteria of argument acceptability. One such criterion is the view that an argument should be accepted only if every attack on it is attacked by an acceptable argument [3, p. 3]. There is general (but not universal) agreement in argumentation studies that there are three standards by which the success of the inference from the premises to the conclusion can be evaluated. This agreement is generally taken to mean that there are three kinds of arguments: deductive, inductive, and defeasible arguments of a kind widely thought not to be inductive (Bayesian) in nature. This third class in- cludes arguments like ‘Birds fly; Tweety is a bird; therefore Tweety flies,’ where exceptions, like ‘Tweety has a broken wing’ are not known in advance and cannot be anticipated statistically. Many of the most common arguments in legal reasoning and everyday conversational argumentation that are of special interest to argumen- tation theorists fall into this class. An example would be an argument from expert opinion of this sort: Experts are generally right about things in their domain of ex- pertise; Dr. Blast is an expert in domain D, Dr. Blast asserts that A, A is in D; therefore an inference can be drawn that A is acceptable, subject to default if any reasonable arguments to the contrary or critical questions are raised. Arguments of this sort are important, for example in legal reasoning, but before the advent of ar- gumentation theory, useful logical tools to identify, analyze and evaluate them were not available.
- 21. 1 Argumentation Theory: A Very Short Introduction 7 4 Argumentation Schemes Argumentation schemes are abstract argument forms commonly used in everyday conversational argumentation, and other contexts, notably legal and scientific argu- mentation. Most of the schemes that are of central interest in argumentation theory are forms of plausible reasoning that do not fit into the traditional deductive and inductive argument forms. Some of the most common schemes are: argument from witness testimony, argument from expert opinion, argument from popular opinion, argument from example, argument from analogy, practical reasoning (from goal to action), argument from verbal classification, argument from sign, argument from sunk costs, argument from appearance, argument from ignorance, argument from cause to effect, abductive reasoning, argument from consequences, argument from alternatives, argument from pity, argument from commitment, ad hominem argu- ment, argument from bias, slippery slope argument, and argument from precedent. Each scheme has a set of critical questions matching the scheme and such a set rep- resents standard ways of critically probing into an argument to find aspects of it that are open criticism. A good example of a scheme is the one for argument from expert opinion, also called appeal to expert opinion in logic textbooks. In this scheme [1, p. 310], A is a proposition, E is an expert, and D is a domain of knowledge. Major Premise: Source E is an expert in subject domain S containing proposition A. Minor Premise: E asserts that proposition A is true (false). Conclusion: A is true (false). The form of argument in this scheme could be expressed in a modus ponens for- mat where the major (first) premise is a universal conditional: If an expert says that A is true, A is true; expert E says that A is true; therefore A is true. The major premise, for practical purposes, however, is best seen as not being an absolute universal gen- eralization of the kind familiar in deductive logic. It is best seen as a defeasible generalization, and the argument is defeasible, subject to the asking of critical ques- tions. If the respondent asks any one of the following six critical questions [1, p. 310], the proponent must give an appropriate reply or the argument defaults. CQ1: Expertise Question. How credible is E as an expert source? CQ2: Field Question. Is E an expert in the field that A is in? CQ3: Opinion Question. What did E assert that implies A? CQ4: Trustworthiness Question. Is E personally reliable as a source? CQ5: Consistency Question. Is A consistent with what other experts assert? CQ6: Backup Evidence Question. Is E’s assertion based on evidence? Some other examples of schemes will be introduced when we come to study the example of extended argumentation in section 2.
- 22. 8 Douglas Walton 5 Enthymemes As indicated in the introduction, an enthymeme is an argument with an implicit premise or conclusion that needs to be made explicit before the argument can be properly understood or evaluated. The classic example is the argument: all men are mortal; therefore Socrates is mortal. As pointed out in many logic textbooks, the premise ‘Socrates is a man’ needs to be made explicit in order to make the argument into a deductively valid argument. Because both premises are needed to support the conclusion adequately, this argument is linked. Consider the following example of an enthymeme. Example 1.1 ((The Free Animals)). Animals in captivity are freer than in nature be- cause there are no natural predators to kill them. The explicit conclusion is clearly the first statement: animals in captivity are freer than in nature. The explicit premise offered to support the conclusion is the statement that there are no natural predators to kill animals that are in captivity. There are two assumptions that play the role of implicit premises in the argument. The first is the statement that there are natural predators to kill animals that are in nature. The second is the conditional statement that if animals are in a place where there are no natural predators to kill them, they are freer than if they are in a place where there are natural predators to kill them. The first implicit premise is a matter of common knowledge. The second one, however, expresses a special way that the arguer is using the word ‘free’ that seems to go against common knowledge, or at any rate, does not seem to be based on it. It seems to represent the arguer’s own special position on the meaning of ‘freedom.’ In the argument diagram in Figure 1.4, the two premises on the right are enclosed in darkened boxes, with a broken line around the border, indicating that both are implicit premises. The one in the middle is labeled as based on common knowledge (CK) and the one on the right is labeled as based on the arguer’s special commitment (COM). The argument shown in Figure 4 is clearly a linked argument, since all three premises are required to adequately support the conclusion. They all function to- gether in support of the conclusion, rather than being separate reasons, each of which supports the conclusion independently of the others. In some cases of enthymemes it is fairly obvious to determine what the missing premise or conclusion should be. In such cases, an argumentation scheme can often be used to apply to the argument given in the text of discourse to see which premise is missing. In other cases, however, there can be different interpretations of the text of discourse, and different judgments about what the missing premise should be taken to be. The more general problem is to judge what an arguer’s commitment is, given some evidence of what the arguer has said and how he has responded to criticisms and other moves in a dialog. If an arguer explicitly asserts a statement and does not retract it, then it is clear that he is committed to that statement. But suppose he explicitly asserts two statements, and a third statement follows from the first two by modus ponens. Is he then committed to the third statement? Logically, it
- 23. 1 Argumentation Theory: A Very Short Introduction 9 Fig. 1.4 Argument Diagram of the Free Animals Example seems that he should be, but when he is confronted with the third statement he may deny that he is committed to it. The other party in the dialog should then challenge him to resolve the inconsistency one way or the other, by either retracting the third statement or one of the premises. 6 An Example Dialog In the following dialog, called the smoking dialog, two participants Ann and Bob, are discussing the issue of whether governments should ban smoking. They take turns making moves, and each move after Ann’s opening move appears to address the prior move of the other party. Thus the dialog has an appearance of being con- nected and continuous in addressing the issue by bringing forward arguments pro and con. The Smoking Dialog: Ann (1): Governments should protect its citizens from harm. There is little doubt that smoking tobacco is extremely harmful to the smoker’s health. Therefore governments should ban smoking. Bob (2): How do you know that smoking tobacco is extremely harmful to the smoker’s health? Ann (3): Smoking leads to many other health problems, including lung cancer and heart disease. According to the American Cancer Society, 3 million people die from smoking each year.
- 24. 10 Douglas Walton Bob (4): The government has a responsibility to protect its citizens, but it also has a responsibility to defend their freedom of choice. Banning smoking would be an intrusion into citizens’ freedom of choice. Ann (5): Smoking is not a matter of freedom of choice. Nicotine is an addictive drug. Studies have shown that once smokers have begun smoking, they become addicted to nicotine. Once they become addicted they are no longer free to choose not to smoke. Bob (6): Governments should not stop citizens from doing things that can be ex- tremely harmful to their health. It is legal to eat lots of fatty foods or drink alcohol excessively, and it makes no sense for governments to try to ban these activities. Examining Ann’s first argument, it is fairly straightforward to put it in a format showing that it has two premises and a conclusion. Premise: Governments should protect its citizens from harm. Premise: Smoking tobacco is extremely harmful to the smoker’s health. Conclusion: Therefore governments should ban smoking. This argument looks to be an instance of the argumentation scheme for argument from negative consequences [1, p. 332]. The reason it offers to support its conclusion that governments should ban smoking is that smoking has negative consequences. An implicit premise is that being extremely harmful to health is a negative conse- quence, but we ignore this complication for the moment.2 Scheme for Argument from Negative Consequences Premise: If A is brought about, then bad consequences will occur. Conclusion: Therefore A should not be brought about. The reason is that a premise in the argument claims that the practice of smoking tobacco has harmful (bad) consequences, and for this reason the conclusion advo- cates something that would make it so that smoking is no longer brought about. However there is another argumentation scheme, one closely related to argument from negative consequences, that could also (even more usefully) be applied to this argument. It is called practical reasoning. The simplest version of this scheme, called practical inference in [1, p. 323] is cited below with its matching set of critical questions. Scheme for Practical Inference Major Premise: I have a goal G. Minor Premise: Carrying out this action A is a means to realize G. Conclusion: Therefore, I ought (practically speaking) to carry out this action A. Critical Questions for Practical Inference CQ1: What other goals do I have that should be considered that might conflict with G? 2 To more fully analyze the argument we could apply a more complex scheme called value-based practical reasoning [2].
- 25. 1 Argumentation Theory: A Very Short Introduction 11 CQ2: What alternative actions to my bringing about A that would also bring about G should be considered? CQ3: Among bringing about A and these alternative actions, which is arguably the most efficient? CQ4: What grounds are there for arguing that it is practically possible for me to bring about A? CQ5: What consequences of my bringing about A should also be taken into ac- count? CQ5 asks if there are negative consequences of the action (side effects) that need to be taken into account, and it can be seen that it covers argumentation from both positive and negative consequences. Applying the argumentation scheme for practical reasoning, we get the following reconstruction of the original argument. Premise 1: Governments have the goal of protecting their citizens from harm. Premise 2: Smoking is harmful to their citizens. Premise 3: Carrying out the action of banning smoking is a means for governments to protect their citizens from this harm. Conclusion: governments should ban smok- ing. This argument can be diagrammed as shown in Figure 1.5. Fig. 1.5 Argument Diagram of the Smoking Example with Practical Inference At his first move, Bob questions one of the premises of Ann’s argument. He asks her to give a reason to support her assertion that smoking tobacco is extremely harmful to the smoker’s health. In response to Bob’s question, Ann offers two such reasons. There could be various ways to represent the structure of her additional argument. The two reasons could perhaps function together as a linked argument, or they could function as two separate reasons having the structure of a convergent argument. But there is another way to analyze her additional argumentation. When Ann puts forward her argument, it appears that she is using her new asser- tion that smoking leads to many other health problems, including lung cancer and heart disease, as additional support for her previous premise that smoking tobacco
- 26. 12 Douglas Walton is extremely harmful to the smoker’s health. What about her next statement that ac- cording to the American Cancer Society, 3 million people die from smoking each year? It appears that this statement is being used to back up her previous statement that smoking leads to many other health problems, including lung cancer and heart disease. This seems to be a plausible reconstruction of her argument. We can produce an even better analysis of her argument using the argumentation scheme for argument from expert opinion. It would appear that she is citing the American Cancer Society as an expert source on health issues relating to cancer and smoking. We could analyze her argument by inserting an implicit premise to this effect, as shown in the argument diagram in Figure 1.6. Fig. 1.6 Argument Diagram of the Smoking Example with Implicit Premise On this analysis, the implicit premise that the American Cancer Society is an expert source is shown in the darkened box with dashed lines around it at the lower right. This premise, when taken with Ann’s explicit premise shown on the left, makes up an argument from expert opinion supporting her previous claim. This example shows how an argumentation scheme can be useful in helping an argument
- 27. 1 Argumentation Theory: A Very Short Introduction 13 analyst to identify an implicit premise that is not explicitly stated in the argument, but that is important for helping us to realize what the basis of the argument is. At move 4, Bob concedes Ann’s claim that the government has a responsibility to protect its citizens, but then he introduces a new argument. This argument is an interesting example of an enthymeme because the implicit statement needed to complete the argument is its conclusion. Premise: Governments have a responsibility to defend citizens’ freedom of choice. Premise: Banning smoking would be an intrusion into citizens’ freedom of choice. Implicit Conclusion: Governments should not ban smoking. Notice that the conclusion of this argument is the opposite of Ann’s previous argument that had the conclusion that governments should ban smoking. Thus Bob’s argument above is meant as a refutation of Ann’s previous argument. It is easy to see that Bob’s argument is connected to Ann’s whole previous argumentation, and is meant to attack it. This observation is part of the evidence that the dialog to this point hangs together in the sense that each move is relevant to previous moves made by one party or the other. There is perhaps one exception to the general pattern in the dialog that each move is connected to the prior move made by the other party. There is something that Bob should perhaps question after Ann’s move 5 when she attacks Bob’s assertion that banning smoking would be an intrusion into citizens’ freedom of choice. She attacks his assertion by arguing that smoking is not a matter of freedom of choice, but does this attack really bear on Bob’s assertion? One might reply that even though it may be true that citizens who have been smoking for a while are addicted to the habit, still, for the government to ban smoking would be an intrusion into citizens’ freedom of choice. It would force them by law to take steps to cure their addiction, and it would even force them by law not to start smoking in the first place. Whether Ann’s argument at move 5 really refutes Bob’s prior argument at move 4 is questionable. Instead of raising these questions about the relevance of Ann’s argument, Bob moves on to a different argument at his move 6. It could be suggested that he might have done better at his move 6 to attack Ann’s prior argument instead of hastily moving ahead to his next argument. 7 Types of Dialog Six basic types of dialog are fundamental to dialog theory – persuasion dialog, the inquiry, negotiation dialog, information-seeking dialog, deliberation, and eristic di- alog. The properties of these six types of dialog are summarized in Table 1. In argumentation theory, each type of dialog is used as a normative model that provides the standards for analyzing a given argument as used in a conversational setting in a given case. Each type of dialog has three stages, an opening stage, an argumentation stage and a closing stage. In a persuasion dialog, the proponent has a particular thesis to be proved, while the respondent has the role of casting doubt
- 28. 14 Douglas Walton Type of Dialog Initial Situation Participant’s Goal Goal of Dialog Persuasion Conflict of Opinions Persuade Other Party Resolve or Clarify Issue Inquiry Need to Have Proof Find and Verify Evi- dence Prove (Disprove) Hy- pothesis Negotiation Conflict of Interests Get What You Most Want Reasonable Settlement Both Can Live With Information- Seeking Need Information Acquire or Give Infor- mation Exchange Information Deliberation Dilemma or Practical Choice Co-ordinate Goals and Actions Decide Best Available Course of Action Eristic Personal Conflict Verbally Hit Out at Op- ponent Reveal Deeper Basis of Conflict Table 1.1 Six Basic Types of Dialog on that thesis or arguing for an opposed thesis. These tasks are set at the opening stage, and remain in place until the closing stage, when one party or the other fulfils its burden of persuasion. The proponent has a burden of persuasion to prove (by a set standard of proof) the proposition that is designated in advance as her ultimate thesis.3 The respondent’s role is to cast doubt on the proponent’s attempts to succeed in achieving such proof. The best known normative model of the persuasion type of dialog in the argumentation literature is the critical discussion [17]. It is not a formal model, but it has a set of procedural rules that define it as a normative structure for rational argumentation. The goal of a persuasion dialog is to reveal the strongest arguments on both sides by pitting one against the other to resolve the initial conflict posed at the opening stage. Each side tries to carry out its task of proving its ultimate thesis to the standard required to produce an argument stronger than the one produced by the other side. This burden of persuasion, as it is called [13], is set at the opening stage. Meeting one’s burden of persuasion is determined by coming up with a strong enough argu- ment using a chain of argumentation in which individual arguments in the chain are of the proper sort. To say that they are of the proper sort means that they fit argumen- tation schemes appropriate for the dialog. ‘Winning’ means producing an argument that is strong enough to discharge the burden of persuasion set at the opening stage. In a deliberation dialog, the goal is for the participants to arrive at a decision on what to do, given the need to take action. McBurney, Hitchcock and Parsons [11] set out a formal model of deliberation dialog in which participants make proposals and counter-proposals on what to do. In this model (p. 95), the need to take action is expressed in the form of a governing question like, “How should we respond to the prospect of global warming?” Deliberation dialog may be contrasted with persuasion dialog. 3 The notions of burden of persuasion and burden of proof have recently been subject to investiga- tion [6, 13]. Here we have adopted the view that in a persuasion dialog, the burden of persuasion is set at the opening stage, while a burden of proof can also shift from one side to the other during the argumentation stage.
- 29. 1 Argumentation Theory: A Very Short Introduction 15 In the model of [11, p. 100], a deliberation dialog consists of an opening stage, a closing stage, and six other stages making up the argumentation stage. Open: In this stage a governing question is raised about what is to be done. A gov- erning question, like ‘Where shall we go for dinner this evening?,’ is a question that expresses a need for action in a given set of circumstances. Inform: This stage includes discussion of desirable goals, constraints on possible actions that may be considered, evaluation of proposals, and consideration of relevant facts. Propose: Proposals cite possible action-options relevant to the governing question Consider: this stage concerns commenting on proposals from various perspec- tives. Revise: goals, constraints, perspectives, and action-options can be revised in light of comments presented and information gathering as well as fact-checking. Recommend: an option for action can be recommended for acceptance or non- acceptance by each participant. Confirm: a participant can confirm acceptance of the recommended option, and all participants must do so before the dialog terminates. Close: The termination of the dialog. The initial situation of deliberation is the need for action arising out of a choice between two or more alternative courses of action that are possible in a given situ- ation. The ultimate goal of deliberation dialog is for the participants to collectively decide on what is the best available course of action for them to take. An important property of deliberation dialog is that an action-option that is optimal for the group considered as a whole may not be optimal from the perspective of an individual participant [11, p. 98]. Both deliberation and persuasion dialogs can be about actions, and common forms of argument like practical reasoning and argument from consequences are often used in both types of dialog. There is no burden of persuasion in a delibera- tion dialog. Argumentation in deliberation is primarily a matter of supporting one’s own proposal for its chosen action-option, and critiquing the other party’s proposal for its chosen action-option. At the concluding stage one’s proposal needs to be abandoned in favor of the opposed one if the reasons given against it are strong enough to show that the opposed proposal is better to solve the problem set at the opening stage. Deliberation dialog is also different from negotiation dialog, which centrally deals with competing interests set at the opening stage. In a deliberation dialog, the participants evaluate proposed courses of action according to standards that may often be contrary to their personal interests.
- 30. 16 Douglas Walton 8 Dialectical Shifts In dialectical shifts of the kind analyzed in [18, pp. 100-116], an argument starts out as being framed in one kind of dialog, but as the chain of argumentation proceeds, it needs to be framed in a different type of dialog. Here is an example. Example 1.2 ((The Dam)). In a debate in a legislative assembly the decision to be made is whether to pass a bill to install a new dam. Arguments are put forward by both sides. One side argues that such a dam will cost too much, and will have bad ecological consequences. The other side argues that the dam is badly needed to produce energy. A lot of facts about the specifics of the dam and the area around it are needed to reasonably evaluate these opposed arguments. The assembly calls in experts in hydraulics engineering, ecology, economics and agriculture, to testify on these matters. Once the testimony starts, there has been a dialectical shift from the original deliberation dialog to an information-seeking dialog that goes into issues like what the ecological consequences of installing the dam would be. But this shift is not a bad thing, if the information provided by the testimony is helpful in aiding the legislative assembly to arrive at an informed and intelligent decision on how to vote. If this is so, the goal of the first dialog, the deliberation, is supported by the advent of the second dialog, the information-seeking interval. A constructive type of shift of this sort is classified as an embedding [18, p. 102], meaning that the advent of the second dialog helps the first type of dialog along toward it goal. An embedding underlies what can be called a constructive or licit shift. Other dialectical shifts are illicit, meaning that the advent of the second dia- log interferes with the proper progress of the first toward reaching its goal [18, p. 107]. Wells and Reed [19] constructed two formal dialectical systems to help judge whether a dialectical shift from a persuasion dialog to a negotiation dialog is licit or illicit. In their model, when a participant is engaged in a persuasion dialog, and proposes to shift to a different type of dialog, he must make a request to ask if the shift is acceptable to the other party. The other party has the option of insisting on continuing with the initial dialog or agreeing to shift to the new type. Wells and Reed have designed dialog rules to allow for a licit shift from persuasion to negoti- ation. Their model is especially useful in studying cases where threats are used as arguments. This type of argument, called the argumentum ad baculum in logic, has traditionally been classified as a fallacy, presumably because making threat to the other party is not a relevant move in a persuasion dialog. What one is supposed to do in a persuasion dialog is to offer evidence to support one’s contention, and mak- ing a threat does not do this, even though it may give the recipient of the threat a prudential reason to at least appear to go along the claim that the other party wants him to accept. The study of dialectical shifts is important in the study of informal fallacies, or common errors of reasoning, of a kind studied in logic textbooks since the time of Aristotle. A good example is provided in the next section.
- 31. 1 Argumentation Theory: A Very Short Introduction 17 9 Fallacious Arguments from Negative Consequences Argument from consequences (argumentum ad consequentiam) is an interesting fal- lacy that can be found in logic textbooks used to help students acquire critical think- ing skills. The following example is quoted from Rescher [15, p. 82]. Example 1.3 ((The Mexican War)). The United States had justice on its side in wag- ing the Mexican war of 1848. To question this is unpatriotic, and would give comfort to our enemies by promoting the cause of defeatism. The argument from consequences in this case was classified as a fallacy for the reason that is not relevant to the issue supposedly being discussed. Rescher (p. 82) wrote that “logically speaking,” it can be “entirely irrelevant that certain undesirable consequences might derive from the rejection of a thesis, or certain benefits accrue from its acceptance.” It can be conjectured from the example that the context is a persuasion dialog in which the conflict of opinions is the issue of which country had justice on its side in the Mexican war of 1848. This issue is a historical or ethical one, and prudential deliberation about whether questioning whether the U.S. had justice on its side would give comfort to our enemies is not relevant to resolving it. We can analyze what has happened by saying that there has been a dialectical shift at the point where the one side argues that questioning that the U.S. was in the right would promote defeatism. Notice that in this case there is nothing wrong in principle with using argumen- tation from negative consequences. As shown above, argument from the negative consequences is a legitimate argumentation scheme and any argument that fits this scheme is a reasonable argument in the sense that if the premises are acceptable, then subject to defeasible reasoning if new circumstances come to be known, the conclusion is acceptable as well. It’s not the argument itself that is fallacious, or structurally wrong as an inference. The problem is the context of dialog in which these instances of argumentation from negative consequences have been used. Such an argument would be perfectly appropriate if the issue set at the opening stage was how to make a decision about how to best support the diplomatic interests of the United States. However, notice that the first sentence of the example states very clearly what the ultimate thesis to be proved is: “The United States had justice on its side in waging the Mexican war of 1848.” The way that this thesis is supposed to be proved is by giving the other side reasons to come to accept it is true. Hence it seems reasonable to conjecture that the framework of the discussion is that of a persuasion dialog. Rescher (1969, 82) classified the Mexican War example as an instance of ar- gument from negative consequences that commits a fallacy of relevance. But what exactly is relevance? How is it to be defined? It can be defined by determining what type of dialog an argument in a given case supposedly belongs to, and then deter- mining what the issue to be resolved is by determining what the goal of the dialog is. The goal is set at the opening stage. If during the argumentation stage, the argumen- tation strays off onto a different path away from the proper kind of argumentation needed to fulfill this goal, a fallacy of relevance may have been committed. Based
- 32. 18 Douglas Walton on this analysis, it can be said that a fallacy of relevance has been committed in the Mexican War example. The dialectical shift to the prudential issue leads to a different type of dialog, a deliberation that interferes with the progress of the origi- nal persuasion dialog. The shift distracts the reader of the argument by introducing another issue, whether arguing this way is unpatriotic, and would give comfort to enemies by promoting the cause of defeatism. That may be more pressing, and it may indeed be true that arguing in this way would have brought about the negative consequences of giving comfort to enemies in promoting the cause of defeatism. Still, even though this argument from negative consequences might be quite reason- able, framed in the context of the deliberation, it is not useful to fulfill the burden of persuasion necessary to resolve the original conflict of opinions. 10 Relevance and Fallacies Many of the traditional informal fallacies in logic are classified under the heading of fallacies of relevance [8]. In such cases, the argument may be a reasonable one that is a valid inference based on premises that can be supported, but the problem is that the argument is not relevant. One kind of fallacy of irrelevance, as shown in the Mexican War example above, is the type of case where there has been a dialectical shift from one type of dialog to another. However, there is also another type of fallacy of relevance, where there is no dialectical shift, but there still is a failure to fulfill the burden of persuasion. In this kind of fallacy, which is very common, the arguer stays within the same type of dialog, but nevertheless fails to prove the conclusion he is supposed to prove and instead goes off in a different direction. The notion of relevance of argumentation can only be properly understood and analyzed by drawing a distinction between the opening stage of a dialog, where the burden of persuasion is set, and the argumentation stage, where arguments, linked into chains of argumentation, are brought forward by both sides. In a persuasion dialog, the burden of persuasion is set at the opening stage. Let’s say, for example, that the issue being discussed is whether one type of light bulb lasts longer than an- other. The proponent claims that one type of bulb lasts longer than another. She has the burden of persuasion to prove that by bringing forward arguments that support it. The respondent takes the stance of doubting the proponent’s claim. He does not have the burden of persuasion. His role is to cast doubt on the proponent’s attempts to prove her claim. He can do this by bringing forward arguments that attack the claim that one type of bulb lasts longer than another. Suppose, however, that during the argumentation stage, he wanders off to different topic by arguing that the one type of bulb is manufactured by a company has done bad things that have led to neg- ative consequences. This may be an emotionally exciting argument, and the claim made in it may even be accurate, but the problem is that it is irrelevant to the issue set at the opening stage. This species of fallacy of relevance is called the red herring fallacy. It occurs where an arguer wanders off the point in a discussion, and directs a chain of argu-
- 33. 1 Argumentation Theory: A Very Short Introduction 19 mentation towards proving some conclusion other than the one he is supposed to prove, as determined at the opening stage. The following example is a classic case of this type of fallacy cited in logics textbook [8]. Example 1.4 ((The Light Bulb)). The Consumers Digest reports that GE light bulbs last longer than Sylvania bulbs.4 But do you realize that GE is this country’s major manufacturer of nuclear weapons? The social cost of GE’s irresponsible behavior has been tremendous. Among other things, we are left with thousands of tons of nuclear waste with nowhere to put it. Obviously, the Consumers Digest is wrong. In the first sentence of the example, the arguer states the claim that he is supposed to prove (or attack) as his ultimate probandum in the discussion. He is supposed to be attacking the claim reported in the Consumers Digest that GE light bulbs last longer than Sylvania bulbs. How does he do this? He launches into a chain of argu- mentation, starting with the assertion that GE is this country’s major manufacturer of nuclear weapons. This makes GE sound very bad, and it would be an emotionally exciting issue to raise. He follows up the statement with another one to the effect that the social cost of GE’s irresponsible behavior has been tremendous. This is an- other serious allegation that would rouse the emotions of readers. Finally he uses argumentation from negative consequences by asserting that because of GE’s irre- sponsible behavior, we are left with thousands of tons of nuclear waste with nowhere to put it. This line of argumentation is a colorful and accusatory distraction. It di- verts the attention of the reader, who might easily fail to recall that the real issue is whether GE light bulbs last longer than Sylvania bulbs. Nothing in all the allega- tions made about GE’s allegedly responsible behavior carries any probative weight for the purpose of providing evidence against the claim reported in the Consumers Digest that GE light bulbs last longer than Sylvania bulbs. In the red herring fallacy the argumentation is directed along a path of argumen- tation other than one leading to proving the conclusion to be proved. The chain of argumentation goes off in a direction that is exciting and distracting for the audience to whom the argument was directed. The red herring fallacy becomes a problem in cases where the argumentation moves away from the proper chain of argument leading to the conclusion to be proved. Sometimes the path leads to the wrong con- clusion (one other than the one that is supposed to be proved), but in other cases it goes nowhere. The general pattern of this type of fallacy is displayed in Figure 1.7. Such a distraction may be harmless if there is plenty of time for discussion. But it can be a serious problem if there is not, because the real issue is not discussed. According to the burden of persuasion, the line of argumentation has as its end point a specific conclusion that needs to be proved. And if the argumentation moves away, it may not do this. 4 In the 1994 edition (p. 127), the first sentence of the light bulb example is, “The Consumers Digest reports that Sylvania light bulbs last longer than GE bulbs.” The example makes more sense if the two light bulb manufacturers names are reversed, and so I have presented the light bulb example this way.
- 34. 20 Douglas Walton Conclusion to be Proved Wrong Conclusion Premises Proper Path of Argumentation Irrelevant Argument Fig. 1.7 General Pattern of the Red Herring Fallacy 11 Basic Problems to be Solved This paper has only touched on the main concepts of argumentation theory and the main techniques used in argumentation studies. It is important to emphasize that the use of such concepts and techniques, while they have proved very valuable for teaching skills of critical thinking, have raised many problems about how to make the concepts and techniques more precise so that they can be applied more pro- ductively to realistic argumentation in natural language texts of discourse. Many of these problems arise from the fact that it can be quite difficult to interpret what is meant in a natural language text of discourse and precisely identify arguments contained in it. Ambiguity and vagueness are extremely common, and in many in- stances, the best one can do is to construct a hypothesis about how to interpret the argument based on the evidence given from the text of discourse. Much of the cur- rent research is indeed directed to studying how to marshal such evidence in an objective manner. For example, applying an abstract argumentation scheme to an argument in a specific case can be very tricky. In some cases, the same argument can fit more than one scheme. A project that needs to be undertaken is to devise criteria that students of critical thinking can use to help them determine in a particular case whether a given argument correctly fits a scheme or not. Another problem is that schemes can vary contextually. For example the scheme for argument from expert opinion used in law has to be different in certain respects from the standard scheme for argument from expert opinion cited above. The reason is that in the law rules have been developed for the admissibility and evaluation of expert opinion evidence. Any argumentation scheme for argument from expert opinion suitable for use in law would have to take these legal developments into account. Similarly, the problem of how to deal with enthymemes in a precise and objective manner has still not been solved, because we lack tools for determining what an
- 35. 1 Argumentation Theory: A Very Short Introduction 21 arguer’s commitments are, and what should properly be taken to constitute common knowledge, in specific cases where we are examining a text of discourse to find implicit statements. While the field has helped to develop objective methods for collecting evidence to deal with these problems in analyzing arguments, much work remains to be done in making them more precise. Of all the types of dialog, the one that has been most carefully and systematically studied is persuasion dialog, and there are formal systems of persuasion dialog [12]. Just recently, deliberation dialog also has come to be formally modeled [11, p. 100]. There is an abundance of literature on negotiation, and there are a software tools for assisting with negotiation argumentation. Comparatively less work is noticeable on information-seeking dialog and on the inquiry model of dialog. There is a scattering of work on eristic dialog, but there appears to be no formal model of this type of dialog that has been put forward, or at least that is well known in the argumentation literature. The notion of dialectical shift needs much more work. In particular, what kinds of evidence are useful in helping an argument analyst to determine when a dialectical shift has taken place during the sequence of argumentation in discourse is a good topic for study. The concepts of burden of proof and presumption are also central to the study of argumentation in different types of dialog. Space has prevented much discussion of these important topics, but the recent work that has been done on them raises some general questions that would be good topics for research. This work [6, 13] suggests that what is called the burden of persuasion is set at the opening stage of a persuasion dialog, and that this burden affects how a different kind of burden, often called the burden of production in law, shifts back and forth during the argumen- tation stage. Drawing this distinction is extremely helpful for understanding how a persuasion dialog should work, and more generally, it helps us to grasp how the critical questions should work as attacks on an argumentation scheme. But is there some comparable notion of burden of proof in the other types of dialog, for exam- ple in deliberation dialog? This unanswered question points a direction for future research in argumentation. References 1. D. Walton, C. Reed and F. Macagno. Argumentation Schemes. Cambridge University Press, Cambridge, UK, 2008. 2. T. J. M. Bench-Capon. Persuasion in practical argument using value-based argumentation frameworks. Logic and Computation, 13:429–448, 2003. 3. T. J. M. Bench-Capon and P. E. Dunne. Argumentation and dialogue in artificial intelligence, IJCAI 2005 tutorial notes. Technical report, Department of Computer Science, University of Liverpool, 2005. 4. P. Besnard and A. Hunter. Elements of Argumentation. MIT Press, Cambridge MA, USA, 2008. 5. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–358, 1995.
- 36. 22 Douglas Walton 6. T. F. Gordon, H. Prakken, and D. Walton. The Carneades model of argument and burden of proof. Artificial Intelligence, 171(10–15):875–896, 2007. 7. C. L. Hamblin. Fallacies. Methuen, London, UK, 1970. 8. P. J. Hurley. A Concise Introduction to Logic. Belmont, Wadsworth, CA, USA, 1994. 9. R. H. Johnson and A. J. Blair. The current state of informal logic. Informal Logic, 9:147–151, 1987. 10. E. C. W. Krabbe. Nothing but objections. In H. V. Hansen and R. C. Pinto, editors, Reason Reclaimed. Vale Press, Newport News, Virginia, USA, 2007. 11. D. Hitchcock, P. McBurney and S. Parsons. The eightfold way of deliberation dialogue. In- ternational Journal of Intelligent Systems, 22(1):95–132, 2007. 12. H. Prakken. Formal systems for persuasion dialogue. The Knowledge Engineering Review, 21(2):163–188, 2006. 13. H. Prakken and G. Sartor. Formalising arguments about the burden of persuasion. In Proceed- ings of the 11th International Conference on Artificial Intelligence and Law, pages 97–106. ACM Press, New York NY, USA, 2007. 14. C. Reed and G. Rowe. Araucaria: Software for argument analysis. International Journal of AI Tools, 14(3–4):961–980, 2004. 15. N. Rescher. Introduction to Logic. St. Martin’s Press, New York NY, USA, 1964. 16. H. Thorsrud. Cicero on his academics predecessors: the fallibilism of Arcesilaus and Carneades. Journal of the History of Philosophy, 40(1):1–18, 2002. 17. F. H. van Eemeren and R. F. Grootendorst. A Systematic Theory of Argumentation. Cambridge University Press, Cambridge, UK, 2004. 18. D. N. Walton and E. C. W. Krabbe. Commitment in Dialogue: Basic Concepts of Interpersonal Reasoning. SUNY Press, Albany NY, USA, 1995. 19. S. Wells and C. Reed. Knowing when to bargain: the roles of negotiation and persuasion in dialogue. In F. Grasso, R. Kibble, and C. Reed, editors, Proceedings of the ECAI workshop on Computational Models of Natural Argument (CMNA), Riva del Garda, Italy, 2006.
- 37. Part I Abstract Argument Systems
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- 39. Chapter 2 Semantics of Abstract Argument Systems Pietro Baroni and Massimiliano Giacomin 1 Abstract Argument Systems An abstract argument system or argumentation framework, as introduced in a semi- nal paper by Dung [13], is simply a pair A,R consisting of a set A whose elements are called arguments and of a binary relation R on A called attack relation. The set A may be finite or infinite in general, however, given the introductory purpose of this chapter, we will restrict the presentation to the case of finite sets of arguments. An argumentation framework has an obvious representation as a directed graph where nodes are arguments and edges are drawn from attacking to attacked arguments. A simple example of argumentation framework AF2.1 = {a,b},{(b,a)} is shown in Figure 2.1. While the word argument may recall several intuitive meanings, like the ones of “line of reasoning leading from some premise to a conclusion” or of “utterance in a dispute”, abstract argument systems are not (even implicitly or indirectly) bound to any of them: an abstract argument is not assumed to have any specific structure but, roughly speaking, an argument is anything that may attack or be attacked by another argument. Accordingly, the argumentation framework depicted in Figure 2.1 is suitable to represent many different situations. For instance, in a context of reasoning about weather, argument a may be associated with the inferential step b a Fig. 2.1 AF2.1: a simple argumentation framework Pietro Baroni Dip. di Elettronica per l’Automazione, University of Brescia, Via Branze 38, 25123 Brescia, Italy e-mail: baroni@ing.unibs.it Massimiliano Giacomin Dip. di Elettronica per l’Automazione, University of Brescia, Via Branze 38, 25123 Brescia, Italy e-mail: giacomin@ing.unibs.it I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 25 DOI 10.1007/978-0-387-98197-0 2, c Springer Science+Business Media, LLC 2009
- 40. 26 Pietro Baroni and Massimiliano Giacomin “Tomorrow will rain because the national weather forecast says so”, while b with “Tomorrow will not rain because the regional weather forecast says so”. In a legal dispute a might be associated with a prosecutor’s statement “The suspect is guilty because an eyewitness, Mr. Smith, says so” while b with the defense reply “Mr. Smith is notoriously alcohol-addicted and it is proved that he was completely drunk the night of the crime, therefore his testimony should not be taken into account”. In a marriage arrangement setting (corresponding to the game-theoretic formulation of stable marriage problem [13]), a may be associated with “The marriage between Alice and John”, while b with “The marriage between Barbara and John”. Similarly, the attack relation has no specific meaning: again at a rough level, if an argument b attacks another argument a, this means that if b holds then a can not hold or, putting it in other words, that the evaluation of the status of b affects the evaluation of the status of a. In the weather example, two intrinsically conflicting conclusions are confronted and the attack relation corresponds to the fact that one conclusion is preferred to the other, e.g. because the regional forecast is considered more reliable than the national one (this kind of attack is often called rebut in the literature). In the legal dispute, the fact that Mr. Smith was drunk is not incompatible per se with the fact that the suspect is guilty, however it affects the reason why s/he should be believed to be guilty (this kind of attack has sometimes been called undercut in the literature, but the use of this term has not been uniform). In the stable marriage problem, the attack from b to a may simply be due to the fact that John prefers Barbara to Alice. Abstracting away from the structure and meaning of arguments and attacks en- ables the study of properties which are independent of any specific aspect, and, as such, are relevant to any context that can be captured by the very terse formalization of abstract argument systems. As a counterpart of this generality, abstract argument systems feature a limited expressiveness and can hardly be adopted as a modeling formalism directly usable in application contexts. In fact the gap between a prac- tical problem and its representation as an abstract argument system is patently too wide and requires to be filled by a less abstract formalism, dealing in particular with the construction of arguments and the conditions for an argument to attack another one. For instance, in a given formalism, attack may be identified at a purely syntac- tic level (e.g. the conclusion of an argument being the negation of the conclusion of another one). In another formalism, attack may not be a purely syntactic notion (e.g. the conclusions of two arguments being incompatible because they give rise to contradiction through some strict logical deduction). Only after the attack relation is defined in these “concrete” terms, it is appropriate to derive its abstract represen- tation and exploit it for the analysis of general properties. 2 Abstract Argumentation Semantics Given that arguments may attack each other, it is clearly the case that they can not stand all together and their status is subject to evaluation. In particular what we are usually interested in is the justification state of arguments: while this notion will
- 41. 2 Semantics of Abstract Argument Systems 27 be given a precise formal meaning1 in the following, intuitively an argument is re- garded as justified if it has some way to survive the attacks it receives, as not justified (or rejected) otherwise. In the following we will use the term argument evaluation to refer to a process aimed at determining the justification state of the arguments in an abstract argument system, i.e. on the basis of the attack relation only. In the case of Figure 2.1 the outcome of the evaluation process is almost undebatable: b should be justified (in particular it does not receive attacks) while a should not be justified (it has no way to survive the attack from b). The case of mutual attack of Figure 2.2 is less obvious: evident symmetry reasons suggest that the evaluation process should assign to b and a the same state, which should correspond neither to “full” justification nor to “full” rejection. An intuitive counterpart to Figure 2.2 is given by the case of two contradicting weather forecasts about rain tomorrow, coming from sources considered equally reliable. It is important to remark that the evaluation process in abstract argument systems concerns the justification state of arguments, not of their conclusions (conclusions “do not exist” in the abstract setting). To clarify this distinction consider a new ar- gument put forward by the prosecutor in the example of legal dispute: “The suspect is guilty because his fingerprints have been found on the crime scene”. The new ar- gument (let say c) has no attack relationships with the previous ones (in particular, the fact that the fingerprints have been found on the crime scene does not conflict with the fact that the witness was drunk that night) and, as a consequence, the cor- responding abstract representation is the one given in Figure 2.3. Then a is still rejected, b is still justified and c (not receiving any attack) is justified too. Note that, at the underlying level, a and c have the same conclusion (“the suspect is guilty”) and that this conclusion should now be intuitively regarded as justified. However, summarizing the justification state of conclusions is outside the scope of the ab- stract representation. An argumentation semantics is the formal definition of a method (either declar- ative or procedural) ruling the argument evaluation process. Two main styles of ar- gumentation semantics definition can be identified in the literature: extension-based and labelling-based. In the extension-based approach a semantics definition specifies how to derive from an argumentation framework a set of extensions, where an extension E of an argumentation framework A,R is simply a subset of A, intuitively representing a set of arguments which can “survive together” or are “collectively acceptable”. b a Fig. 2.2 AF2.2: a mutual attack b a c Fig. 2.3 AF2.3: presence of an isolated argument 1 Actually the term “justification” is used informally and with different meanings in the literature.
- 42. 28 Pietro Baroni and Massimiliano Giacomin Putting things in more formal terms, given an extension-based semantics S and an argumentation framework AF = A,R we will denote the set of extensions pre- scribed by S for AF as ES(AF) ⊆ 2A . The justification state of an argument a ∈ A according to an extension-based semantics S is then a derived concept, defined in terms of the membership of a to the elements of ES(AF). In the labelling-based approach a semantics definition specifies how to derive from an argumentation framework a set of labellings, where a labelling L is the as- signment to each argument in A of a label taken from a predefined set L, which corresponds to the possible alternative states of an argument in the context of a sin- gle labelling. Putting things in formal terms, given a labelling-based semantics S with set of labels L, a labelling of an argumentation framework A,R is a mapping L : A → L. We denote the set of all possible labellings, i.e. of all possible map- pings from A to L, as L(A,L), and the set of labellings prescribed by S for AF as LS(AF) ⊆ L(A,L). Again, the justification state of an argument a according to a labelling-based semantics S turns out to be a derived concept, defined in terms of the labels assigned to a in the various elements of LS(AF). Let us exemplify the semantics definition styles with reference to the simple case of Figure 2.2. Let S1 ext be a hypothetical extension-based semantics whose underly- ing principle consists in identifying as extensions the largest sets of non-conflicting arguments which reply to the attacks they receive. Intuitively, both the sets {a} and {b} satisfy this principle, yielding ES1 ext (AF2.2) = {{a},{b}}. Alternatively, let S2 ext be a hypothetical extension-based semantics whose underlying principle consists in identifying as extensions the sets of arguments which are unattacked. Clearly, no argument features this property in AF2.2, yielding as unique extension the empty set: ES2 ext (AF2.2) = {/ 0}. Let us turn to the labelling-based style and adopt the set of labels L = {in,out,undecided}. Let S1 lab be a hypothetical labelling-based seman- tics whose underlying principle consists in labelling in the elements of the largest sets of non-conflicting arguments which reply to the attacks they receive and la- belling out the arguments attacked by arguments labelled in. Then two labellings are prescribed: LS1 lab (AF2.2) = {{(a,in),(b,out)},{(a,out),(b,in)}}. On the other hand, let S2 lab be a hypothetical labelling-based semantics whose underlying princi- ple consists in labelling in only unattacked arguments, labelling out the arguments attacked by arguments labelled in, and labelling undecided all the remaining ones. Clearly this would yield LS2 lab (AF2.2) = {{(a,undecided),(b,undecided)}}. As it is also evident from the above examples, for a given argumentation frame- work one or more extensions (labellings) may be prescribed by a given semantics. It has to be remarked that also the case where no extensions (labellings) are prescribed for an argumentation framework AF is in general possible, namely ES(AF) = / 0 (LS(AF) = / 0). This corresponds to the case where the semantics S is undefined in AF since no extensions (labellings) compliant with the definition of S exist. For extension-based semantics, note in particular that the case ES(AF) = / 0 is very differ- ent from ES(AF) = {/ 0}. In the following we will denote as DS the set of argumenta- tion frameworks where a semantics S is defined, namely, DS = {AF | ES(AF) = / 0} or DS = {AF | LS(AF) = / 0}. A semantics S is called universally defined if any
- 43. 2 Semantics of Abstract Argument Systems 29 argumentation framework belongs to DS. Further, an important terminological con- vention concerning the cardinality of the set of extensions (labellings) is worth intro- ducing. If a semantics S always prescribes exactly one extension (labelling) for any argumentation framework belonging to DS then S is said to belong to the unique- status (or single-status) approach, otherwise it is said to belong to the multiple-status approach. While the adoption of the labelling or extension-based style is a matter of subjec- tive preference by the proponent(s) of a given semantics, a natural question concerns the expressiveness of the two styles. It is immediate to observe that any extension- based definition can be equivalently expressed in a simple labelling-based formula- tion, where a set of two labels is adopted (let say L = {in,out}) corresponding to extension membership. On the other hand, an arbitrary labelling can not in general be formulated in terms of extensions. It is however a matter of fact that labellings considered in the literature typically include a label called in which naturally cor- responds to extension membership, while other labels correspond to a partition of other arguments, easily derivable from extension membership and the attack rela- tion. As a consequence, equivalent extension-based formulations of labelling-based semantics are typically available. Given this fact and the definitely prevailing cus- tom of adopting the extension-based style in the literature, this chapter will focus on extension-based semantics. 3 Principles for extension-based semantics While alternative semantics proposals differ from each other by the specific notion of extension they endorse, one may wonder whether there are reasonable general properties which are shared by all (or, at least, most) existing semantics and can be regarded as evaluation principles for new semantics. We discuss these principles in the following, distinguishing between properties of individual extensions and prop- erties of the whole set of extensions. The reader may refer to [2] for a more extensive analysis of this issue. The first basic requirement for any extension E corresponds to the idea that E is a set of arguments which “can stand together”. Consequently, if an argument a attacks another argument b, then a and b can not be included together in an extension. This corresponds to the following conflict-free principle, which, as to our knowledge, is satisfied by all existing semantics in the literature. Definition 2.1. Given an argumentation framework AF = A,R, a set S ⊆ A is conflict-free, denoted as cf(S), if and only if ∄a,b ∈ S such that aRb. A semantics S satisfies the conflict-free principle if and only if ∀AF ∈ DS, ∀E ∈ ES(AF) E is conflict-free. A further requirement corresponds to the idea that an extension is a set of ar- guments which “can stand on its own”, namely is able to withstand the attacks it receives from other arguments by “replying” with other attacks. Formally, this has a counterpart in the property of admissibility, which lies at the heart of all seman- tics discussed in [13] and is shared by many more recent proposals. The definition
- 44. 30 Pietro Baroni and Massimiliano Giacomin is based on the notions of acceptable argument and admissible set. In words, a is acceptable wrt. (or, equivalently, defended by) a set S if S “counterattacks” all at- tackers of a and a set S is admissible if it is conflict-free and defends all its elements. Definition 2.2. Given an argumentation framework AF = A,R, an argument a ∈ A is acceptable wrt. a set S ⊆ A if and only if ∀b ∈ A bRa ⇒ SRb.2 The function FAF : 2A → 2A which, given a set S ⊆ A, returns the set of the acceptable arguments wrt. S, is called the characteristic function of AF. Definition 2.3. Given an argumentation framework AF = A,R, a set S ⊆ A is admissible if and only if cf(S) and ∀a ∈ S a is acceptable wrt. S. The set of all the admissible sets of AF is denoted as A S(AF). Definition 2.4. A semantics S satisfies the admissibility principle if and only if ∀AF ∈ DS ES(AF) ⊆ A S(AF), namely ∀E ∈ ES(AF) it holds that: a ∈ E ⇒ (∀b ∈ A,bRa ⇒ ERb). (2.1) The property of reinstatement is somewhat dual with respect to the notion of defense. Intuitively, if the attackers of an argument a are in turn attacked by an ex- tension E one may assume that they have no effect on a: then a should be, in a sense, reinstated, therefore it should belong to E. This leads to the following reinstatement principle which turns out to be the converse of the implication (2.1). Definition 2.5. A semantics S satisfies the reinstatement principle if and only if ∀AF ∈ DS, ∀E ∈ ES(AF) it holds that: (∀b ∈ A,bRa ⇒ ERb) ⇒ a ∈ E. (2.2) To exemplify these properties consider the argumentation framework AF2.4 rep- resented in Figure 2.4. It is easy to see that A S(AF2.4) = {/ 0,{a},{b},{a,c}}, in particular c needs the “help” of a in order to be defended from the attack of b. The set {b} obviously satisfies the reinstatement condition (2.2), and so does {a,c}, while the set {a} does not, since it attacks all attackers of c, but does not include it. Let us turn now to principles concerning sets of extensions. A first fundamen- tal principle corresponds to the fact that the set of extensions only depends on the attack relation between arguments while it is totally independent of any property of arguments at the underlying language level. Formally, this principle corresponds to the fact that argumentation frameworks which are isomorphic have the “same” (modulo the isomorphism) extensions, as stated by the following definitions. b a c Fig. 2.4 AF2.4: not just a mutual attack 2 With a little abuse of notation we define SRb ≡ ∃a ∈ S : aRb. Similarly, bRS ≡ ∃a ∈ S : bRa.
- 45. 2 Semantics of Abstract Argument Systems 31 Definition 2.6. Two argumentation frameworks AF1 = A1,R1 and AF2 = A2,R2 are isomorphic if and only if there is a bijective mapping m : A1 → A2, such that (a,b) ∈ R1 if and only if (m(a),m(b)) ∈ R2. This is denoted as AF1 ⊜m AF2. Definition 2.7. A semantics S satisfies the language independence principle if and only if ∀AF1 ∈ DS,∀AF2 ∈ DS such that AF1 ⊜m AF2,ES(AF2) = {M(E) | E ∈ ES(AF1)}, where M(E) = {b | ∃a ∈ E,b = m(a)}. All argumentation semantics we are aware of adhere to this principle. Another principle concerns possible inclusion relationships between extensions. Considering for instance the admissible sets of the example in Figure 2.4, the empty set is of course included in all other ones, moreover the set {a} is included in {a,c}. In such a situation, the question arises whether an extension may be a strict subset of another one. The answer is typically negative, for reasons which are related in par- ticular with the notion of skeptical justification, to be discussed later. Accordingly, the I-maximality principle can be introduced. Definition 2.8. A set of extensions E is I-maximal if and only if ∀E1,E2 ∈ E, if E1 ⊆ E2 then E1 = E2. A semantics S satisfies the I-maximality principle if and only if ∀AF ∈ DS ES(AF) is I-maximal. A further principle is related with the notion of attack which is intrinsically di- rectional: if a attacks b this corresponds to the fact that a has the power to affect b, while not vice versa (unless, in turn, b attacks a). Generalizing this consideration, one may require the evaluation of an argument a to be only affected by its attackers, the attackers of its attackers and so on, i.e. by its ancestors in the attack relationship. In other words, an argument a may affect another argument b only if there is a di- rected path from a to b. For instance, in Figure 2.4 the evaluations of a and b affect each other and both affect c, while the evaluation of c should not affect those of a and b but rather depend on them. It is reasonable to require this notion of direction- ality to be reflected by the set of extensions prescribed by a semantics. This can be formalized by referring to sets of arguments not receiving attacks from outside. Definition 2.9. Given an argumentation framework AF = A,R, a non-empty set S ⊆ A is unattacked if and only if ∃a ∈ (AS) : aRS. The set of unattacked sets of AF is denoted as U S(AF). We also need to introduce the notion of restriction of an argumentation frame- work to a subset S of its arguments. Definition 2.10. Let AF = A,R be an argumentation framework. The restriction of AF to S ⊆ A is the argumentation framework AF↓S = S,R∩(S×S). The directionality principle can then be defined by requiring an unattacked set to be unaffected by the remaining part of the argumentation framework as far as extensions are concerned. Definition 2.11. A semantics S satisfies the directionality principle if and only if ∀AF ∈ DS,∀S ∈ U S(AF),A ES(AF,S) = ES(AF↓S), where A ES(AF,S) {(E ∩S) | E ∈ ES(AF)} ⊆ 2S.
- 46. 32 Pietro Baroni and Massimiliano Giacomin In words, the intersection of any extension prescribed by S for AF with an unattacked set S is equal to one of the extensions prescribed by S for the re- striction of AF to S, and vice versa. Referring to the example of Figure 2.4, U S(AF2.4) = {{a,b},{a,b,c}}. Then, the restriction of AF2.4 to its unattacked set {a,b} coincides with AF2.2 shown in Figure 2.2. A hypothetical semantics S1 such that ES1 (AF2.4) = {{a,c},{b}} and ES1 (AF2.2) = {{a},{b}} would satisfy the di- rectionality principle in this case. On the other hand, a hypothetical semantics S2 such that ES2 (AF2.4) = {{a,c}} and ES2 (AF2.2) = {{a},{b}} would not. 4 The notion of justification state At a first level, the justification state of an argument a can be conceived in terms of its extension membership. A basic classification encompasses only two possible states for an argument, namely justified or not justified. In this respect, two alterna- tive types of justification, namely skeptical and credulous can be considered. Definition 2.12. Given a semantics S and an argumentation framework AF ∈ DS, an argument a is: • skeptically justified if and only if ∀E ∈ ES(AF) a ∈ E; • credulously justified if and only if ∃E ∈ ES(AF) a ∈ E. Clearly the two notions coincide for unique-status approaches, while, in general, credulous justification includes skeptical justification. To refine this relatively rough classification, a consolidated tradition considers three justification states. Definition 2.13. [18] Given a semantics S and an argumentation framework AF ∈ DS, an argument a is: • justified if and only if ∀E ∈ ES(AF), a ∈ E (this is clearly the “strongest” possible level of justification, corresponding to skeptical justification); • defensible if and only if ∃E1,E2 ∈ ES(AF) : a ∈ E1,a / ∈ E2 (this is a weaker level of justification, corresponding to arguments which are credulously but not skeptically justified); • overruled if and only if ∀E ∈ ES(AF), a / ∈ E (in this case a can not be justified in any way and should be rejected). Some remarks are worth about the above classification, which is largely (and sometimes implicitly) adopted in the literature: on one hand, while (two) differ- ent levels of justified arguments are encompassed, no distinctions are drawn among rejected arguments; on the other hand, the attack relation plays no role in the deriva- tion of justification state from extensions. These points are addressed by a different classification of justification states introduced by Pollock [16] in the context of a unique-status approach. Again, three cases are possible for an argument a: • a belongs to the (unique) extension E: then it is justified, or, using Pollock’s terminology, undefeated;
- 47. 2 Semantics of Abstract Argument Systems 33 • a does not belong to the (unique) extension E and is attacked by (some member of) E: then, using Pollock’s terminology, it is defeated outright, corresponding to a strong form of rejection; • a does not belong to the (unique) extension E but does not receive attacks from E: then it is provisionally defeated, corresponding to a weaker form of rejection. Both these classifications of justification states are unsatisfactory in some respect. It is however possible to combine the intuitions underlying both of them, obtaining a systematic classification of seven possible justification states [6]. As a starting point, considering the relationship between an argument a and a specific extension E, three main situations3, as in Pollock’s classification, can be envisaged: • a is in E, denoted as in(a,E), if a ∈ E; • a is definitely out from E, denoted as do(a,E), if a / ∈ E ∧ERa; • a is provisionally out from E, denoted as po(a,E), if a / ∈ E ∧¬ERa. Then, taking into account the possible existence of multiple extensions, an argu- ment can be in any of the above three states with respect to all, some or none of the extensions. This gives rise to 27 hypothetical combinations. It is however easy to see that some of them are impossible: for instance, if an argument is in a given state with respect to all extensions this clearly excludes that it is in another state with respect to any extension. Directly applying this kind of considerations, seven possible justification states emerge. Definition 2.14. Given an argumentation framework AF = A,R and a non-empty set of extensions E the possible justification states of an argument a ∈ A according to E are defined by the following mutually exclusive conditions: • ∀E ∈ E in(a,E), denoted as JSI; • ∀E ∈ E do(a,E), denoted as JSD; • ∀E ∈ E po(a,E), denoted as JSP; • ∃E ∈ E : do(a,E), ∃E ∈ E : po(a,E), and ∄E ∈ E : in(a,E), denoted as JSDP; • ∃E ∈ E : in(a,E), ∃E ∈ E : po(a,E), and ∄E ∈ E : do(a,E), denoted as JSIP; • ∃E ∈ E : in(a,E), ∃E ∈ E : do(a,E), and ∄E ∈ E : po(a,E), denoted as JSID; • ∃E ∈ E : in(a,E), ∃E ∈ E : do(a,E), and ∃E ∈ E : po(a,E), denoted as JSIDP. Correspondences with “traditional” definitions of justification states are easily drawn. An argument is skeptically justified if and only if it is in the JSI state, while credulous justification corresponds to the disjunction of the states JSI, JSIP, JSID, and JSIDP. As to Definition 2.13, the state of justified corresponds to JSI, the state of overruled to the disjunction of JSD, JSP, and JSDP, while the state of defensible to the disjunction of JSIP, JSID, and JSIDP. Turning to Pollock’s classification, it is easy to see that in the case of a unique-status semantics only JSI, JSD and JSP may hold, which correspond to the state of undefeated, defeated outright and provisionally defeated, respectively. Other meaningful ways of defining aggregated justification states are investigated in [6]. 3 The case a ∈ E ∧ERa is prevented by the conflict-free principle.
- 48. 34 Pietro Baroni and Massimiliano Giacomin 5 A review of extension-based argumentation semantics Turning from general notions to actual approaches, we now examine several argu- mentation semantics proposed in the literature. From a historical point of view, it is possible to distinguish between: • four “traditional” semantics, considered in Dung’s original paper [13], namely complete, grounded, stable, and preferred semantics; • subsequent proposals introduced by various authors in the literature, often to overcome some limitation or improve some undesired behavior of a traditional approach: we consider stage, semi-stable, ideal, CF2, and prudent semantics. It is important to note that while the definitions of the semantics we will de- scribe are formulated in the context of purely abstract argumentation frameworks, the underlying intuitions have commonalities with other (somewhat more concrete) formalizations in related contexts. In fact, as already mentioned, one of the main results of Dung’s paper is showing that abstract argumentation frameworks are able to capture the properties of a large variety of more specific formalisms. This means that it is possible to define mappings from entities defined in a more specific formal- ism into an argumentation framework. In [13] mappings of this kind are provided for stable marriage problems, default theories in Reiter’s default logic, logic programs with negation as failure, and Pollock’s theory of defeasible reasoning. Let AF = A,R be an argumentation framework obtained through a mapping from a more specific formalism (e.g. a logic program). We have now two ways of deriving a “meaningful” set of subsets of A: on one hand, we can apply a purely abstract semantics S to AF, obtaining the relevant set of extensions ES(AF). On the other hand, we can start from a meaningful concept in the underlying formalism (e.g. the set of models of a logic program) and then map it into a set M of subsets of A (continuing the example, by deriving the set of arguments corresponding to each model). An important question then arises: can interesting relationships be identified between ES(AF) and M, given a suitable choice of the semantics S? A strikingly affirmative answer is provided in [13]: it is shown that properly selecting S among the traditional grounded, preferred, and stable semantics one obtains sets of extensions ES(AF) which coincide with the sets of arguments corresponding to meaningful notions in the formalisms mentioned above. Specific indications about these coincidences will be given in the review of individual semantics. At a general level, they confirm that abstract argumentation semantics is a powerful analysis tool, able to focus on essential properties of a variety of formalisms and to shed light on their (possibly hidden) significant common features. 5.1 Complete semantics We start our review by the notion of complete extension, as it lies at the heart of all traditional Dung’s semantics. Actually the notion of complete extension is not asso- ciated with a notion of complete semantics in [13], but the term complete semantics has subsequently gained acceptance in the literature and will be used in the present
- 49. 2 Semantics of Abstract Argument Systems 35 analysis to refer to the properties of the set of complete extensions. Complete se- mantics is denoted as CO. The notion of complete extension is based on the principles of admissibility and reinstatement: a complete extension is a set which is able to defend itself and in- cludes all arguments it defends, as stated by the following definition. Definition 2.15. Given an argumentation framework AF = A,R, a set E ⊆ A is a complete extension if and only if E is admissible and every argument of A which is acceptable wrt. E belongs to E, i.e. E ∈ A S(AF)∧FAF(E) ⊆ E. It is worth noting here that the empty set is admissible and that arguments not receiving attacks in an argumentation framework AF (called initial arguments and denoted as IN(AF) in the sequel) are acceptable wrt. the empty set (in fact IN(AF) = FAF (/ 0)). It can be shown that the following properties hold: • ∀AF ECO(AF) = / 0 (namely CO is universally defined); • / 0 ∈ ECO(AF) if and only if IN(AF) = / 0; • ∀E ∈ ECO(AF) IN(AF) ⊆ E. Due to reinstatement, any complete extension not only includes the initial ar- guments, but also the arguments they defend, those which are in turn defended by them, and so on. More formally, for any argumentation framework AF = A,R, given a set S ⊆ A, let F1 AF (S) FAF (S) and for i 1, Fi AF (S) FAF (Fi−1 AF (S)). Then, it turns out that ∀AF, ∀E ∈ ECO(AF), ∀i ≥ 1, Fi AF (/ 0) ⊆ E. In the example of Figure 2.1, {b} = IN(AF2.1) is clearly the only complete exten- sion. Similarly in the example of Figure 2.3 the only complete extension is {b,c}. In the example of Figure 2.2 there are three complete extensions: / 0 (as there are no initial arguments), {a}, and {b} (each one defending itself against the other). By similar considerations and the property of reinstatement it is easy to see that in the example of Figure 2.4 ECO(AF2.4) = {/ 0,{a,c},{b}}. The example of Fig- ure 2.5 requires some more articulated considerations. First note that IN(AF2.5) = / 0 hence / 0 ∈ ECO(AF2.5). Then note that all singletons except {c} are admissible: to check whether they are complete extensions or not we have to resort to the reinstate- ment property. In particular, a defends c from b but not from d, so {a} stands as a complete extension. On the other hand b defends d from its only attacker c, there- fore {b,d} is a complete extension, while {b} is not. Argument d does not defend any other argument apart itself, thus {d} is a complete extension. We have now to consider possibly larger admissible sets. It is easy to see that {a,c} and {a,d} are admissible (and attack all arguments they do not include). Summarizing we have ECO(AF2.5) = {/ 0,{a},{d},{b,d},{a,c},{a,d}}. We complete the treatment of CO by considering its ability to satisfy the proper- ties which are not common to all semantics: we note that admissibility and reinstate- ment are enforced by definition, while the satisfaction of directionality is proved in b a c d Fig. 2.5 AF2.5: two mutual attacks.
- 50. 36 Pietro Baroni and Massimiliano Giacomin [2]. As the examples above abundantly show, CO does not satisfy I-maximality, since a complete extension E1 may be a proper subset of a complete extension E2. 5.2 Grounded semantics Grounded semantics, denoted as GR, is a traditional unique-status approach whose formulation in argumentation-based reasoning (see e.g. the one proposed by Pollock in [16]) predates the following quite technical definition given in [13]. Definition 2.16. The grounded extension of an argumentation framework AF, de- noted as GE(AF), is the least fixed point of its characteristic function FAF . The underlying and preexisting informal intuition is however rather simple and can be directly put in relationship with some notions we already discussed in the context of complete semantics. The basic idea is that the (unique) grounded exten- sion can be built incrementally starting from the initial unattacked arguments. Then the arguments attacked by them can be suppressed, resulting in a modified argumen- tation framework where, possibly, the set of initial arguments is larger. In turn the arguments attacked by the “new” initial arguments can be suppressed, and so on. The process stops when no new initial arguments arise after a deletion step: the set of all initial arguments identified so far is the grounded extension. An example with a graphical illustration of this incremental process is given in Figure 2.6, resulting in EGR(AF) = {{a,c}}. To put it in other words, the grounded extension includes those and only those arguments whose defense is “rooted” in initial arguments (see [2] for a formal treatment of this notion, called strong defense). If there are no initial arguments the grounded extension is the empty set. The reader should have noticed that the above construction corresponds to the iterated application of FAF (/ 0) already met in previous section: the set of arguments obtained up to each step i above coincides with Fi AF (/ 0) and the process stops when Fi AF (/ 0) = Fi+1 AF (/ 0) (i.e. a fixed point of FAF is reached). As a counterpart to this intuitive correspondence, it is proved in [13] that: i) the grounded extension is the least (wrt. set inclusion) complete extension: for any AF GE(AF) ∈ ECO(AF) and ∀E ∈ ECO(AF) GE(AF) ⊆ E; ii) for any finite (and, more generally, finitary [13]) AF GE(AF) = ∪i=1,...,∞Fi AF (/ 0). Given the above explanations it should be now immediate to see that GE(AF2.1) = {b}, GE(AF2.3) = {b,c}, GE(AF2.2) = GE(AF2.4) = GE(AF2.5) = / 0. c d b f a e c d f a e c f a e Fig. 2.6 AF2.6: an example to illustrate grounded semantics
- 51. 2 Semantics of Abstract Argument Systems 37 Besides the relationship with Pollock’s approach [16], the grounded extension has been put in correspondence in [13] with the well-founded semantics of logic pro- grams [20]. Turning to general semantics properties it is immediate to see that GR is universally defined and satisfies admissibility, reinstatement and I-maximality. It is proved in [2] that GR also satisfies directionality. 5.3 Stable semantics Stable semantics, denoted as S T, relies on a very simple (and easy to formalize) intuition: an extension should be able to attack all arguments not included in it. This leads to the notion of stable extension [13]. Definition 2.17. Given an argumentation framework AF = A,R, a set E ⊆ A is a stable extension of AF if and only if E is conflict-free and ∀a ∈ A, a / ∈ E ⇒ ERa. By definition, any stable extension E is also a complete extension (thus in par- ticular GE(AF) ⊆ E) and a maximal conflict-free set of AF. Referring to the ex- amples seen so far, identifying stable extensions is straightforward: ES T(AF2.1) = {b}, ES T(AF2.2) = {{a},{b}}, ES T(AF2.3) = {b,c}, ES T(AF2.4) = {{a,c},{b}}, ES T(AF2.5) = {{a,c},{a,d},{b,d}}, ES T(AF2.6) = {{a,c,e},{a,c, f}}. The simple intuition underlying stable semantics has significant counterparts in several contexts: it is proved in [13] that stable extensions can be put in correspon- dence with solutions of cooperative n-person games, solutions of the stable mar- riage problem, extensions of Reiter’s default logic [19], and stable models of logic programs [15]. Stable semantics however has also a significant drawback: it is not universally defined as there are argumentation frameworks where no stable exten- sions exist. A simple example is provided in Figure 2.7: no conflict-free set is able to attack all other arguments in this case. While it has sometimes been claimed by supporters of stable semantics that situations where stable extensions do not exist are “pathological” in some sense, it has been shown in [13] that perfectly reasonable problems may be formalized with argumentation frameworks such that ES T(AF) = / 0. As to general properties, it is easy to see that I-maximality is enforced by defi- nition and, since stable extensions are a subset of complete extensions, also admis- sibility and reinstatement are satisfied. S T is not directional, due to the fact that it is not universally defined. To see this, consider the example in Figure 2.8: we have that ES T(AF2.8) = {{b}}, however, considering the unattacked set S = {a,b}, we have ES T(AF2.8↓S) = {{a},{b}}. Therefore A ES T(AF2.8,S) = {{b}} = ES T(AF2.8↓S). a b c Fig. 2.7 AF2.7: a three-length cycle
- 52. 38 Pietro Baroni and Massimiliano Giacomin 5.4 Preferred semantics The “aggressive” requirement that an extension must attack anything outside it may be relaxed by requiring that an extension is as large as possible and able to defend itself from attacks. This is captured by the notion of preferred extension [13]. Definition 2.18. Given an argumentation framework AF = A,R, a set E ⊆ A is a preferred extension of AF if and only if E is a maximal (wrt. set inclusion) element of A S(AF). According to this definition every preferred extension E is also a complete ex- tension (entailing that GE(AF) ⊆ E); indeed, preferred extensions may be equiv- alently defined as maximal complete extensions. It follows that preferred seman- tics, denoted as PR, is universally defined (as complete semantics is) and that, taking into account the treatment of the examples in Section 5.1, EPR(AF2.1) = {{b}}, EPR(AF2.2)={{a},{b}},EPR(AF2.3)={{b,c}},EPR(AF2.4)={{a,c},{b}}, EPR(AF2.5) = {{a,c},{a,d},{b,d}}, and EPR(AF2.6) = {{a,c,e},{a,c, f}}. The reader may notice that in all these examples the set of preferred extensions coincides with the set of stable extensions. In general any stable extension is also a preferred extension but not vice versa. In the example of Figure 2.7 we have EPR(AF2.7) = {/ 0} while ES T(AF2.7) = / 0. In the example of Figure 2.8 there are two preferred exten- sions, namely {a}, {b}, but only one of them, namely {b}, is also stable. The intuition underlying preferred semantics has a correspondence with pref- erential semantics of logic programs [12]. As to general semantics properties it is immediate to see that PR satisfies I-maximality, admissibility, reinstatement, while it is proved in [2] that it is directional. Given these facts, PR has often been regarded as the most satisfactory semantics in the context of Dung’s framework. 5.5 Stage and semi-stable semantics Stage [21] and semi-stable [8] semantics, denoted respectively as S TA and S S T, are based on the idea of prescribing the maximization not only of the arguments in- cluded in an extension but also of those attacked by it. Definition 2.19. Given an argumentation framework AF = A,R and a set E ⊆ A the range of E is defined as E ∪ E+, where E+ {a ∈ A : ERa}. E is a stage extension if and only if E is a conflict-free set with maximal (wrt. set inclusion) range. E is a semi-stable extension if and only if E is a complete extension with maximal (wrt. set inclusion) range. As evident from Definition 2.19, the two semantics differ in the requirement on the sets whose range is maximal: being conflict-free for stage semantics, complete b a c Fig. 2.8 AF2.8: S T is not directional
- 53. 2 Semantics of Abstract Argument Systems 39 extensions (or equivalently, admissible sets) for semi-stable semantics. Both S TA and S S T are clearly universally defined (differently from stable semantics), while co- inciding with stable semantics (differently from preferred semantics) when stable extensions exist. In fact, for any stable extension E it holds that E ∪ E+ = A. It follows that any complete extension (conflict-free set) which is not stable does not satisfy the range maximization requirement for argumentation frameworks where stable extensions exist, hence the coincidence of S T and S S T (S TA) in these cases. If stable extensions do not exist, S S T selects anyway as extensions some complete extensions and the maximization requirement restricts the choice to preferred ex- tensions: ∀AF ES S T(AF) ⊆ EPR(AF). On the other hand, S TA does not necessarily select admissible sets as extensions. Figure 2.9 shows an argumentation framework where S S T and S TA agree and do not coincide neither with S T nor with PR: in fact ES T(AF2.9) = / 0, EPR(AF2.9) = {{a},{b}}, ES S T(AF2.9) = ES TA(AF2.9) = {{b}}. On the other hand, in the case of Figure 2.7 EPR(AF2.7) = ES S T(AF2.7) = {/ 0} while ES TA(AF2.7) = {{a},{b},{c}}. It is easy to see that both S S T and S TA satisfy I-maximality, while only S S T satisfies admissibility and reinstatement. S S T and S TA do not satisfy directionality, as S T does not. 5.6 Ideal semantics Ideal semantics [14], denoted as I D, provides a unique-status approach allowing the acceptance of a set of arguments possibly larger than in the case of GR. Definition 2.20. Given an argumentation framework AF = A,R a set S ⊆ A is ideal if and only if S is admissible and ∀E ∈ EPR(AF) S ⊆ E. The ideal extension, denoted as ID(AF), is the maximal (wrt. set inclusion) ideal set. The definition of ideal set prescribes admissibility and skeptical justification un- der preferred semantics. From Section 5.2 we know that the grounded extension sat- isfies both requirements and is therefore an ideal set (this in particular implies that ideal semantics is universally defined). Ideal sets strictly larger than the grounded extension may exist, as shown by the example in Figure 2.10 where it holds that EPR(AF2.10) = {{a,d},{b,d}}: in this case the grounded extension is empty while ID(AF2.10) = {d}. By definition I D satisfies I-maximality and admissibility. It is proved in [2] that I D also satisfies reinstatement and directionality. d b a c Fig. 2.9 AF2.9: S S T may not coincide with P R or S T
- 54. 40 Pietro Baroni and Massimiliano Giacomin 5.7 CF2 semantics CF2 semantics is defined in the frame of the SCC-recursive scheme [7] which is a general pattern for the definition of argumentation semantics based on the graph- theoretical notion of strongly connected components (SCCs) of an argumentation framework, i.e. the equivalence classes of arguments under the relation of mutual reachability via attack links. CF2 semantics can be roughly regarded as selecting its extensions among the maximal conflict-free sets of AF, on the basis of some topological requirements related to the decomposition of AF into strongly connected components. Examining in detail the definition of CF2 semantics is beyond the scope of this chapter: the interested reader may refer to [7]. Definition 2.21. Given an argumentation framework AF = A,R, a set E ⊆ A is an extension of CF2 semantics, i.e. E ∈ ECF2(AF), if and only if • E ∈ MCF(AF) if |SCCSAF | = 1 • ∀S ∈ SCCSAF (E ∩S) ∈ ECF2(AF↓UPAF (S,E)) otherwise where MCF(AF) denotes the set of maximal conflict-free sets of AF, SCCSAF denotes the set of strongly connected components of AF, and, for any E,S ⊆ A, UPAF (S,E) = {a ∈ S | ∄b ∈ E : b / ∈ S,bRa}. The underlying idea consists in relying only on I-maximality and the conflict- free principle within a unique strongly connected component S, where all argu- ments (if more than one) both receive and deliver at least an attack. In particular it turns out that when AF consists of exactly one strongly connected component, the set of extensions prescribed by CF2 semantics exactly coincides with the set of maximal conflict-free sets of AF. This yields a uniform multiple-status treatment of cycles, which is not achieved by other semantics. In fact, it is easy to see that for any argumentation framework AF consisting of an even-length attack cycle it holds that ES T(AF) = EPR(AF) = ES S T(AF) = {/ 0}. For instance we already know that ES T(AF2.2) = EPR(AF2.2) = ES S T(AF2.2) = MCF(AF2.2) = {{a},{b}}. Similarly, in the case of a four-length cycle two extensions arise, each consisting of two argu- ments. On the other hand, for any argumentation framework AF consisting of an b a c d Fig. 2.10 AF2.10: an example to illustrate ideal semantics a b c e d Fig. 2.11 AF2.11: an example to illustrate CF2 semantics
- 55. 2 Semantics of Abstract Argument Systems 41 odd-length attack cycle it holds that ES T(AF) = / 0 and EPR(AF) = ES S T(AF) = {/ 0}. This disuniformity in the treatment of cycles gives rise to counterintuitive behav- iors in some simple examples and has therefore been regarded as problematic in the literature [17]. Clearly CF2 semantics is able to overcome this limitation since ECF2(AF) = MCF(AF) for any argumentation framework AF consisting of an at- tack cycle, independently of its length. For instance, ECF2(AF2.7) = {{a},{b},{c}}. In a generic argumentation framework, CF2 extensions may be computed fol- lowing the decomposition of AF into strongly connected components, which, due to a well-known graph-theoretical property, can be partially ordered according to the attack relation. Consider the example in Figure 2.11, which consists of two SCCs, namely S1 = {a,b,c} and S2 = {d,e}. According to Definition 2.21 it must be the case that E ∩ S1 is a maximal conflict-free set of AF2.11↓{a,b,c}. Thus we get three possible starting points for the construction of extensions, namely {a}, {b}, {c}. It has then to be noted that in any of these three cases d is attacked. Therefore in all cases UPAF (S2,E) consists of the only argument e, yielding E ∩S2 = {e} as the only possibility. In summary ECF2(AF2.11) = {{a,e},{b,e},{c,e}}. It can be seen that in all other examples considered so far CF2 extensions coincide with preferred extensions. As to general properties, CF2 semantics is I-maximal, since its extensions are maximal conflict-free sets of AF, directional [2], and universally defined, since, for any AF, MCF(AF) = / 0. On the other hand, the “desired” treatment of odd-length cycles entails that CF2 semantics gives up the traditional properties of admissibil- ity and reinstatement as shown, for instance, by the example of Figure 2.7 where no extension is an admissible set and all violate the reinstatement property. It is proven however in [2] that the departure from the notion of reinstatement is not radical, since CF2 semantics satisfies two weaker versions of this property called CF-reinstatement and weak reinstatement. As another confirmation that CF2 se- mantics has significant relationships with traditional semantics it can be recalled that any preferred extension is included in a CF2 extension, any stable extension is a CF2 extension and the grounded extension is included in any CF2 extension. 5.8 Prudent semantics The family of prudent semantics [10] is introduced by considering a more extensive notion of attack in the context of traditional semantics. In particular, an argument a indirectly attacks an argument b if there is an odd-length attack path from a to b. For instance, in Figure 2.5 a indirectly attacks d, while in Figure 2.6 a indirectly attacks d and f, b indirectly attacks e, and c indirectly attacks f. The odd-length path need not be the shortest path and may include cycles: in Figure 2.10 both a and b indirectly attack d, while in Figure 2.11 a indirectly attacks e. A set S of arguments is free of indirect conflicts, denoted as icf(S), if ∄a,b ∈ S such that a indirectly attacks b. The prudent version of the admissibility property and of several traditional notions of extension can then be defined. Definition 2.22. Given an argumentation framework AF = A,R, a set of argu- ments S ⊆ A is p(rudent)-admissible if and only if ∀a ∈ S a is acceptable wrt. S
- 56. 42 Pietro Baroni and Massimiliano Giacomin and icf(S). A set of arguments E ⊆ A is: a preferred p-extension if and only if E is a maximal (wrt. set inclusion) p-admissible set; a stable p-extension if and only if icf(E) and ∀a ∈ (AE) ERa; a complete p-extension if and only if E is p- admissible and there is no a / ∈ E such that a is acceptable wrt. E and icf(E ∪{a}). Definition 2.23. Given an argumentation framework AF = A,R, the function Fp AF : 2A → 2A such that for a given a set S ⊆ A, Fp AF (S) = {a | a is acceptable with respect to S ∧ icf(S ∪ {a})} is called the p-characteristic function of AF. Let j be the lowest integer such that the sequence (Fp AF )i(/ 0) is stationary for i ≥ j: (Fp AF )j(/ 0) is the grounded p-extension of AF, denoted as GPE(AF). The prudent versions of grounded, complete, preferred and stable semantics are denoted as GRP, COP, PRP and S TP, respectively. It is possible to note that the adoption of indirect conflict has a different impact on the various notions of traditional semantics. The definition of stable prudent ex- tension corresponds to the one of stable extension with an additional requirement. This means that, when S TP is defined, its extensions are also stable (and thus have the same properties) but DS TP DS T. For instance in the example of Figure 2.12 S TP is not defined since the only stable extension, namely {a,d,e} is not prudent due to the indirect attack from a to d. Analogously to the traditional version, the grounded prudent extension, denoted as GPE(AF), can be conceived as the result of the incre- mental application of a (p-)characteristic function starting from initial arguments. As Fp AF is more restrictive than FAF it follows that the grounded prudent extension is a possibly strict subset of the traditional grounded extension. This entails in par- ticular that reinstatement is given up by GRP, as it can be seen in the example of Figure 2.12 where GPE(AF2.12) = {a,e} GE(AF2.12) = {a,d,e} and d is not rein- stated. In the context of complete and preferred prudent semantics a notable effect is that an initial argument may not be included in an extension. This is the case of a in the example of Figure 2.12 where ECOP(AF2.12) = EPRP(AF2.12) = {{a,e},{d,e}}. Besides reinstatement this gives up also the directionality property, showing the neat departure of COP and PRP from the track of traditional semantics. 6 Advanced topics This final section provides a quick overview and literature references for some ad- vanced topics in the field of abstract argumentation, namely semantics agreement, skepticism relations, and semantics principles. As to the first point, while different semantics proposals correspond to alternative intuitions which manifest themselves in distinct behaviors in some argumentation c d b a e Fig. 2.12 AF2.12: an example to illustrate prudent semantics
- 57. 2 Semantics of Abstract Argument Systems 43 frameworks, there are also argumentation frameworks where the sets of extensions prescribed by different semantics coincide, as evident from several of the examples seen above. Topological properties of argumentation frameworks providing suffi- cient conditions for semantics agreement have been investigated in Dung’s original paper: for instance the absence of attack cycles is sufficient to ensure agreement among grounded, stable and preferred semantics, while the absence of odd-length attack cycles is sufficient to ensure agreement between stable and preferred seman- tics. Subsequent results concerning topological families of argumentation frame- works relevant to agreement properties are reported in [11, 1], while a systematic set-theoretical analysis of agreement classes is provided in [5]. As to the notion of skepticism, it has often been used in informal ways to dis- cuss semantics behavior, e.g. by observing that a semantics is “more skeptical” than another one. Intuitively, a semantics is more skeptical than another if it makes less committed choices about the justification of the arguments: a skeptical behavior tends to leave arguments in an “undecided” justification state, while a non-skeptical behavior corresponds to more “resolute” choices about acceptance or rejection of arguments. The issue of formalizing skepticism relations between sets of extension in terms of set theoretical properties and then comparing semantics according to skepticism has been first addressed in [6] and subsequently developed in [4]. Turning finally back to general properties of argumentation semantics, some of them, like admissibility and reinstatement, have regularly been considered in the lit- erature [18], while the task of systematically defining a set of criteria for semantics evaluation and comparison has been undertaken only recently. Besides the princi- ples we have explicitly discussed in Section 3, two families of adequacy properties (based on skepticism relations) are introduced in [2], where it is shown that, consid- ering also these properties, none of the literature semantics discussed in this chapter is able to comply with all the desirable criteria. A novel semantics able to satisfy all of them has been proposed in [3]. At a different abstraction level, where argument structure and construction are explicitly dealt with, general rationality postulates for argumentation systems have been introduced in [9]. Exploring the definition of gen- eral principles for argumentation at different abstraction levels, investigating their relationships and analyzing their suitability for different application domains appear to be open and fruitful research directions. A related research issue concerns the identification of a generic definition scheme able to encompass into a unifying view a large variety of semantics. In this perspective, it is shown in [7] that all traditional semantics adhere to the parametric SCC-recursive scheme, where they differ sim- ply by a base function which only specifies the sets of extensions of argumentation frameworks consisting of a single SCC. The above mentioned results suggest that though there is a wide corpus of lit- erature on abstract argumentation semantics, providing a rich variety of alternative approaches, the field is far from being “mature” and there is still large room for investigating both fundamental theoretical issues and their potential impact on prac- tical applications.
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- 59. Chapter 3 Abstract Argumentation and Values Trevor Bench-Capon and Katie Atkinson 1 Introduction Abstract argumentation frameworks, as described in Chapter 2, are directed towards determining whether a claim that some statement is true can be coherently main- tained in the context of a set of conflicting arguments. For example, if we use preferred semantics, that an argument is a member of all preferred extensions es- tablishes that its claim must be accepted as true, and membership of at least one preferred extension shows that the claim is at least tenable. In consequence, that admissible sets of arguments are conflict free is an important requirement under all the various semantics. For many common cases of argument, however, this is not appropriate: two ar- guments can conflict, and yet both be accepted. For an example suppose that Trevor and Katie need to travel to Paris for a conference. Trevor offers the argument “we should travel by plane because it is quickest”. Katie replies with the argument “we should travel by train because it is much pleasanter”. Trevor and Katie may con- tinue to disagree as to how to travel, but they cannot deny each other’s arguments. The conclusion will be something like “we should travel by train because it is much pleasanter, even though travelling by plane is quicker”. The point concerns what Searle [24] calls direction of fit. For matters of truth and falsity, we are trying to fit what we believe to the way the world actually is. In contrast, when we consider what we should do we are trying to fit the world to the way we would like it to be. Moreover, because people may have different preferences, values, interests and aspirations, people may rationally choose different options: if Katie prefers comfort Trevor Bench-Capon Department of Computer Science, University of Liverpool, UK, e-mail: tbc@liverpool.ac. uk Katie Atkinson Department of Computer Science, University of Liverpool, UK, e-mail: katie@liverpool. ac.uk I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 45 DOI 10.1007/978-0-387-98197-0 3, c Springer Science+Business Media, LLC 2009
- 60. 46 Trevor Bench-Capon and Katie Atkinson to speed she will rationally choose the train, but this does not mean that Trevor can- not rationally choose the plane if he prefers speed to comfort. We will return to this example throughout this chapter. Within standard abstract argumentation frameworks one approach to recognising the importance of direction of fit [22] is to require sceptical acceptance for epis- temic arguments but only credulous acceptance for practical arguments. This does successfully model the existence of a choice with respect to practical arguments, but it does not motivate the choice, nor does it allow us to predict choices on the basis of choices made in the past. Value based argumentation frameworks (VAFs) [6], described in this chapter, are an attempt to address issues about the rational justification of choices systematically. Value based justification of choices is common in many important areas: in poli- tics where specific policies are typically justified in terms of the values they promote, and where politicians’ values are advanced as reasons to vote for them; in law, where differences in legal jurisdictions and decisions over time can be explained in terms of the values of the societies in which the judgements are made [11]; in matters of morality where individual and group ethical perspectives play a crucial role in reasoning and action [2]; as well as more everyday examples, such as given above. In this chapter we will first give some philosophical background, in particular introducing the notion of audience, and some of the features that we require from practical reasoning. Section 3 will discuss the nature of values in more detail, in particular the distinction between values and goals. Section 4 will introduce the formal machinery of Value Based Argumentation Frameworks, and discuss some of their more important properties. Section 5 describes some applications of value based argumentation. Section 6 discusses some recent developments, and section 7 concludes the chapter with a summary. 2 Audiences One of the first people to stress the importance of the audience in determining whether an argument is persuasive or not was Chaim Perelman [20], [19]: “If men oppose each other concerning a decision to be taken, it is not because they commit some error of logic or calculation. They discuss apropos the applicable rule, the ends to be considered, the meaning to be given to values, the interpretation and characterisation of facts.” [[19] p.150, italics ours]. A similar point was made by John Searle [24]: “Assume universally valid and accepted standards of rationality, assume perfectly rational agents operating with perfect information, and you will find that rational disagreement will still occur; because, for example, the rational agents are likely to have different and incon- sistent values and interests, each of which may be rationally acceptable.” [[24], xv] Both Perelman and Searle recognise that there may be complete agreement on facts, logic, which arguments are valid, which arguments attack one another and the
- 61. 3 Abstract Argumentation and Values 47 rules of fair debate, and yet still disagreement as to the correct decision. This was true when Trevor and Katie were thinking about how to travel to Paris, and there are many other examples. Consider an example from politics. One choice that any government must make is to decide on an appropriate rate of income tax. Typically there will be an argument in favour of increasing the rate of taxation, since this progressive form of taxation will reduce income inequalities. Against this, it can be argued that a decrease in taxation will promote more en- terprise, increasing Gross National Product, and so raising the absolute incomes of everyone. It is possible to see both these arguments as valid, since both supply a rea- son to act: and yet a choice must be made, since the actions are incompatible. Which choice is made will depend on whether the chooser prefers equality or enterprise in the particular circumstances with which he is confronted. Two parties may be in agreement as to the consequences of a movement in the tax rate, and yet disagree as to the choice to be made because they differ in their fundamental aspirations. Different people will prize social values differently, and one may prefer equality to enterprise, while another prefers enterprise to equality. Thus while both arguments are agreed to be valid, one audience will ascribe more force to one of the arguments, while a different audience will make a different choice. In such cases these different audiences will rationally disagree, and agreement can only be reached by coming up with additional arguments which convince all audiences in terms of their own pref- erences, or by converting those who disagree to a different appraisal of social values. This will often require that different arguments be presented to different audiences. Thus when in the 1980s the UK Conservative Party under Margaret Thatcher were attempting to justify dramatic cuts in income tax for the highly paid, one argument was simply that fairness meant that people deserved to keep a larger proportion of their “earnings”. This argument was quite acceptable at Party Conferences where the audience comprised predominately high earners, but was not persuasive to the country at large, since most people were not subject to higher rate taxation. To con- vince the nation at large a different argument was needed: namely that there would be a “trickle down” effect, benefitting everyone, whatever their level of income. This was clearly persuasive as Thatcher was twice re-elected. Thus whether an argument is persuasive depends not only on the intrinsic merits of the argument – of course, it needs to be based on plausible premises and must be sound – but also on the audience to which it is addressed. Moreover, for practical reasoning, what is important about the audience is what they want to see happen, and this seems to turn on how they rank the various values that accepting the arguments promote. In the next section we will consider values, and their relation to practical reasoning, in more detail. 3 Values This far we have seen that whether a particular audience is persuaded by an argu- ment depends on the attitude of that audience to the values on which the argument is
- 62. 48 Trevor Bench-Capon and Katie Atkinson founded. Values are used in the sense of fundamental social or personal goods that are desirable in themselves, and should never be confused with any numeric measure of the strength, certainty or probability of an argument. Liberty, Equality and Fra- ternity, the values of the French Revolution, are paradigmatic examples of values. Values are widely recognised as the basis for persuasive argument. For example, the National Forensic League, which conducts debating competitions throughout the USA uses the “Lincoln-Douglas” (LD) debate format which is based on the notion of a clash of values. In an LD debate the resolution forces each side to take on com- peting values and argue about which one is supreme. For example, if the resolution is, “Resolved: An oppressive government is better than no government at all,” the affirmative side might value “order” and the negative side might value “freedom”. Such a debate would revolve around whether order is more valuable than freedom. In the original debate between Abraham Lincoln and Stephen Douglas on which LD debates are based, Douglas championed the rights of states to legislate for their par- ticular circumstances, whereas Lincoln argued on the basis that there were certain inviolable human rights that all states had to respect, even though this constrained state autonomy. But what is the role of values in practical reasoning? Historically the basis for treatments of practical reasoning has been the practical syllogism, first discussed by Aristotle. A standard modern statement is given in [16]: K1 I’m to be in London at 4.15 If I catch the 2.30, I’ll be in London at 4.15 So, I’ll catch the 2.30. The first premise is a statement of some desired state of affairs, the second an action which would bring about that state of affairs, and the conclusion is that the action should be performed. There are, however, problems with the formulation: it is abductive rather than deductive, does not consider alternative, possibly better, ways of achieving the desired state of affairs, or possibly undesirable side effects of the action. Walton [26] addresses these issues by regarding the practical syllogism as an argumentation scheme, which he calls the sufficient condition scheme for practical reasoning, which provides a presumptive reason to perform the action, but which can be critiqued on the basis of alternatives and undesirable consequences. He states the sufficient condition practical reasoning scheme as: W2 G is a goal for agent a Doing action A is sufficient for agent a to carry out goal G Therefore agent a ought to do action A. This, however, still does not explain why G is a goal for the agent, nor indi- cate how important bringing about G is to the agent. Neither K1 nor W1 make any mention of values: rather that the agent has certain values is implicit in calling the desired state of affairs a “goal” for that agent. Accordingly, to make this role of values explicit, Walton’s scheme was developed by Atkinson and her colleagues [4] into the more elaborated scheme:
- 63. 3 Abstract Argumentation and Values 49 A1 In the circumstances R, we should perform action A to achieve new circumstances S, which will realise some goal G which will promote some value V. What this scheme does in particular is to distinguish three aspects which are conflated into the notion of goal in K1 and W1. These aspects are: the state of affairs which will result from the action; the goal, which is those aspects of the new state of affairs for the sake of which the action is performed; and the value, which is the reason why the agent desires the goal. Making these distinctions opens up several distinct types of alternative to the recommended action. We may perform a different action to realise the same state of affairs; we may act so as to bring about a different state of affairs which realises the same goal; or we may realise a different goal which promotes the same value. Alternatively, since the state of affairs potentially realises several goals, we can justify the action in terms of promoting a different value. In coming to agreement this last possibility may be of particular importance: we may want to promote different values, and so agree to perform the action on the basis of different arguments. Our contention is that, in the spirit of the notion of audience developed in section 2, what is important, what is the appropriate comparison for choosing between alternatives, is the value. In order to see the distinction between a goal and a value, consider again Trevor and Katie’s journey to Paris. The goal is to be in Paris for the conference, and this is not in dispute: the dispute is how that goal should be realised and turns on the values promoted by the different methods of travel. What is important is not the state reached, but the way in which the transition is made. The style of argumentation represented by A1 has been formalised in [1] in terms of a particular style of transition system, Alternating Action Based Transition sys- tems (AATS) [27]. An AATS consists of a set of states and a set of agents and the transitions between the states are in terms of the joint actions of the agents, that is, actions composed from the actions available to the agents individually. In terms of A1 the circumstances R and S are represented by the states of the system, the goal G is realised if G holds in S (of course, G may hold in several of the states), and the action is the particular agent’s component of a joint action which is a transition from R to S. The value labels the transition, indicating that it is the movement from R to S using that particular transition that promotes the value. A fragment of the AATS for Trevor and Katie’s travel dilemma is shown in Figure 3.1, t/kt/p is the action of Trevor/Katie travelling by train/plane, C/St/k means that Comfort/Speed is promoted in respect of Trevor/Katie, 00 that both are in Liverpool and 11 that both are in Paris. 00 11 ttkt +Ct +Ck tpkt +St +Ck tpkp +St +Sk ttkp +Ct +Sk Fig. 3.1 AATS for travel to Paris example
- 64. 50 Trevor Bench-Capon and Katie Atkinson Although there is only one destination state, each of the four potential ways of reaching it promotes different values, and hence give rise to different arguments in their favour. Which arguments will succeed will depend on the preferences between the values of Comfort and Speed of the two agents concerned. Essentially then, in this problem there will be a number of possible audiences, depending on how the values are ordered. Suppose that Trevor values his own speed over his own comfort and Katie her comfort over her speed, and that neither consider values promoted in respect of the other. Then Trevor will choose to go by plane and Katie by train. Here the agents can choose independently, as their values are affected only by their own actions: in later sections we will introduce a third value which requires them to consider what the other intends to do also. The basic idea underlying Value Based Argumentation Frameworks is that it is possible to associate practical arguments with values, and that in order to determine which arguments are acceptable we need to consider the audience to which they are addressed, characterised in terms of an ordering on the values involved. We need, however, to recognise that not all the arguments relevant to a practical decision will be practical arguments. For example, if there is a train strike (or it is a UK Bank Holiday when there are often no trains from Liverpool), the argument that the train should be used cannot be accepted no matter how great the audience preference is for Comfort over Speed. In order to recognise that such epistemic arguments con- strain choice, such arguments are associated with the value Truth, and all audiences are obliged to rank Truth above all other values. In the next section we will give a formal presentation of Value Based Argumentation Frameworks. 4 Value Based Argumentation Frameworks We present the Value Based Framework as an extension of Dung’s original Argu- mentation Framework [14], defined in Chapter 2 of this book. We do this by extend- ing the standard pair to a 5 tuple. Definition 3.1. A value-based argumentation framework (VAF) is a 5-tuple: VAF = A, R, V, val, P where A is a finite set of arguments, R is an irreflexive binary relation on A (i.e. A, R is a standard AF), V is a non-empty set of values, val is a function which maps from elements of A to elements of V and P is the set of possible audiences (i.e. total orders on V). We say that an argument a relates to value v if accepting A promotes or defends v: the value in question is given by val(a). For every a ∈ A, val(a) ∈ V. When the VAF is considered by a particular audience, the ordering of values is fixed. We may therefore define an Audience Specific VAF (AVAF) as: Definition 3.2. An audience specific value-based argumentation framework (AVAF) is a 5-tuple: VAFa = A, R, V, val, Valprefa
- 65. 3 Abstract Argumentation and Values 51 where A, R, V and val are as for a VAF, a is an audience, a ∈ P, and Valprefa is a preference relation (transitive, irreflexive and asymmetric) Valprefa ⊆ V x V, reflecting the value preferences of audience a. The AVAF relates to the VAF in that A, R, V and val are identical, and Valpref is the set of preferences derivable from the ordering a ∈ P in the VAF. Our purpose in introducing VAFs is to allow us to distinguish between one ar- gument attacking another, and that attack succeeding, so that the attacked argument may or may not be defeated. Whether the attack succeeds depends on the value or- der of the audience considering the VAF. We therefore define the notion of defeat for an audience: Definition 3.3. An argument A ∈ AF defeatsa an argument B ∈ AF for audience a if and only if both R(A,B) and not (val(B),val(A)) ∈ Valprefa. We can now define the various notions relating to the status of arguments: Definition 3.4. An argument a ∈ A is acceptable-to-audience-a (acceptablea) with respect to set of arguments S, (acceptablea(A,S)) if: (∀ x)((x ∈ A defeatsa(x,A)) → (∃ y)((y ∈ S) defeatsa(y,x))). Definition 3.5. A set S of arguments is conflict-free-for-audience-a if: (∀ x) (∀ y)((x ∈ S y ∈ S) → (¬ R(x,y) ∨ valpref(val(y),val(x)) ∈ Valprefa))). Definition 3.6. A conflict-free-for-audience-a set of arguments S is admissible-for- an-audience-a if: (∀ x)(x ∈ S → acceptablea(x,S)). Definition 3.7. A set of arguments S in a value-based argumentation framework VAF is a preferred extension for-audience-a (preferreda) if it is a maximal (with respect to set inclusion) admissible-for-audience-a subset of A. Now for a given choice of value preferences valprefa we are able to construct an AF equivalent to the AVAF, by removing from R those attacks which fail because they are faced with a superior value. Thus for any AVAF, vafa = A, R, V, val, Valprefa there is a corresponding AF, afa = A, defeats, such that an element of R, R(x,y) is an element of defeats if and only if defeatsa(x,y). The preferred extension of afa will contain the same arguments as vafa, the preferred extension for audience a of the VAF. Note that if vafa does not contain any cycles in which all arguments pertain to the same value, afa will contain no cycles, since the cycle will be broken at the point at which the attack is from an inferior value to a superior one. Hence both afa and vafa will have a unique, non-empty, preferred extension for such cases. A proof is given in [6]. Moreover, since the AF derived from an AVAF contains no cycles, the grounded extension coincides with the preferred extension for this audience, and so there is a straightforward polynomial time algorithm to compute it, also given in [6]. For the moment we will restrict consideration to VAFs which do not contain any cycles in a single value.
- 66. 52 Trevor Bench-Capon and Katie Atkinson For such VAFs, the notions of sceptical and credulous acceptance do not apply, since any given audience will accept only a single preferred extension. These pre- ferred extensions may, and typically will, however, differ from audience to audience. We may therefore introduce two useful notions, objective acceptance, arguments which are acceptable to all audiences irrespective of their particular value order, and subjective acceptance, arguments which can be accepted by audiences with the appropriate value order. Definition 3.8. Objective Acceptance. Given a VAF, A, R, V, val, P an argument a ∈ A is objectively acceptable if and only if for all p ∈ P, a is in every preferredp. Definition 3.9. Subjective Acceptance. Given a VAF, A, R, V, val, P an argu- ment a ∈ A is subjectively acceptable if and only if for some p ∈ P, a is in some preferredp. An argument which is neither objectively nor subjectively acceptable (such as one attacked by an objectively acceptable argument with the same value) is said to be indefensible. All arguments which are not attacked will, of course, be objectively acceptable. Otherwise objective acceptance typically arises from cycles in two or more values. For example, consider a three cycle in two values, say two arguments with V1 and one with V2. The argument with V2 will either resist the attack on it when it is preferred to V1, or, when V1 is preferred, fail to defeat the argument it attacks which will, in consequence, be available to defeat its attacker. Thus the argument in V2 will be objectively acceptable, and both the arguments with V1 will be subjectively acceptable. For a more elaborate example consider Figure 3.2. a blue b c d red red blue e f g h red red blue blue Fig. 3.2 VAF with values red and blue There will be two preferred extensions, according to whether red blue, or blue red. If red blue, the preferred extension will be {e,g,a,b}, and if blue red, {e,g,d,b}. Now e and g and b are objectively acceptable, but d, which would have been objectively acceptable if e had not attacked d, is only subjectively acceptable (when blue red), and a, which is indefensible if d is not attacked, is also subjec- tively acceptable (when red blue). Arguments c, f and h are indefensible. Results characterising the structures which give rise to objective acceptance are given in [6].
- 67. 3 Abstract Argumentation and Values 53 4.1 VAF Example We will illustrate VAFs using our running example of Trevor and Katie’s conference travel arrangements. Recall that VAFa = A, R, V, val, Valprefa. We therefore need to instantiate the five elements of this tuple. From Figure 3.1 above we get four arguments: A1: Katie should travel by train (Kt) to promote her comfort (Ck). A2: Katie should travel by plane (Kp) to promote her speed (Sk). A3: Trevor should travel by train (Tt) to promote his comfort (Ct). A4: Trevor should travel by plane to (Tp) to promote his speed (St). But there are other considerations: it is far more boring to travel alone than in company. This gives two other arguments: A5: Both Katie and Trevor should travel by train (KtTt) to avoid boredom (B). A6: Both Katie and Trevor should travel by plane (KpTp) to avoid boredom (B). Thus Ae = {A1,A2,A3,A4,A5,A6} and val = {A1→Ck, A2→Sk, A3→Ct, A4→St, A5→B, A6→B}. We can now identify attacks between these arguments. Since neither Katie nor Trevor can travel by both train and plane, A1 attacks A2, and vice versa, and A3 attacks and is attacked by A4. Moreover A1and A3 attack and are attacked by A6, and A2 and A4 attack and are attacked by A5. Thus Re = {A1,A2, A2,A1, A3,A4, A4,A3, A1,A6, A3,A6, A6,A1, A6,A3, A2,A5, A4,A5,A5,A2,A5,A4, A5,A6, A6,A5}. The values are given by the values used in the arguments, but for the present we will make no distinction at first between values promoted in respect of Trevor and values promoted in respect of Katie. Thus Ve = {B, C, S}. Finally the audiences P will be every possible ordering of the elements in Ve, so P = {BCS, BSC, SBC, SCB, CBS, CSB} We can represent the VAF diagrammatically as a directed graph, as shown in Figure 3.3. A1 Kt Ck A2 Kp Sk A6 KpTp B A5 KtTt B A4 Tp St A3 Tt Ct Fig. 3.3 VAF for travel example
- 68. 54 Trevor Bench-Capon and Katie Atkinson Note that here we do have a cycle of two arguments with the same value, namely B. This means that some audiences will not have a unique preferred extension. This does not pose any serious problem in this small example. Now consider specific audiences. Suppose that Katie, who very much dislikes flying, ranks C as her highest value, and S as her least important. Now AVAFkatie = Ae, Re, Ve, val, {C,B,C,S,B,S}. When we use Katie’s preferences to eliminate unsuccessful attacks, this produces the corresponding AFkatie = Ae, {A1,A2,A3,A4, A1,A6, A3,A6, A5,A2,A5,A4, A5,A6}. This AF has a unique preferred extension, PEkatie = {A1,A3,A5}, which means that she will be in favour of both Trevor and herself travelling by train. Suppose, however, Trevor, who has no objection to flying, prefers speed to com- fort, but dislikes travelling alone, so that he is a member of the audience {BSC}. Now AVAFtrevor = Ae, Re, Ve, val, {B,S,B,C,S,C}. And AFtrevor = Ae, {A2,A1, A4,A3, A6,A1, A6,A3, A5,A4, A5,A6, A6,A5}. This contains a cycle for the two arguments in B, and so Trevor will have two pre- ferred extensions: {A1,A3,A5}, and {A2,A4,A6}. Trevor could solve this dilemma by considering that A3 also promotes C and A4 also promotes S, and so choose {A2,A4,A6}. But what is required is a joint decision: neither Trevor nor Katie can act independently so as to ensure that A5 or A6 is followed. We therefore need to consider the joint audience, and to distinguish between values promoted in respect of Trevor and values promoted in respect of Katie. Ck B St Ct Ck B Sk Ct B Ck St Sk Ct Sk St ) c ( ) b ( ) a ( Fig. 3.4 Partial Orders representing combined audiences: (a) Katie CBS and Trevor BSC; (b) Katie CBS and Trevor SBC; (c) Katie BCS and Trevor BSC Katie’s order is Ck B Sk, while Trevor’s is B St Ct. Since they have B in common – either both are bored or neither are bored – we can merge their orderings on B to get the partial order shown in Figure 3.4(a). The AVAF for the combined audience is thus Ae, Re,Ve, val, {B,St, B,Ct, B,Sk, St,Ct, Ck,B, Ck,Sk, Ck,St, Ck,Ct}. This gives rise to the AF shown in Figure 3.5.
- 69. 3 Abstract Argumentation and Values 55 A1 Kt Ck A2 Kp Sk A6 KpTp B A5 KtTt B A4 Tp St A3 Tt Ct Fig. 3.5 AF for Combined Audience We can use this VAF to illustrate the algorithm for finding the Preferred Exten- sion given in [6]. First we include the arguments with no attacker: in this case A1. A1 attacks A2 and A6 and so they are excluded. Now A5 has no attacker and so it is included. A5 excludes A4, leaving A3 without an attacker, and so A3 is included to give the preferred extension of the combined audience as {A1,A3,A5}. This case is straightforward, because the combined audience yields a single pre- ferred extension. The same is true if Trevor preferred S to B, the combined order being shown in Figure 3.4(b). This would cause A5,A4 to be replaced in R by A4,A5. Now both A1 and A4 are not attacked, and so they defeat the remaining arguments yielding the preferred extension {A1,A4}. This is possible: they simply agree to travel separately by their preferred means. More complicated is the situation where Katie prefers B to C, so that the merged order is as shown in Figure 3.4(c), and A6,A1 replaces A1,A6 in R. Now there is no longer any argument which has no attackers, and the algorithm must be applied twice; first including A5 and then including A6, so that are two pre- ferred extensions, {A1,A3,A5}, and {A2,A4,A6}, both of which are acceptable to them both. Now, since Katie will lean towards the former and Trevor the latter they must find a way to decide between Ck and St. This might depend on who had the strongest opinions, or who is the more altruistic or conciliatory. Alternatively one person might change their preferences: if Katie moved back to her original ordering of CBS, Trevor would either have to decide to prefer S to B or to agree to travel by train. This possibility shows how preferences can emerge from the reasoning process: although initially Katie might express a preference for B to C, and Trevor for B to S, when the consequences are realised she may decide that C is actually more important than B, and he may decide S is more important than B. 5 Example Applications As noted in Section 1, reasoning with values is common to many application do- mains. In previous work [2, 3, 5, 9] we have shown how the application of abstract argumentation with values can be applied to problems in law, medicine, ethics and
- 70. 56 Trevor Bench-Capon and Katie Atkinson e-democracy, and we will briefly discuss these applications here. We begin by con- sidering legal reasoning with values. 5.1 Law Reasoning with legal cases has often been viewed as a decision being deduced about a particular case through the application of a set of rules, given the facts of the case, e.g. [25]. However, the facts of cases are not set in stone as they can be open to interpretation from different lawyers. Additionally, the rules used to reach decisions are defeasible by their nature and many are derived from precedent cases, so they too may be open to interpretation. Thus, within the AI and Law literature it has been recognised that when considering arguments in legal cases, the purposes of the law – the values intended to be promoted or upheld through the application of the law – must be represented and accounted for, e.g. [8] [21]. In the literature on legal case-based reasoning the issue was first brought to attention in Berman and Hafner’s seminal paper on the topic [8] arguing that legal case-based reasoning needs to recognise teleological as well as factual aspects. This is so since the law is not composed arbitrarily, rather it is constructed to serve social ends, so when conflicts in the application of rules occur in legal cases they can be resolved more effectively by considering the purposes of these rules and their relative applicability to the particular case in question. This enables preferences amongst purposes to be revealed, and then the argument can be presented appropriately to the audience through an appeal to the social values that the argument promotes or defends. In order to demonstrate how the values of the law can be represented and rea- soned about within a case, we have previously presented a reconstruction [3] of a famous case in property law by simulating the opinion and dissent in that case. The case is that of Pierson vs Post1 which concerned a dispute about ownership of a hunted fox. The said fox was being pursued by Post who was hunting with hounds on unoccupied waste-land. Whilst Post was in pursuit of the fox another man, Pier- son, came along and intercepted the chase, killing and carrying off the fox. Central to the arguments considered in the case was whether ownership of a wild animal can be attributed through mere pursuit. However, there are numerous other argu- ments that need to be considered which draw out the emphasis placed on the values considered within the case. Firstly, the value ‘public benefit’ was considered as it was argued that fox hunting is of benefit to the public because it assists farmers, so it should be encouraged by giving the sportsman such as Post protection of the law. There are of course counter arguments to this based on the humane treatment of animals. Furthermore, there are arguments concerning consideration of public benefit based on the desire to punish malicious behaviour as allegedly shown by Pierson in intercepting the fox that he could see Post was chasing. 1 3 Cai R 1752 Am Dec 264 (Supreme Court of New York, 1805)
- 71. 3 Abstract Argumentation and Values 57 Secondly, there were arguments set forth about the need for the law to be clear: in attributing ownership without bodily possession this would encourage a climate of litigation based on similar claims related to pursuit alone. Thirdly, the value of ‘economic benefit’ was considered in relation to the protec- tion of property rights where the claimant is engaged in a profitable enterprise. Given the facts of the case and the values stated above that have been recognised as pertinent to the reasoning in the case, the argument scheme for practical reasoning can be applied to generate the competing arguments about who to decide for in the case. Once generated these arguments can be organised into a VAF and evaluated in the usual manner. In the actual case the court found for Pierson, thus holding that clarity was more important than the values promoted by finding for Post. Preference orderings of values that led to this decision are reflected in our full representation of the case, which can be found in [3]. Explicitly representing the values promoted by the arguments put forward in the case helps to clarify the justifications for the arguments advanced and ground those justifications within the purposes that law is intended to capture and uphold. 5.2 Medicine A second example scenario that has been considered in terms of value-based ar- gumentation is one concerning a system for reasoning about the medical treatment of a patient [5]. Decision making in this domain often requires consideration of a wide range of options, some of which may conflict, and may also be uncertain. Thus, value-based argumentation can play a role in supporting the decision making process in this domain. The scenario modelled in [5] illustrates a running example of a patient whose health is threatened by blood clotting. In deciding which particular treatment to ad- minister to the patient there are a number of policies and concerns that affect the decision, and each must be given its due weight. In the computational model of the scenario a number of different perspectives are represented that are given as val- ues of individual agents. The arguments and subsequent conclusions drawn by the individual agents are then adjudicated by a central agent which comes to a deci- sion based on an evaluation of the competing arguments. Concerning the individual agents’ values, these represent perspectives such as: the treatment of the patient based on general medical policy; the safety of the patient concerning knowledge of contraindications of the various drugs; the efficacy of the treatments in reference to specific medical knowledge; and, the cost of the different treatments available. Given the above agent perspectives (and others that we do not detail here), the practical reasoning argument scheme can be used to generate arguments about which drug should be used to treat a particular patient. These arguments can be critiqued by agents other than those that generated the recommendation, based on their individual knowledge, through the posing of the appropriate critical questions. This may lead to different agents recommending different treatments, one of which
- 72. 58 Trevor Bench-Capon and Katie Atkinson must be chosen. In order to decide between the competing choices, the arguments justifying each are organised into a VAF and evaluated according to the preference given over the values represented by the individual agents. For example, it may be the case that the treatment agent recommends a particular drug that is known to be highly effective (since no critique from the efficacy agent indicates otherwise) and has no contraindications (according to the safety agent), yet the cost agent has an argument that the drug cannot be used on monetary grounds. The question then is whether treatment is to be preferred to cost (which may be the case if there are no suitable alternatives are identified). Resolution of this issue will be determined by the central adjudicating agent who provides the value ordering to decide upon the winning argument and subsequent treatment recommended, in accordance with the policy of the relevant health authority at the time. Whilst a key motivation for the example application described above was the representation of the different perspectives within the situation, there are other ad- vantages worthy of note. Firstly, the reasoning involved in medical scenarios is often highly context dependant and relative to specific individuals so there is a high de- gree of uncertainty. Thus any ordering of preferences must take the specific context into account and the argumentation based approach enables this. Secondly, the ar- gumentation element is effected inside a single agent and the information that it uses is distributed across different information sources, which need not themselves consider every eventuality, and play no part in the evaluation. This simplifies their construction and facilitates their reuse in other applications. Finally, the critiques that are posed against putative solutions are made only as and when they can affect the evaluation status of arguments already advanced. This means that all reasoning undertaken is potentially relevant to the solution. 5.3 Moral Reasoning The running example that we have presented in this paper concerning travel to a conference is represented in terms of an AATS. We now turn to briefly discussing another example scenario, concerning moral reasoning, that has been modelled in these terms. The scenario is a particular ethical dilemma discussed by Coleman [12] and Christie [11], amongst others, and it involves two agents, called Hal and Carla, both of whom are diabetic. The situation is that Hal, through no fault of his own, has lost his supply of insulin and urgently needs to take some to stay alive. Hal is aware that Carla has some insulin kept in her house, but Hal does not have permission to enter Carla’s house. The question is whether Hal is justified in breaking into Carla’s house and taking her insulin in order to save his life. By taking Carla’s insulin, Hal may be putting her life in jeopardy, since she will come to need that insulin herself. One possible response is that if Hal has money, he can compensate Carla so that her insulin can be replaced before she needs it. Alternatively if Hal has no money but Carla does, she can replace her insulin herself, since her need is not immediately
- 73. 3 Abstract Argumentation and Values 59 life threatening. There is, however, a serious problem if neither have money, since in that case Carla’s life is really under threat. Coleman argued that Hal may take the insulin to save his life, but should compensate Carla. Christie’s argument against this was that even if Hal had no money and was unable to compensate Carla he would still be justified in taking the insulin by his immediate necessity, since no one should die because of poverty. In [2] we have represented this scenario in terms of an AATS and considered the arguments that can be generated concerning how the agents could justifiably act. Following our methodology, we take the arguments generated and organise them into a VAF to see the attack relations between them and evaluate them in accor- dance with the particular value preference orderings. An interesting point that can be taken from this particular example concerns the nuances between different ‘lev- els’ of morality that can be drawn out by distinguishing the individual agents within the value orderings. For example, prudential reasoning takes account of the different agents, with the reasoning agent preferring values relating to itself, whereas strict moral reasoning ignores the individual agents and treats the values equally. For ex- ample, in the insulin scenario two values are recognised: life, which is demoted when Hal or Carla ceases to be alive, and freedom, which is demoted when Hal or Carla ceases to have money. Thus, an agent may rank life over freedom, but within this value ordering it may discriminate between agents; for example, the agent may place equal value on its own and another’s life, or it may be that it prefers its own life to another’s (or vice versa). This leads to distinctions such as selfish agents who prefer their own interests above all those of other agents, and noble agents whose values are ordered, but within a value the agent prefers another’s interests. In addition to the AATS representation set out in [2], simulations have also been run, which are reported in [10], that confirm the reasoning as set out. 5.4 e-Democracy The final application area that we discuss is an e-Democracy setting whose focus is more on the support given by value based argumentation within a system to facilitate the collection and analysis of human arguments within political debates. The application is presented as a discussion forum named Parmenides whose underlying structure is based upon the practical reasoning argument scheme and the latest version of the system is described in [9]. The system is intended as a forum by which the government is able to present policy proposals to the public so users can submit their opinions on the justification presented for the particular policy. The justification for action is structured in the form of the practical reasoning argument scheme, though this imposed structure is hidden from the user. Within a particular topic of debate, a justification upholding a proposed government action is presented to users of the system in the form of the argument scheme. Users are then led in a structured fashion through a series of web pages that pose the appropriate critical questions to determine which parts of the justification the users agree or disagree
- 74. 60 Trevor Bench-Capon and Katie Atkinson with (the circumstances, the action, the consequences or the value). Users are not aware (and have no need to be aware) of the underlying structure for argument representation but it is, nevertheless, imposed on the information they submit. This enables the collection of information which is structured in a clear and unambiguous fashion from a system which does not require users to gain specialist knowledge before being able to use it. In addition to collecting arguments, Parmenides also has analysis facilities that make use of AFs. All the information that the users submit through the system is stored in a back-end database. This information is then organised into an argu- mentation framework to show the attacking arguments between the positions ex- pressed. Associated with the arguments in the AF is statistical information con- cerning a breakdown of support for the arguments, i.e. the number of users agree- ing/disagreeing with a particular element of the justification. Thus, arguments can be assessed by considering which ones are the most controversial to the users. The Parmenides system is intended to overcome some of the problems faced by existing discussion forum formats, such as unstructured blogs and e-petitions. In such systems where there is no structuring of the information, it is undoubtedly very difficult for the policy maker to adequately address each person’s concerns since he or she is not aware of users’ specific reasons for disagreeing. Furthermore, it may be difficult to recognise agreement and disagreement between multiple user replies. In contrast, the structure imposed by Parmenides allows the administrator of the system to see exactly which particular part of the argument is disagreed with by the majority of users, e.g. arguments based on a description of the circumstances, or arguments based on a disagreement about the importance of promoting a particular value. Identifying these different sources of disagreement allows the policy maker to see why his policy is disliked, so he may be able to better respond to the criticisms made, or indeed change the policy. In particular, it can indicate whether the values motivating the policy are shared by the respondents. Parmenides has been tested on a number of different political debates, including: the UK debate about banning fox hunting2; the justification for the 2003 war in Iraq; and, a debate about the proposal to increase the number of speed cameras on UK roads. Work on the Parmenides system is ongoing to further extend its representation facilities, through the use of schemes additional to the practical reasoning scheme, and to further extend the facilities for analysing the arguments through the use of argumentation frameworks. 6 Developments of Value Based Argumentation In this section we will mention some developments of Value Based Argumentation. In [13] there is an interesting exploration of the relation between neural networks, in particular neural-symbolic learning systems, and value based argumentation sys- 2 For this particular debate on the system see: http://cgi.csc.liv.ac.uk/∼parmenides/foxhunting/
- 75. 3 Abstract Argumentation and Values 61 tems, including an extensive discussion of the insulin example described in the last section. In [15] there is a formal generalisation of VAFs to allow for arguments that promote multiple values, and in which preferences among values can be specified in various ways. In [7] a method is given to determine which audiences can accept a particular set of arguments. Here, however, we will look in detail only at the appli- cation of Modgil’s extended argumentation frameworks (EAF) [17] to VAFs. For a preliminary exploration of the relation between EAFs and VAFs see [18]. The core idea of EAFs is, like VAFs, to enable a distinction between an argu- ment attacking an argument, and an argument defeating another argument. Whereas VAFs, however, rely on a comparison of properties of the arguments concerned, EAFs achieve this in an entirely abstract manner by allowing arguments to attack not only other arguments, but also attacks. EAFs thus enable arguments to resist an attack for a number of reasons. In VAFs arguments resist attacks solely in virtue of a preference between the values concerned. This enables VAFs to be rewritten as standards AFs, by introducing some auxiliary arguments to articulate the notion of an attack on an attack. These auxiliary arguments represent the status of arguments, value preferences, and arguments representing particular audiences. Suppose we have a VAF with two arguments, A and B which attack one another. A is associated with value V1 and B with Value V2. A will be defeated if B defeats it, and B will be defeated if A defeats it. Defeat is only possible if the attacking argument is not defeated, and if the value of A is not preferred to that of B. Thus the attack on the attack of A on B in an EAF becomes an attack on the argument that A defeats B. This enables us to represent a VAF as a standard AF, with preferred extensions depending on the choices made regarding value preferences. We can extend the AF to include audiences as well. Suppose Audience X prefers V1 to V2 and Audience Y prefers V2 to V1. This can be shown as in Figure 3.6. A is V1 justified V1 V2 V2 V1 is X Audience Audience is Y B A defeats B is defeated B is justified V2 B A defeats A is defeated Fig. 3.6 AF representing VAF with audiences The rewriting of VAFs in this way is shown to be sound and complete with re- spect to EAFs in [18]. When we rewrite VAFs in this way, subjective acceptance in the VAF is equivalent to credulous acceptance in the rewritten AF, and objective acceptance in the VAF is equivalent to sceptical acceptance in the rewritten AF. This
- 76. 62 Trevor Bench-Capon and Katie Atkinson Property A7 A5 Life Property A6 A6 Property not A6 A6 A5 def A5 Life not A5 A5 def A7 A7 Property def A6 A7 A7 not L P P L b) a) Fig. 3.7 3 cycle and re-write can be seen by considering the three cycle in the two value case shown in Figure 3.7. There will be two preferred extensions depending on which preference is chosen: {L P, A5defA7,not A7,A6,A5} and {PL,A7,A7defA6,notA6,A5}. Thus A5 is correctly sceptically acceptable in 7b, and objectively acceptable in 7a, and the re- maining arguments, other than notA5, are credulously acceptable in 7b, and A6 and A7 are subjectively acceptable in 7a. A1 Kt Ck A2 Kp Sk A3 Tt Ct A4 Tp St A6 B KpTp A5 B KtTt Ck B B Ck Ck Sk Sk Ck Sk B B Sk S C C S B St St B B Ct Ct B Ct St St Ct Katie CBS Trevor BSC Fig. 3.8 Value based EAF for travel example. Finally, we apply this to our running example of Trevor and Katie travelling to Paris. The rewritten framework is given in Figure 3.8: note that we have used the EAF style of attacks on attacks rather than the rewrite, for clarity in the diagram.
- 77. 3 Abstract Argumentation and Values 63 We have added the audiences Trevor and Katie. Note that, although their preferences differ, these arguments do not conflict, as Trevor and Katie must be allowed to have different preferences: although they are trying to come to a consensus of which argu- ments to accept, they are free to maintain their own value orders. Trevor’s audience attacks the preferences between values in respect of Trevor, and Katie’s audience attacks preferences in respect of Katie. Both audiences attack preferences in com- mon. We evaluate the framework in Figure 3.8 by first removing the arguments at- tacked by the audiences, and then the attacks attacked by surviving arguments. Re- flecting the impact of audiences in this way gives a standard AF, the connected com- ponent of which is the same as that shown in Figure 3.5. Now A1 is not attacked, and so the preferred extension will contain the two audiences, the consequent pref- erences (note that both SC and CS have been defeated as Trevor and Katie dis- agree), together with A1, (which is not attacked), A3 and A5 (whose attackers are defeated). Thus, as before, given these preferences both Trevor and Katie choose to travel by train. 7 Summary Just as deduction is a natural paradigm for justifying beliefs, argumentation is the natural paradigm for explaining and justifying why one course of action is preferred to another, since the notions of defeasibility and individual preference are central to argumentation. We can be coercive about what is the case, but need to be persuasive about what should be the case. But in order to exploit this aspect of argumentation, it is necessary to extend the purely abstract notion of argumentation proposed by Dung to enable individual preferences to explain the choices made in determining which arguments will be accepted by an agent in a particular context. We have discussed such an extension, representing the individual interests and aspirations as values, and individual preferences as orderings on these values. Using this extension we have shown how different agents can rationally make different choices in accordance with their value orderings, and how in turn these value orderings can emerge from particular situations. In particular we have dis- cussed examples where two agents with different value orderings must agree collec- tively on what they should do. The range of applications in which reasoning of this sort is required is wide, and we have discussed a number of application areas: law, medicine, politics and moral dilemmas, and an everyday situation. In this chapter we have shown how this important style of reasoning, central to the notion of an au- tonomous agent, can be captured in a particular form of argumentation framework which, while permitting the expression of individual preferences, retains all the ben- efits of the clean semantics associated with abstract argumentation frameworks.
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- 79. Chapter 4 Bipolar abstract argumentation systems Claudette Cayrol and Marie-Christine Lagasquie-Schiex 1 Introduction In most existing argumentation systems, only one kind of interaction is considered between arguments. It is the so-called attack relation. However, recent studies on argumentation [23, 34, 35, 4] have shown that another kind of interaction may exist between the arguments. Indeed, an argument can attack another argument, but it can also support another one. This suggests a notion of bipolarity, i.e. the existence of two independent kinds of information which have a diametrically opposed nature and which represent repellent forces. Bipolarity has been widely studied in different domains such as knowledge and preference representation [10, 31, 25, 6]. Indeed, in [6] two kinds of preferences are distinguished: the positive preferences representing what the agent really wants, and the negative ones referring to what the agent rejects. This distinction has been supported by studies in cognitive psychology which have shown that the two kinds of preferences are completely independent and are processed separately in the mind. Another application where bipolarity is largely used is that of decision making. In [3, 19], it has been argued that when making decision, one generally takes into account some information in favour of the decisions and other pieces of information against those decisions. In [20], a nomenclature of three types of bipolarity has been proposed using par- ticular characteristics like exclusivity (can a piece of information be at the same time positive and negative), duality (can negative information be computed using posi- tive information), exhaustivity (can information be neither positive, nor negative), computation of positive and negative information on the same data, computation of positive and negative information with the same process, existence of a consistency constraint between positive and negative information. IRIT-UPS, 118 route de Narbonne, 31062 Toulouse, France, e-mail: {ccayrol,lagasq}@ irit.fr I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 65 DOI 10.1007/978-0-387-98197-0 4, c Springer Science+Business Media, LLC 2009
- 80. 66 Claudette Cayrol and Marie-Christine Lagasquie-Schiex The first type of bipolarity proposed by [20] (symmetric univariate bipolarity) expresses the fact that the negative feature is a reflection of the positive feature (so, they are mutually exclusive and a single bipolar univariate scale is enough for representing them). The second one (dual bivariate bipolarity) expresses the fact that we need two separate scales in order to represent both features, although they stem from the same data (so, an information can be positive and negative at the same time and there is no exclusivity). However a duality must exist between both features. And the third one (heterogeneous bipolarity) expresses the fact that both features do not stem from the same data though there is some minimal consistency require- ment between both features. In this chapter, we focus on the use of bipolarity in the particular domain of argumentation. In all the disparate cases, an argumentation process follows different steps: i) building the arguments and the interactions between them, ii) valuating the arguments and accounting for their interactions or not, iii) finally selecting the most acceptable arguments (or sets of arguments) and using them in order to draw a conclusion or choose a decision. Bipolarity can appear under different forms in each step of this process. In this chapter we are only concerned by the use of bipolarity at the interaction level (a more complete study of bipolarity in each step of the argumentation process is proposed in [4, 5]). At this level, the main point is the definition of the interactions between argu- ments. As already said, due for instance to the presence of inconsistency in knowl- edge bases, arguments may be conflicting. Indeed, in all argumentation systems, an attack relation is considered in order to capture the conflicts. However, most logical theories of argumentation assume that: if an argument a1 attacks an argument a3 and a3 attacks an argument a2, then a1 supports a2. In this case, the notion of support does not have to be formalized in a way really different from the notion of attack. It is the case of the basic argumentation framework defined by Dung, in which only one kind of interaction is explicitly represented by the attack relation. In this context, the support of an argument a by another argument b can be represented only if b defends a in the sense of [21]. So, support and attack are dependent notions. It is a parsimonious strategy, but it is not a correct description of the process of argumentation. Let us take several examples for illustrating the difference between “defence” and “support”: Ex. 1 We want to begin a hike. We prefer a sunny weather, then a sunny and cloudy one, then a cloudy but not rainy weather, in this order. We will cancel the hike only if the weather is rainy. But clouds could be a sign of rain. We look at the sky early in the morning. It is cloudy. The following exchange of informal arguments occurs between Tom, Ben and Dan: t1 Today we have time, we begin a hike. b The weather is cloudy, clouds are sign of rain, we had better cancel the hike. t2 These clouds are early patches of mist, the day will be sunny, without clouds, so the weather will be not cloudy (and we can begin the hike).
- 81. 4 Bipolar abstract argumentation systems 67 d These clouds are not early patches of mist, so the weather will be not sunny but cloudy; however these clouds will not grow, so it will not rain (and we can begin the hike). In this exchange, we can identify the following path of conflicts between ar- guments: argument d attacks argument t2 which attacks argument b which in turn attacks argument t1. So, with Dung’s framework, argument t2 is a defender of ar- gument t1, and argument d is a defeater of argument t1. Nevertheless, arguments t2 and d support the hike project. So, the idea of a chain of arguments and counter- arguments in which we just have to count the links and take the even one as defeaters and the odd ones as supporters is an oversimplification. So, the notion of defence proposed by [21] is not sufficient to represent support. The following example also illustrates the need for a new kind of interaction between arguments; the following arguments are exchanged during the meeting of the editorial board of a newspaper: Ex. 2 a: Assuming agreement and no right of censor, information I concerning X will be published. b1: X is the prime minister who may use the right of censor. c0: We are in democracy and even a prime minister cannot use the right of censor. c1: I believe that X has resigned. So, X is no longer the prime minister. d: The resignation has been announced officially yesterday on TV Channel 1. b2: I is private information so X denies publication. e: I is an important information concerning X’s son. c2: Any information concerning a prime minister is public information. repetition of c1 and d: . . . c3: But I is of national interest, so I cannot be considered as private information. In this example, some conflits appear: for instance, b1 (resp. b2) is in conflict with a. But we may also consider that the argument d given by an agent Ag1 supports the argument c1 given by another agent Ag2. It is not only a “dialogue-like speech act”: a new piece of information is really given and it is given after the production of the argument c1. So taking d into account leads either to modify c1, or to find a more intuitive solution for representing the interaction between d and c1. In this case, we adopt an incremental point of view, considering that pieces of information given by different agents enable them to provide more and more arguments. We do not want to revise already advanced arguments. In contrast, we intend to represent as much as possible all the kinds of interaction between these arguments. The last example shows how a notion of support between two arguments can be formalized with a logical representation of the structure of the arguments. Ex. 3 A murder has been performed and the suspects are Liz, Mary and Peter. The following pieces of information have been gathered: The type of murder suggests us that the killer is a female. The killer is certainly small. Liz is tall and Mary and Peter are small. The killer has long hair and uses a
- 82. 68 Claudette Cayrol and Marie-Christine Lagasquie-Schiex lipstick. A witness claims that he saw the killer who was tall. Moreover, we are told that the witness is short-sighted, so he is no more reliable. We use the following propositional symbols: sm (the killer is small), fem (the killer is a female), mary (the killer is Mary), lglip (the killer has long hair and uses a lipstick), wit (the witness is reliable), bl (the witness is short-sighted). Here, an argument takes the form of a set of premises which entails a conclusion. So the following arguments can be formed: a1 in favour of mary (with premises {sm, fem,(sm ∧ fem) → mary}), a2 in favour of ¬sm (with premises {wit,wit → ¬sm}), a3 in favour of ¬wit (with premises {bl,bl → ¬wit}), a4 in favour of fem (with premises {lglip,lglip → fem}). a3 attacks a2 which attacks a1. So a3 defends a1 against a2. Moreover, a4 confirms the premise fem of a1. So, a4 supports a1 (in the sense that a4 strengthens a1). Contrastedly, a3 defends a1 against a2 means that a3 weakens the attack on a1 brought by a2. So, on one side, a1 gets a support and on the other side a1 suffers a weakened attack. The above examples show that the argumentation process uses arguments and counter-arguments, support and attack relations, but not always in the same way. The arguments which are available in a dynamic argumentation process rely upon premises which are not always pieces of evidence. If we accept that a new fact can undermine one of the premises (thus forming an attack), we must also accept that a new fact can enforce, or confirm a premise (thus forming a support interaction). Following all these remarks, and in order to formalise realistic examples, a more powerful tool than the abstract argumentation framework proposed by Dung is needed. In particular, we are interested in modelling situations where two inde- pendent kinds of interactions are available: a positive and a negative one (see for example in the medical domain the work [23]). So, following [23, 35], we present a new argumentation framework: an abstract bipolar argumentation framework. The chapter is organized as follows: Section 2 introduces the formal definitions of an abstract Bipolar Argumentation Framework (BAF). Then, we consider the fun- damental problem of determining which arguments (or sets of arguments) can be considered as acceptable. The formal way to handle this problem is to define argu- mentation semantics. Section 3 introduces extension-based acceptability semantics for a BAF. These new semantics rely upon criteria which make explicitly use of both support and attack relations. In Section 4, another way to define extension-based semantics for a BAF is followed. First, a transformation of a BAF into a Dung’s meta-argumentation framework is given. The support relation is used to form meta- arguments (called coalitions) in such a way that at the meta-level only conflict in- teractions may appear. Extensions of a BAF can then be defined from Dung’s ex- tensions of the meta framework. Section 5 addresses the question of labelling-based semantics in a BAF. Some labelling functions are proposed for a BAF. Section 6 is devoted to the related issues and to some concluding remarks. Note that the proofs of the properties given in this chapter can be found in the associated original papers.
- 83. 4 Bipolar abstract argumentation systems 69 2 Abstract bipolar frameworks An abstract bipolar argumentation framework is an extension of the basic abstract argumentation framework introduced by [21] in which a new kind of interaction between arguments is represented by the support1 relation2. This new relation is assumed to be totally independent of the attack relation (i.e. it is not defined using the attack relation). So, we have a bipolar representation of the interactions between arguments. Def. 1 An abstract bipolar argumentation framework (BAF) A,Ratt,Rsup consists of: a set A of arguments, a binary relation Ratt on A called the attack relation and another binary relation Rsup on A called the support relation. These binary relations must verify the following consistency constraint: Ratt ∩Rsup = ∅3. Consider ai and aj ∈ A, aiRattaj (resp. aiRsupaj) means that ai attacks (resp. sup- ports) aj. Let a ∈ A, Ratt − (a) (resp. Rsup − (a)) denotes the set of attackers (resp. supporters) of a. In the following, we assume that A represents the set of arguments proposed by rational agents at a given time, so we will assume that A is finite. A BAF can be represented by a directed graph Gb called the bipolar interaction graph, with two kinds of edges, one for the attack relation (→) and another one for the support relation (❀). See for instance the following representations: For Ex. 1 For Ex. 2 For Ex. 3 (hiking project) (editorial meeting) (murder) b // t1 t2 oo o/ o/ o/ ff d __ ?? OO O O O c0 // b1 // a b2 oo e oo o/ o/ o/ d // /o /o /o c1 OO // c2 ?? ~ ~ ~ ~ ~ ~ ~ ~ c3 OO a3 // a2 // a1 a4 ~ ~ ~ ~ In the following, we abstract from the structure of the arguments and we consider arbitrary independent relations Ratt and Rsup. Def. 2 Let BAF = A,Ratt,Rsup be a bipolar argumentation framework and Gb be the associated interaction graph. Let a, b ∈ A. A path from a to b in Gb is a sequence (a1,...,an) of elements of A s.t. n ≥ 2, a = a1, b = an, a1Ra2, . . . , an−1Ran, with R = Ratt or Rsup. Such a path has length n−1. Note that if n = 2 and a = b then the path is a loop and if the relation R used in the loop is Ratt then a is said self-attacking. The use of bipolarity suggests new kinds of interaction between arguments: in Ex. 2, the fact that d supports an attacker of b1 may be considered as a kind of 1 Note that the term “support” refers to a relation between two arguments and not a relation between premises and conclusion, as in Toulmin [32]. 2 If the support relation is removed, we retrieve Dung’s framework. 3 In the context of the argumentation, this consistency constraint is essential: it does not seem ratio- nal to advance an argument which simultaneously attacks and supports the same other argument.
- 84. 70 Claudette Cayrol and Marie-Christine Lagasquie-Schiex negative interaction between d and b1, which is however weaker than a direct at- tack. From a cautious point of view, such arguments cannot appear together in a same extension. In order to address this problem, a new kind of attack has been introduced [13, 14] which combines a sequence of supports with a direct attack. Def. 3 Let a, b ∈ A. There is a sequence of supports for b by a (or for short a supports b) iff there exists a sequence (a1,...,an) of elements of A s.t. n ≥ 2, a = a1, b = an, a1Rsupa2, . . . , an−1Rsupan. Def. 4 A supported attack for an argument b by an argument a is a sequence (a,x,b) of arguments of A s.t. a supports x4 and xRattb. In Ex. 2, there is a supported attack for b1 by d. Then, taking into account attacks and sequences of supports leads to the following definitions applying to sets of arguments: Def. 5 Let S ⊆ A, let a ∈ A. S set-attacks a iff there exists a supported attack or a direct attack for a from an element of S. S set-supports a iff there exists a sequence of supports for a from an element of S. The above definitions are illustrated on the following example: Ex. 4 Consider the following graph: a // /o /o /o b // /o /o /o c // d j g @@ h @@ i // /o /o /o e OO f OO O O O // k OO In this graph, the paths a−b−c−d and i−c correspond to supported attacks. The set {a,h} set-attacks d and b and set-supports b and c. 3 Extension-based semantics for acceptability In Dung’s framework, the acceptability of an argument depends on its membership to some sets, called acceptable sets or extensions. These extensions are characterised by particular properties. It is a collective acceptability. Following Dung’s method- ology, we propose characteristic properties that a set of arguments must satisfy in order to be an output of the argumentation process, in a bipolar framework. We re- call that such a set of arguments must be in some sense coherent and must enable to win a dispute. Maximality for set-inclusion is also often required. Considering a BAF A,Ratt,Rsup and using the notion of “set-attack” and “set- support” given by Def. 5, we first investigate the notion of coherence, then we pro- pose new semantics for acceptability in bipolar argumentation frameworks. 4 In the sense of Def. 3.
- 85. 4 Bipolar abstract argumentation systems 71 3.1 Managing the conflicts In the basic argumentation framework, whatever the considered semantics, selected acceptable sets of arguments are constrained to be coherent in the sense that they must be conflict-free. In a bipolar argumentation framework, the concept of coher- ence can be extended along two different lines: • forbidding not only direct attacks but also supported attacks enforces a kind of internal coherence: we do not accept a set S of arguments which set-attacks one of its elements (this a generalization of Dung’s notion of conflict-free). • extending the consistency constraint between support and attack relations leads to define a kind of external coherence: we do not accept a set S of arguments which set-attacks and set-supports the same argument. Def. 6 Let S ⊆ A. S is +conflict-free5 iff ∃a,b ∈ S s.t. {a} set-attacks b. S is safe6 iff ∃b ∈ A s.t. S set-attacks b and either S set-supports b, or b ∈ S. Ex. 4 (cont’d) The set {h,b} is not +conflict-free (there is a direct attack). The set {b,d} is not +conflict-free since d suffers a supported attack from b. Contrastedly, {a,h} and {b, f} are +conflict-free. The set {a,h} is not safe since a supports b and h attacks b. The set {b, f} is not safe since d suffers a supported attack from b and f supports d. Contrastedly, {g,i,h} is safe. Note that the notion of safe set encompasses the notion of +conflict-free set: Prop. 1 ([14]) Let S ⊆ A. If S is safe, then S is +conflict-free. If S is +conflict-free and closed for Rsup then S is safe. Ex. 4 (cont’d) The set {g,h,i,e} is +conflict-free and closed for Rsup. So it is safe. 3.2 New acceptability semantics According to the methodology proposed by [21], two notions play an important role in the definition of extension-based semantics: the notion of coherence, and the notion of defence (that is for short attack against attack). In a BAF, several notions of coherence, and two kinds of attack (direct and supported) are available. So several extensions of the notion of defence could be proposed. However, we have chosen to restrict to the classical defence, for the following reasons. First, the purpose of this chapter is to present some principles governing bipolar frameworks, rather than an exhaustive survey. Secondly, most of the works talking about bipolarity consider that 5 This notation means that checking if a set is +conflict-free needs to consider more conflicts than with the basic notion of conflict-free suggested by Dung. 6 This definition is inspired by [35] and by the notion of a controversial argument given in [21].
- 86. 72 Claudette Cayrol and Marie-Christine Lagasquie-Schiex a support does not have the same strength as an attack. In that sense, an argument can be considered as defended if and only if its direct attackers are directly attacked. The above remark is illustrated by the following example: a1 // a2 ~~ a3 oo o/ o/ o/ There is a supported attack for a1 by a3 and no attack for a3. However, a1 directly attacks a2 and it seems sufficient to resinstate a1. Let us recall the definition of defence given in [21]. Def. 7 Let S ⊆ A. Let a ∈ A. S defends a iff ∀b ∈ A, if bRatta then ∃c ∈ S s.t. cRattb. In the following, the concept of admissibility is first extended. The idea is to reinforce the coherence of the admissible sets. Then, extensions under the preferred semantics will be defined as maximal (for ⊆) admissible sets of arguments. Three different definitions for admissibility can be given, from the most general one to the most specific one. First, a direct translation of Dung’s definition gives the definition of d-admissibility (“d” means “in the sense of Dung”). Taking into account external coherence leads to s(afe)-admissibility. Finally, external coherence can be strengthened by requiring that an admissible set is closed for Rsup. So, we obtain the definition of c(losed)-admissibility. Def. 8 Let S ⊆ A. S is d-admissible iff S is +conflict-free and defends all its elements. S is s-admissible iff S is safe and defends all its elements. S is c-admissible iff S is +conflict-free, closed for Rsup and defends all its elements. From the above definitions, it follows that each c-admissible set is s-admissible, and each s-admissible set is d-admissible. Def. 9 A set S ⊆ A is a d-preferred (resp. s-preferred, c-preferred) extension iff S is maximal for ⊆ (or for short ⊆-maximal) among the d-admissible (resp. s- admissible, c-admissible) subsets of A. Ex. 1 (cont’d) In this case, the three semantics give the same result: {d,t1} is the unique d-preferred, s-preferred and c-preferred extension. Ex. 4 (cont’d) The set {g,h,i,e, f,d, j} is the unique c-preferred extension. Ex. 5 Consider the BAF represented by a ❀ b ← h. The set {a,h} is the unique d-preferred extension. There are two s-preferred extensions {a} and {h}. And there is only one c-preferred extension {h}. One of the most important issues with regard to extensions concerns their ex- istence. The existence of d-preferred (resp. s-preferred, c-preferred) extensions is guaranteed since the empty set is d-admissible (resp. s-admissible, c-admissible), and each d-admissible (resp. s-admissible, c-admissible) is included in a d-preferred (resp. s-preferred, c-preferred) extension. Note that analogous definitions for admis- sibility could be proposed using a stronger notion of defence (a stronger defence would be defined for instance by replacing attack with set-attack in Def. 7).
- 87. 4 Bipolar abstract argumentation systems 73 Considering another well-known semantics, the stable semantics, nice results can be obtained if we keep the basic definition of a stable extension, but replace attack with set-attack. It is a straightforward way to extend the stable semantics in a BAF. Def. 10 S is a stable extension iff S is +conflict-free and ∀a ∈ S, S set-attacks a. In the following, we restrict to acyclic BAF, in the sense that the associated in- teraction graph is acyclic. In Dung’s basic framework, it has been proved that, in the case of an acyclic attack graph, there is always a unique stable (which is also preferred) extension. So, Def. 10 ensures the existence of a unique stable extension in an acyclic BAF7. However, the unique stable extension is not always safe. Ex. 5 (cont’d) The set {a,h} is the unique stable extension, and it is not safe. Indeed, the following result can be proved: Prop. 2 ([14]) Let S be a stable extension. S is safe iff S is closed for Rsup. The following results enable to characterize d-preferred, s-preferred and c- preferred extensions when the BAF is acyclic: Prop. 3 ([14]) Let S be the unique stable extension of an acyclic BAF. 1. S is also the unique d-preferred extension. 2. The s-preferred extensions and the c-preferred extensions are subsets of S. 3. Each s-preferred extension which is closed for Rsup is c-preferred. 4. If S is safe, then S is the unique c-preferred and the unique s-preferred extension. 5. If A is finite, each c-preferred extension is included in a s-preferred extension. 6. If S is not safe, the s-preferred extensions are the subsets of S which are ⊆- maximal among the s-admissible sets. 7. If S is not safe, and A is finite, there is only one c-preferred extension. Ex. 5 (cont’d) {h} is the only s-preferred extension which is also closed for Rsup. So, {h} is the unique c-preferred extension. Ex. 6 Consider the BAF represented by: a1 // /o /o /o A A A A A A A A a2 // /o /o /o b c h oo o/ o/ o/ OO {a1,a2,h} is the only d-preferred extension. {a1,a2} and {h} are the only two s-preferred extensions. None of them is closed for Rsup. ∅ is the unique c-preferred extension. If we add an isolated argument a3 (for which no interaction exists with the other available arguments), then we obtain: {a1,a2,a3,h} is the only d-preferred extension. {a1,a2,a3} and {h,a3} are the only two s-preferred extensions. None of them is closed, and {a3} is the unique c-preferred extension. 7 We instantiate Dung’s AF with the relation set-attacks and the resulting graph is still acyclic.
- 88. 74 Claudette Cayrol and Marie-Christine Lagasquie-Schiex The above discussion enables to draw the following conclusions. In the particular case of an acyclic BAF, two semantics present nice features: the stable semantics and the c-preferred semantics. If we are interested in internal coherence only, we will have to determine the unique stable extension, which is also the unique d-preferred extension. If we are interested in a more constrained concept of coherence, we will compute the unique c-preferred extension. 4 Turning a bipolar framework into a Dung meta-framework The extension-based acceptability semantics introduced in Section 3 rely upon cri- teria which make explicitly use of support and attack relations, through the concept of supported attack. Here, we follow another way to define extension-based seman- tics for a BAF. First, a transformation of a BAF into a Dung’s meta-argumentation framework is given. This meta-argumentation framework consists only of a set of meta-arguments (called coalitions), and a conflict relation between these meta- arguments. The attack relation of the initial BAF will appear only at the meta-level. As a consequence, a meta-argument will gather arguments which are not in conflict. The support relation of the initial BAF will not appear at the meta-level, but will be used to gather arguments in a coalition. The idea is that a meta-argument makes sense only if its members are somehow related by the support relation. So, the two fundamental principles governing the definition of a coalition are: the Coherence principle (there is no direct attack between two arguments of a same coalition) and the Support principle (if two arguments belong to a same coalition, they must be somehow, directly or indirectly, related by the support relation). 4.1 The concept of coalition Consider BAF = A,Ratt,Rsup represented by the graph Gb. Gb sup will denote the partial graph representing the partial system A,Rsup8. AF will denote the partial argumentation system A,Ratt associated with BAF and represented by the partial graph denoted by Gb att. Def. 11 C ⊆ A is a coalition of BAF iff: (i) The subgraph of Gb sup induced by C is connected; (ii) C is conflict-free9 for AF; (iii) C is ⊆-maximal among the sets satisfying (i) and (ii). 8 We consider that the reader knows the basic concepts of graph theory (chain, connexity,...). See for instance [7] for a background on graph theory. 9 In the basic sense proposed by Dung.
- 89. 4 Bipolar abstract argumentation systems 75 Note that when Ratt is empty, the coalitions are exactly the connected compo- nents10 of the partial graph Gb sup. Prop. 4 ([17]) An argument which is not self-attacking is in at least one coalition. Ex. 7 Consider the BAF: a O O O c oo O O O // e // /o /o /o O O O f // /o /o /o g // /o /o /o h b @@ @ @ @ @ @ d i hh The coalitions are: C1 = {b,c,d}, C2 = {i}, C3 = {a,b}, C4 = {e, f,g,h} The following result shows that coalitions can be restated in terms of connected components of an appropriate subgraph. By the way, it gives a constructive way for computing coalitions. Prop. 5 ([17]) C ⊆ A is a coalition of BAF iff: (i) There exists S ⊆ A ⊆-maximal conflict-free for AF s.t. C is a connected component of the subgraph of Gb sup induced by S and (ii) C is ⊆-maximal among the subsets of A satisfying (i). Prop. 5 suggests a procedure for computing the coalitions of BAF: Step 1: Consider AF and determine the maximal conflict-free sets for AF. Step 2: For each set of arguments Si obtained at Step 1, determine the connected components of the subgraph of Gb sup induced by Si. Step 3: Keep the ⊆-maximal sets obtained at Step 2. The notion of conflict-free set is related to the notion of independent set: Prop. 6 ([17]) Let S ⊆ A. S is conflict-free for AF iff S is an independent subset of A in the graph Gb att. So, S is ⊆-maximal conflict-free for AF iff S is a ⊆-maximal independent set of vertices in the graph Gb att and Step 1 of the computational procedure consists in determining all the ⊆-maximal independent subsets of Gb att. Remark that the time complexity of the best algorithms providing all the ⊆-maximal independent sets is exponential. Note also that there exist several algorithms in the literature for finding all the ⊆-maximal independent sets (see for instance the work of J.M. Nielsen [26]). We also know that: • For Step 2, a depth-first exploration of a graph provides the connected compo- nents in linear time O(number of vertices + number of edges). • And for Step 3, maximization with respect to ⊆ is also an exponential process. 10 Let G = (V,E) be a graph. Let S ⊆ V. S is a connected component of G iff the subgraph of G induced by S is connected and there exists no S′ ⊆ V s.t. S ⊂ S′ and the subgraph of G induced by S′ is connected.
- 90. 76 Claudette Cayrol and Marie-Christine Lagasquie-Schiex 4.2 A meta-argumentation framework Let C(A) denote the set of coalitions of BAF. We define a conflict relation on C(A) as follows. Def. 12 Let C1 and C2 be two coalitions of BAF. C1 C-attacks C2 iff there exists an argument a1 in C1 and an argument a2 in C2 s.t. a1Ratta2. It can be proved that: Prop. 7 ([17]) Let C1 and C2 be two distinct coalitions of BAF. If C1 ∩C2 = ∅ then C1 C-attacks C2 or C2 C-attacks C1. So a new argumentation framework CAF = C(A),C-attacks can be defined, re- ferred to as the coalition framework associated with BAF. Ex. 7 (cont’d) In this example, CAF can be represented by (by abusing notations, → represents the attack relation in BAF and also the C-attack relation in CAF): C3 C1 oo // C4 C2 oo Dung’s definitions apply to CAF, and it can be proved that: Prop. 8 ([17]) Let {C1,...,Cp} be a finite set of distinct coalitions. {C1,...,Cp} is conflict-free for CAF iff C1 ∪...∪Cp is conflict-free for AF. So, CAF is a “meta-argumentation” framework with a set of “meta-arguments” (the set of coalitions C(A)) and a “meta-attack” relation on these coalitions (the C- attacks relation). A coalition gathers arguments which are close in some sense and can be produced together. However, as coalitions may conflict, following Dung’s methodology, preferred and stable extensions of CAF can be computed. Such exten- sions will contain coalitions which are collectively acceptable. The last step consists in gathering the elements of the coalitions of an extension of CAF. By this way, the best groups of arguments (w.r.t. the given interaction relations) will be selected. Def. 13 Let S ⊆ A. S is a Cp-extension (Cp means “Coalition-preferred”) of BAF iff there exists {C1,...,Cp} a preferred extension of CAF s.t. S = C1 ∪...∪Cp. S is a Cs-extension (Cs means “Coalition-stable”) of BAF iff there exists {C1,...,Cp} a stable extension of CAF s.t. S = C1 ∪...∪Cp. When the only preferred extension of CAF is the empty set, we define the empty set as the unique Cp-extension of BAF. Ex. 7 (cont’d) There is only one preferred extension of CAF, which is also stable: {C1,C2}. So, S = {b,c,d,i} is the Cp-extension (and also the Cs-extension) of BAF. Some nice properties of Dung’s basic framework are preserved: • A BAF has always a (at least one) Cp-extension. It is a consequence of Def. 13. • In contrast, there does not always exist a Cs-extension of BAF. The reason is that there may be no stable extension of CAF. • Each Cs-extension is also a Cp-extension. The converse is false.
- 91. 4 Bipolar abstract argumentation systems 77 • There cannot exist two Cp-extensions s.t. one strictly contains the other one. It follows from Def. 11 and 13. However, other properties are lost. A Cp-extension is not always admissible for AF, and a Cs-extension is not always a stable extension of AF: Ex. 8 Consider the BAF represented by: a // b c oo o/ o/ o/ // /o /o /o d // e The coalitions are: C1 = {a}, C2 = {b,c,d}, C3 = {e}. And the associated CAF can be represented by: C1 // C2 // C3 There is only one preferred extension of CAF, which is also stable: {C1,C3}. So, S = {a,e} is the Cp-extension (and also the Cs-extension) of BAF. We have dRatte, but a does not defend e against d (neither by a direct attack, nor by a supported attack, though a attacks an element of the coalition which attacks e). So, S is not admissible for AF. S does not contain c, but there is no attack (no supported attack) of an element of S against c. So, S is not a stable extension of AF. Note that a coalition is considered as a whole and its members cannot be used separately in an attack process. Ex. 8 suggests that admissibility is lost due to the size of the coalition {b,c,d}, and that it would be more fruitful to consider two independent coalitions {c,b} and {c,d}. A new formalization of coalitions in terms of conflict-free maximal support paths has been proposed in [17]. However, it does not enable to recover Dung’s properties. Note that the lost of admissibility in Dung’s sense is not surprising: admissibility is lost because it takes into account “individual” attack and defence, whereas, with meta-argumentation and coalitions, “collective” attack and defence are considered. 5 Labellings in bipolar frameworks This section addresses the question of labelling-based semantics in a BAF. A labelling-based semantics relies upon a set of labels and is defined by specifying the criteria for assigning labels to arguments. An example of labelling-based seman- tics in a basic argumentation framework is given by [22] with the robust semantics. More generally, several approaches have been proposed for valuing the arguments in a classical argumentation framework (for example [24, 29, 30, 8, 15, 1, 33]). In some of them, the value of an argument depends on its interactions with the other ar- guments; in other ones, it depends on an intrinsic strength of the argument. Besides, Karacapilidis Papadias [23] have proposed an argumentation web-tool, named HERMES, for decision making in the medical field. Taking into account notions of attacks and supports between arguments, this system permits the expression and the weighting of arguments. The basic elements of this system are: a solution (an answer to the question which is discussed) and a position (expressing the support for, or the opposition to a solution or to another position). HERMES can label the solutions and the positions by the status “active” or “inactive”. At the end of the discussion, the “active” (resp. “inactive”) solutions are accepted (resp. rejected). An
- 92. 78 Claudette Cayrol and Marie-Christine Lagasquie-Schiex “active” solution is a recommended choice among the other solutions concerning a same question. Different recursive labellings are proposed in HERMES. But, the value of a position p depends only on the active positions which are linked to p in the acyclic discussion graph, and the value of a position is always binary. In this section, we propose a limited use of the notion of labelling-based seman- tics for a BAF: we show how bipolar interactions can be used for defining valuations over the set of arguments, i.e. functions which assign a value to each argument of the BAF (a further step would be to use such a valuation in order to select arguments, that is to completely define labelling-based semantics in a BAF, in an analogous way as what has been done in [16] for basic argumentation frameworks). The approach presented here (see [13]) has the following features: the valuation process takes place before the selection process; the valuation process makes use of a rich set of values and not only two as in HERMES (so, it is called a gradual valu- ation); the value assigned to an argument takes into account all the direct attackers and supporters of this argument (it is not the case in HERMES in which the value of an argument only depends on the active positions); so it is called a local valuation. This proposition extends the works [22, 8, 16] to bipolar argumentation frame- works as defined in Section 2. It follows the same principles as those already de- scribed in [15] augmented with new principles corresponding to the “support” in- formation. So, the three underlying principles for a gradual interaction-based local valuation are: • P1: The value of an argument depends on the values of its direct attackers and of its direct supporters. • P2: If the quality of the support (resp. attack) increases then the value of the argument increases (resp. decreases). • P3: If the quantity of the supports (resp. attacks) increases then the quality of the support (resp. attack) increases. The value of an argument is obtained with the composition of several functions: • one for aggregating the values of all the direct attackers; this function computes the value of the “attack” • one for aggregating the values of all the direct supporters; this function computes the value of the “support” • one for computing the effect of the attack and of the support on the value of the argument. In the respect of the previous principles, we assume that there exists a completely ordered set V with a minimum element Vmin and a maximum element Vmax. The following formal definition for a gradual local valuation can be given. Def. 14 Let A,Ratt,Rsup be a bipolar argumentation framework. Let a ∈ A with Ratt − (a) = {b1,...,bn} and Rsup − (a) = {c1,...,cp}. A local gradual valuation on A,Ratt,Rsup is a function v : A → V s.t.: v(a) = g(hsup(v(c1), . . . , v(cp)),hatt(v(b1), . . . , v(bn))) with
- 93. 4 Bipolar abstract argumentation systems 79 the function hatt (resp. hsup): V∗ → Hatt (resp. V∗ → Hsup)11 valuing the quality of the attack (resp. support) on a, and the function g: Hsup ×Hatt → V with g(x,y) increasing on x and decreasing on y. The function h, h = hatt or hsup, must satisfy: 1. if xi ≥ x′ i then h(x1,...,xi ...,xn) ≥ h(x1,...,x′ i ...,xn), 2. h(x1,...,xn,xn+1) ≥ h(x1,...,xn), 3. h() = α ≤ h(x1,...,xn) ≤ β, for all x1,...,xn 12. Def. 14 produces a generic local gradual valuation. Let us give two instances of the generic definition, to illustrate the different principles. • A first instance is defined by Hatt = Hsup = V = [−1,1] interval of the real line, hatt(x1,...,xn) = hsup(x1,...,xn) = max(x1,...,xn), and g(x,y) = x−y 2 (so, we have α = −1, β = 1 and g(α,α) = 0). • Another one is defined by V = [−1,1] interval of the real line, Hatt = Hsup = [0,∞[ interval of the real line, hatt(x1,...,xn) = hsup(x1,...,xn) = Σn i=1 xi+1 2 , and g(x,y) = 1 1+y − 1 1+x (so, we have α = 0, β = ∞ and g(α,α) = 0). The following table shows the values computed with both instances on some simple examples: Example with 1st instance with 2nd instance No attack, no support: a v(a) = 0 v(a) = 0 Direct attack: a // b v(b) = −0.5 v(b) = −0.33 Direct support: a // /o /o /o b v(b) = 0.5 v(b) = 0.33 Defence: a // b // c v(c) = −0.25 v(c) = −0.25 Sequence of supports: a // /o /o /o b // /o /o /o c v(c) = 0.75 v(c) = 0.4 Supported attack: a // /o /o /o b // c v(c) = −0.75 v(c) = −0.4 Ex. 1 (cont’d) With the first (resp. second) instance, v(t1) = 1 4 (resp. 37 154 ). Ex. 5 (cont’d) With the first and the second instances, v(b) = 0. In this case, there is a perfect equilibrium13 between support and attack. A local gradual valuation defined as above satisfies the following properties [13]: • If Ratt − (a) = Rsup − (a) = ∅ then v(a) = g(α,α). • If Ratt − (a) = ∅ and Rsup − (a) = ∅ then v(a) = g(α,y) ≤ g(α,α) for y ≥ α. • If Ratt − (a) = ∅ and Rsup − (a) = ∅ then v(a) = g(x,α) ≥ g(α,α) for x ≥ α. 11 V∗ denotes the set of the finite sequences of elements of V, including the empty sequence. Hatt and Hsup are ordered sets. 12 So, α is the minimal value for an attack (resp. a support) – i.e. there is no attack (resp. no support) –, and β is the maximal value for an attack (resp. a support). 13 Note that it is not necessarily the case, and an appropriate choice of the function g enables to give more importance to the attack than to the support.
- 94. 80 Claudette Cayrol and Marie-Christine Lagasquie-Schiex And we have the following comparative scale14: Vmin ≤ g(α,y) ≤ g(α,α) ≤ g(x,α) ≤ Vmax (for y ≥ α) (for x ≥ α) Moreover the valuation proposed in Def. 14 satisfies the principles P1 to P3 (see [13] for a more detailed discussion). 6 Related issues and conclusion In this chapter, an extension of [21]’s abstract argumentation framework has been proposed in order to take into account two kinds of interaction between arguments modelled with a support relation and an attack relation. In this abstract BAF, two issues have been considered: • taking into account bipolarity for defining acceptability semantics: either by en- forcing the coherence of the admissible sets, or by turning a BAF into a meta- argumentation framework using the concept of coalition; • taking into account bipolar interactions for proposing gradual labellings for the arguments. 6.1 Related issues about acceptability and bipolarity Deflog [35]: DEFLOG argumentation system enables to express a support or an at- tack between sentences in the language, with a new sentence using specific connec- tors (one for each kind of interaction). Examples of sentences (with → for the attack relation and ❀ for the support relation) are: a, b, (a ❀ b), (a → b), (c ❀ (a ❀ b)), (d → (a ❀ b)). In DEFLOG, the notions of sequence of supports and of supported attacks can be retrieved but at the language level (between sentences). Moreover, the notion of conflict-free set proposed in DEFLOG corresponds to the notion of safe set (no sentence which is, at the same time, supported and attacked by the set). DEFLOG enables to define the dialectical interpretations (or extensions) of a given set of sentences S: an extension is built from a partition (J,D) of S such that J is conflict-free and attacks the sentences of D. Note that the attack relation and the support relation are explicitly expressed in the sentences. So, one can have an extension of a set S s.t. some supported sentences by J do not belong to S. DEFLOG extensions correspond to [21]’s stable extensions for DEFLOG theories that do not go beyond the expressiveness of Dung’s argu- mentation frameworks, and note that a Dung’s AF can always be expressed in DE- FLOG. So in this precise sense, DEFLOG’s extensions are a faithful generalization of Dung’s stable extensions, allowing more expressiveness. Moreover, [35] gives also a faithful generalization of Dung’s preferred extensions. 14 Using this scale, the values ≤ (resp. ≥) to g(α,α) are considered as negative (resp. positive) ones even if g(α,α) = 0.
- 95. 4 Bipolar abstract argumentation systems 81 Evidence-based argumentation [28]: In this work, the fundamental claim is that an argument cannot be accepted unless it is supported by evidence. So, special ar- guments are distinguished: the prima-facie arguments (which do not require any support to stand). Arguments may be acceptable only if they are supported (indirectly) by prima- facie arguments. This is evidential support. Moreover, only supported arguments may attack other arguments. Then, the notion of defence is rather complex: A set of arguments S defends an argument a if S provides evidential support for a and S invalidates each attack on a (either by a direct attack on the attacker of a or by rendering this attack unsup- ported). Following our definitions, a BAF is an abstract framework, where arguments may stand and attack with or without support. However, evidential reasoning as proposed by [28] could also be handled in a BAF in the following way: Given X a set of arguments (which are considered as prima-facie arguments in a given application), a notion of evidential support can be defined via a sequence of supports from an argument of X. Then, the notion of attack can be restricted so that attackers be elements of X, or receive evidential support from X. Finally, instead of choosing the classical definition for “S defends a” (as presented in Def. 7), it can be required first that S provides support for a and secondly that for each supported attack on a, one argument of the sequence of supports is directly attacked by S. 6.2 Related issues about coalitions of arguments Another way for defining acceptability semantics in a bipolar framework is to turn a bipolar argumentation framework into a meta-argumentation framework . This transformation has the following characteristics: the support relation is used in or- der to identify “coalitions” (sets of arguments which can be used together without conflict and which are related by the support relation) and the attack relation is used in order to identify conflicts between coalitions and then to define new acceptability semantics as in Dung’s framework. The concept of coalition has already been related to argumentation. Collective argumentation framework [9, 27]: A collective argumentation frame- work is an abstract framework where the initial data are a set of arguments and a binary “attack” relation between sets of arguments. The key idea is the follow- ing: a set of arguments can produce an attack against other arguments, which is not reducible to attacks between particular arguments. That is in agreement with our notion of coalition, since in our work, a coalition is considered as a whole and its members cannot be used separately in an attack process. The proposal by Nielsen and Parsons is similar to Bochman’s proposal. Both proposals take the attacks be- tween sets of arguments as initial data, and define semantics by properties on subsets of arguments. However, Nielsen and Parsons propose an abstract framework which allows sets of arguments to attack single arguments only, and they stick as close
- 96. 82 Claudette Cayrol and Marie-Christine Lagasquie-Schiex as possible to the semantics provided by Dung. In contrast, Bochman departs from Dung’s methodology and give new specific definitions for stable and admissible sets of arguments. Our proposal essentially differs from collective argumentation in two points. First, we keep exactly Dung’s construction for defining semantics, but we ap- ply this construction in a meta-argumentation framework (the coalition framework). The second main difference lies in the meaning of a coalition: we intend to gather as many arguments as possible in a coalition, and a coalition cannot be broken in the defence process. Generation of coalition structures in MAS [18, 2]: In multi-agent systems (MAS), the coalition formation is a process in which independent and autonomous agents come together to act as a collective. A coalition structure (CS) is a partition of the set of agents into coalitions. Each coalition has a value (the utility that the agents in the coalition can jointly get minus the cost which this coalition induces for each agent). So the value of a CS is obtained by aggregating the values of the different coalitions in the structure. One of the main problems is to generate a preferred CS, that is a structure which maximizes the global value. Recently, [2] has proposed an abstract system where the initial data are a set of coalitions equipped with a conflict relation. A preferred CS is a subset of coalitions which is conflict-free and defends itself against attacks. Coalitions may conflict for instance if they are non-disjoint or if they achieve a same task. However, the generation of the coalitions is not studied in [2]. So, one perspective is to apply our work to the formation of coalitions taking into account interactions between the agents. Arguments represent agents in that case. Indeed, it is very im- portant to put together agents which want to cooperate (“supports” relation) and to avoid gathering agents who do not want to cooperate (“attacks” relation). Then, the concept of Cp-extension provides a tool for selecting the best groups of agents (w.r.t. the given interaction relations). More generally, the work reported here is generic and takes place in abstract frameworks, since no assumption is made on the nature of the arguments. Argu- ments may have a logical structure such as a pair explanation, conclusion, may just be positions advanced in a discussion, or may be agents interacting in a multi-agent system. All that we need is the bipolar interaction graph describing how the argu- ments under consideration are interrelated. We think that this generic work should stimulate discussion across boundaries. 6.3 Related issues about valuation and bipolarity Most works about valuations of arguments take place in the basic framework. Some of them consider intrinsic valuations, which express to what extent an argument increases the confidence in the statement it promotes. Other approaches consider interaction-based valuations. These approaches usually differ in the set of values which are available.
- 97. 4 Bipolar abstract argumentation systems 83 However, very few works have been interested in valuations which handle both support and attack interactions. Most of these works have been developed for spe- cific applications. Medical applications: The most influential work has been proposed in HERMES system [23]. But there is no graduality (only two possible values with HERMES), and some parts of the interacting arguments are not taken into account for the com- putation of the value. See in Section 5. Valued maps of argumentations: The bipolar valuation in argumentation has been used for a collective annotation of documents. Collective annotation models supporting exchange through discussion threads. A discussion thread is initiated by an annotation about a given document. Then, users can reply with annotations which confirm or refute the previous ones. Annotations are associated with a social validation which provides a synthetic view of the dis- cussions. The purpose of this validation is to identify annotations which are globally confirmed by the discussion thread. It can also take into account an intrinsic value of the annotations. In [11, 12] a discussion thread is modelled by a BAF. The set of arguments con- tains the nodes of the thread. Pairs of the support (resp. attack) relation correspond to replies in the thread of the confirm (resp. refute) type. The social validation of a given annotation is computed with the local bipolar valuation. Moreover, the bipo- lar valuation procedure has been slightly modified in order to take into account an intrinsic value of each annotation. References 1. L. Amgoud. Contribution à l’intégration des préférences dans le raisonnement argumentatif. PhD thesis, Université Paul Sabatier, Toulouse, July 1999. 2. L. Amgoud. Towards a formal model for task allocation via coalition formation. In Proc. of AAMAS, pages 1185–1186, 2005. 3. L. Amgoud, J.-F. Bonnefon, and H. Prade. An argumentation-based approach to multiple criteria decision. In Proc. of ECSQARU, pages 269–280, 2005. 4. L. Amgoud, C. Cayrol, and M. Lagasquie-Schiex. On the bipolarity in argumentation frame- works. In Proc. of the 10th NMR-UF workshop, pages 1–9, 2004. 5. L. Amgoud, C. Cayrol, M.-C. Lagasquie-Schiex, and P. Livet. On bipolarity in argumentation frameworks. International Journal of Intelligent Systems, 23:1062–1093, 2008. 6. S. Benferhat, D. Dubois, S. Kaci, and H. Prade. Bipolar representation and fusion of prefer- ences in the possibilistic logic framework. In Proc. of KR, pages 158–169, 2002. 7. C. Berge. Graphs and Hypergraphs. North-Holland Mathematical Library, 1973. 8. P. Besnard and A. Hunter. A logic-based theory of deductive arguments. Artificial Intelligence, 128 (1-2):203–235, 2001. 9. A. Bochman. Collective argumentation and disjunctive programming. Journal of Logic and Computation, 13 (3):405–428, 2003. 10. C. Boutilier. Towards a logic for qualitative decision theory. In Proc. of KR, pages 75–86, 1994. 11. G. Cabanac, M. Chevalier, C. Chrisment, and C. Julien. A social validation of collaborative annotations on digital documents. In Proc. of IWAC, pages 31–40, 2005.
- 98. 84 Claudette Cayrol and Marie-Christine Lagasquie-Schiex 12. G. Cabanac, M. Chevalier, C. Chrisment, and C. Julien. Collective annotation: Perspectives for information retrieval improvement. In Proc. of RIAO, 2007. 13. C. Cayrol and M. Lagasquie-Schiex. Gradual valuation for bipolar argumentation frameworks. In Proc of the 8th ECSQARU, pages 366–377, 2005. 14. C. Cayrol and M. Lagasquie-Schiex. On the acceptability of arguments in bipolar argumenta- tion frameworks. In Proc of the 8th ECSQARU, pages 378–389, 2005. 15. C. Cayrol and M.-C. Lagasquie-Schiex. Gradual handling of contradiction in argumentation frameworks. In Intelligent Systems for Information Processing: From representation to Appli- cations, chapter Reasoning, pages 179–190. Elsevier, 2003. 16. C. Cayrol and M.-C. Lagasquie-Schiex. Graduality in argumentation. Journal of Artificial Intelligence Research, 23:245–297, 2005. 17. C. Cayrol and M.-C. Lagasquie-Schiex. Coalitions of arguments in bipolar argumentation frameworks. In Proc. of CMNA, pages 14–20, 2007. 18. V. Dang and N. Jennings. Generating coalition structures with finite bound from the optimal guarantees. In Proc. of AAMAS, pages 564–571, 2004. 19. D. Dubois and H. Fargier. On the qualitative comparison of sets of positive and negative affects. In Proc. of ECSQARU, pages 305–316, 2005. 20. D. Dubois and H. Prade. A bipolar possibilitic representation of knowledge and preferences and its applications. In Proc. of WILF (LNCS 3849), pages 1–10, 2006. 21. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995. 22. H. Jakobovits and D. Vermeir. Robust semantics for argumentation frameworks. Journal of logic and computation, 9(2):215–261, 1999. 23. N. Karacapilidis and D. Papadias. Computer supported argumentation and collaborative deci- sion making: the HERMES system. Information systems, 26(4):259–277, 2001. 24. P. Krause, S. Ambler, M. Elvang, and J. Fox. A logic of argumentation for reasoning under uncertainty. Computational Intelligence, 11 (1):113–131, 1995. 25. J. Lang, L. Van der Torre, and E. Weydert. Utilitarian desires. Journal of Autonomous Agents and Multi-Agents Systems, 5(3):329–363, 2002. 26. J. Nielsen. On the number of maximal independent sets in a graph. Technical Report RS 02-15, Center for Basic Research in Computer Science (BRICS), April 2002. 27. S. Nielsen and S. Parsons. A generalization of Dung’s abstract framework for argumentation. In Proc. of the 3rd WS on Argumentation in multi-agent systems, 2006. 28. N. Oren and T. J. Norman. Semantics for evidence-based argumentation. In Proc. of COMMA, pages 276–284, 2008. 29. S. Parsons. Normative argumentation and qualitative probability. In Proc. of ECSQARU, pages 466–480, 1997. 30. H. Prakken and G. Sartor. Argument-based extended logic programming with defeasible pri- orities. Journal of Applied Non-Classical Logics, 7:25–75, 1997. 31. S. W. Tan and J. Pearl. Specification and evaluation of preferences under uncertainty. In Proc. of KR, pages 530–539, 1994. 32. S. Toulmin. The Uses of Arguments. Cambridge University Press, Mass., 1958. 33. B. Verheij. Two Approaches to Dialectical Argumentation: Admissible Sets and Argumenta- tion Stages. In Proc. of Dutch Conference on Artificial Intelligence, 357–368, 1996. 34. B. Verheij. Automated argument assistance for lawyers. In Proc. of International Conference on Artificial Intelligence and Law, 43–52, 1999. 35. B. Verheij. Deflog: on the logical interpretation of prima facie justified assumptions. Journal of Logic in Computation, 13:319–346, 2003.
- 99. Chapter 5 Complexity of Abstract Argumentation Paul E. Dunne and Michael Wooldridge 1 Introduction The semantic models discussed in Chapter 2 provide an important element of the formal computational theory of abstract argumentation. Such models offer a variety of interpretations for “collection of acceptable arguments” but are unconcerned with issues relating to their implementation. In other words, the extension-based seman- tics described earlier distinguish different views of what it means for a set, S, of arguments to be acceptable, but do not consider the procedures by which such a set might be identified. This observation motivates the study of natural questions relating to the actual implementation of different semantics, e.g., using semantics s what can be stated regarding methods that: decide if S ∈ Es(A,R) for S ⊆ A; or determine if x ∈ S for at least one (alternatively every) S ∈ Es(A,R), etc? Such questions raise two separate issues: that of algorithms by which upper bounds can be obtained; and that of mechanisms by which lower bounds can be established. Some discussion of the former will be given in Chapter 6; the field of Computational Complexity Theory provides a number of approaches by which the latter issue can be addressed: these methods and their application within abstract argument systems are the subject of the current chapter. In the next section we give an overview of some basic notions in complexity theory and continue with a review of some fundamental results on complexity in abstract argument systems in Section 3. In Section 4 we consider analogous results within deductive frameworks, assumption-based frameworks and a number of complexity-theoretic properties of value-based frameworks. Section 5 Paul E. Dunne Dept. of Computer Science, University of Liverpool Liverpool UK, e-mail: ped@csc.liv.ac. uk Michael Wooldridge Dept. of Computer Science, University of Liverpool Liverpool UK, e-mail: mjw@csc.liv.ac. uk I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 85 DOI 10.1007/978-0-387-98197-0 5, c Springer Science+Business Media, LLC 2009
- 100. 86 Paul E. Dunne and Michael Wooldridge summarises some recent developments concerning novel semantics. Conclusions and selected open questions are discussed in the final section. 2 Elements of Computational Complexity Theory In crude terms, computational complexity theory deals with classifying computa- tional problems with respect to the resources needed for their solution, e.g., the time required by the fastest program that will solve the problem. In this section we intro- duce some of the basic concepts in the field of computational complexity. 2.1 Languages and Decision Problems We think of “computational problems” in terms of recognising objects, (e.g., propo- sitional formulae, argumentation frameworks, etc.), which have some property of interest, (e.g., instantiations that make the formula true (⊤), non-empty subsets of arguments that define preferred extensions). For such problems one has a set of problem instances, and the goal is to decide whether or not a given instance should be accepted, i.e., has the property of interest. With this approach, decision prob- lems are defined by describing the form taken by instances and the question asked of these instances, i.e., the property we want to check. The subset of instances for which positive answers are given is often referred to as a language. For example, the decision problem (language) 3-CNF Satisfiability (3-SAT) has, Instance: Propositional formula, ϕ(x1,x2,...,xn) over the variables {x1,...,xn} in conjunctive normal form with at most three literals in each clause, i.e., ϕ is specified by a set of m clauses, {C1,C2,...,Cm}, with Cj = yj,1 ∨ yj,2 ∨ yj,3 where yj,k is a literal from {x1,...,xn,¬x1,...,¬xn}, so that ϕ = ∧m j=1 Cj. Question: Is there an instantiation, α = a1,a2,...,an ∈ ⊥,⊤n for which setting xi := ai results in every clause, Cj having at least one literal given the value ⊤? Notice that this approach allows us to distinguish ideas of problem size. Although this could be captured in terms of the number of bits used to encode an instance, there is often some natural parameter that can be used as an alternative, e.g., the size of a 3-CNF formula is usually measured as the number of propositional variables in its definition (n). In general we use |x| to denote the size of a problem instance x. 2.2 Complexity Classes – P, NP, coNP and PH The concept of complexity class is used to describe problems whose resource re- quirements are similar. Given a language, L, the problem of deciding whether x ∈ L
- 101. 5 Complexity of Abstract Argumentation 87 is viewed as having an efficient algorithm if there is a constant value, k, and program M, for which • if x ∈ L, then M returns “accept” else M returns “reject”. • M returns its answer after at most |x|k steps. The program M is said to provide an algorithm for L with run-time nk, so leading to the complexity class, P, (of polynomial time decidable languages) as P = ∞ k=0 { L : There is an algorithm with run-time nk deciding x ∈ L.} Note that we generally regard problems as being computationally easy or tractable if they are polynomial time decidable, although of course, if k is very large, polyno- mial time decidability may not in fact imply the existence of a practicable algorithm to solve the problem. It is often the case that the question x ∈?L can be phrased in terms of identi- fying some auxiliary structure (or witness) that x is indeed a member of L, e.g., ϕ ∈ 3− SAT, is witnessed by any instantiation, α, for which ϕ(α) = ⊤. In general, associated with x one may have a set of possible witnesses, W(x) to x ∈ L, any such witness having size at most |x|r, for some constant r. Suppose that LW is the lan- guage of pairs x,y defined by instances, x of L and witnesses y ∈ W(x) to x ∈ L. Concentrating on languages, LW in P, we get the complexity classes NP and coNP – L ∈ NP if (x ∈ L) ⇔ ∃ y ∈ W(x) : x,y ∈ LW L ∈ coNP if (x ∈ L) ⇔ ∀ y ∈ W(x) : x,y ∈ LW So, for the complementary problem to 3-SAT, called 3-UNSAT, ϕ ∈ 3 − UNSAT if and only if ϕ has no satisfying instantiation: 3− SAT∈NP while 3− UNSAT∈coNP. Looking at the requirements for L ∈ P, L ∈ NP, L ∈ coNP, we note the follow- ing pattern: polynomial time decidable languages are characterised by unary pred- icates – PL – over instances of L, i.e., tests x ∈ L equate to evaluating the predi- cate PL(x) ≡ (x ∈ L); languages in NP and coNP are characterised by polynomial time decidable binary predicates – PL(x,y) over instances and possible witnesses so that NP languages are those expressible as ∃ y PL(x,y) and coNP expressible as ∀ y PL(x,y). This view naturally suggests extending to (k +1)-ary polynomial time decidable predicates PL(x,y1,y2,...,yk) and the languages characterised as (x ∈ L) ⇔ Qk yk Qk−1 yk−1 ··· Q2 y2 Q1 y1 PL(x,y1,y2,...,yk) where Qi ∈ {∃,∀}, Qi = Qi+1. When Qk = ∃ (respectively ∀) the corresponding class is denoted by Σ p k (respectively, Π p k ). The collection ∪∞ k=0 Σ p k (= ∪∞ k=0 Π p k ) is called the Polynomial Hierarchy (PH) . As examples of languages in Σ p k and Π p k we have the so-called quantified satisfia- bility problems – QSATΣ k and QSATΠ k – whose instances are 3-CNF formulae defined on k disjoint sets of n propositional variables – X1,X2,...,Xk – so that
- 102. 88 Paul E. Dunne and Michael Wooldridge ϕ ∈ QSATΣ k ⇔ ∃α1∀α2 ···∃αk−1∀αk ϕ(α1,α2,...,αk) = ⊥ (k even) ∃α1∀α2 ···∀αk−1∃αk ϕ(α1,α2,...,αk) = ⊤ (k odd) ϕ ∈ QSATΠ k ⇔ ∀α1∃α2 ···∀αk−1∃αk ϕ(α1,α2,...,αk) = ⊤ (k even) ∀α1∃α2 ···∃αk−1∀αk ϕ(α1,α2,...,αk) = ⊥ (k odd) It is immediate from the formal definitions of P, NP, coNP and PH that the corre- sponding sets (of languages) satisfy P ⊆ NP = Σ p 1 coNP = Π p 1 ⊆ ··· ⊆ Σ p k Π p k ⊆ Σ p k+1 Π p k+1 ⊆ ··· It is conjectured that all of these containments are strict and that Σ p k = Π p k for any k ≥ 1: these generalize the well-known P = NP conjecture and, to date, are unproven. 2.3 Hardness, Completeness, and Reducibility The forms presented in Section 2.2 allow problems to be grouped together via upper bounds: expressing membership in L in terms of some polynomial time decidable finite arity predicate Pk+1 places L (at worst) in one of Σ p k or Π p k . The relationship of polynomial time many one reducibility between languages is one key technique underlying arguments that such upper bounds are “optimal”. Suppose, given some complexity class, C, we can show that L has the following property: For every L′ ∈ C there is a polynomial time procedure, τ, that transforms instances x of L′ to instances τ(x) of L in such a way that x ∈ L′ if and only if τ(x) ∈ L. We write L′ ≤p m L to describe this relationship. What may be said of L in such cases? Certainly, were L ∈ P then we could deduce C ⊆ P: given an instance x of L′ ∈ C, construct the instance τ(x) of L (polynomial time) and then use the polynomial time method to decide τ(x) ∈ L. We can thereby deduce that the complexity of L is at least as high as the complexity of any language in C. A language L for which every L′ ∈ C has L′ ≤p m L is said to be C–hard . If, in addition, L ∈ C then L is called C–complete . Just as P is considered as encapsulating all efficiently decidable languages, so the classes of NP–hard, coNP–hard, Σ p k –hard and Π p k –hard languages are viewed as progressively more and more demanding in terms of their time requirements. Noting the long-standing conjecture about the relationship between these classes a proof that L ∈ P is a positive statement that L is tractable; a proof that L is C-hard for some C ∈ PH (other than P) provides a strong indication that L ∈ P, i.e., that L is intractable.1 While the condition L′ ≤p m L for every L′ ∈ C may seem somewhat demanding, noting that the relation ≤p m is transitive we can replace “∀ L′ ∈ C L′ ≤p m L” by “For 1 It should be noted that C ∈ PH is, almost without exception a class such as Σ p k or Π p k for some fixed k 0. There are a number of technical consequences which suggest it is extremely unlikely the class PH itself has complete languages, i.e., L such that ∀L′ ∈ PH L′ ≤p m L.
- 103. 5 Complexity of Abstract Argumentation 89 some C-hard, L′ : L′ ≤p m L”. For the classes introduced in Section 2.2 we have the following results of Cook [10] and Wrathall [45]. Theorem 5.1. a. 3-SAT is NP–complete; 3-UNSAT is coNP–complete. b. QSATΣ k is Σ p k –complete; QSATΠ k is Π p k –complete. For the results discussed later in this Chapter with very few exceptions the complex- ity classifications use reductions from 3-SAT or 3-UNSAT. 2.4 More Advanced Ideas The topics outlined above provide sufficient background for the majority of com- plexity analyses on decision problems for AFs. There are, however, a number of developments – in particular in the related frameworks discussed in Section 4.1 – which occur in more recent work on complexity of abstract argumentation: here we briefly introduce the basic notions of oracle-based complexity classes and the models proposed in work of Cook and Reckhow [11] in order to capture relative complexity of proof systems. Oracle computations, are defined in terms of the availability of a device (or ora- cle) that at the cost of a single step in the algorithm provides the answer to a given language membership query, e.g., oracle computations using 3-SAT may construct a 3-CNF ϕ, query the 3-SAT oracle as to whether ϕ ∈ 3 − SAT with the answer determining subsequent steps taken. This notion, when coupled with oracles for NP-complete languages, gives rise to a range of complexity classes differentiated by the particular restrictions placed on the manner in which such calls are made. One important representative of such classes is the so-called difference class, Dp formally defined as those languages, L whose members are the intersection of a language L1 ∈ NP with a language L2 ∈ coNP: a language L ∈ Dp can thus be de- cided by a polynomial time algorithm that is allowed to make at most two calls on an NP oracle (which, by virtue of Thm 5.1(a) can be assumed to be 3-SAT): given an instance x of L, test x ∈ L1 by forming the appropriate 3-SAT instance, τ1(x), and calling the oracle; if the response is positive, then x ∈ L2 is tested in the same way, forming the instance τ2(x) using a second oracle call to verify τ2(x) ∈ 3−SAT, i.e. τ2(x) ∈ 3−UNSAT. More generally, the class of languages for which polynomial time algorithms using at most f(|x|) calls on some oracle for a language complete in a class C is denoted by PC[ f(|x|)] (so that Dp ⊆ PNP[2]). Where no restriction is placed on the oracle invocation the notation PC is used. As will be seen from the results reviewed in Sect. 4.1, the algorithmic “base class” can be defined within arbitrary complexity classes, not simply P: this leads to classes such as NPC , etc. The formalism from [11] has been adopted in order to relate the efficiency of proof procedures for credulous reasoning in AFs to more widely known proof pro- cedures in propositional logic, e.g., resolution, tableau-based, sequents, etc. The model starts from an abstraction of “proof system” Π for a (coNP) language, L as
- 104. 90 Paul E. Dunne and Michael Wooldridge a procedure which given x ∈ L admits a formal derivation of this fact: the relative efficiency of two processes Π1 and Π2 being viewed as the number of derivation steps each requires.2 This approach precisely formalises two systems as equivalent whenever derivations in one can be simulated by polynomially longer derivations in the other. 3 Fundamental Complexity Results in Argument Frameworks Faced with a particular semantics and framework there are a number of questions which one may wish to decide: whether a given collection of arguments satisfies the conditions specified by the semantics; whether a particular argument belongs to at least one or every such set; whether there is any (non-empty) collectively ac- ceptable subset, etc.. We shall refer to these subsequently as Verification (VERs) ; Credulous Acceptance (CAs) ; Sceptical Acceptance (SAs) ; Existence (EXs); and Non-emptiness (NEs). Table 5.1 presents the formal definitions. Table 5.1 Decision Problems in AFs Problem Instance Question VERs G(A,R); S ⊆ A Is S ∈ Es(G)? CAs G(A,R); x ∈ A Is there any S ∈ Es(G) for which x ∈ S? SAs G(A,R); x ∈ A Is x a member of every T ∈ Es(G)? EXs G(A,R) Is Es(G) non-empty? NEs G(A,R) Is there any S ∈ Es(G) for which S = / 0? Before discussing the intractability results that form the main concern of this chapter, we briefly review the cases for which efficient methods are known. Theorem 5.2. a. For s = GR (the grounded semantics), all of the decision problems in Table 5.1 are in P. Furthermore the unique subset S for which S ∈ EGR(A,R) can be constructed in polynomial time. b. Given A,R and S ⊆ A deciding if S is conflict-free, admissible, or stable, i.e., the decision problem VERST (A,R,S), are all in P. 2 In carrying out such comparisons it is presumed that basic derivations in each system are compa- rable, e.g., can be implemented in polynomial time.
- 105. 5 Complexity of Abstract Argumentation 91 3.1 Intractability Results in Preferred and Stable Semantics We now turn to the problems defined in Table 5.1 with respect to the other extension based semantics - Preferred and Stable - introduced in Dung [22]. Our main aim is to outline the constructions of Dimopoulos and Torres [21] and Dunne and Bench- Capon [28] from which the classifications shown in Table 5.2 result. Table 5.2 Complexity of Decision Problems in Preferred (PR) and Stable Semantics (ST) A VERPR coNP–complete [21] B CAPR NP–complete [21] C SAPR Π p 2 –complete [28] D NEPR NP–complete [21] E CAST NP–complete [21] F SAST coNP–complete/Dp–complete See discussion below. G EXST NP–complete attributed to Chvatal in [33]; also [16, 21, 32]. All of the lower bound results in Table 5.2 are obtained as variations on what we shall refer to as the standard translation from 3-CNF formulae to AFs. Definition 5.1. Given ϕ(z1,...,zn) a 3-CNF with clauses {C1,...,Cm} the AF, Gϕ(Aϕ,Rϕ) constituting the standard translation from ϕ has Aϕ = {ϕ}∪{C1,...,Cm}∪{z1,...,zn}∪{¬z1,...,¬zn} Rϕ = {Cj,ϕ : 1 ≤ j ≤ m} ∪ {zi,¬zi,¬zi,zi : 1 ≤ i ≤ n} ∪ {yi,Cj : yi is a literal (i.e., zi or ¬zi) of the clause Cj} The AF described in Defn 5.1 is, modulo some minor simplifications, identical to that originally used in [21]. This framework provides an extremely versatile mecha- nism that underpins almost all the complexity analyses of extension based semantics in abstract argumentation frameworks.3 The basic form of the standard translation suffices to establish (B) and (E) of Table 5.2, whereas (A) and (D) follow from quite simple modifications to it. For example consider the claim in Table 5.2(B) that CAPR is NP–complete. First note that CAPR ∈ NP since we may use the set of all admissible subsets of A con- taining x as witnesses to G,x ∈ CAPR. We may then use the standard translation to prove 3-SAT ≤p m CAPR: given ϕ form the instance Gϕ,ϕ of CAPR. If ϕ ∈ 3-SAT then the literals instantiated to ⊤ by a satisfying assignment indicate a subset S of the arguments {z1,...,zn,¬z1,...,¬zn} for which S ∪ {ϕ} is admissible. On the other hand if Gϕ,ϕ ∈ CAPR then an admissible set containing ϕ must include a conflict- free subset of literals arguments that collectively attack all of the clause arguments: instantiating the corresponding literals to ⊤ produces a satisfying assignment of ϕ. 3 Exceptions are a specialized case of CAPR and a select number of reductions dealing with value- based argumentation frameworks, cf. [25, Thm. 8(a), Thms. 23–25], and later in this chapter.
- 106. 92 Paul E. Dunne and Michael Wooldridge Adapting the standard translation by adding a new argument, ψ that is attacked by ϕ and attacks all of the literal arguments yields an AF, Hϕ for which (both) Hϕ ∈ NEPR and Hϕ ∈ EXST hold if and only if ϕ ∈ 3-SAT. Table 5.2(A) is an immediate consequence of the former property using the special case of verifying if the empty set is a preferred extension. The discussion above accounts for all the cases given in Table 5.2 with the ex- ceptions of SAPR and SAST . We first address the apparent ambiguity in the clas- sification of SAST – Table 5.2(F). A further modification to the framework Hϕ by which the new argument ψ now attacks every argument in Gϕ, gives an AF in which ψ belongs to every stable extension if and only if ϕ ∈ 3-UNSAT so giving a coNP-hardness lower bound.4 In principle, one appears to have a coNP method via “A,R,x ∈ SAST ⇔ ∀ S ⊆ A (VERST (A,R,S) ⇒ (x ∈ S)”. There is, however, a possible objection: G,x ∈ SAST even when G has no stable exten- sion whatsoever, i.e., G ∈ EXST .5 In order to deal with this objection one might require as a precondition of G,x ∈ SAST that G ∈ EXST leading to an easy Dp upper bound: positive instances are characterised as those in CAST ∩{A,R,x : ∀ S ⊆ A (VERST (A,R,S) ⇒ (x ∈ S)} The matching Dp–hardness lower bound provides another illustration of the flex- ibility of the standard translation: instances ϕ1,ϕ2 of the canonical Dp–hard prob- lem 3-SAT,3-UNSAT being transformed to an instance K,ψ2 of SAST . The con- struction is illustrated in Fig. 5.1. We leave the reader to verify that this framework satisfies both EXST and has ψ2 a member of every stable extension if and only if ϕ1 ∈ 3-SAT and ϕ2 ∈ 3-UNSAT.6 α Φ2 Φ1 Ψ2 attacks all arguments Ψ1 Ψ2 Φ1 H Φ2 H Ψ1 attacks all arguments Fig. 5.1 The reduction from ϕ1,ϕ2 ∈ 3-SAT,3-UNSAT to K,ψ2 ∈ SAST 4 In [28] the coNP-hardness result is attributed to Dimopolous and Torres who do not explicitly consider this problem: the commentary of [28, p. 189] observes the lower bound follows from an easy modification to a construction in [21]. 5 Resulting in AFs for which A,R,x ∈ SAST and A,R,x ∈ CAST for every x ∈ A, i.e. every argument is sceptically accepted but none credulously so. 6 The construction in Fig. 5.1 has not previously appeared in the literature. The issues concerning precise formulations of SAST appear first to have been raised in [31].
- 107. 5 Complexity of Abstract Argumentation 93 Although the remaining case in Table 5.2 – Π p 2 -completeness of SAPR deals with a, notionally, harder class of languages, again the lower bound follows by adapting the standard translation: in this case to instances ϕ(y1,...,yn,z1,...,zn) of QSATΠ 2 .7 It is worth noting that the decision problem actually considered via this reduction, in [28], is the so-called coherence property of AFs, i.e., whether G is such that EPR(G) = EST (G): this problem is shown to be Π p 2 -complete with the classification of SAPR an immediate consequence of the reduction used. 3.2 Dialogue and Relationships to Proof Complexity The standard translation from 3-CNF (which easily generalises to arbitrary CNF) gives rise to one concrete interpretation of argumentation process in terms of log- ical proof: in particular, the concept of dialogue based procedures by which two parties attempt to reach agreement on the (credulous) acceptability status of some argument, will be discussed in Chapter 6. If one considers applying such procedures to determining the status of ϕ (in the standard translation) then a demonstration that ϕ is not admissible corresponds to a formal logical proof that the propositional formula ¬ϕ is a tautology. There are, of course, a number of widely used and well- studied proof mechanisms for propositional logic, the question of interest in terms of complexity in argumentation, is what one can state about the efficiency of dia- logue based argumentation processes: that is to say, using the comparative schema proposed in [11], how does the use of dialogue approaches compare to other tech- niques? This question has been examined with respect to one particular credulous reasoning process: the two-party immediate response protocol (TPI) introduced in [44]. Informally, TPI-dialogues involve two protagonists (PRO and CON) debating the acceptability of a given argument x: PRO claiming x to be acceptable and CON adopting the opposite stance. The dialogue is set in the context of an AF where each player takes turns advancing arguments in A: PRO starts by putting forward x. A re- quirement of the game is that, whenever possible to do so, a player must put forward an argument that attacks the most recent argument put forward by their opponent: where this is not possible the player must backtrack to a (specified) earlier point in the discussion or concede. In [29] the number of moves required in this game when played on the standard translation of an unsatisfiable CNF is considered. The TPI procedure turns out to be equivalent to a standard propositional proof theory – the so-called CUT-free Gentzen calculus [34] and, as a consequence of [42], there are TPI-disputes requiring exponentially many moves in order to resolve the status of particular arguments. 7 The reduction originally presented in [28] is not restricted to 3-CNF formulae but describes a general translation from arbitrary propositional formulae over the logical basis {∧,∨,¬}.
- 108. 94 Paul E. Dunne and Michael Wooldridge 4 Complexity in Related Abstract Frameworks As described in several chapters there are a number of abstract treatments of argu- mentation that build on the basic structures and semantics proposed in Dung [22]. Among such are assumption-based frameworks (ABFs) discussed in Chapter 10; the closely related deductive systems considered in Chapter 7; and the value-based ar- gumentation frameworks (VAFs) whose elements have been presented in Chapter 3. Our aim in this section is to review the range of complexity-theoretic results that have been proved within these models. In general we will not give detailed defini- tions of relevant ideas and refer the reader to the appropriate chapter for these. 4.1 Complexity in Assumption-based Argumentation Assumption-based frameworks [5], can be interpreted as specific concrete interpre- tations of abstract argumentation frameworks, i.e. as mechanisms for constructing the structure A,R by generating arguments in A and attacks between these. This approach starts from some deductive system – (L,R) in which L is a formal lan- guage, e.g., well-formed propositional sentences, and R a set of inference rules of the form α ← {α1,...,αn} describing the conclusions (α ∈ L) that are supported by the premises {α1,...,αn} ⊆ L. Such systems have an associated derivability re- lation, ⊢ : 2L → L; Δ ⊢ α holds whenever α may be obtained (via R) from Δ. It should be noted that attention is restricted to theories in which the underlying derivability relation is monotonic, i.e., (Δ ⊢ α) ⇒ (Δ′ ⊢ α) for any Δ′ ⊇ Δ. The key elements added are assumption sets, A ⊆ L and the contrary function, – which is a (total) mapping from α ∈ A to its contrary α ∈ L. In very simplified terms, (sets of) assumptions define the basis for atomic arguments (in AFs), and the con- trary mapping provides the reasons underpinning attacks between arguments. Just as the semantics of “collection of acceptable arguments” in AFs is given by different notions of extension, so too in ABFs the objects of interest are subsets of assump- tions defining extensions. In total, viewing an argument as “a statement derivable from some set of assumptions”, leads to the notion of attack between arguments as (the argument) Δ ⊢ α attacks (the argument) Δ′ ⊢ β if Δ ⊢ γ for an assumption γ ∈ Δ′. In this way we can take any basic semantics w.r.t. AFs, and define analogues w.r.t. ABFs, e.g., a set of assumptions, Δ, is conflict-free if for every α ∈ Δ it is not the case that Δ ⊢ α. Similarly, one may formulate each of the decision problems of Table 5.1 in ABF settings. There are, however, a number of important distinctions: as a result complexity-theoretic treatments of ABFs use significantly different techniques to those discussed in Section 3. In particular, a. In AFs both arguments, A, and the attack relation, R, are specified explicitly. In ABFs these are implicit and dependent on the underlying set of assumptions A and the precise deductive theory embodied within (L,R).
- 109. 5 Complexity of Abstract Argumentation 95 b. The deductive system (L,R) is not limited to classical propositional logic with the contrary being simply logical negation, e.g., (L,R) and – could be instantiated in terms of a number of non-monotonic logics such as the default logic of [39]. Since the attack relation is defined between assumption sets, in principal one may express any ABF (L,R),A,– as an AF, A,R: A = 2A, R = {Δ,Δ′ : Δ ⊢ γ for some γ ∈ Δ′}. There is, however, one complication: in practice not every subset of A is of interest, only those that satisfy the technical requirement of be- ing closed, i.e., Δ ⊆ A is closed if and only if α ∈ A Δ ⇒ ¬(Δ ⊢ α). While this provides a starting point for algorithms and upper bound constructions, for lower bounds such approaches yield little of benefit: |(L,R),A,–| is exponentially smaller than the corresponding AF. A detailed investigation of complexity-theoretic issues within ABFs has been pre- sented in a series of papers by Dimopolous, Nebel and Toni [18, 19, 20]. Using the notation LABF to distinguish ABF instantiations of decision problems L as presented in Table 5.1 and LABF,LT with reference to different formal theories LT = (L,R), a key element in exact complexity characterisations is the computational complexity of the derivability relation for the underlying logic, i.e., the derivability problem (DER) for the formal theory (L,R), has instances Δ,α – Δ ⊆ L, α ∈ L – accepted if and only if Δ ⊢ α. For example, in standard propositional logic the derivability problem is coNP–complete: Δ ⊢ α if and only if the formula α ∨ ∨ϕ∈Δ ¬ϕ is a tautology. Combining the notion of “oracle complexity classes” as described in Section 2.4 using oracles for DER(Δ,α) provides a generic approach to obtaining upper bounds on the complexity of decision problems within ABFs. For example, consider the decision problem VER ABF,LT ST of verifying that a given set of assumptions defines a stable extension within (L,R),A,–, where the derivability problem for LT is in some class C. In order to decide if Δ is accepted: 1. Check that Δ is closed. 2. Check that Δ is conflict-free, i.e., ∀ α ∈ Δ ¬(Δ ⊢ α ). 3. Check that Δ attacks every assumption α ∈ Δ, i.e., Δ ⊢ α for each α ∈ AΔ. All of these stages can be carried out using |A| calls to an oracle for DER: (1) tests (Δ,α) ∈ DER for α ∈ A Δ; (2) involves a further |Δ| calls; and (3) a final set of |AΔ| calls. In consequence, VER ABF,LT ST ∈ PC . Concentrating on upper bounds for Table 5.1 within the most general settings8 the upper bounds obtained in [20] are stated in Table 5.3. It may be noted that with the exception of upper bounds on stability related prob- lems, those relating to preferred and admissible sets of assumptions are rather higher than might be expected having allowed for the additional overhead associated with calls to the DER oracle, e.g., CAADM ∈ NP whereas CAABF ADM ∈ coNPNPC rather than 8 That is to say, no specific properties of the underyling frameworks are assumed, e.g., the property “flatness” described in [5].
- 110. 96 Paul E. Dunne and Michael Wooldridge Table 5.3 Upper bounds for main decision problems in ABFs Problem Semantics Instance (ABF) ABF bound (DER ∈ C) Instance (AF) AF bound VERs Admissible (L,R),A,–, Δ ⊆ A coNPC A,R, S ⊆ A P VERs Preferred (L,R),A,–, Δ ⊆ A coNPNPC A,R, S ⊆ A coNP VERs Stable (L,R),A,–, Δ ⊆ A PC A,R, S ⊆ A P CAs Admissible (L,R),A,–, ϕ ∈ L NPNPC A,R, x ∈ A NP CAs Preferred (L,R),A,–, ϕ ∈ L NPNPC A,R, x ∈ A NP CAs Stable (L,R),A,–, ϕ ∈ L NPC A,R, x ∈ A NP SAs Preferred (L,R),A,–, ϕ ∈ L coNPNPNPC A,R, x ∈ A Π p 2 SAs Stable (L,R),A,–, ϕ ∈ L NPC A,R, x ∈ A coNP/Dp NPC . That the reasoning problems exhibit “higher than expected” complexity in ABFs is not on account of the (additional) closure checking stage, despite the fact this does not feature in corresponding AF algorithms.9 The increased complexity arises from the nature of the attack relation, e.g. deciding G,x ∈ CAPR involves: guess S ⊆ A, confirm that x ∈ S and S is conflict-free; check S attacks each y that at- tacks S. Suppose, however, we consider the analogous version for CA ABF,LT PR : guess Δ ⊆ A; confirm that Δ is closed, Δ ⊢ ϕ, and Δ does not attack itself; finally check that any closed assumption set attacking Δ is itself attacked by Δ. This final stage requires tests involving Δ and all other sets of assumptions, rather than (as is effec- tively the case in AFs and suffices for stability) checking a property of Δ in relation to single assumptions. We conclude this overview of complexity in ABFs by noting that for the credu- lous and sceptical reasoning variants, the classifications of Table 5.3 turn out to be optimal for a wide range of instantiations of (L,R),A,– modelling non-classical logics such as DL [39], AEL [38], etc. The typical approach to lower bound proofs, e.g., as illustrated in the specific examples of DL and AEL, uses bounds on the com- plexity of DER: the cases DL and AEL being coNP–complete. While for DL, one can show CA ABF,DL PR ∈ NPNP = Σ p 2 , no reduction in the generic upper bound is possi- ble for AEL: CA ABF,AEL PR ∈ NPNPNP , i.e Σ p 3 . The Σ p 2 (resp. Σ p 3 ) hardness reductions use instances of QSATΣ 2 (resp. QSATΣ 3 ) to define ABFs instantiated as DL (resp. AEL) systems: detailed constructions may be found in [20]. 4.2 Complexity in Value-based Argumentation Frameworks We recall, from Chapter 3, that value-based argumentation frameworks (VAFs) aug- ment the basic A,R abstraction of Dung’s AFs by introducing a finite set of values, V, and a mapping η : A → V describing the abstract value, η(x) endorsed by x ∈ A, so a VAF is described via a four tuple, G(V) = A,R,V,η. In VAFs the underlying structures are the completely abstract frameworks of [22]: whereas ABFs provide 9 In fact this stage is redundant in a number of ABF models, e.g DL.
- 111. 5 Complexity of Abstract Argumentation 97 a basis for argument construction and attacks between arguments, the motivation behind VAFs is to offer an explanatory mechanism accounting for choices between distinct justifiable collections, S and T, which are not collectively acceptable, i.e., S and T may be admissible under Dung’s semantics, however, S ∪T fails to be. Such occurrences raise the question of the supporting reasons as to which of S or T is adopted: as developed in Chapter 3, VAFs rationalize these choices in terms of value orderings on V. Any commitment to a preference of vi ∈ V over vj ∈ V (written vi ≻ vj) induces a simplification of A,R,V,η whereby every attack x,y ∈ R for which η(x) = vj and η(y) = vi can be removed. Under the restrictions discussed in Chapter 3, applying this refinement of R, any total ordering, α of V, will result in an acyclic framework, G (V) α : as has been noted elsewhere such a framework will have EGR(G (V) α ) = EPR(G (V) α ) = EST (G (V) α ). Value orderings thus motivate the two principal decision problems that have been reviewed in algorithmic and complex- ity studies of VAFs: Subjective Acceptance (SBA) and Objective Acceptance (OBA) . Both take as an instance a VAF G(V) and argument x ∈ A. G(V),x ∈ SBA ⇔ ∃ α a total ordering of V : CAPR(G (V) α ,x) G(V),x ∈ OBA ⇔ ∀ α total orderings of V : CAPR(G (V) α ,x) The acyclic form of G (V) α gives CAPR(G (V) α ,x) ∈ P hence SBA∈NP and OBA∈coNP. Both bounds turn out to be exact, as shown in [30, 3]: these again use variants of the standard translation from Defn 5.1. This may appear surprising given that although the standard translation is well-suited to relating subsets (of arguments) to instantiations of propositional variables, it is less clear how it could be applied to deal with relating orderings of values to such instantiations. The device used in [30] associates a “neutral” value with the formula and clause arguments in the VAF defined from ϕ(Zn) and replaces the mutually attacking pairs {zi,¬zi} with a cycle of four arguments pi → qi → ri → si → pi. Two arguments (pi and ri) are assigned the value posi to promote “zi = ⊤ in a satisfying assignment of ϕ” while the others (qi and si) are given the value negi in order to promote “zi = ⊥ in a satisfying assignment of ϕ”. Although their definition and these classifications suggest that SBA (resp. OBA) are closely related to CAPR (resp. SAPR for coherent AFs) , recent work, discussed in Section 5 highlights several differences between the nature of decision problems in VAFs and, what appear to be analogous problems in AFs. 5 Recent Developments The computational complexity of the standard semantics (preferred, stable, grounded) in AFs settings is, in the most general case, now well understood: ex- act complexity bounds having been established for each of the canonical decision problems given in Table 5.1 with respect to these semantics. There has, however, continued to be extensive development of this aspect of the formal theory of ar-
- 112. 98 Paul E. Dunne and Michael Wooldridge gumentation, driven by a number of reasons. Among these – and forming the top- ics reviewed in this section – one has: the various proposals for novel extension- based semantics, some of which have been discussed in Chapter 2, e.g., Ideal se- mantics [23, 24], Semi-stable semantics [6, 7], Prudent semantics [12]. A second consideration concerns the extent to which intractability issues may be alleviated by constructing efficient algorithmic approaches applicable to AFs which are restricted in some way, e.g., by analogy with the known tractable case of acyclic topologies. 5.1 Novel extension-based semantics and their complexity In this section we outline recent treatments of complexity in three of the develop- ments of Dung’s standard AF semantics: prudent, ideal, and semi-stable semantics. We recall that the rationale underlying prudent semantics stems from the po- tential problematic side effects that might eventuate by regarding as collectively acceptable, arguments {x,y} for which x “indirectly attacks” y. An indirect attack by x on y is present in A,R if “there exists a finite sequence x0,...,x2n+1 such that (1) x = x0 and y = x2n+1 and (2) for each 0 ≤ i 2n, xi,xi+1 ∈ R” [22, p. 332]. It should be noted that this formulation, which we have quoted verbatim, presents some ambiguity, which is significant from complexity-theoretic and semantic per- spectives: it fails to distinguish “indirect attacks” in which no argument is repeated, i.e., “simple paths”; from those in which arguments but not attacks may be repeated; from those in which attacks may be repeated, e.g. the cases in Fig. 5.2. x3 x1x4 x0 x2x5 x6x7 x7 x3 x0 x1x4 x5 x2 x3 x2 x1 x0 (a) (b) (c) Fig. 5.2 Three possible forms of “indirect” attack – (a) Simple; (b) x1 = x4; (c) x1,x2 = x4,x5 The concept of conflict-free set from [22], is replaced under the prudent seman- tics by that of prudently conflict-free set, i.e., one in which there is no indirect attack between any two members. There are evident interpretative issues with cases (b) and
- 113. 5 Complexity of Abstract Argumentation 99 (c) in Fig. 5.2, however, the most natural interpretation (where an indirect attack is a simple path) has one significant computational drawback. Fact 1 Given A,R and S ⊆ A deciding if S is prudently conflict-free is coNP– complete even if S contains only two arguments. Proof. Immediate from the result of Lapaugh and Papadimitriou [35] which shows deciding the existence of a simple even length path between two specified arguments in a directed graph to be NP–complete. The extension to odd length simple path is trivial, so the lemma follows by observing that a prudently conflict-free set is one in which no simple odd length path is present between two arguments. Noting the definitions of admissible set, preferred and stable extensions from [22] and the fact that conflict-freeness is an integral part of these, the result of Fact 1 immediately allows us to deduce that under the prudent semantics (so that “conflict- free set” becomes “prudently conflict-free”) the respective verification problems are all coNP–complete. Complexity of credulous and sceptical acceptance under the prudent semantics has yet to be studied in depth. The intractability status of key decision problems in this semantics is predicated on the interpretation of “indirect attack” given by (a), i.e., as a simple path. These do not hold if repeated attacks – Fig 5.2(c) – are used: here polynomial time methods are available. The status of allowing repeated arguments – Fig 5.2(b) – is, to the authors’ knowledge still open. The ideal semantics were originally proposed with respect to ABFs, but have a natural formulation in AFs: S ⊆ A is an ideal set within A,R if S is both admis- sible and a subset of every set in EPR(A,R); S is an ideal extension if it a maximal ideal set. Detailed studies of the complexity of ideal semantics in AFs are presented in [26, 27]. The treatment of complexity issues presented in these papers exploit a number of more advanced techniques, however, the hardness proofs continue to be built on the standard translation of CNF formulae to AFs. In terms of the canoni- cal problems in Table 5.1, the verification problem (for ideal sets) is shown to be coNP-complete, placing this decision problem at the same level of complexity as the verification problem for preferred extensions. Arguably the most radical technique exploited – although widely applied in a number of earlier complexity-theoretic analyses – is the use of randomized reductions coupled with structural complexity results from [8, 9], as opposed to standard many-one reducibility (≤p m) which fea- tures in all of the results discussed earlier.10 Combining these elements, the verifica- tion problem (for ideal extensions) and credulous acceptance problems are shown to be complete for the (conjectured to be) subclass of PNP in which oracle queries are non-adaptive (denoted PNP || ). We note that the upper bounds from [26, 27] do not use randomized elements. This complexity class also arises in the known lower bounds for both credulous and sceptical acceptance in semi-stable semantics presented in [31]. These lower bounds again apply the structural characterizations of [8] (using ≤p m reducibility, i.e., not randomized). Upper bounds, however, are Σ p 2 and Π p 2 , i.e., exact classifications of reasoning problems in semi-stable semantics is open. 10 Relevant background is outlined in [27] and described in full in [26]. The actual “randomized” element is not explicit but arises from results of [43] for the satisfiability variant used.
- 114. 100 Paul E. Dunne and Michael Wooldridge 5.2 Properties of restricted frameworks The complexity lower bounds discussed above describe worst-case scenarios, i.e., the fact that, for example CAPR is NP–hard, does not imply that every algorithm on every instance will entail unrealistic computational overheads. As will be seen in Chapter 6, if the AF is acyclic, then all of the canonical decision problems of Table 5.1 have polynomial time solutions. In consequence a natural question to con- sider is whether other graph-theoretic restrictions also result in frameworks with efficient decision processes. Examining this question leads to two classes of results: positive outcomes of the form “decision problem L has a polynomial time algorithm in AFs satisfying some property P”; and negative classifications of the form “deci- sion problem L in frameworks satisfying property P are no easier than the general case”. Results of the first type extend (beyond acyclic frameworks) the range of AFs for which tractable solutions exist. Recent work has added to the class of such frameworks: symmetric AFs – those for which x,y ∈ R ⇔ y,x ∈ R – in work of Coste-Marquis et al. [13]; bipartite AFs (those for which A may be partitioned into two conflict-free sets) [25]. Using the notion of “treewidth decomposition”, see e.g., [4] a select number of problems whose instances are single AFs such as EXST , NEPR admit linear time algorithms given a treewidth decomposition of width k as part of the instance: the construction of these algorithms rely on a deep result (Cour- celle’s Theorem [14, 15]) demonstrating how efficient algorithms for testing graph- theoretic properties may be obtained given an appropriate logical description of the property (the so-called Monadic Second Order Logic) and a bounded treewidth de- composition of the graph. For more details on this approach and its application in AF settings we refer the reader to [2] and [25]. There are, however, a number of natural properties that fail to yield any reduction in complexity. Typically the approach adopted in proving such results is to demon- strate that frameworks with the property of interest are general enough to effectively “simulate” any framework, e.g., if S is admissible in A,R then S∪T is admissible in A ∪ B,R′ where the latter AF has a particular property. Using such methods, [25] shows that no reduction in complexity arises in: k-partite AFs (k ≥ 3); planar systems; and those in which no argument attacks or is attacked by more than two other arguments. This remains the case when all restrictions hold simultaneously. We conclude this overview of recent work by returning to the issue of complexity in VAFs. In contrast to the class of positive cases that have been identified with AFs, the situation with VAFs turns outs out to be far more negative. The most extreme indication of this status is the following result of [25]. Fact 2 a. SBA is NP–complete and OBA is coNP–complete even if the underlying graph is a binary tree and every v ∈ V is associated with at most three arguments. b. For every ε 0 SBA is NP–complete and OBA is coNP–complete even if the un- derlying graph is a binary tree and |V| ≤ |A|ε. Both constructions use reductions from variants of 3-SAT/3-UNSAT, however, these differ in a number of ways from the standard translation (which is clearly is not
- 115. 5 Complexity of Abstract Argumentation 101 a binary tree). The situation highlighted by results such as Fact 2 provides further indications that the nature of SBA/OBA in VAFs, while superficially similar to, is in fact radically different from that of CA/SA in AFs. 6 Conclusions and Further Research In this final section we outline some areas of research which offer a variety of chal- lenging directions through which the algorithmic and complexity foundations of abstract argumentation may be further advanced. We stress that our aim is to focus on general areas rather than particular open questions as such: the reader who has followed the earlier exposition will have noted that a number of specific open issues have already been raised in the text. 6.1 Average case properties As discussed in Section 5.2, the lower bounds on problem complexity are worst- case, so leaving open the possibility that feasible algorithms may be available in suitable contexts. In addition to the use of restrictions on the form of instances one other approach that has been widely considered in the theory of algorithms is the study of average-case complexity. Underpinning this approach one considers a prob- ability distribution, µ, on instances of a decision problem – often, but not invariably so, µ is the uniform distribution whereby each instance is equally likely, proceeding to define the average-case run time of an algorithm P on instances of size n of L as ∑x∈I(n) µ(x)ρ(P,x) where ρ(P,x) is the run-time of P on instance x. Formal defini- tions of average-case complexity classes may be found in [36]. To date surprisingly little work has been carried out concerning the application of average-case methods to decision problems in AFs either in terms of algorithmic development or in consid- ering the limitations of such approaches. It remains open to what extent techniques such as those applied to other intractable problems, e.g., [1] for the NP–complete Hamiltonian cycle problem, or [46] for CNF satisfiability could be replicated in AF settings. Of some relevance to such approaches are so-called “phase-transition” ef- fects, which received much attention in the mid-late 1990s as potential indicators of factors separating tractable and intractable classes of problem instances, e.g., the studies of random CNF-SAT from [37, 40]. Analytic studies of such effects appears to indicate connections between suitable witnessing structures, e.g., satisfying as- signment, being present “almost certainly” and the performance of algorithms to identify such structures. Of some interest in the context of AF semantics are the results of [41, 17] which give conditions ensuring that a random AF “almost cer- tainly” has a stable extension. There has as yet, however, been no detailed study of the implications of these results for fast on average methods for identifying or enumerating stable extensions. In the same way that the analyses of [41, 17] relate
- 116. 102 Paul E. Dunne and Michael Wooldridge to the existence of stable extensions in AFs, it would be of some interest to exam- ine to consider existence properties of other solution structures in random AFs and algorithmic consequences. 6.2 Approaches to dynamic updates An important feature of the argumentation forms discussed so far is that, in practice, these are not static systems: typically an AF, A,R, represents only a “snapshot” of the environment, and, as further facts, information and opinions emerge the form of the initial view may change significantly in order to accommodate these. For example, additional arguments may have to be considered so changing A; existing attacks may cease to apply and new attacks (arising from changes to A) come into force. It is clear that accounting for such dynamic aspects raises a number of issues in terms of assessing the acceptability status of individual arguments. As with the study of average-case properties, the treatment of algorithms and complexity issues relating to determining argument status in dynamically changing environments has been somewhat neglected. Thus, given A,R and S ⊆ A for which S ∈ Es(A,R) according to some semantics s, natural decision questions are: does x ∈ S continue to be credulously accepted (w.r.t. to semantics s) in the AF B,S where B results by removing some arguments from A and replacing these; similarly T modifies the attack relation R. Summary Complexity issues provide an important foundational element of the formal com- putational theory of abstract argumentation. Our review of the preceding pages is intended to give a flavour of the class of questions of interest and an appreciation of the techniques that have been brought to bear in addressing these. While some notable progress has been achieved since the appearance of [22] – particularly in understanding of decision properties of the standard semantics and the canonical problems of Table 5.1, nevertheless a significant number of areas and potential ana- lytic tools originating from complexity-theoretic studies, remain unexplored. References 1. D. Angluin and L. Valiant. Fast probabilistic algorithms for hamiltonian circuits and match- ings. Jnl. of Comp. and System Sci., 18:82–93, 1979. 2. S. Arnborg, J. Lagergren, and D. Seese. Easy problems for tree-decomposable graphs. Jnl. of Algorithms, 12:308–340, 1991.
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- 119. Chapter 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks Sanjay Modgil and Martin Caminada 1 Introduction Previous chapters have focussed on abstract argumentation frameworks and prop- erties of sets of arguments defined under various extension-based semantics. The main focus of this chapter is on more procedural, proof-theoretic and algorithmic aspects of argumentation. In particular, Chapter 2 describes properties of extensions of a Dung argumentation framework A,R under various semantics. In this context a number of questions naturally arise: 1. For a given semantics s, “global” questions concerning the existence and con- struction of extensions can be addressed: a. Does an extension exist? b. Give an extension (it does not matter which, just give one) c. Give all extensions. 2. For a given semantics s, “local” questions concerning the existence and construc- tion of extensions, relative to a set A ⊆ A can be addressed. Note that it is often the case that |A| = 1, in which case the member of A is called the query argument. a. Is A contained in an extension ? (Credulous membership question.) b. Is A contained in all extensions ? (Sceptical membership question.) c. Is A attacked by an extension? d. Is A attacked by all extensions? e. Give an extension containing A. Sanjay Modgil Department of Computer Science, King’s College London, e-mail: sanjay.modgil@kcl.ac. uk Martin Caminada Interdisciplinary Lab for Intelligent and Adaptive Systems, University of Luxembourg e-mail: martin.caminada@uni.lu I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 105 DOI 10.1007/978-0-387-98197-0 6, c Springer Science+Business Media, LLC 2009
- 120. 106 Sanjay Modgil and Martin Caminada f. Give all extensions containing A. g. Give an extension that attacks A. h. Give all extensions that attack A. In this chapter, procedures will be described for answering a selection of the above questions with respect to finite argumentation frameworks A,R (in which A is finite). Notice that for some semantics, such as the grounded and preferred semantics, extensions always exist, so that 1a will be answered in the positive for any framework. Also, for the grounded semantics, at most one extension exists, so that questions distinguished by reference to ‘an’ or ‘all’ extensions are equivalent (e.g., questions 2a and 2b). Sections 2 and 3 will introduce some key concepts underpinning the approaches that we will use in the description of proof theories and algorithms. Sections 4 - 6 will then focus on application of these approaches to the core semantics defined by Dung [13]; namely grounded, preferred and stable. Broadly speaking, two approaches will be presented. Firstly, Section 2 formally describes the argument graph labelling approach that was originally proposed by Pollock [23], and has more recently been the subject of renewed analysis and in- vestigation [5, 6, 25, 27]. The basic idea is that the status assignment to arguments defined by the extension-based approach (see Chapter 2), can be directly defined through assignment of labels to the arguments (nodes) in the framework’s corre- sponding argument graph. Section 2 provides formal underpinnings for the defini- tion of argument graph labelling algorithms that are used to address a selection of the above questions in Sections 4 - 6. Section 3 then describes a framework for argument game based proof theories [9, 16, 17, 28]. The inherently dialectical nature of argumentation lends itself to for- mulation of argument games in which a proponent starts with an initial argument to be tested, and then an opponent and the proponent successively attack each other’s arguments. The initial argument provably has a certain status if the proponent has a winning strategy whereby he can win irrespective of the moves made by the oppo- nent. In Sections 4 - 6 we describe specific games, emphasising the way in which the rules of each specific game correspond to the semantics they are meant to capture. 2 Labellings In this section the labelling approach (based on its formulation in [5, 6]) is briefly reviewed. Given an argumentation framework AF = A,R, a labelling assigns to each argument exactly one label, which can be either IN, OUT or UNDEC. The la- bel IN indicates that the argument is justified, OUT indicates that the argument is overruled, and UNDEC indicates that the status of the argument is undecided. Definition 6.1. Let A,R be an argumentation framework. • A labelling is a total function L : A → {IN,OUT,UNDEC}
- 121. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 107 • We define: in(L) = {x|L(x) = IN}; out(L) = {x|L(x) = OUT}; undec(L) = {x|L(x) = UNDEC} Notice that from hereon, we may represent a labelling L as a triple of the form (in(L), out(L), undec(L)). We now define what it is for an argument to be assigned a legal labelling: Definition 6.2. Let L be a labelling for A,R and x ∈ A • x is legally IN iff x is labelled IN and every y that attacks x (yRx) is labelled OUT • x is legally OUT iff x is labelled OUT and there is at least one y that attacks x and y is labelled IN • x is legally UNDEC iff x is labelled UNDEC, there is no y that attacks x such that y is labelled IN, and it is not the case that: for all y, y attacks x implies y is labelled OUT. The rules defining legal labelling assignments encode one’s intuitive understand- ing of the status assignments defined by the extension-based semantics and their use of the reinstatement principle, as described in Chapter 2. An argument x is IN only if all its attackers are OUT, and each attacker is OUT only if it is itself attacked by an argument that is IN. Thus, the arguments that are IN in a legal labelling correspond to a single extension. It is sometimes not possible to obtain a labelling where each argument is either legally IN or legally OUT; consider for example an argumenta- tion framework with just a single argument that attacks itself. This is why we need a third label UNDEC, which basically means that there is insufficient ground to explic- itly justify the argument and insufficient ground to explicitly overrule the argument. Notice that from Definition 6.2 it follows that x is legally UNDEC iff it is labelled UNDEC, and at least one y that attacks x is labelled UNDEC, and no y attacking x is labelled IN. Definition 6.3. For l ∈ {IN,OUT,UNDEC} an argument x is said to be illegally l iff x is labelled l, and it is not legally l. • An admissible labelling L is a labelling without arguments that are illegally IN and without arguments that are illegally OUT. • A complete labelling L is an admissible labelling without arguments that are illegally UNDEC Notice that the additional requirement on complete labellings corresponds intu- itively to Chapter 2’s characterisation of a complete extension as a fixed point of a framework AF’s characteristic function FAF . Since the grounded and preferred ex- tensions of a framework are the minimal, respectively maximal, fixed points (com- plete extensions) of a framework, then as one would expect, grounded and preferred labellings are given by complete labellings that minimise, respectively maximise, the arguments that are made legally IN. A stable labelling is a complete labelling in which all arguments are either legally IN or legally OUT, and hence no argument is UNDEC. Definition 6.4. Let L be a complete labelling. Then:
- 122. 108 Sanjay Modgil and Martin Caminada • L is a grounded labelling iff there there does not exist a complete labelling L′ such that in(L′) ⊂ in(L) 1 • L is a preferred labelling iff there there does not exist a complete labelling L′ such that in(L′) ⊃ in(L) • L is a stable labelling iff undec(L) = / 0 In [6], the following theorem is shown to hold: Theorem 6.1. Let AF = A,R be an argumentation framework, and E ⊆ A. For s ∈ {admissible, complete, grounded, preferred, stable}: E is an s extension of AF iff there exists an s labelling L with in(L) = E 2 In Sections 4 - 6 we will describe algorithms that compute labellings and so address a subset of the questions enumerated in Section 1. We conclude this section with an example: Example 6.1. Consider the framework in Figure 6.1. There exists three complete labellings: 1. (/ 0, / 0, {a,b,c,d,e}); 2. ({a}, {b}, {c,d,e}); and 3. ({b,d}, {a,c,e}, / 0). 1 is the grounded labelling, 2 and 3 are preferred, and 3 is also stable. a b c d e Fig. 6.1 An argumentation framework 3 Argument Games In general, proof theories license the way in which pieces of information can be articulated in order to prove a fact. They therefore provide a basis for algorithm de- velopment, and proofs constructed according to these theories provide explanations as to why a given fact is believed to be true. For example, a proof that argument x is in an admissible extension, would consist of showing how one can establish the exis- tence of such an extension, rather than simply identifying the extension. Intuitively, 1 Since every framework has a unique minimal fixed point, one could alternatively define L to be a grounded labelling iff for each complete labelling L′ it holds that in(L) ⊂ in(L′) 2 Note that for s = admissible there is a 1-1 mapping between s extensions and s labellings. An admissible extension may have more than one admissible labelling. For example, the admissible extension {c}, of c → b ,c → a, has two admissible labellings: ({c},{b},{a}) and ({c},{b,a}, / 0).
- 123. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 109 one would need to show how to defend x by showing that for every argument y that is put forward (moved) as an attacker of x, one must move an argument z that attacks y, and then subsequently show how any such z can be reinstated against attacks (in the same way that z reinstates x). The arguments moved can thus be organised into a graph of attacking arguments that constitutes an explanation as to why x is in an admissible extension. The process of moving arguments and counter-arguments can be implemented as an algorithm [27]. In this chapter we follow the approach of [9, 14, 16, 17, 26, 28] and present the moving of arguments as 2-person dialogue games that provide a natural way in which to lay out and understand the algorithms that implement them. To be sure, the actual algorithms themselves, should, except for didactic purposes, not be implemented as dialogue games, but rather as monological procedures (or methods in OO-languages) that are called recursively. A dialogue game is played by two players, PRO (for “proponent”) and OPP (for “opponent”), each of which are referred to as the other’s ‘counterpart’. A game begins with PRO moving an initial argument x that it wants to put to the test. OPP and PRO then take turns in moving arguments that attack their counterpart’s last move. From hereon: a sequence of moves in which each player moves against its counterpart’s last move is referred to as a dispute. If the last move in a dispute is by player Pl, and Pl’s counterpart cannot respond to this last move, then Pl is said to win the dispute. If a dispute with initial argument x is won by PRO, we call the dispute a line of defense for x. The rules of the game encode restrictions on the legality of moves in a dispute, and different sets of rules capture the different semantics under which justification of the initial argument x is to be shown, by effectively establishing when OPP or PRO run out of legal moves. In general, however, a player can backtrack to a counterpart’s previous move and initiate a new dispute. Consider the dispute aPRO −bOPP −cPRO − dOPP −ePRO − fOPP won by OPP (xPl −yPl′ denotes player Pl′ moving argument y against counterpart Pl’s argument x). PRO must then try and backtrack to move an argument against either bOPP or dOPP and establish an alternative line of defense for a. Suppose such a line of defense aPRO −bOPP −gPRO. Then OPP can backtrack and try an alternative line of attack moving h against a, so that PRO must now try and win the newly initiated dispute aPRO −hOPP. Thus, the ‘playing field’ of a game — the data structure on the basis of which argument games are played — can be represented by an argumentation framework’s induced dispute tree, in which every branch from root to leaf is a dispute: Definition 6.5. Let AF = A,R be an argumentation framework, and let a ∈ A. The dispute tree induced by a in AF is a tree T of arguments, such that T’s root node is a, and ∀x,y ∈ A: x is a child of y in T iff xRy. Figure 6.2i) shows an argumentation framework, and part of the tree induced by a is shown in Figure 6.2ii). Notice that multiple instances of arguments are indi- viduated by numerical indicies. Any game played by PRO and OPP in which PRO
- 124. 110 Sanjay Modgil and Martin Caminada c a b i) a1 b2 a3 b4 c5 a6 c7 a8 c9 b10 a11 PRO OPP PRO OPP PRO a1 b2 a3 c5 a6 c7 a8 b10 a11 ii) iii) Fig. 6.2 i) shows an argumentation framework, and ii) shows the dispute tree induced in a. iii) shows the dispute tree induced under the assumption that OPP cannot repeat moves in the same dispute (branch of the tree) attempts to show that a is justified, must necessarily involve the submission of at- tacking arguments conforming to some sub-tree of the induced tree in Figure 6.2ii). In particular, PRO must show that it fully fulfills its burden of proof, in response to OPP who fully fulfills its burden of attack. In other words, OPP moves all ys that attack an x moved by PRO, and each such y must in turn be responded to by PRO moving at least one x′ that attacks y. This does of course capture the reinstatement principle used to define the extensions of an argumentation framework, and corre- lates with Section 2’s definition of legal labellings and the extensions they define3. In the context of a game, this is captured by the notion of a winning strategy for an argument. Notice that in the following definition we refer to the notion of a sub- dispute d′ of a dispute d, which, intuitively is any sub-sequence of d that starts with the same initial argument as d. Definition 6.6. Let AF = A,R, T the dispute tree induced by a in AF, and T′ a sub-tree of T. Then T′ is a winning strategy for a iff: 1. The set DT′ of disputes in T′ is a non-empty finite set such that each dispute d ∈ DT′ is finite and is won by PRO (terminates in an argument moved by PRO) 2. ∀d ∈ DT′ , ∀d′ such that d′ is some sub-dispute of d and the last move in d′ is an argument x played by PRO, then for any y such that yRx, there is a d′′ ∈ DT′ such that d′ −yOPP is a sub-dispute of d′′. If PRO plays moves as described in a winning strategy sub-tree, then PRO is guaranteed to win. As stated earlier, the rules of a game encode restrictions on the arguments a player can legally move in a dispute in order to attack its counterpart’s argument. These restrictions vary according to the semantics of interest, and are encoded in a legal move function: 3 Recall that x is legally IN iff all ys that attack an x are legally OUT, and each such y is legally OUT iff there is at least one x′ attacking y that is legally IN
- 125. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 111 Definition 6.7. Let AF = A,R, T the dispute tree induced by a in AF. Let DT be the set of all disputes in T. Then φ is a legal move function such that φ : DT → 2A . Given a dispute tree T induced by a, the legal move function φ for a semantics s, prunes T to obtain the sub-tree T′ of T that we call the φ tree induced by a. T′ is the playing field of the game for semantics s. Thus, we define a φ-winning strategy for a [9, 17] as a sub-tree of the φ dispute tree induced by a, in the same way as Definition 6.6, except that we replace ‘for any x such that xRy’ in condition 2, with ‘for any x that OPP can φ legally move against y’. Intuitively, φ is defined such that a is in an admissible extension that conforms to the semantics s iff there is a φ-winning strategy for a in the φ tree induced by a, where the arguments moved by PRO in the φ-winning strategy are conflict free (recall that an admissible extension must contain no arguments that attack each other). For example, consider games whose legal move function φ prohibits OPP from repeating arguments in the same dispute. Figure 6.2iii) shows the φ-dispute tree that is a sub-tree of the dispute tree induced by a (Figure 6.2ii)). After PRO plays a6, OPP cannot backtrack and extend the dispute d = a1 −b2 −a3 by moving b against a3, since b has already been moved by OPP in d. Similarly, OPP cannot backtrack to move c against a8 in order to extend d′ = a1 −c7 −a8. Note also that both DT1 = {d1 = a1 −b2 −a3 −c5 −a6} and DT2 = {d2 = a1 −c7 −a8 −b10 −a11} are winning strategies. In the former case, consider the sub-dispute d′ 1 = a1 −b2 −a3 of d1. OPP can legally move c against a3, but there is a dispute in DT1 that extends d′ 1 (d1 itself) in which PRO moves against OPP’s move of c. To summarise, suppose PRO wishes to show that x is a member of an extension E under the semantics s. The associated legal move function φ for s defines some φ-dispute tree T that is a sub-tree of the dispute tree induced by x, and defines all possible disputes the players can play in a game. The φ-dispute tree T should be such that: x ∈ E iff there is a φ-winning strategy T′ in T, such that the arguments moved by PRO in T′ do not attack each other (are conflict free). A φ-winning strat- egy is a set of disputes won by PRO in which PRO has fulfilled its burden of proof by countering all possible φ-legal moves of OPP. 4 Grounded Semantics For any argumentation framework, there is guaranteed to be exactly one grounded extension. Hence, questions 1b, 1c and 2a - 2h can all be addressed by construc- tion of a framework’s grounded extension. In Section 4.1 we present an algorithm that generates the grounded labelling of an argumentation framework. Section 4.2 then describes an argument game for deciding whether a given argument is in the grounded extension, thus providing an alternative way for addressing the questions 2a and 2b. The grounded semantics places the highest burden of proof on membership of the extension that it defines. This equates with Chapter 2’s definition of the exten- sion as the least fixed point of a framework AF’s characteristic function FAF (i.e., the smallest admissible E that contains exactly those arguments that are acceptable
- 126. 112 Sanjay Modgil and Martin Caminada w.r.t. E). The extra burden of proof is intuitively captured by the fact that in de- fending x’s membership of the grounded extension E, one must ‘appeal to’ some argument other than x itself. That is to say, for any y such that y attacks x, y is attacked by at least one z1 ∈ E such that z1 = x, and in turn, z1 must be reinstated against any attack, by some z2 ∈ E such that z2 = x, z2 = z1, and so on. This property is exploited by both the algorithm for generating the grounded labelling, and argu- ment games for the grounded semantics. The property is relatively straightforward to show given Chapter 2’s description of how, starting with the empty set, iteration of the characteristic function yields the grounded extension. We have that x ∈ Fi AF iff for every attack on x, x is reinstated by some z ∈ Fj AF , where z = x and j i. 4.1 A labelling algorithm for the Grounded Semantics An algorithm for generating the grounded labelling starts by assigning IN to all ar- guments that are not attacked, and then iteratively: OUT is assigned to any argument that is attacked by an argument that has just been made IN, and then IN to those arguments all of whose attackers are OUT. Thus, the arguments assigned IN on each iteration, are those that are reinstated by the arguments assigned IN on the previ- ous iteration. The iteration continues until no more new arguments are made IN or OUT. Any arguments that remain unlabelled are then assigned UNDEC. One can straightforwardly show that the algorithm is sound and complete since it effectively mimics construction of the grounded extension through iteration of a framework’s characteristic function. The algorithm for generating the grounded labelling LG of a framework A,R is presented more formally below, in which we use Section 2’s representation of a labelling L as a triple (in(L), out(L), undec(L)). Algorithm 6.1 Algorithm for Grounded Labelling 1: L0 = (/ 0,/ 0,/ 0) 2: repeat 3: in(Li+1) = in(Li) ∪ {x | x is not labelled in Li, and ∀y : if yRx then y ∈ out(Li) } 4: out(Li+1) = out(Li) ∪ {x | x is not labelled in Li, and ∃y : yRx and y ∈ in(Li+1) } 5: until Li+1 = Li 6: LG = (in(Li), out(Li), A− (in(Li) ∪ out(Li) ) Consider the following example framework: a → b → c , d ⇄ e L1 = ({a},{b}, / 0), L2 = ({a,c},{b}, / 0), L3 = L2 and so LG = ({a,c},{b},{d,e}). Finally, notice that the algorithm presented here can be made more efficient in a number of ways. For example, when assigning IN to arguments in line 3, checking whether all attackers are OUT can be made more efficient by giving each argument
- 127. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 113 a counter attackers-out that represents the number of attackers that are la- belled OUT. Since all arguments are initially unlabelled, this counter is set to zero before the actual labelling begins. Every time that an argument is labelled OUT, it sends a message to each of the arguments that it attacks to increase its variable attackers-out. Evidently, if this variable equals the number of attackers, the attacked argument can be labelled IN. 4.2 Argument games for the Grounded Semantics We have discussed how, in defending an argument x’s membership of the grounded extension, one must not loop back to x itself, and how the same restriction applies to any argument moved in x’s line of defence. Intuitively, this is captured by a le- gal move function φG1 that prohibits PRO from repeating arguments it has already moved in a dispute. Definition 6.8. Given A,R, a dispute d such that x is the last argument in d, and PRO(d) the arguments moved by PRO in d, then φG1 is a legal move function such that: • If d is of odd length (next move is by OPP) then φG1 (d) = {y | yRx } • If d is of even length (next move is by PRO) then: φG1 (d) = {y | 1. yRx 2. y / ∈ PRO(d) } Theorem 6.2. Let AF = A,R be a finite argumentation framework. Then, there exists a φG1 -winning strategy T for x such that the set PRO(T) of arguments moved by PRO in T is conflict free, iff x is in the grounded extension of AF. One can give an intuitive proof of Theorem 6.2 by appealing to the correspon- dence between the grounded extension and grounded labelling of an argumentation framework (see Theorem 6.1). That is to say, by showing that: 1. Let T be a φG1 -winning strategy for x such that PRO(T) is conflict free. Then there is a grounded labelling L with L(x) = IN. 2. Let L be a labelling with L(x) = IN. Then there exists a φG1 -winning strategy for x such that PRO(T) is conflict free. Proof of the above correspondences can be found in [21]. Consider the example framework in Figure 6.3i). Part of the dispute tree induced by a is shown in Figure 6.3ii), and the φG1 dispute tree induced by a is shown in Figure 6.3iii). Observe that {(a1 −b2 −c3 −d4 −e6)} is a φG1 -winning strategy for a (a is in the grounded extension {a,c,e}). Finally, consider the example framework in Figure 6.3iv). In this case the φG1 -winning strategy for a consists of two disputes:
- 128. 114 Sanjay Modgil and Martin Caminada d c e i) a1 c3 d4 e6 c5 d7 d8 c10 d11 PRO OPP PRO OPP PRO ii) iii) b a b2 OPP e9 a1 c3 d4 e6 d8 c10 b2 e9 iv) a b d c e Fig. 6.3 i) shows an argumentation framework and ii) shows the dispute tree induced in a. iii) shows the φG1 -dispute tree induced by a and the φG1 winning strategy encircled. The φG1 winning strategy for a, in the framework in iv), consists of two disputes. {(aPRO −bOPP −dPRO),(aPRO −cOPP −ePRO)}. Some gain in efficiency can be obtained by a legal move function φG2 that ad- ditionally prohibits PRO from moving a y that is itself attacked by the x that PRO moves against (i.e., augmenting 1 and 2 in Definition 6.8 with ¬(xRy)). This is be- cause if PRO moves such a y against x, then OPP can simply repeat x and move against y, and then PRO will be prevented from repeating y. The φG2 game is in- stantiated by Prakken and Sartor for their argument-based system of prioritized ex- tended logic programming [24]. Amgoud and Cayrol [1] do the same for their argu- ment based system for inconsistency handling in propositional logic. The following soundness and completeness result can be proved as a straightforward generalisa- tion of proofs for the specific systems in [24, 1]. Such a generalised proof can be found in [4]. Theorem 6.3. Let AF = A,R be a finite argumentation framework. Then, there exists a φG2 -winning strategy T for x such that the set PRO(T) of arguments moved by PRO in T is conflict free, iff x is in the grounded extension of AF
- 129. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 115 Since the arguments moved by PRO in a winning strategy are required to be con- flict free, it is obvious to see that shorter proofs may also be obtained by preventing PRO from moving arguments in a dispute d that attack themselves or attack or are attacked by arguments that PRO has already moved in d. Definition 6.9. Let POSS(d) = {y | ¬(yRy) and ∀z ∈ PRO(d), ¬(zRy) and ¬(yRz)}. One can then further restrict PRO’s moves in Definition 6.8, by adding the condition that y ∈ POSS(d), thus obtaining the legal move function φG3 . Finally, further gains in efficiency can be obtained by noticing that if T is a φG1 , φG2 or φG3 winning strategy, then PRO(T) is conflict free. Thus, one need not instigate the conflict free check on winning strategies suggested by the above soundness and completeness results. To see why, notice that φG1 , φG2 and φG3 make no restrictions on moves by OPP, and one can show that the following theorem holds: Theorem 6.4. Let T be a φ winning strategy such that φ makes no restrictions on moves by OPP. Then PRO(T) is conflict free. We refer the reader to [21] for a proof of the above theorem. 5 Preferred Semantics For any argumentation framework, existence of a preferred extension is guaranteed, and there can be more than one preferred extension. Hence, the decision questions 2a (credulous membership) and 2b (sceptical membership) are distinct. In Section 5.2 we describe argument games for addressing the credulous membership ques- tion. Section 5.3 then describes argument games for addressing the more difficult sceptical membership question. Solution-orientated questions 2c - 2h require proce- dures for identifying one or all preferred extensions. Such questions become rele- vant when end-users would like to be informed about the reasons as to how and why an argument is justified or overruled, and can be addressed by labelling algorithms that compute one or all preferred labellings. We describe labelling algorithms in the following section. 5.1 A Labelling Algorithm for the Preferred Semantics In this Section we review Caminada’s work on labelling algorithms [6]. Theorem 6.1 in Section 2 establishes an equivalence between an argumentation framework’s preferred extensions and the framework’s preferred labellings. In [5] it is shown that:
- 130. 116 Sanjay Modgil and Martin Caminada L is a preferred labelling iff L is an admissible labelling such that for no ad- missible labelling L′ is it the case that in(L′) ⊃ in(L). (R1) Hence, a framework’s preferred extensions can be identified by algorithms that compute admissible labellings that maximise the number of arguments that are legally IN. In [6], admissible labellings are generated by starting with a labelling that labels all arguments IN and then iteratively, selects arguments that are illegally IN and applies a transition step to obtain a new labelling, until a labelling is reached in which no argument is illegally IN. Definition 6.10. Let L be a labelling for A,R and x an argument that is illegally IN in L. A transition step on x in L consists of the following: 1. the label of x is changed from IN to OUT 2. for every y ∈ {x} ∪ {z|xRz}, if y is illegally OUT, then the label of y is changed from OUT to UNDEC (i.e., any argument made illegally OUT by 1 is changed to UNDEC) In what follows, we assume a function transition step that takes as input x and L, and applies the above operations to yield a labelling L′. We then define a transition sequence as follows: A transition sequence is a list [L0, x1, L1, x2, ..., xn, Ln] (n ≥ 0), where for i = 1...n, xi is illegally IN in Li−1, and Li = transition step(Li−1, xi). A transition sequence is said to be terminated iff Ln does not contain any argument that is illegally IN. Let us examine a transition sequence that starts with the initial labelling L0 in which all arguments are labelled IN (from hereon any such labelling is referred to as an ‘all-in’ labelling and we assume that any initial labelling L0 is an all-in labelling). Any labelling containing an argument x that is illegally IN cannot be a candidate admissible labelling (since not all of x’s attackers are OUT and so x is not reinstated against all attackers), and so must be relabelled OUT. One might expect that the second part of the transition step relabels to IN, those arguments that are made illegally OUT by the first step. However this may not only result in a loop, but would also ‘overcommit’ arguments to membership of an admissible labelling (and so extension); just because an argument may be acceptable w.r.t. an admissible extension E does not mean that it must be in E. For finite frameworks it can be shown that: For any terminated transition sequence [L0, x1, L1, x2, ..., xn, Ln], it holds that Ln is an admissible labelling. (R2) To see why, observe that L0 contains no arguments that are illegally OUT, and it is straightforward to show that a transition step preserves the absence of arguments that are illegally OUT. Hence, since the terminated sequence contains no arguments that are illegally IN, then by Definition 6.3, Ln is admissible. In [6], it is also shown that:
- 131. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 117 For any preferred labelling L, it holds that there exists a terminated transition sequence [L0, x1, L1, x2, ..., xn, Ln], where Ln = L. (R3) The above results R1, R2 and R3, imply that terminated transition sequences whose final labellings maximise the arguments labelled IN are exactly the preferred la- bellings. Before presenting the algorithm for generating such sequences, let us con- sider how admissible labellings are generated for the argumentation framework in Figure 6.4i). Starting with the initial all-in labelling L0 = ({a,b,c}, / 0, / 0), then se- lecting a on which to perform a transition step obtains L1 = ({b,c},{a}, / 0). Now only c is illegally IN, and relabelling it to OUT results in a being illegally OUT, and so L2 = ({b},{c},{a}). Now b is illegally IN, and relabelling b to OUT results in both b and c being illegally OUT, so that they are both labelled UNDEC. Thus, the transition sequence terminates with the labelling L3 = (/ 0, / 0,{a,b,c}) in which all arguments are UNDEC. It is easy to verify that irrespective of whether a, b or c is selected on the first transition step, every terminated transition sequence will result in L3. Consider now the framework in Figure 6.4ii). Starting with the initial all-in la- belling L0 = ({a,b,c}, / 0, / 0), we observe that B and C are illegally IN: 1. Selecting b for the first transition step obtains the terminated sequence [L0, b, L1 = ({a,c},{b}, / 0)]. L1 is an admissible and complete labelling, yielding the admissible and complete extension {a,c}. 2. Selecting c for the first transition step obtains the terminated sequence [L0, c, L1 = ({a,b},{c}, / 0), b, L2 = ({a},{b},{c})], yielding the admissible extension {a} a b ii) c a b c i) Fig. 6.4 Two argumentation frameworks Notice that in the second sequence, the label of c is changed from OUT to UNDEC since c is made illegally OUT by the second transition step’s assignment of OUT to the illegally IN b. L2 is an admissible but not complete labelling, since c is illegally UNDEC. To help avoid non-complete labellings, one can guide the choice of arguments on which to perform transition steps: choose an argument that is super- illegally IN, if such an argument is available. Definition 6.11. An argument x in L that is illegally IN, is also super-illegally IN iff it is attacked by a y that is legally IN in L, or UNDEC in L.
- 132. 118 Sanjay Modgil and Martin Caminada Thus, b would preferentially be selected according to the above strategy, since b and not c is super-illegally IN in ({a,b,c}, / 0, / 0). As shown in [6], both the results R2 and R3 are preserved under such a strategy. Algorithm 6.2 Algorithm for Preferred Labellings 1: candidate-labellings := / 0; 2: find labellings(all-in); 3: print candidate-labellings; 4: end. 5: . 6: . 7: procedure find labellings(L) 8: . 9: # if L is worse than an existing candidate labelling then prune the search tree 10: # and backtrack to select another argument for performing a transition step 11: if ∃L′ ∈ candidate-labellings: in(L) ⊂ in(L′) then return; 12: . 13: # if the transition sequence has terminated 14: if L does not have an argument that is illegally IN then 15: for each L′ ∈ candidate-labellings do 16: # if L′’s IN arguments are a strict subset of L’s IN arguments 17: # then remove L′ 18: if in(L′) ⊂ in(L) then 19: candidate-labellings := 20: candidate-labellings − {L′}; 21: end if 22: end for 23: # add L as a new candidate 24: candidate-labellings := candidate-labellings ∪ {L}; 25: return; # we are done, so try the next possibility 26: else 27: if L has an argument that is super-illegally IN then 28: x := some argument that is super-illegally IN in L; 29: find labellings(transition step(L, x)); 30: else 31: for each x that is illegally IN in L do 32: find labellings(transition step(L, x)) 33: end for 34: end if 35: end if 36: endproc We now describe the above listed algorithm for generating preferred labellings. The main procedure find labellings starts with the all-in labelling, and then iteratively applies transitions steps in an attempt to generate terminated transition sequences that update the global variable candidate-labellings. The algo- rithm preferentially selects from amongst super-illegal arguments for performing transition steps, if such arguments are available. If at any stage in the generation of a transition sequence, the arguments that are IN in the labelling Li thus far obtained
- 133. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 119 are a strict subset of in(L′) for some L′ ∈ candidate-labellings, then no further transition steps on Li can result in a preferred labelling (that maximises the arguments that are IN). This follows from the result that during the course of a tran- sition sequence, the set of IN labelled arguments monotonically decreases (as shown in [6]). Thus, any further transition steps on Li will only reduce the arguments that are IN. In such cases, the algorithm backtracks to Li−1 and, if possible, selects an- other argument on which to perform a transition step. In the case that a transition sequence terminates, the obtained labelling L is compared with all labellings L′ in candidate-labellings. If for any L′, in(L′) is a strict subset of in(L), then L′ is removed from candidate-labellings. Thus, given a finite argu- mentation framework A,R, the algorithm calculates the preferred labellings and so preferred extensions. 5.2 Argument Games for the Credulous Preferred Semantics Since the admissible extensions of a framework form a complete partial order with respect to set inclusion (and so every admissible extension is a subset of a preferred extension), then for argument games addressing the credulous membership question, it suffices to show an admissible extension containing the argument in question. In contrast with the grounded semantics, x’s membership of an admissible extension E can now ‘appeal to’ x itself, in the sense that in defending x’s membership of E, and membership of all subsequent defenders, one can loop back to x itself. This then means, that to prevent infinite disputes, it is now OPP, rather than PRO, that should not be allowed to repeat an argument y it has already moved in a dispute, since y can then be attacked by PRO repeating the argument it moved against OPP’s first move of y. Consider the framework in Figure 6.5i), and the dispute tree induced in a in Fig- ure 6.5ii). In both disputes (branches) PRO is allowed to repeat its arguments (c5 and d9). OPP repeats its arguments, and the disputes continue with PRO repeatedly fulfilling its burden of proof w.r.t. c (d). It is of course sufficient that PRO fulfill its burden of proof only once. Hence, as well as preventing PRO from introduc- ing a conflict into a dispute, the following legal move function prohibits OPP from repeating arguments. Definition 6.12. Given A,R, a dispute d such that x is the last argument in d, and OPP(d) the arguments moved by OPP in d, then φPC1 is a legal move function such that: • If d is of odd length (next move is by OPP) then: φPC1 (d) = {y | 1. yRx 2. y / ∈ OPP(d) }
- 134. 120 Sanjay Modgil and Martin Caminada d c i) a1 c3 d4 c5 d6 d7 c8 d9 b2 c10 PRO OPP PRO OPP PRO ii) iii) a1 c3 d4 c5 d7 c8 d9 b2 b a OPP iv) a1 c3 d7 b2 Fig. 6.5 i) shows an argumentation framework and ii) shows the dispute tree induced in a. iii) and iv) respectively shows the φPC1 and φPC2 dispute trees induced by a. • If d is of even length (next move is by PRO) then: φPC1 (d) = {y | 1. yRx 2. y ∈ POSS(d) } Notice that φPC1 mirrors Section 4.2’s grounded game function φG3 (that aug- ments φG1 to restrict PRO to moving arguments in POSS(d)). They differ only in that φPC1 prevents repetition by OPP, and φG3 prevents repetition by PRO. Consider again the framework in Figure 6.5i). The φPC1 dispute tree induced by a is shown in Figure 6.5iii), and both disputes in the tree individually constitute φPC1 winning strategies. Notice that for the example framework in Figure 6.3iv), the φPC1 - winning strategy for a consists of two disputes: {(aPRO − bOPP − dPRO),(aPRO − cOPP −ePRO)}. The following theorem states the soundness and completeness result for φPC1 games: Theorem 6.5. Let AF = A,R be a finite argumentation framework. Then, there exists a φPC1 -winning strategy T for x such that the set PRO(T) of arguments moved by PRO in T is conflict free, iff x is in an admissible (and hence preferred) extension of AF. One can give an intuitive proof of the above by using the correspondence be- tween admissible extensions and admissible labellings of an argumentation frame- work (see Theorem 6.1). That is, it suffices to prove that: 1. Let T be a φPC1 -winning strategy for x such that PRO(T) is conflict free. Then there exists an admissible labelling L with L(x) = IN.
- 135. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 121 2. Let L be an admissible labelling with L(x) = IN. Then there exists a φPC1 - winning strategy for x such that PRO(T) is conflict free. Proof of the above correspondences can be found in [21]. Observe that the spectrum of outcomes would not be changed by a function φPC2 that augments φPC1 by prohibiting OPP from moving any argument y (and not just a y already moved by OPP) that is attacked by an argument x in PRO(d). This is because PRO can then simply move x against y, and if yRx, prohibiting repetition by OPP will mean that y cannot be moved against this second move of x by PRO. Notice that if this prohibition on OPP is in place, then one cannot have a dispute of the form (...yPRO ...xOPP − yPRO ...) in which PRO repeats an argument, since the prohibition on OPP would prevent the move xOPP. Hence, shorter proofs can be obtained by a function φPC2 that augments φPC1 by prohibiting OPP from moving any argument attacked by an argument in PRO(d), and prohibiting repetition by PRO. Indeed, [9] prove that the following theorem holds: Theorem 6.6. Let AF = A,R be a finite argumentation framework. Then, there exists a φPC2 -winning strategy for x such that the set PRO(T) of arguments moved by PRO in T is conflict free, iff a is in a preferred extension of AF. Figure 6.5iv) shows the φPC2 dispute tree induced by a for the framework in Figure 6.5i), where both disputes in the tree are φPC2 winning strategies. However, notice that neither dispute fully fulfills the remit of a proof to explain the credulous membership of a, since neither demonstrates the reinstatement of c, respectively d, against its attacker d, respectively c, and so provides an explanation for the ad- missibility of {a,c}, respectively {a,d}. This illustrates a more general point that efficiency gains often come at the expense of explanatory power. Finally, note that unlike games for the grounded semantics, checking that the arguments moved by PRO in a φPC1 (or φPC2 ) winning strategy are conflict free, is required. This is because φPC1 and φPC2 games place restrictions on moves by OPP (and hence the result concluding Section 4.2 does not hold). For example, consider that a is not in an admissible, and hence preferred, extension of the framework in Figure 6.6. Now, {(aPRO − bOPP − cPRO − dOPP − gPRO),(aPRO − eOPP − fPRO − gOPP − dPRO)} is a φPC1 winning strategy since OPP cannot legally extend either dispute. However, the arguments moved by PRO are not conflict free (PRO has moved g and d). c b a d e f g Fig. 6.6 Argument a is not in an admissible and so preferred extension of the above framework.
- 136. 122 Sanjay Modgil and Martin Caminada 5.3 Argument Games for the Sceptically Preferred Semantics The question of whether an argument is sceptically preferred is much harder to an- swer than the credulously preferred membership problem. To understand why, it may first help to realise that the credulous membership problem only requires us to point at one extension, while the sceptical membership problem requires us to prove something about all possible extensions. Thus, the credulously preferred member- ship problem is an existence problem while the sceptically preferred membership problem is a verification problem. To understand better why verification is hard in this case, we recall the definition of sceptically preferred membership: an argument a is sceptically preferred iff it is a member of all preferred extensions. The crux of the problem is that we have to verify whether there exists preferred extensions that do not contain a. In so doing, it is not immediately clear where to begin to search for such extensions. The following result establishes a connection between a and preferred extensions that might possibly exclude a (we refer the reader to [21] for a proof of this result). It basically ensures that the search space for the sceptical decision problem is confined to elements that are indirectly connected to defense sets of a. Theorem 6.7 (Complement lemma). An argument a is sceptically preferred if and only if for every admissible extension B, there is an admissible extension A, contain- ing a, that is consistent with B. Thus, conversely, a is not sceptically preferred if there exists an admissible exten- sion B that conflicts with all admissible extensions around a. Because such an exten- sion B blocks sceptically preferred membership, such an extension is called a block. With the help of Theorem 6.7 we may now formulate an abstract and inefficient, but conceptually correct proof procedure to determine sceptical membership. This pro- cedure works by falsification, as follows. Try to construct a block B. If this attempt fails, we may, with the help of Theorem 6.7 conclude that a is sceptically preferred. The procedure to block a can be described as an argument game that we infor- mally describe here. The difference with the games described earlier, is that the play- ers exchange entire admissible extensions rather than single arguments. The game works as follows. Suppose PRO’s goal is to show that a is sceptically preferred. To this end, PRO starts by constructing an admissible extension, A{1} around a. Since A{1} is the only admissible extension known at this stage, it follows that at this stage a is sceptically preferred. To invalidate this temporary conclusion, the bur- den of proof shifts to OPP who must show that a is not sceptically preferred. By virtue of Theorem 6.7 it suffices for OPP to show that there exists an admissible extension that conflicts with A{1}. If OPP does not manage to construct such an extension, the procedure ends and OPP has lost. Suppose OPP manages to produce A{1,1} as a response to A{1}. Thus, A{1,1} is an admissible extension that con- flicts with A{1}. Once A{1,1} is advanced, a is no longer sceptically preferred, be- cause A{1,1} conflicts with every admissible extension around a constructed thus far, viz. A{1}. To invalidate this temporary conclusion, the burden of proof shifts back to PRO who must now show that there exists another admissible extension
- 137. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 123 around a that does not conflict with A{1,1}. If PRO fails to do so (and PRO’s search was adequate and exhaustive), it follows that A{1,1} conflicts with all admissible extensions around A, so that a is not sceptically preferred. Suppose otherwise, i.e, suppose that PRO is able to construct an admissible extension, A{1,1,1}, that does not conflict with A{1,1}. OPP must now either extend A{1,1} such that it also con- flicts with A{1,1,1} or else drop A{1,1} to start all over to attack another member of A{1}. Continuing this way (including backtracking), OPP is busy with extending an admissible extension until either PRO is unable to produce another admissible extension around a, or else until OPP’s admissible extension cannot be further ex- tended (on pain of becoming inconsistent). More generally, we may suppose that A{1},...,A{n} are possible begin moves of PRO, and A{i1,...,ik,m}, k ≥ 1 is the mth possible response of either PRO or OPP to A{i1,...,ik}. Naturally, all the A{ī} are admissible extensions. The following constraints hold: 1. Every extension advanced by PRO must contain the main argument, a. 2. Every response of PRO must be consistent with the extension that is previously advanced by OPP. 3. Every response of OPP must attack PRO’s immediately preceding extension. 4. Within one branch, every extension advanced by OPP must be an extension of OPP’s previous extension in the same branch. 5. Both parties may backtrack and construct alternative replies. 6. OPP has won if it is able to move last; else PRO has won. If OPP has won this means that OPP was able to create a block B = A{i1,...,i2k}, where k ≥ 1 (note that we have ‘2k’ since all moves by OPP have an even number of indices). With B, OPP is able to move last in the particular branch where that block was created and all sub-branches emanating from the main branch. It must be noted that all this only works in finitary argument systems, i.e., argument systems where all arguments have a finite number of attackers. Algorithms for non-finitary argu- ment systems require additional constraints such as fairness which must guarantee that every possibility is enumerated eventually. The above ideas are taken from earlier work on the sceptically preferred mem- bership problem, notably that of Doutre et al. [12] and Dung et al. [15]. In [12], the procedure to find a possible block is presented as a so-called meta-acceptance dialogue. As above, moves in this dialogue are extensions (hence the meta), and a dialogue is won by OPP if it is able to move last in at least one branch. In Dung et al. [15] the procedure to construct a “fan” of admissible extensions around A that together represent all preferred extensions is called generating a complete base for a. A base for a is a set of admissible extensions, B, such that every preferred exten- sion around a includes at least one element of B. A complete base for a, then, is a set of admissible extensions, B, such that every preferred extension includes at least one element of B. In line with Theorem 6.7, Dung et al. proceed to show that a base B is incomplete if and only if there exists a preferred extension that attacks every element of B. Their proof procedure is a combination of a so-called BG-derivation (base generation derivation) followed by a CB-verification (complete base verifica-
- 138. 124 Sanjay Modgil and Martin Caminada tion). With BG a base for a is generated, such that every preferred extension around a contains an element of B. Such a base always exists, but not every base may serve as a representant of sceptical membership. To check whether B indeed represents sceptical membership, it is checked for completeness, which effectively means that it must hold out against every candidate block that might undermine B. Again, all decision procedures only work in finitary argument systems. 6 Stable and Semi-Stable Semantics Stable semantics are, what one might call ‘xenophobic’, since every argument out- side of a stable extension is attacked by an argument in the stable extension. Unlike the preferred semantics, existence of a stable extension is not guaranteed; consider that a framework consisting of a single argument that attacks itself has no stable ex- tension. However, as in the case of the preferred semantics, there may be more than one extension, and so decision questions 2a (credulous membership) and 2b (scep- tical membership) are distinct. These questions, questions 1b, 1c, and the solution- orientated questions 2a - 2h can be addressed by an algorithm (taken from [6]) that generates all stable extensions of a framework. Since a stable labelling makes all arguments either OUT or IN, one can straightforwardly adapt the algorithm for preferred labellings in Section 5.1, so as to only yield labellings without UNDEC labelled arguments. Thus, line 11 in the algorithm is replaced by: if undec(L) = / 0 then return; Furthermore, we do not have to compare the arguments made IN by other candidate labellings, and so we can remove lines 15 to 22. The result is an algorithm that cal- culates all stable extensions of a finite framework. Argument games for stable semantics have only recently been studied. In [28], the authors study coherent argumentation frameworks, in which every preferred ex- tension is also stable (meaning that the preferred and stable extensions coincide, since each stable extension is by definition also a preferred extension). Thus, for coherent argumentation frameworks, one can simply apply existing games for the preferred semantics to decide membership under stable semantics. For the general case, where one is not restricted to coherent argumentation frame- works, the situation is more complex, but can still be expressed in terms of the cred- ulous games defined in Section 5.2. Given a framework A,R, and letting PRO(T), respectively OPP(T), denote the arguments moved by PRO, respectively OPP, in a dispute tree T, then an argument x is in a stable extension iff there exists a set S of φPC1 winning strategies such that: 1. at least one winning strategy in S is for x. 2. {PRO(T)|T ∈ S} is conflict free. 3. {PRO(T)∪OPP(T)|T ∈ S} = A
- 139. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 125 This can be seen as follows. First of all, each φPC1 winning strategy corresponds to an admissible labelling. A set of winning strategies that do not attack each other (point 2) again corresponds to an admissible labelling. If this resulting admissible labelling spans the entire argumentation framework (each argument is either IN or OUT) then this labelling is also stable (point 3). Then, if x is IN in this labelling, then x is labelled IN in at least one stable labelling (point 1). It is also possible to define a single dispute game that determines credulous ac- ceptance w.r.t. stable semantics. Such a game has recently been stated by Caminada and Wu [7]. One particular feature of their approach, which builds on the work of Vreeswijk and Prakken [28], is that they do not use the concept of a winning strat- egy. Instead, for an argument x to be in a stable extension, it suffices to have at least one game for x that is won by PRO. Caminada and Wu are able to do this by first defining a game for credulous preferred in which PRO may repeat its own moves, but not the moves of OPP, and in which OPP may repeat PRO’s moves but not its own moves. Moreover, PRO has to react to the directly preceding move of OPP, whereas OPP is free to react either to the directly preceding move of PRO, or to a previous PRO move. A dispute is won by PRO iff OPP cannot move. A dispute is won by OPP iff PRO cannot move, or if OPP managed to repeat one of PRO’s moves. Basically, the game can be understood in terms of PRO and OPP building an admissible labelling in which PRO makes IN moves, and OPP makes OUT moves. This game can be altered to implement stable semantics by introducing a third kind of move, which is called QUESTION. By uttering QUESTION x, OPP asks PRO for an explicit opinion on argument x. PRO is then obliged to reply with either IN x or with IN y, where y is an attacker of x. Caminada and Wu show that this game indeed models credulous acceptance under the stable semantics. Once a procedure for credulous acceptance w.r.t. stable semantics has been de- fined, the issue of sceptical acceptance w.r.t. stable semantics becomes relatively straightforward: an argument x is in all stable extensions iff one fails to establish credulous membership of any attacker of x. For the left to right half, observe that if x is in all stable extensions, then all attackers of x are attacked by all such ex- tensions (an argument y is attacked by an extension if it is attacked by an argument in that extension), and so no attacker of x can be in any such extension, since each such extension is conflict free. For the right to left half, observe that if any attacker of x does not belong to any stable extension, then it is attacked by all such exten- sions. Thus every extension contains an argument that reinstates x, and so contains x. Caminada has recently proposed semi-stable semantics [5, 6], that unlike the sta- ble semantics, guarantees that every (finite) framework has at least one semi-stable extension. In the case that there exists at least one stable extension for a frame- work, semi-stable semantics yield the same extensions as stable semantics. From the perspective of argument labellings, semi-stable semantics select those labellings in which the set of UNDEC arguments is minimal. Referring to Definition 6.4, this can be expressed as follows:
- 140. 126 Sanjay Modgil and Martin Caminada Let L be a complete labelling. Then L is a semi-stable labelling iff there does not exist a complete labelling L′ such that undec(L′) ⊂ undec(L) For example, consider the framework in Figure 6.4ii) augmented by an additional argument d that attacks itself. The augmented framework has no stable extension, but {a,c} is the single semi-stable extension equating with the semi-stable labelling ({a,c},{b},{d}). Notice that although {a,c} is also the single preferred extension, in general not every preferred extension is a semi-stable extension since not every preferred extension minimises UNDEC. However, every semi-stable extension is a preferred extension, which suggests that we can adapt Section 5.1’s algorithm for preferred labellings in order to compute semi-stable labellings. In [6] it is also shown that: L is a semi-stable labelling iff L is an admissible labelling such that for no admissible labelling L′ is it the case that undec(L′) ⊂ undec(L). (R1′) Since every semi-stable extension is a preferred extension then R3 in Section 5.1 also holds for semi-stable labellings L. This result, together with R1′ and R2 in Section 5.1, implies that terminated transition sequences whose final labellings min- imise the arguments labelled UNDEC are exactly the semi-stable labellings. Hence, one can adapt Section 5.1’s algorithm by replacing line 11 by: if ∃L′ ∈ candidate-labellings: undec(L′) ⊂ undec(L) then return; In other words, if at any stage in the generation of a transition sequence, the UNDEC arguments of the labelling Li thus far obtained, are a strict superset of undec(L′) for some L′ ∈ candidate-labellings, then no further transition steps on Li can result in a semi-stable labelling, and so one can backtrack to per- form a transition step on another choice of argument. This follows from the result that during the course of a transition sequence, the set of UNDEC labelled arguments monotonically increases (as shown in [6]). Finally, we replace line 18 with: if undec(L) ⊂ undec(L′); and we are done. We have an algorithm that calculates the semi-stable labellings of a finite argumentation framework. 7 Conclusions In this chapter we have described labelling algorithms and argument game proof theories for various argumentation semantics. Labellings and argument games can be seen as alternatives to the extension-based approach to specifying argumentation
- 141. 6 Proof Theories and Algorithms for Abstract Argumentation Frameworks 127 semantics described in Chapter 2. We conclude with some further reflections on these different ways of specifying argumentation semantics. One of the original motivations for developing the labelling approach was to pro- vide an easy and intuitive account of formal argumentation. After all, principles like “In order to accept an argument, one has to be able to reject all its counterargu- ments” and “In order to reject an argument, one has to be able to accept at least one counterargument” are easy to explain and have therefore been used as the basis of the labelling approach. Also, our teaching experiences indicate that students who are new to argumentation tend to find it easier to understand the labelling approach rather than the extension-based approach to argumentation. In fact, it is often easier for them to understand the extension-based approach after having been introduced to the labelling approach. Another advantage of the labelling approach is that it allows one to specify a number of relatively small and simple properties, each of which can be individu- ally satisfied or not, and that collectively define the argumentation semantics. This modular approach can be of assistance when constructing formal proofs. Also, by explicitly distinguishing between IN, OUT and UNDEC (instead of merely speci- fying the set of IN-labelled arguments as in the extension-based approach) one is provided with more detailed information. For instance, Section 5.1’s algorithm for generating all preferred extensions, would be much more difficult to specify using the extension-based approach. Finally, we note that the labelling approach essentially identifies a graph or ‘net- work’ labelling problem, suggesting that the approach more readily lends itself to extensions of argument frameworks that accommodate: different types of relation between arguments (e.g. support [10] and collective attack [22]); attacks on attacks [20]; multi-valued and quantitative valuations of arguments [2, 11], and so on. In essence, these extensions of Dung’s abstract argumentation framework can be un- derstood as instantiating a more general network reasoning model in which the val- uations of nodes (arguments) is determined by propagating the valuations of the connected nodes, as mediated by the semantics of the connecting arcs. Algorithms for determining these valuations will thus generalise the three value labelling algo- rithms described in this chapter. With regard to the argument game approach, we recall that Dung’s abstract ar- gumentation semantics can be understood as a semantics for a number of non- monotonic and defeasible logics [3, 13], in the sense that: α is an inference from a theory Δ in a logic L, iff α is the conclusion of a justified argument of the argumentation framework A,R defined by Δ and L. The argument game approach places an emphasis on the dialectical nature of argumentation, in the sense that the approach appeals more directly to an inter- subjective notion of truth: truth becomes that which can be defended in a rational exchange and evaluation of interacting arguments. Thus, what accounts for the cor- rectness of an inference is that it can be shown to rationally prevail in the face of
- 142. 128 Sanjay Modgil and Martin Caminada arguments for opposing inferences, where it is application of the reinstatement prin- ciple that encodes logic neutral, rational means for establishing such standards of correctness. This account of argumentation as a semantics, contrasts with model- based semantics for formal entailment that appeal to an objective notion of truth: true is that which holds in every possible model. Notice that dialectical semantics are not unique to formal argumentation. For instance, Lorenzen and Lorenz [18, 19] have proposed dialectical devices as a method of demonstration in formal logic. An advantage of dialectical semantics is that they are able to relate formal en- tailment to something most people are familiar with in everyday life: debates and discussions. Argument games of the type described in this chapter are therefore use- ful not only for providing guidelines and principles for the design of algorithms, but also for bridging the gap between formal and informal reasoning. Finally, we note that the dialectical view also accords with our understanding of reasoning as an incremental process. Rather than have all the arguments and their attacks defined from the outset, we incrementally acquire knowledge in order to con- struct arguments required to counter-argue existing arguments. At any stage in this incremental process we can evaluate the status of arguments, which in turn motivates acquisition of further knowledge for construction and submission of arguments. Ar- gument games allow one to model such processes. Provided that there is a well understood notion of what constitutes an attack between any two arguments, one can then formalise the games described in this chapter, without reference to a pre- existing framework. This also allows one to acknowledge that reasoning agents are resource bounded, and suggests that bounds on reasoning resources may be charac- terised by bounds on the breadth and depth of the dispute trees constructed in order to prove the claim of the argument under test. Acknowledgements The authors would like to thank Gerard Vreeswijk for his con- tributions to the contents of this chapter. Thanks also to Nir Oren for commenting on a draft of the chapter. References 1. L. Amgoud and C. Cayrol. A Reasoning Model Based on the Production of Acceptable Argu- ments. Annals of Mathematics and Artificial Intelligence, 34(1–3),197–215, 2002. 2. H. Barringer, D. M. Gabbay and J. Woods. Temporal Dynamics of Support and Attack Net- works: From Argumentation to Zoology. Mechanizing Mathematical Reasoning, 59–98, 2005. 3. A. Bondarenko and P.M. Dung and R.A. Kowalski and F. Toni. An abstract, argumentation- theoretic approach to default reasoning. Artificial Intelligence, 93:63–101, 1997. 4. M. Caminada. For the sake of the Argument. Explorations into argument-based reasoning. Doctoral dissertation Free University Amsterdam, 2004. 5. M. Caminada. On the Issue of Reinstatement in Argumentation. In European Conference on Logic in Artificial Intelligence (JELIA), 111–123, 2006. 6. M. Caminada. An Algorithm for Computing Semi-stable Semantics. In European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU), 222– 234, 2007.
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- 147. Chapter 7 Argumentation Based on Classical Logic Philippe Besnard and Anthony Hunter 1 Introduction Argumentation is an important cognitive process for dealing with conflicting infor- mation by generating and/or comparing arguments. Often it is based on construct- ing and comparing deductive arguments. These are arguments that involve some premises (which we refer to as the support of the argument) and a conclusion (which we refer to as the claim of the argument) such that the support deductively entails the claim. In order to formalize argumentation, we could potentially use any logic to define the logical entailment of the claim from the support. Possible logics include defea- sible logics, description logics, paraconsistent logics, modal logics, and classical logic. In this chapter, we focus on deductive arguments in the setting of classical logic. Hence, our starting position is that a deductive argument consists of a claim entailed by a collection of statements such that the claim as well as the statements are denoted by formulae of classical logic and entailment is deduction in classical logic. Classical logic is a well-known formalism. It is widely used in philosophy, mathematics, and computer science for capturing deductive reasoning. It has a sim- ple and intuitive syntax and semantics, and it is supported by a proof theory and extensive foundational results. By using classical logic, we can provide a simple and efficient formalization of argument and counterargument. So in our framework, an argument is simply a pair Φ,α where the first item in the pair is a minimal consistent set of formulae that proves the second item. That is, we account for the support and the claim of an argument though we do not indicate the method of inference since it does not differ from one argument to another: We Philippe Besnard IRIT, Universite Paul Sabatier, Toulouse, France Anthony Hunter Department of Computer Science, University College London, London, UK I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 133 DOI 10.1007/978-0-387-98197-0 7, c Springer Science+Business Media, LLC 2009
- 148. 134 Philippe Besnard and Anthony Hunter only consider deductive arguments, hence the method of inference for each and every argument is always entailment according to classical logic. A counterargument for an argument Φ,α is an argument Ψ,β where the claim β contradicts the support Φ. Furthermore, we identify a particular kind of counterargument called a canonical undercut Ψ,β where β is equivalent to ¬(φ1 ∧ .. ∧ φn) and {φ1,...,φn} is the support of the argument being undercut. This is a valuable form of undercut since it subsumes many other kinds of undercut, and hence focusing on only canonical undercuts renders the presentation and evaluation of counterarguments as a more manageable process. Each undercut to an argument is itself an argument, and so may be undercut, and hence by recursion each undercut needs to be considered for its undercuts. Exploring systematically the universe of arguments in order to present an exhaustive synthesis of the relevant chains of undercuts for a given argument is the basic principle of our approach. Following on from the idea that we can capture undercuts, and by recursion un- dercuts to undercuts, our notion of an argument tree is that it is a synthesis of all the arguments that challenge the argument at the root of the tree, and it also contains all counterarguments that challenge these arguments and so on recursively. In each instance, the only counterarguments we consider are the canonical undercuts. In the rest of this chapter, we formalize and illustrate arguments and counterar- guments (including canonical undercuts), and show how these can be collected into argument trees. We conclude the chapter with a comparison with other approaches to formalising argumentation. Since the aim of this chapter is to just introduce some of the basic ideas to argumentation based on classical logic, the interested reader is requested to refer to [3, 6] for more details including formal results. 2 Preliminaries We assume the reader has some knowledge of classical logic. We will represent atoms by lower case roman letters (a,b,c,d,...), formulae by greek letters (α,β,γ, ....), and use ∧,∨,→, and ¬ to denote the logical connectives conjunction, disjunc- tion, negation, and implication (respectively). We use ⊢ to denote the classical con- sequence relation, and so if Δ is a knowledgebase, and α is a formula, then Δ ⊢ α denotes that Δ entails α (or equivalently α is a consequence of Δ). We also use ⊥ to denote a contradiction, and so Δ ⊢ ⊥ denotes that Δ is contradictory (or equivalently inconsistent). For the knowledgebase, we first assume a fixed Δ (a finite set of formulae) and use this Δ throughout. So when we consider arguments and counterarguments, they will be formed from this Δ. For examples, we will explicitly give the elements of the knowledgebase. We further assume that every subset of Δ is given an enumeration α1,...,αn of its elements, which we call its canonical enumeration. This really is not a demanding constraint: In particular, the constraint is satisfied whenever we impose an arbitrary
- 149. 7 Argumentation Based on Classical Logic 135 total ordering over Δ. Importantly, the order has no meaning and is not meant to represent any respective importance of formulae in Δ. It is only a convenient way to indicate the order in which we assume the formulae in any subset of Δ are conjoined to make a formula logically equivalent to that subset. The paradigm for our approach is a large repository of information, represented by Δ, from which arguments can be constructed for and against arbitrary claims. Apart from information being understood as declarative statements, there is no a priori restriction on the contents and the pieces of information in the repository can be arbitrarily complex. Therefore, Δ is not expected to be consistent. It need not even be the case that individual formulae in Δ are consistent. The formulae in Δ can represent certain or uncertain information, and they can represent objective, subjective, or hypothetical statements. So Δ can represent facts, beliefs, views, ... Furthermore, the items in Δ can be beliefs from different agents who need not even have the same opinions. It can indeed be the case that an argu- ment formed from such a Δ takes advantage of partial views from different agents. In any case, it is quite possible for Δ to have two or more formulae which are logi- cally equivalent (Δ can be such that it contains both α ∨β and β ∨α for example). But wherever they come from, all formulae in Δ are on a par and treated equitably. Note, we do not assume any meta-level information about formulae. In particular, we do not assume some preference ordering or “certainty ordering” over formulae. This is in contrast to numerous proposals for argumentation which do assume some form of ordering over formulae. Such orderings can be useful to resolve conflicts by, for example, selecting formulae from a more reliable source. However, this, in a sense, pushes the problem of dealing with conflicting information to one of finding and using orderings over formulae, and as such raises further questions such as: Where does the knowledge about reliability of the sources come from? How can it be assessed? How can it be validated? Besides, reliability is not universal, it usually comes in specialized domains. This is not to say priorities (or indeed other forms of meta-level information) are not useful. Indeed it is important to use them in some situations when they are available, but we believe that to understand the elements of argumentation, we need to avoid drawing on them — we need to have a comprehensive framework for argumentation that works without recourse to priorities over formulae. 3 Arguments We adopt a very common intuitive notion of an argument and consider some of the ramifications of the definition. Essentially, an argument is a set of appropriate formulae that can be used to classically prove some claim, together with that claim (formulae represent statements, including claims). Definition 7.1. An argument is a pair Φ,α such that 1. Φ ⊢ ⊥.
- 150. 136 Philippe Besnard and Anthony Hunter 2. Φ ⊢ α. 3. Φ is a minimal subset of Δ satisfying 2. If A = Φ,α is an argument, we say that A is an argument for α (which in general is not an element of Δ) and we also say that Φ is a support for α. We call α the claim of the argument and we call Φ the support of the argument. Example 7.1. Let Δ = {a,a → b,c → ¬b,c,d,d → b,¬a,¬c}. Some arguments are: {a,a → b},b {c → ¬b,c},¬b {d,d → b},b {¬a},¬a {¬c},¬c {a → b},¬a∨b {¬c},d → ¬c The need for the first condition of Definition 7.1 can be illustrated by means of the next example. Example 7.2. Consider the following atoms. a The office phone number is 020 4545 8721 b I am a billionaire Now let {a,¬a} ⊆ Δ, and so by classical logic, we have {a,¬a} ⊢ b However, we do not want to have {a,¬a} as the support for an argument with claim b. If we were to allow that as an argument, then we would have an argument with this support and with any claim in the language. Hence, if we were to allow inconsistent supports, then we would have an overwhelming number of useless arguments. The second condition of Definition 7.1 aims at ensuring that the support is suffi- cient for the consequent to hold, as is illustrated in the next example. Example 7.3. Consider the informal argument which is acceptable. It is an even number, and therefore we can infer it is not an odd number. Now consider the following atoms. e It is an even number o It is an odd number So we can represent the premise of informal argument by the set {e}. However, by classical logic we have that {e} ⊢ ¬o, and hence the following is not an argument.
- 151. 7 Argumentation Based on Classical Logic 137 {e},¬o If we want to turn the informal argument (which is an enthymeme) into an argument, we need to make explicit all the premises. So we can represent the above informal argument by the following formal argument. {e,¬e∨¬o},¬o An enthymeme is a form of reasoning in which some premises are implicit, most often because they are obvious. As another example, “The baby no longer has her parents, therefore she is an orphan” (in symbols, ¬p hence o) is an enthymeme: The reasoning is correct despite omitting the trivial premise stating that “if a baby no longer has her parents, then she is an orphan” (in symbols, {¬p,¬p → o} ⊢ o). Minimality (i.e., condition 3 Definition 7.1) is not an absolute requirement, al- though some properties depend on it. Importantly, the condition is not of a mere technical nature. Example 7.4. Consider the following formulae. p I like paprika r It is raining r → q If it is raining, then I should use my umbrella It is possible to argue that “I should use my umbrella, because I should use my umbrella, if it is raining, and indeed it is”, to be captured formally by the argument {r,r → q},q In contrast, it is counter-intuitive to argue that “I should use my umbrella, because I like paprika and I should use my umbrella, if it is raining, and indeed it is”, to be captured formally by {p,r,r → q},q which fails to be an argument because condition 3 is not satisfied. The underlying idea for condition 3 is that an argument makes explicit the con- nection between reasons for a claim and the claim itself. But that would not be the case if the reasons were not exactly identified. In other words, if reasons incorpo- rated irrelevant information and so included formulae not used in the proof of the claim. Arguments are not necessarily independent. In a sense, some encompass others (possibly up to some form of equivalence), which is the topic we now turn to. Definition 7.2. An argument Φ,α is more conservative than an argument Ψ,β iff Φ ⊆Ψ and β ⊢ α. Example 7.5. {a},a∨b is more conservative than {a,a → b},b. Roughly speaking, a more conservative argument is more general: It is, so to speak, less demanding on the support and less specific about the consequent.
- 152. 138 Philippe Besnard and Anthony Hunter Example 7.6. Consider the following atoms. p The number is divisible by 10 q The number is divisible by 2 r The number is an even number We use these for the following set of formulae. Δ = {p, p → q,q → r} Hence, the following is an argument with the claim “The number is divisible by 2”. {p, p → q},q Similarly, the following is argument with claim “The number is divisible by 2 and the number is an even number”. {p, p → q,q → r},r ∧q However, the first argument {p, p → q},q is more conservative than the second argument {p, p → q,q → r},r ∧q which can be retrieved from it: {p, p → q},q {q,q → r} |= r ∧q ⇒ {p, p → q,q → r},r ∧q We will use the notion of “more conservative” to help us identify the most useful counterarguments amongst the potentially large number of counterarguments. 4 Counterarguments Informally, an argument that disagrees with another argument is described as a coun- terargument. So counterarguments are an important part of the argumentation pro- cess. They highlight points of contention. In logic-based approaches to argumentation, an intuitive notion of counterargu- ment is captured with the idea of defeaters, which are arguments whose claim refutes the support of another argument [23, 14, 19, 25, 24, 22]. This gives us a general way for an argument to challenge another. Definition 7.3. A defeater for an argument Φ,α is an argument Ψ,β such that β ⊢ ¬(φ1 ∧...∧φn) for some {φ1,...,φn} ⊆ Φ. Example 7.7. Let Δ = {¬a,a ∨ b,a ↔ b,c → a}. Then, {a ∨ b,a ↔ b},a ∧ b is a defeater for {¬a,c → a},¬c. A more conservative defeater for {¬a,c → a},¬c is {a∨b,a ↔ b},a∨c.
- 153. 7 Argumentation Based on Classical Logic 139 The notion of assumption attack to be found in the literature is less general than the above notion of defeater, of which special cases are undercut and rebuttal as discussed next. Some arguments directly oppose the support of others, which amounts to the notion of an undercut. Definition 7.4. An undercut for an argument Φ,α is an argument Ψ,¬(φ1 ∧ ...∧φn) where {φ1,...,φn} ⊆ Φ. Example 7.8. Let Δ = {a,a → b,c,c → ¬a}. Then, {c,c → ¬a},¬(a ∧ (a → b)) is an undercut for {a,a → b},b. A less conservative undercut for {a,a → b},b is {c,c → ¬a},¬a. The most direct form of a conflict between arguments is when two arguments have opposite claims. This case is captured in the literature through the notion of a rebuttal. Definition 7.5. An argument Ψ,β is a rebuttal for an argument Φ,α iff β ↔ ¬α is a tautology. Example 7.9. Consider a discussion in a newspaper editorial office about whether or not to proceed with the publication of some indiscretion about a prominent politi- cian. Suppose the key bits of information are captured by the following five state- ments. p Simon Jones is a Member of Parliament p → ¬q If Simon Jones is a Member of Parliament then we need not keep quiet about details of his private life r Simon Jones just resigned from the House of Commons r → ¬p If Simon Jones just resigned from the House of Commons then he is not a Member of Parliament ¬p → q If Simon Jones is not a Member of Parliament then we need to keep quiet about details of his private life The first two statements form an argument A whose claim is that we can publicize details about his private life. The next two statements form an argument whose claim is that he is not a Member of Parliament (contradicting an item in the support of A) and that is a counterargument against A. The last three statements combine to give an argument whose claim is that we cannot publicize details about his private life (contradicting the claim of A) and that, too, is a counterargument against A. In symbols, we obtain the following argument (below left), and counterarguments (below right). {p, p → ¬q},¬q An undercut is {r,r → ¬p},¬p A rebuttal is {r,r → ¬p,¬p → q},q Trivially, undercuts are defeaters but it is also quite simple to establish that rebut- tals are defeaters. Furthermore, if an argument has defeaters then it has undercuts,
- 154. 140 Philippe Besnard and Anthony Hunter naturally. It may happen that an argument has defeaters but no rebuttals as illustrated next. Example 7.10. Let Δ = {a∧b,¬b}. Then, {a∧b},a has at least one defeater but no rebuttal. There are some important differences between rebuttals and undercuts that can be seen in the following examples. In the first, we see how an undercut for an argument need not be a rebuttal for that argument, and in the second, we see how rebuttal for an argument need not be an undercut for that argument. Example 7.11. {¬a},¬a is an undercut for {a,a → b},b but is not a rebuttal for it. Clearly, {¬a},¬a does not rule out b. Actually, an undercut may even agree with the claim of the objected argument: {b∧¬a},¬a is an undercut for {a,a → b},b. In this case, we have an argument with an undercut that conflicts with the support of the argument but implicitly provides an alternative way to deduce the claim of the argument. This should make it clear that an undercut need not question the claim of an argument but only the reason(s) given by that argument to support its claim. Of course, there are also undercuts that challenge an argument on both counts: Just consider {¬a ∧ ¬b},¬a which is such an undercut for the argument {a,a → b},b. Example 7.12. {¬b},¬b is a rebuttal for {a,a → b},b but is not an undercut for it because b is not in {a,a → b}. Observe that there is not even an argument equivalent to {¬b},¬b which would be an undercut for {a,a → b},b: In order to be an undercut for {a,a → b},b, an argument should be of the form Φ,¬a, Φ,¬(a → b) or Φ,¬(a ∧ (a → b)) but ¬b is not logically equivalent to ¬a, ¬(a → b) or ¬(a∧(a → b)). Both undercuts and rebuttals are useful kinds of counterargument. However, we will see in the next section that we can effectively capture all we need to know about the counterarguments to an argument by just using a special kind of undercut called a canonical undercut. 5 Canonical undercuts A particularly useful kind of undercut is the maximally conservative undercut which we define next. Definition 7.6. Ψ,β is a maximally conservative undercut of Φ,α iff for all undercuts Ψ′,β′ of Φ,α, if Ψ′ ⊆Ψ and β ⊢ β′ then Ψ ⊆Ψ′ and β′ ⊢ β. Evidently, Ψ,β is a maximally conservative undercut of Φ,α iff Ψ,β is an undercut of Φ,α such that no undercuts of Φ,α are strictly more conservative than Ψ,β.
- 155. 7 Argumentation Based on Classical Logic 141 The next example shows that a collection of counterarguments to the same argu- ment can sometimes be summarized in the form of a single maximally conservative undercut of the argument, thereby avoiding some amount of redundancy among counterarguments. Example 7.13. Consider the following formulae concerning who is going to a party. r → ¬p∧¬q If Rachel goes, neither Paul nor Quincy go p Paul goes q Quincy goes Hence both Paul and Quincy go (initial argument) {p,q}, p∧q Now assume the following additional piece of information r Rachel goes Hence Paul does not go (a first counterargument) {r,r → ¬p∧¬q},¬p Hence Quincy does not go (a second counterargument) r,r → ¬p∧¬q},¬q A maximally conservative undercut (for the initial argument) that subsumes both counterarguments above is {r,r → ¬p∧¬q},¬(p∧q) The fact that the maximally conservative undercut in Example 7.13 happens to be a rebuttal of the argument is only accidental. Actually, the claim of a maximally conservative undercut for an argument is exactly the negation of the full support of the argument. In other words, if Ψ,¬(φ1 ∧ ... ∧ φn) is a maximally conservative undercut for an argument Φ,α, then Φ = {φ1,...,φn}. Note that if Ψ,¬(φ1 ∧...∧φn) is a maximally conservative undercut for an ar- gument Φ,α, then so are Ψ,¬(φ2 ∧...∧φn ∧φ1) and Ψ,¬(φ3 ∧...∧φn ∧φ1 ∧ φ2) and so on. However, they are all identical (in the sense that each is more conser- vative than the others). We can ignore the unnecessary variants by just considering the canonical undercuts defined as follows. Definition 7.7. An argument Ψ,¬(φ1 ∧ ... ∧ φn) is a canonical undercut for Φ,α iff it is an undercut for Φ,α and φ1,...,φn is the canonical enumera- tion of Φ. Recall (from the Preliminaries section) that the ordering given by the canonical enumeration has no meaning and is not meant to represent any respective importance
- 156. 142 Philippe Besnard and Anthony Hunter of formulae in Δ. It is only a convenient way to indicate the order in which we assume the formulae in any subset of Δ are conjoined to make a formula logically equivalent to that subset. Example 7.14. Returning to Example 7.13, suppose the canonical enumeration is as follows. r, p,r → ¬p∧¬q,q Then both the following are maximally conservative undercuts, but only the first is a canonical undercut. {r,r → ¬p∧¬q},¬(p∧q) {r,r → ¬p∧¬q},¬(q∧ p) The nice feature of canonical undercuts is that they are all maximally conser- vative undercuts. In other words, an argument Ψ,¬(φ1 ∧ ... ∧ φn) is a canoni- cal undercut for Φ,α iff it is a maximally conservative undercut for Φ,α and φ1,...,φn is the canonical enumeration of Φ. Clearly, an argument may have more than one canonical undercut. This raises the question of how do the canonical undercuts for the same argument look like, and how do they differ from one another? In response to the first question, any two different canonical undercuts for the same argument have the same claim, but dis- tinct supports, and in response to the second question, given two different canonical undercuts for the same argument, none is more conservative than the other. Example 7.15. Let Δ = {a,b,¬a,¬b}. Both the following are canonical undercuts for {a,b},a ↔ b, but neither is more conservative than the other. {¬a},¬(a∧b) {¬b},¬(a∧b) A further important property of canonical undercuts is the following which shows how they give us the useful information concerning counterarguments for an ar- gument: For each defeater Ψ,β of an argument Φ,α, there exists a canonical undercut for Φ,α that is more conservative than Ψ,β. Therefore, the set of all canonical undercuts of an argument represent all the defeaters of that argument (informally, all its counterarguments). This is to be taken advantage of in the next section. 6 Argument trees How does argumentation usually take place? Argumentation starts when an initial argument is put forward, making some claim. An objection is raised, in the form of a counterargument. The latter is addressed in turn, eventually giving rise to a counter-counterargument, if any. And so on. However, there often is more than one counterargument to the initial argument, and if the counterargument actually raised
- 157. 7 Argumentation Based on Classical Logic 143 in the first place had been different, the counter-counterargument would have been different, too, and similarly the counter-counter-counterargument, if any, and so on, and hence the argumentation would have taken a possibly quite different course. So do we find all the alternative courses which could take place from a given initial argument? And is it possible to represent them in a rational way? Let alone the most basic question: How do we make sure that no further counterargument can be expressed from the information available? Example 7.16. Let the following be our knowledgebase. {a,b,c,¬a∨¬b∨¬c} Suppose we start with the following argument. {a,b,c},a∧b∧c Now we have numerous undercuts to this argument including the following. {b,c,¬a∨¬b∨¬c},¬a {a,c,¬a∨¬b∨¬c},¬b {a,b,¬a∨¬b∨¬c},¬c {a,¬a∨¬b∨¬c},¬b∨¬c {b,¬a∨¬b∨¬c},¬a∨¬c {c,¬a∨¬b∨¬c},¬a∨¬b {¬a∨¬b∨¬c},¬a∨¬b∨¬c All these undercuts say the same thing which is that the set {a,b,c} is inconsistent together with the formula ¬a∨¬b∨¬c. As a result, this can be captured by the last undercut listed above. Note this is the maximally conservative undercut amongst the undercuts listed, and moreover it is a canonical undercut. This example therefore illustrates how the canonical undercuts are the undercuts that represent all the other undercuts. Often each undercut may itself be undercut, as illustrated by the next example. Example 7.17. Consider the following atoms that concern the debate on addressing the shortage of oil with biofuel and thereby addressing the effect of the shortage on inflation. However, when grain is used to produce biofuel, then this can cause inflation because it causes shortages of grain for food. But, biofuel can be produced from the large unexploited sources of biowaste (such as the remainder of the wheat plant after the grain has been removed). o there is decreased availability of oil b there is increased production of biofuel g there is decreased availability of food grain i there is increased inflation of prices w there is unexploited availability of biowaste
- 158. 144 Philippe Besnard and Anthony Hunter We can use these for the following formulae. {o,o → b,b → g,b → ¬i,g → i,w,w → b∧¬g} Now we can form the the following argument to capture the first part of the above discussion. {o,o → b,b → ¬i},¬i The following canonical undercut captures the counterargument to the above. {o,o → b,b → g,g → i},¬(o∧(o → b)∧(b → ¬i)) We can capture the counterargument to the above undercut by the following canon- ical undercut. {w,w → b∧¬g},¬(o∧(o → b)∧(b → g)∧(g → i)) So we can represent the above discussion by three arguments, where we start with an argument, we have an undercut to that argument, and an undercut to the undercut. It is also common to have more than one canonical undercut for an argument. Multiple undercuts are illustrated by the following example. Example 7.18. Suppose we have the following knowledgebase. {(¬r → ¬p) → ¬q,(p → s) → ¬¬q,r,s,¬p} Also suppose we construct the following argument. {r,(¬r → ¬p) → ¬q},¬q Then the following two arguments are each a canonical undercut to the above argu- ment. {s,(p → s) → ¬¬q},¬(r ∧(¬r → ¬p) → ¬q) {¬p,(p → s) → ¬¬q},¬(r ∧(¬r → ¬p) → ¬q) Finally, the following argument is a canonical undercut to the first of the above canonical undercuts. {¬p,(¬r → ¬p) → ¬q},¬(s∧(p → s) → ¬¬q) We now turn to one final issue before we formalize the notion of argument trees. Example 7.19. Argumentation sometimes falls into a repetitive and uninformative cycle as illustrated below with a case of the “Chicken and Egg dilemma”. Dairyman: – Egg was first Farmer: – Chicken was first Dairyman: – Egg was first Farmer: – Chicken was first . . . – . . .
- 159. 7 Argumentation Based on Classical Logic 145 The following propositional atoms are introduced: p Egg was first q Chicken was first r The chicken comes from the egg s The egg comes from the chicken It can be assumed that the chicken was first and that the egg was first are not equivalent. (i.e. ¬(p ↔ q)). Also, it can be assumed that the egg comes from the chicken (i.e. s) and the chicken comes from the egg (i.e. r). Moreover, if the egg comes from the chicken then the egg was not first. (i.e. s → ¬p). Similarly, if the chicken comes from the egg then the chicken was not first (i.e. r → ¬q). Then, the above dispute can be represented as follows: {s → ¬p,s,¬(p ↔ q)},q ↑ {r → ¬q,r,¬(p ↔ q)}, p ↑ {s → ¬p,s,¬(p ↔ q)},q ↑ {r → ¬q,r,¬(p ↔ q)}, p ↑ . . . We are now ready for our definition (below) of an argument tree in which the root of the tree is an argument of interest, and the children for any node are the canonical undercuts for that node. In the definition, we avoid the circularity seen in the above example by incorporating an intuitive constraint. Definition 7.8. An argument tree for α is a tree where the nodes are arguments such that 1. The root is an argument for α. 2. For no node Φ,β with ancestor nodes Φ1,β1,...,Φn,βn is Φ a subset of Φ1 ∪···∪Φn. 3. The children nodes of a node N consist of all canonical undercuts for N that obey 2. We illustrate the definition of an argument tree in the following examples. Example 7.20. Returning to Example 7.9, we have the following five formulae. p Simon Jones is a Member of Parliament p → ¬q If Simon Jones is a Member of Parliament then we need not keep quiet about details of his private life r Simon Jones just resigned from the House of Commons r → ¬p If Simon Jones just resigned from the House of Commons then he is not a Member of Parliament ¬p → q If Simon Jones is not a Member of Parliament then we need to keep quiet about details of his private life
- 160. 146 Philippe Besnard and Anthony Hunter These can be used to construct the argument tree below. {p, p → ¬q},¬q ↑ {r,r → ¬p},¬(p∧(p → ¬q)) Example 7.21. Given Δ = {a,a → b,c,c → ¬a,¬c ∨ ¬a}, we have the following argument tree. {a,a → b},b ր տ {c,c → ¬a},¬(a∧(a → b)) {c,¬c∨¬a},¬(a∧(a → b)) Note the two undercuts are equivalent. They do count as two arguments because they are based on two different items of the knowledgebase (even though these items turn out to be logically equivalent). For the rest of this chapter, we adopt a lighter notation, writing Ψ,✸ for a canonical undercut of Φ,β. Clearly, ✸ is ¬(φ1 ∧...∧φn) where φ1,...,φn is the canonical enumeration for Φ. Example 7.22. Consider the following atoms concerning the safety of mobiles phones for children. q1 Mobile phones are safe for children q2 Mobile phones have a health risk q3 Mobile phones heat the brain q4 Mobile phones emit strong electromagnetic radiation q5 There is a high density of phone masts q6 Mobile phones can be used hands-free q7 Hot baths heat the brain Now suppose we have the following knowledgebase {¬q2,¬q2 → q1,q4,q4 → q3,q3 → q2,q5,q5 → ¬q4,q6,q6 → ¬q3,q7,q7 → ¬q2} From this knowledgebase, we can obtain the following argument tree. {¬q2,¬q2 → q1},q1 ↑ {q4,q4 → q3,q3 → q2},✸ ր ↑ տ {q5,q5 → ¬q4},✸ {q6,q6 → ¬q3},✸ {q7,q7 → ¬q2},✸ We motivate the conditions of Definition 7.8 as follows: Condition 2 is meant to avoid the situation illustrated by Example 7.23; and Condition 3 is meant to avoid the situation illustrated by Example 7.24.
- 161. 7 Argumentation Based on Classical Logic 147 Example 7.23. Let Δ = {a,a → b,c → ¬a,c}. {a,a → b},b ↑ {c,c → ¬a},✸ ↑ {a,c → ¬a},✸ This is not an argument tree because Condition 2 is not met. The undercut to the undercut is actually making exactly the same point (that a and c are incompatible) as the undercut itself does, just by using modus tollens instead of modus ponens. Example 7.24. Given Δ = {a,b,a → c,b → d,¬a∨¬b}, consider the following tree. {a,b,a → c,b → d},c∧d ր տ {a,¬a∨¬b},¬b {b,¬a∨¬b},¬a This is not an argument tree because the two children nodes are not maximally conservative undercuts. The first undercut is essentially the same argument as the second undercut in a rearranged form (relying on a and b being incompatible, as- sume one and then conclude that the other doesn’t hold). If we replace these by the maximally conservative undercut {¬a∨¬b},✸, we obtain an argument tree. The form of an argument tree is not arbitrary. It summarizes all possible courses of argumentation about the argument in the root node. Each node except the root node is the starting point of an implicit series of related arguments. What happens is that for each possible course of argumentation (from the root), an initial sequence is provided as a branch of the tree up to the point that no subsequent countern-argument needs a new item in its support (where new means not occurring somewhere in that initial sequence). Also, the counterarguments in a course of argumentation may somewhat differ from the ones in the corresponding branch of the argument tree: Example 7.25. We return to Example 7.18 which has the following knowledgebase. {(¬r → ¬p) → ¬q,(p → s) → ¬¬q,r,s,¬p} The argument tree with {r,(¬r → ¬p) → ¬q},¬q as its root is {r,(¬r → ¬p) → ¬q},¬q ր տ {s,(p → s) → ¬¬q},✸ {¬p,(p → s) → ¬¬q},✸ ↑ {¬p,(¬r → ¬p) → ¬q},✸ Example 7.26. We return to the “Chicken and Egg dilemma” presented in Example 7.19
- 162. 148 Philippe Besnard and Anthony Hunter Dairyman: – Egg was first Farmer: – Chicken was first Dairyman: – Egg was first Farmer: – Chicken was first . . . – . . . Here are the formulae again: p Egg was first q Chicken was first r The chicken comes from the egg s The egg comes from the chicken ¬(p ↔ q) That the egg was first and that the chicken was first are not equivalent r → ¬q The chicken comes from the egg implies that the chicken was not first s → ¬p The egg comes from the chicken implies that the egg was not first So, Δ = {¬(p ↔ q),r → ¬q,s → ¬p,r,s}. The argument tree with the dairyman’s argument as its root is {r → ¬q,r,¬(q ↔ p)}, p ↑ {s → ¬p,s,¬(q ↔ p)},✸ but it does not mean that the farmer has the last word nor that the farmer wins the dispute! The argument tree is merely a representation of the argumentation (in which the dairyman provides the initial argument). Although the argument tree is finite, the argumentation here is infinite and unresolved. We now consider a widely used criterion in argumentation theory for determining whether the argument at the root of the argument tree is warranted (e.g. [21, 15]). For this, each node is marked as either U for undefeated or D for defeated. Definition 7.9. The judge function, denoted Judge, assigns either Warranted or Unwarranted to each argument tree T such that Judge(T) = Warranted iff Mark(Ar) = U where Ar is the root node of T. For all nodes Ai in T, if there is child Aj of Ai such that Mark(Aj) = U, then Mark(Ai) = D, otherwise Mark(Ai) = U. As a direct consequence of the above definition, the root is undefeated iff all its children are defeated. Example 7.27. Returning to Example 7.22, we see that the root of the tree T is un- defeated, and hence Judge(T) = Warranted. Example 7.28. Returning to Example 7.26, we see that the root of the tree T is de- feated, and hence Judge(T) = Unwarranted. In general, a complete argument tree (i.e. an argument tree with all the canonical undercuts for each node as children of that node) provides an efficient representation of the arguments and counterarguments. Furthermore, if Δ is finite, there is a finite number of argument trees with the root being an argument with claim α that can be formed from Δ, and each of these trees has finite branching and a finite depth.
- 163. 7 Argumentation Based on Classical Logic 149 7 Discussion In this chapter, we have reviewed a framework for argumentation based on classical logic. The key features of this framework are the clarification of the nature of ar- guments and counterarguments, the identification of canonical undercuts which we argue are the only undercuts that we need to take into account, and the representa- tion of argument trees which provide a way of exhaustively collating arguments and counterarguments. This chapter is based on a particular proposal for logic-based ar- gumentation for the propositional case [3, 6]. In comparison with the previous pro- posals based on classical logic (e.g. [1, 20]), our proposal provides a much more detailed analysis of counterarguments, and ours is the first proposal to consider canonical undercuts. We believe that without the notion of canonical undercuts, the number of undercuts available is unmanageable, and so restricting to canonical un- dercuts reduces the number of undercuts to consider, and renders it manageable. It is interesting to compare our approach with other logic-based approaches to argumentation that use logics other than classical logic for the definition of deduc- tion. The most common alternative is a form of defeasible logic such as defeasi- ble logic programming [15], defeasible argumentation with specificity-based pref- erences [23], and argument-based extended logic programming [21]. For a general coverage of defeasible logics in argumentation see [10, 22, 6]. Whilst there are various positive features of these proposals based on defeasible logic, including computational viability, and modeling of intuitive features of de- feasible reasoning, there are complications with these proposals with respect to the interplay between strict rules and defeasible rules, and the interplay between these rules and the use of priorities over rules. This can render the deductive process to be less than transparent. In comparison, we believe that using classical logic provides a simpler and clear notion of deduction, of argument, and of counterargument. Furthermore, various questions of rationality have been raised concerning the use of defeasible logic for deduction in argumentation [8]. One of the proposals made for rationality is that contrapositive reasoning needs to be supported. Introducing contrapositive reasoning is controversial in defeasible logic [9], but it is an intrinsic feature of classical logic. In other words, classical logic offers a nice solution to the problems raised in [8]. A more general approach to logic-based argumentation is to leave the logic for deduction as a parameter. This was proposed in abstract argumentation systems [25], and developed in assumption-based argumentation (ABA) [11]. A substantial part of the development of the theory and implement of ABA is focused on defeasible logic. However, ABA is a general framework allowing for the use classical logic for deduction, and thereby we could instantiate ABA with our proposal. It is also interesting to compare our approach with abstract argument systems. Su- perficially, an argument tree could be viewed as an argument framework in Dung’s system. An argument in an argument tree could be viewed as an argument in a Dung argument framework, and each arc in an argument tree could be viewed as an attack relation. However, the way sets of arguments are compared is different.
- 164. 150 Philippe Besnard and Anthony Hunter Some differences between Dung’s approach and our approach can be seen in the following examples. Example 7.29. Consider a set of arguments {A1,A2,A3,A4} with the attack relation R s.t. A2RA1, A3RA2, A4RA3, and A1RA4. Here there is an admissible set {A1,A3}. We can try to construct an argument tree with A1 at the root. As a counterpart to the attack relation, we regard that A1 is undercut by A2, A2 is undercut by A3, and so on. However, the corresponding sequence of nodes A1,A2,A3,A4,A1 is not an argument tree because A1 occurs twice in the branch (violating Condition 2 of Definition 7.8). So, the form of the argument tree for A1 fails to represent the fact that A1 attacks A4. Example 7.30. Let Δ = {b,b → a,d ∧¬b,¬d ∧¬b}, giving the following argument tree for a. {b,b → a},a ր տ {d ∧¬b},✸ {¬d ∧¬b},✸ ↑ ↑ {¬d ∧¬b},✸ {d ∧¬b},✸ For this let A1 be {b,b → a},a, A2 be {d ∧ ¬b},✸ and A3 be {¬d ∧ ¬b},✸. Disregarding the difference between the occurrences of ✸, this argument tree rewrites as A2RA1, A3RA1, A3RA2, and A2RA3 where A1 denotes the root node {b,b → a},a. In this argument tree, each defeater of the root node is defeated. Yet no admissible set of arguments contains A1. Our proposal has been extended in a number of ways. As covered in [6], we have proposed techniques that take into account intrinsic aspects of the arguments and counterargument such as the degree of conflict between them, and the degree of similarity between them, and extrinsic aspects such as the impact on the audience and the empathy or antipathy the audience may have for individual arguments. Further developments include formalization of enthymemes in our logic-based framework [16, 7], a generalization of the framework to also consider the proponent of each argument, and thereby argue about whether a proponent is an appropriate proponent for that argument [17], and a refinement of the proposal to reason with temporal knowledge [18]. We have also generalised the proposal reviewed in this chapter to handle first- order logic [4]. Indeed, this is straightforward since all we need to do is to use a first-order language and to use the classical first-order consequence relation instead of the classical propositional logic. The key finiteness results still hold: So for a finite number of formulae in the knowledgebase, there is a finite number of argument trees for a claim, and each of these trees contains a finite number of nodes. We have developed algorithms for generating arguments [12, 13], formaliza- tions of decision problems concerning arguments and counterarguments in quan- tified Boolean formaulas [2], and compilation techniques with the aim of improving the viability of argumentation based on classical logic [5].
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- 166. 152 Philippe Besnard and Anthony Hunter 19. D. Nute. Defeasible logics. In Handbook of Logic in Artificial Intelligence and Logic Pro- gramming, Volume 3: Nonmonotonic Reasoning and Uncertainty Reasoning, pages 355–395. Oxford University Press, 1994. 20. J. Pollock. How to reason defeasibly. Artificial Intelligence, 57:1–42, 1992. 21. H. Prakken and G. Sartor. Argument-based extended logic programming with defeasible pri- orities. Journal of Applied Non-classical Logic, 7:25–75, 1997. 22. H. Prakken and G. Vreeswijk. Logical systems for defeasible argumentation. In D. Gabbay, editor, Handbook of Philosophical Logic, pages 219–318. Kluwer, 2002. 23. G. Simari and R. Loui. A mathematical treatment of defeasible reasoning and its implemen- tation. Artificial Intelligence, 53:125–157, 1992. 24. B. Verheij. Automated argument assistance for lawyers. In Proceedings of the 7th Interna- tional Conference on Artificial Intelligence and Law (ICAIL 1999), pages 43–52. ACM Press, 1999. 25. G. Vreeswijk. Abstract argumentation systems. Artificial Intelligence, 90:225–279, 1997.
- 167. Chapter 8 Argument-based Logic Programming Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari 1 Introduction In this chapter we describe several formalisms for integrating Logic Programming and Argumentation. Research on the relation between logic programming and ar- gumentation has been and still is fruitful in both directions: Some argumentation formalisms were used to define semantics for logic programming and also logic programming was used for providing an underlying representational language for non-abstract argumentation formalisms. One of the first attempts dates back to 1987, when Donald Nute [19] introduced a formalism called LDR (Logic for Defeasible Reasoning) with a simple represen- tational language consisting of three types of rules: Strict, defeasible and defeaters. Although LDR is not a defeasible argumentation formalism in itself, its implemen- tation, d-Prolog, defined as an extension of PROLOG, was the first language that introduced defeasible reasoning programming with specificity as a comparison cri- terion between rules. Another important step was taken in the nineties, when Phan M. Dung [10] em- phasized that “there are extremely interesting relations between argumentation and logic programming”. Dung showed that argumentation can be viewed as a special form of logic programming with negation as failure. He introduced a general logic Alejandro J. Garcı́a Consejo Nacional de Investigaciones Cientı́ficas y Técnicas (CONICET) Department of Computer Science and Engineering, Universidad Nacional del Sur e-mail: ajg@cs.uns.edu.ar Guillermo R. Simari Department of Computer Science and Engineering, Universidad Nacional del Sur e-mail: grs@cs.uns.edu.ar Jürgen Dix Department of Informatics Clausthal University of Technology e-mail: dix@tu-clausthal.de I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 153 DOI 10.1007/978-0-387-98197-0 8, c Springer Science+Business Media, LLC 2009
- 168. 154 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari programming based method for generating meta-interpreters for argumentation sys- tems. A few years later, in 1997, inspired by legal reasoning, Prakken and Sartor [24] introduced an argument-based formalism for extended logic programming with de- feasible priorities. In their formalism, arguments are expressed in a logic program- ming language with both strong and default negation. Conflicts between arguments are decided with the help of priorities on the rules. These priorities can be defeasibly derived as conclusions within the system. The semantics of the system is given by a fixed point definition, while its proof theory is stated in dialectical style. A proof takes the form of a dialogue between a proponent and an opponent: An argument is shown to be justified if the proponent can make the opponent run out of moves in whatever way the opponent attacks. Another formalism for combining logic programming and argumentation, Defea- sible Logic Programming (DeLP) has been introduced by Garcı́a and Simari in [15]. The representational language of DeLP is defined as an extension of a logic pro- gramming language that considers two types of rules: Strict and defeasible, and allows for both strong and default negation. A DeLP-query succeeds, i.e., it is war- ranted from a DeLP-program, if it is possible to build an argument that supports the query and this argument is found to be undefeated by a warrant procedure. This process implements an exhaustive dialectical analysis that involves the construction and evaluation of arguments that either support or interfere with the given query. The DeLP dialectical analysis is based on previous works on defeasible argumentation conducted by Simari and Loui [28], and Simari, Chesñevar, and Garcı́a [27]. In [26], Schweimeier and Schroeder formulate a variety of notions of attack for extended logic programs from combinations of undercut and rebuttal. As shown in Section 4, their language corresponds to that used by Prakken and Sartor [24] without strict rules and any priorities. They also define a general hierarchy of argu- mentation semantics parameterized by the notions of attack chosen by the proponent and the opponent. They prove the equivalence and subset relationships between the semantics and examine some essential properties concerning consistency and the coherence principle, which relates default negation and explicit negation. The rest of the chapter is organized as follows. Section 2 describes Defeasible Logic Programming. Section 3 introduces Prakken’s approach of argument-based extended logic programming with defeasible priorities. Section 4 gives an overview of argumentation semantics for extended logic programming by Schweimeier and Schroeder. Section 5 introduces Dung’s approach and Section 6 illustrates Nute’s framework. Finally, in Section 7 we discuss more recent developments before con- cluding with Section 8. 2 Defeasible Logic Programming Defeasible Logic Programming (DeLP), as introduced in [15], is a formalism that combines techniques of both logic programming and defeasible argumentation. As
- 169. 8 Argument-based Logic Programming 155 in logic programming, in DeLP knowledge is represented using facts and rules; however, DeLP also provides the possibility of representing information in the form of weak rules in a declarative manner. These weak rules are the key element for in- troducing defeasibility [21] and they are used to represent a relation between pieces of knowledge that could be defeated when all things have been considered. DeLP uses a defeasible argumentation inference mechanism for warranting the entailed conclusions. A defeasible logic program (DeLP-program for short), is a set of facts, strict rules and defeasible rules, defined as follows. • Facts are ground literals representing atomic information or the negation of atomic information using strong negation “∼”, (e.g., chicken(little) and ∼scared(little)). • Strict rules, denoted by L0 ← L1,...,Ln, represent non-defeasible information. The head of the rule, L0, is a ground literal and the body {Li}i0 is a non-empty set of ground literals, (e.g., bird ← chicken and ∼innocent ← guilty). • Defeasible rules, denoted by L0 — L1,...,Ln, represent tentative information. The head L0 is a ground literal and the body {Li}i0 is a non-empty set of ground literals (e.g., ∼flies — chicken or flies — chicken,scared). Syntactically, the symbol “— ” is all that distinguishes a defeasible rule from a strict one. Pragmatically, a defeasible rule is used to represent tentative information that may be used if nothing could be posed against it. A defeasible rule “H — B” is understood as expressing that “reasons to believe in the antecedent B provide reasons to believe in the consequent H” [28]. For instance, “lights on — switch on” expresses that reasons to believe that the switch of a room is on, provide reasons to believe that lights will be on. Note that this rule represents tentative information because it may happen that the switch is on and the lights are not on, for example, if light bulbs are broken, or there is no electricity (as represented by the following rule: “∼lights on — ∼electricity”). The information that represents a strict rule is not tentative. For instance, given “∼innocent ← guilty” and the fact guilty, then it will not be possible to infer innocent. That is, if guilty has a strict derivation then ∼innocent will also have a strict derivation and innocent cannot be concluded. However, as it will be shown below, if guilty is a tentative conclusion, then ∼innocent will also be a tentative conclusion, and innocent may be concluded if there is information that supports it. When required, a DeLP-program is denoted by the pair (Π,Δ) distinguishing the subset Π of facts and strict rules (that represents non-defeasible knowledge), and the subset Δ of defeasible rules. Observe that strict and defeasible rules are ground. However, following the usual convention [17], some examples will use “schematic rules” with variables. To distinguish variables, they are written with an uppercase letter. In examples, a period may be used as a delimiter between rules or facts. As in logic programming, queries can be posed to a program. A DeLP-query is a ground literal that DeLP will try to warrant. For example, dark(a) or illuminated(b) are DeLP-queries. Example 8.1. Consider the DeLP-program (Π8.1,Δ8.1) where:
- 170. 156 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari Π8.1= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ night. switch on(a). ∼day ← night. switch on(b). ∼dark(X) ← illuminated(X). switch on(c). sunday. ∼electricity(b). deadline. ∼electricity(c). emergency lights(c). ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ Δ8.1= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ lights on(X) — switch on(X). ∼lights on(X) — ∼electricity(X). lights on(X) — ∼electricity(X),emergency lights(X). dark(X) — ∼day. illuminated(X) — lights on(X),∼day. working at(X) — illuminated(X). ∼working at(X) — sunday. working at(X) — sunday,deadline. ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ The set Π8.1 has information about three rooms: a, b and c. For instance, there are facts expressing that in room a the light switch is on, in room b there is no electricity and in room c there are emergency lights. There are also facts expressing that it is Sunday night and that people working there have a deadline. Observe that Π8.1 has also two strict rules. For instance, the second one express that an illuminated room is not dark. The set Δ8.1 provides defeasible rules representing tentative information, that can be used to infer, for instance, which room is illuminated, or if someone is working in a particular room. For example, the first defeasible rule states that “reasons to believe that the switch of a room is on, provides reasons to believe that the lights of that room are on”. The second rule express that “usually if there is no electricity then lights of a room are not on”. The third defeasible rule states that “ normally, if there is no electricity but there are emergency lights, then lights will be on”. The last two rules state that “normally there is nobody working in a room on a Sunday”, however, “if they have a deadline, people may be working on a Sunday”. As it will be explained below, there are several queries that succeed with respect to this program because they are warranted (e.g., illuminated(a), illuminated(c), ∼dark(c), dark(b) and working at(a)), whereas other queries are not warranted (e.g., illuminated(b), ∼illuminated(c)). Defeasible rules allow to infer tentative conclusions. A defeasible derivation of a literal Q from a DeLP-program (Π,Δ), denoted by (Π,Δ)|∼ Q, is a finite sequence of ground literals L1,L2,...,Ln = Q, where either: 1. Li is a fact in Π, or 2. there exists a rule Ri in (Π,Δ) (strict or defeasible) with head Li and body B1,B2,...,Bk and every literal of the body is an element Lj of the sequence ap- pearing before Li ( j i.) The sequence: switch on(b), lights on(b), night, ∼day, illuminated(b), is a de- feasible derivation for illuminated(b) from (Π8.1,Δ8.1). In particular, a derivation
- 171. 8 Argument-based Logic Programming 157 from (Π, / 0) is called a strict derivation, (e.g., night,∼day in Example 8.1). That is, a derivation is defeasible if at least one defeasible rule is used, otherwise it is strict. Two literals are considered contradictory if one is the complement of the other with respect to strong negation, (e.g., day and ∼day). Since Π represents non-defeasible information, then it is natural to require certain internal coherence. Therefore, in DeLP, two contradictory literals cannot have a strict derivation from a valid pro- gram. Nevertheless, as expected, from a valid DeLP-program, there can result contra- dictory literals that have defeasible derivations (e.g., lights on(b) and ∼lights on(b) from (Π8.1,Δ8.1)). In these situations, in order to decide which literal is accepted as warranted, DeLP incorporates a defeasible argumentation formalism that allows the identification of the pieces of knowledge that are in contradiction. Then, a dialectical process is used for deciding which information prevails as warranted. This process involves the construction and evaluation of arguments and counter-arguments, as described next. Definition 8.1 (Argument Structure). Let H be a ground literal, (Π,Δ) a DeLP- program, and A⊆Δ. The pair A,H is an argument structure if: 1. there exists a defeasible derivation for H from (Π,A), 2. there is no defeasible derivation from (Π,A) of contradictory literals, 3. and there is no proper subset A′ of A such that A′ satisfies (1) and (2). In an argument structure A,H, H is the claim and A is the argument support- ing that claim. For instance, from the DeLP-program of Example 8.1 the follow- ing argument structures can be obtained: A1,dark(b), A2,illuminated(b), and A3,∼lights on(b), where A1={dark(b) — ∼day} A2={(illuminated(b) — lights on(b),∼day), (lights on(b) — switch on(b)} A3={∼lights on(b) — ∼electricity(b)} Observe that an argument is a set of defeasible rules; however, strict rules may be used for the defeasible derivation of its claim. For instance, A1 uses the strict rule ∼day ← night for the derivation of ∼day. A DeLP-query Q succeeds, i.e., it is warranted from a DeLP-program, if it is possible to build an argument A that supports Q and A is found to be undefeated by a warrant procedure. This process implements an exhaustive dialectical analysis that involves the construction and evaluation of arguments that either support or interfere with the query under analysis. That is, given an argument A that supports Q, the warrant procedure will evaluate if there are other arguments that counter- argue or attack A or a sub-argument of A. An argument C,P is a sub-argument of A,Q if C⊆A. Definition 8.2 (Counter-Argument). An argument B,S is a counter-argument for A,H at literal P, if there exists a sub-argument C,P of A,H such that P and S disagree, that is, there exist two contradictory literals that have a strict derivation from Π ∪ {S,P}. The literal P
- 172. 158 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari is referred to as the counter-argument point and C,P as the disagreement sub- argument. Two contradictory literals trivially disagree (e.g., day and ∼day), however, two literals that are non-contradictory can also disagree. For instance, in Ex- ample 8.1, the literals dark(b) and illuminated(b) disagree because from Π8.1∪ {illuminated(b), dark(b)} there exists a strict derivation for ∼dark(b). Hence, following the example above, A2,illuminated(b) is a counter-argument for A1,dark(b) (and vice versa). Observe that the argument A3,∼lights on(b) is a counter-argument for A2,illuminated(b) at (the inner point) lights on(b) and the disagreement sub-argument in A2 is {lights on(b) — switch on(b)},lights on(b). Given an argument A,H, there could be several counter-arguments attacking different points in A, or different counter-arguments attacking the same point in A. In DeLP, in order to verify whether an argument is non-defeated, all of its associ- ated counter-arguments B1, B2, ..., Bk have to be examined, each of them being a potential reason for rejecting A. If any Bi is (somehow) “better” than, or unrelated to, A, then Bi is a candidate for defeating A. However, if for some Bi, the argument A is “better” than Bi, then Bi will not be a defeater for A. To compare arguments and counter-arguments a preference relation among arguments is needed. In DeLP the argument comparison criterion is modular, and thus the most ap- propriate criterion for the domain that is being represented can be introduced. In the literature of DeLP different criteria have been defined. For example in [15] a crite- rion that uses rule priorities was introduced. Recently, in [11] a comparison criterion based on priorities among selected literals of the program was defined. In the follow- ing examples we will use a syntactic criterion called generalized specificity [15, 29]. This criterion favors two aspects in an argument: It prefers (1) a more precise argu- ment or (2) a more concise argument. For instance, A2,illuminated(b), is more specific (more precise) than A1,dark(b). For the following definitions we will abstract away from the comparison criterion, assuming there exists one (denoted “≻”). Definition 8.3 (Defeaters: Proper and Blocking). Let B,S be a counter-argument for A,H at point P, and C,P the disagreement sub-argument. If B,S ≻ C,P (i.e., B,S is “better” than C,P) then B,S is a proper defeater for A,H. If B,S is unrelated by the preference relation to C,P, (i.e., B,S ≻ C,P, and C,P ≻ B,S) then B,S is a blocking defeater for A,H. Finally, B,S is a defeater for A,H, if B,S is either a proper or blocking defeater for A,H. In a previous example, we have shown that A2,illuminated(b) is a counter- argument for A1,dark(b) (and vice versa). Considering generalized specificity as the comparison criterion, then A2 ≻ A1. Therefore, A2,illuminated(b) is a proper defeater for A1,dark(b). The argument A3,∼lights on(b) is a blocking defeater for A2,illuminated(b) because A3 and the disagreement sub-argument D1 ={ lights on(b) — switch on(b)} of A2 are unrelated by this comparison criterion. If an argument A1,H1 is defeated by A2,H2, then A2,H2 represents a rea- son for rejecting A1,H1. Nevertheless, a defeater A3,H3 for A2,H2 may also
- 173. 8 Argument-based Logic Programming 159 exist, rejecting A2,H2 and reinstating A1,H1. Note that the argument A3,H3 may be in turn defeated, reinstating A2,H2, and so on. In this manner, a sequence of arguments (called argumentation line) can arise, where each element is a defeater of its predecessor. Definition 8.4 (Argumentation Line). An argumentation line for A1,H1 is a se- quence of argument structures, Λ = [A1,H1, A2,H2, A3,H3, . . . ], where each element of the sequence Ai,Hi, i 1, is a defeater of its predecessor Ai−1,Hi−1. In an argumentation line Λ = [A1,H1, A2,H2, A3,H3, . . . ], the first ele- ment, A1,H1, becomes a supporting argument for H1, A2,H2 an interfering ar- gument, A3,H3 a supporting argument, A4,H4 an interfering one, and so on. Thus, an argumentation line can be split into two disjoint sets: ΛS = {A1,H1, A3,H3, A5,H5, ...} of supporting arguments, and ΛI = {A2,H2, A4,H4, ...} of interfering arguments. Considering the arguments generated above from (Π8.1,Δ8.1) of Example 8.1, Λ1 = [A1,dark(b), A2,illuminated(b), A3,∼lights on(b)] is an argumentation line, with two supporting arguments and an interfering one. Note that an infinite argumentation line may arise if an argument structure is rein- troduced in the sequence. For instance, Λi = [A3,∼lights on(b), D1,lights on(b), A3,∼lights on(b), D1,lights on(b), ... ]. This is an example of circular argu- mentation where an argument structure is reintroduced again in the argumentation line to defend itself. Clearly, this situation is undesirable as it leads to the construc- tion of an infinite sequence of arguments. Circular argumentation and other forms of fallacious argumentation were studied in detail in [27, 15]. In DeLP, argumentation lines have to be acceptable, that is, they have to be finite, an argument cannot appear twice, and supporting (resp. interfering) arguments have to be non-contradictory. Definition 8.5 (Acceptable Argumentation Line). An argumentation line Λ = [A1,H1,...An,Hn] is acceptable iff: 1. Λ is a finite sequence. 2. The set ΛS of supporting arguments (resp. ΛI), is concordant (a set {Ai,Hi}n i=1 is concordant iff Π ∪ n i=1 Ai is non-contradictory.). 3. No argument Ak,Hk in Λ is a disagreement sub-argument of an argument Ai,Hi appearing earlier in Λ (i k.) 4. For all i, such that Ai,Hi is a blocking defeater for Ai−1,Hi−1, if Ai+1,Hi+1 exists, then Ai+1,Hi+1 is a proper defeater for Ai,Hi. A single acceptable argumentation line for A1,H1 may not be enough to es- tablish whether A1,H1 is an undefeated argument. The reason is that there can be several defeaters (B1,Q1, B2,Q2, ..., Bk,Qk) for A1,H1, and therefore k argumentation lines for A1,H1 need to be considered. Since for each defeater Bi,Qi there can be in turn several defeaters, then a tree structure (called dialec- tical tree) is defined. In this tree, the root is labeled with A1,H1 and every node (except the root) represents a defeater (proper or blocking) of its parent. Each path from the root to a leaf corresponds to a different acceptable argumentation line. The definition follows.
- 174. 160 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari Definition 8.6 (Dialectical Tree). A dialectical tree for A1,H1, TA1,H1, is de- fined as follows: 1. The root of the tree is labeled with A1,H1. 2. Let N be a non-root node labeled An,Hn, and [A1,H1,...,An,Hn] be the sequence of labels of the path from the root to N. Let {B1,Q1, B2,Q2, ..., Bk,Qk} be the set of all the defeaters for An,Hn. For each defeater Bi,Qi (1≤i≤k), such that the argumentation line Λ′ = [A1,H1,...,An,Hn,Bi,Qi] is acceptable, the node N has a child Ni labeled Bi,Qi. If there is no defeater for An,Hn or there is no Bi,Qi such that Λ′ is acceptable, then N is a leaf. In Example 8.1, C1,working at(c) has two defeaters: C2,∼lights on(c), and C3,∼working at(c), where: C1= {(working at(c) — illuminated(c)), (illuminated(c) — lights on(c),∼day), (lights on(c) — switch on(c))}; C2= {∼lights on(c) — ∼electricity(c)} and C3={∼working at(c)— sunday}. Since C2 and C3 have defeaters, then the following acceptable argumentation lines arise: [C1, C2, C4] where C4={ working at(c) — sunday,deadline}, and [C1, C3, C5] where C5={lights on(c)— ∼electricity(c), emergency lights(c)}. These two lines are con- sidered in the dialectical tree T 1 ∗ of Figure 8.1. A dialectical tree provides a structure integrating all the possible acceptable ar- gumentation lines that can be generated for deciding whether an argument is unde- feated. In order to compute whether the status of the root is defeated or undefeated, a recursive marking process is introduced next. Let T be a dialectical tree, a marked dialectical tree, T∗, can be obtained marking every node in T as follows: (1) Each leaf in T is marked as U in T∗. (2) An inner node N of T is marked as D in T∗ iff N has at least a child marked as U, otherwise N is marked as U in T∗. Thus, the root A,H of a dialectical tree is marked with U if all its children are marked as D (i.e., all the defeaters for A,H are defeated). Figure 8.1 shows three different marked dialectical trees. Nodes are depicted as triangles with marks (D or U) on their right. D-nodes are also grey colored. The marked dialectical tree T 1 ∗ has also the argument names of the example introduced above as labels inside the triangles. A ground literal H is considered to be warranted if an argument A for H exists, and all the defeaters for A,H are defeated. Therefore, this marking procedure provides an effective way of determining if a DeLP-query H is warranted. Note that given a DeLP-query H, there can be several arguments that support H. Therefore, H is warranted if there is at least one argument A for H such that the root of a dialectical tree for A,H is marked as U. Definition 8.7 (Warranted Literals). Let P be a DeLP-program, H a ground literal, A,H an argument from P and T∗ a marked dialectical tree for A,H. Literal H is warranted from P iff the root of T∗ is marked as U. Example 8.2. Consider again the DeLP-program (Π8.1,Δ8.1) of Example 8.1. The set ω includes some literals that are warranted from the program (Π8.1,Δ8.1) ω ={working at(c), illuminated(c), illuminated(a), dark(b), ∼dark(a)}, whereas the set ω′ includes literals that are not warranted from (Π8.1,Δ8.1) ω′ ={dark(a),
- 175. 8 Argument-based Logic Programming 161 U D U D D U D D D D C2 C3 U U D D U U U U C4 C1 U C5 U U U T1 * T2 * T3 * Fig. 8.1 Marked Dialectical Trees illuminated(b), ∼lights on(b), ∼empty(a),illuminated(r)}. Observe that the last two literals of ω′ have no supporting argument at all. DeLP Implementation and Extensions An interpreter for DeLP is available (http://lidia.cs.uns.edu.ar/DeLP). This interpreter takes a DeLP-program P, and a DeLP-query Q as input. It then returns one of the following four possible answers: YES, if Q is warranted from P; NO, if the complement of Q is warranted from P; UNDECIDED, if neither Q nor its complement are warranted from P; or UNKNOWN, if Q is not in the language of the program P. For instance, considering the program (Π8.1,Δ8.1) of Example 8.1 the answer for illuminated(c), is YES, the answer for dark(a) is NO, the answer for ∼lights on(b) is UNDECIDED, and the answer for ∼empty(a) is UNKNOWN. Observe that for all the literals of the set ω in Example 8.2 the answer is YES. However, as shown above, literals of ω′ may have different answers. The reader may have noticed in Figure 8.1 that in some dialectical trees it is not necessary to explore the whole tree in order to mark the root. Thus, some pruning is possible. For instance, in the tree T 2 ∗ of Figure 8.1 as one of the children of the root is marked U, then the mark of the other two children does not affect the mark of the root. As an optimization, the DeLP interpreter implements a marking procedure with pruning that is similar to the classic α-β pruning algorithm (see [15] for details). It is interesting to note that changes in the definition of acceptable argumentation line may produce a different behavior of the formalism. Therefore, Definition 8.5 could be modified by providing a way of tuning the system in order to have a different behavior. In [15] an extension of DeLP that allows default negation in the body of defeasi- ble rules was defined (e.g., a — not b,c,not ∼d). In DeLP ‘not F’ is assumed when the literal F is not warranted. The definition of argument was adapted for this ex-
- 176. 162 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari tension and a new form of defeat (attack to an assumption) was introduced (see [15] for details). Thus, default negated literals become new points of attack, and when the dialectical analysis is carried out, default negated literals can be defeated by arguments. In [14] dialectical explanations for DeLP-queries (called δ-Explanations) were introduced. δ-Explanations allow to visualize the reasoning carried out by the sys- tem, and the support for the answer. It is clear that without this information at hand it is very difficult to understand the returned answer. δ-Explanations were also defined for schematic queries, i.e., a DeLP-query with at least one variable that represents the set of DeLP-queries that unify with the schematic one (e.g., illuminated(X).) Possibilistic Defeasible Logic Programming (P-DeLP) represents an important extension of DeLP in which the elements of the language have the form (ϕ,α), where ϕ is of the form L0 ← L1,...,Ln or just L0, where the Li,(0 ≤ i ≤ n) are ground literals as in DeLP, and α is a certainty weight associated with ϕ. A P-DeLP-program Γ is a set of clauses. To define the possibilistic entailment we only need the triviality axiom of Possibilistic Gödel Logic (PGL) and the general- ized modus ponens rule (GMP). The triviality axiom says that we can add (ϕ,0) for any formula ϕ in the language and the GMP says that from (L0 ← L1,...,Ln,γ) and (L1,βn),...,(L1,βn) we can obtain (L0,min(γ,β1,...,βn)). A derivation of a weighted literal (Q,α) from Γ , denoted Γ ⊢ (Q,α), is a sequence of clauses C1,...,Cm such that Cm = (Q,α), and for each 1 ≤ i ≤ m it holds that Ci ∈ Γ , or Ci is an instance of the triviality axiom, Ci is obtained by using GMP over its preceding clauses in the sequence. The set of clauses Γ of a P-DeLP program can be split, like in DeLP, as (Π,Δ) distinguishing the subset Π of facts and strict rules with certainty weight α = 1 (rep- resenting non-defeasible knowledge), and the subset Δ of of clauses with certainty weight α 1 (representing defeasible knowledge). An argument for a literal (Q,α) built from a P-DeLP program (Π,Δ) is a subset A ⊆ Δ such that: Π ∪A ⊢ (Q,α), Π ∪A is non-contradictory, and A is minimal with respect to set inclusion satisfying the two previous conditions; and it is denoted A,Q,α. From these extended notions, a complete argumentation system has been devel- oped which incorporates possibilistic reasoning. Complete details of P-DeLP can be found in [1, 2]. 3 Argument-based extended logic programming with defeasible priorities Inspired by legal reasoning, Prakken and Sartor have introduced in [24] an argument- based formalism for extended logic programming with defeasible priorities. In their formalism, arguments are expressed in a logic programming language with both strong and default negation. Conflicts between arguments are decided with the help of priorities on the rules. These priorities can be defeasibly derived as conclusions within the system. Its semantics is given with a fixed point definition, while its proof
- 177. 8 Argument-based Logic Programming 163 theory is stated in dialectical style, where a proof takes the form of a dialogue be- tween a proponent and an opponent: an argument is shown to be justified if the proponent can make the opponent run out of moves in whatever way the opponent attacks. We give a brief description of this formalism and refer the interested reader to [24, 22, 25] for details. Their proposed system assumes input information in the form of an ordered theory (S,D) where S and D are sets of, respectively, strict and defeasible rules. In their language, ∼ represents default negation and ¬ strong nega- tion, and rules are denoted by r : L0 ∧...∧Lj ∧∼Lk ∧...∧∼Lm ⇒ Ln where r, a term, is the name of the rule, each Li (0 ≤ i ≤ n) is a strong literal, and each ∼Lk is a weak literal. Only defeasible rules can contain weak literals. An argument is a finite sequence A = [r0,...,rn] of ground instances of rules such that 1. for every i (0 ≤ i ≤ n), for every strong literal Lj in the antecedent of ri there is a k i such that Lj is the consequent of rk; and 2. no two distinct rules in the sequence have the same consequent. Each consequent of a rule in A is a conclusion of A, and every literal L is an assump- tion of A iff ∼L occurs in some rule in A. An argument is strict iff it does not contain a defeasible rule; it is defeasible otherwise. Note that arguments are not assumed to be either minimal or consistent. The presence of assumptions in a rule gives rise to two kinds of conflicts between arguments, conclusion-to-conclusion attack and conclusion-to-assumption attack. An argument A1 attacks A2 iff there are sequences of strict rules S1 and S2, such that the concatenation A1 + S1 is an argument with conclusion L and (1) A2 + S2 is an argument with conclusion L, or (2) L is an assumption of A2. In order to compare conflicting arguments, priorities among rules (that are relevant to the conflict) are used. For any two sets R and R′ of defeasible rules, R R′ iff for some r ∈ R and all r′ ∈ R′ it holds that r r′. Defeat among arguments is built up from two other notions, rebutting and un- dercutting an argument. An argument A defeats an argument B iff (1) A=[ ] and B attacks itself, or else if (2a) A undercuts B or (2b) A rebuts B and B does not undercut A. As explained in [25], an argument A rebuts an argument B iff A conclusion-to- conclusion attacks B and either A is strict and B is defeasible, or A’s rules involved in the conflict have no lower priority than B’s rules involved in the conflict (see [24, 22] for details). An argument A undercuts an argument B precisely in case of the second kind of conflict (attack on an assumption). Note that it is not necessary that the rules responsible for the attack on the assumption have no lower priority than the one containing the assumption. An argument A strictly defeats B iff A defeats B and B does not attack A. As stated above, priorities between rules can be defeasibly derived as conclu- sions within the system. Hence, the language is extended with a special predicate ≺, where r1 ≺ r2 means that rule r2 has priority over r1. This new predicate symbol
- 178. 164 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari denotes a strict partial order, and therefore the set S must contain the axioms of a strict partial order. Thus, the rebut and defeat relations must be made relative to an ordering relation that might vary during the reasoning process. Since priorities are not fixed, in order to determine if an argument A is acceptable with respect to a set T of arguments, the relevant defeat relations are verified relative to the priority conclusions drawn by the arguments in T. Therefore, given a set of arguments T, priorities between rules are defined as T ={r r′ | r ≺ r′ is a conclu- sion of some A ∈ T}. Hence, an argument A (strictly) T-defeats B iff, assuming the ordering T , A (strictly) defeats B. An argument A is acceptable with respect to a set T of arguments iff all argu- ments T-defeating A are strictly T-defeated by some argument in T. Thus, accept- ability of an argument with respect to a set T depends on the priority conclusions of the arguments in T. Let Γ be an ordered theory (S,D), and T be any set of arguments from Γ ; the characteristic function of Γ is FΓ (T)={A | A is acceptable with respect to T}. Then, the status of an argument A is defined as follows: A is justified iff A is in the least fixed point of FΓ , A is overruled iff A is attacked by a justified argument, and A is defensible iff A is neither justified nor overruled. For example, consider a theory (S,D) where D has the following rules: r0 : ⇒ a r2 : ∼b ⇒ c r4 : ⇒ r0 ≺ r3 r6 : ⇒ r5 ≺ r4 r1 : a ⇒ b r3 : ⇒ ¬a r5 : ⇒ r3 ≺ r0 The set of justified arguments is constructed as follows. The only / 0-undefeated argument is [r6], then, FΓ (/ 0) = F1 = {[r6]} and F1 = {r5 r4}. Now, the con- flict between [r4] and [r5] can be solved because [r4] strictly F1-defeats [r5]. Then, FΓ (F1) = F2 = {[r6], [r4]} and F2 = {r5 r4,r0 r3}. Hence, the conflict between [r0] and [r3] can be solved because [r3] strictly F2-defeats [r0]. Thus, FΓ (F2) = F3 = {[r6], [r4], [r3]} and F3 = F2 . Finally, FΓ (F3) = F4 = {[r6], [r4], [r3], [r2]} and FΓ (F4) = F4. The proof theory of this formalisms is stated in dialectical style, where a proof takes the form of a dialogue between a proponent (P) and an opponent (O). An argument is shown to be justified if P can make O run out of moves. A priority dialogue is a finite sequence of moves (Playeri,Ai), where: 1. Playeri = P iff i is odd, and Playeri = O iff i is even; 2. If Playeri = Playerj = P and i = j, then Ai = Aj; 3. If Playeri = P(i 1) then Ai is a minimal argument such that Ai strictly {Ai}- defeats Ai−1; or Ai−1 does not {Ai}-defeat Ai−2; 4. If Playeri = O(i 1) then Ai strictly / 0-defeats Ai−1; A priority dialogue tree is a finite tree of moves such that (1) each branch is a dialogue and (2) if Playeri = P then the children of movei are all / 0-defeat Ai. A player wins a dialogue if the other player cannot move, and a player wins a dialogue tree iff it wins all branches of the tree. Finally, an argument A is provably justified iff there is a dialogue tree with A as its root, and won by the proponent. The semantics of this formalism was very much inspired by Dung’s work. If it is restricted to static priorities (given as input instead of derived within the system) then it instantiates Dung’s grounded semantics [10]. However, the semantics slightly
- 179. 8 Argument-based Logic Programming 165 differs from Dung’s when dynamic priorities are considered. In [23] some detailed legal examples are given, and a third way of attacking arguments, namely by arguing that a rule is not applicable, is introduced. 4 Argumentation Semantics for Extended Logic Programming In [26], Schweimeier and Schroeder formulate a variety of notions of attack for ex- tended logic programs from combinations of undercutting and rebuttal. A general hierarchy of argumentation semantics is defined, parameterized by the notions of attack chosen by proponent and opponent, and the equivalence and subset relation- ships between the semantics is shown. The placement of existing semantics in this hierarchy is determined. The language corresponds to [24] without strict rules, and either without priori- ties, or, equivalently, if all rules have the same priority. An extended logic program (ELP) P is a (possibly infinite) set of rules of the form L0 ← L1, ..., Lm, not Lm+1, ...,not Lm+n (m,n ≥ 0) where each Li (0 ≤ i ≤ m+n) is an objective literal, i.e., an atom M or its explicit negation ¬M. An argument associated with P is a finite sequence A = [r0,...,rn] of ground instances of rules ri ∈ P such that for every 0 ≤ i ≤ n, for every objective literal Lj in the body of ri there is a k i such that Lj=head(rk). The head of a rule in A is called a conclusion of A, and a default literal not L in a rule of A is called an assumption of A. An argument A with a conclusion L is a minimal argument for L if there is no subargument of A with conclusion L. Given an ELP P, the set of minimal arguments associated with P is denoted by ArgsP. From the two basic notions of attack (undercut u, and rebut r), they introduce further notions: Attacks (a), defeats (d), strongly attacks (sa) and strongly undercuts (su). Here are the precise definitions. Let A1 and A2 be arguments. • A1 undercuts A2 if there is an objective literal L such that L is a conclusion of A1 and not L is an assumption of A2. • A1 rebuts A2 if there is an objective literal L such that L is a conclusion of A1 and ¬L is a conclusion of A2. • A1 attacks A2 if A1 undercuts or rebuts A2. • A1 defeats A2 if A1 undercuts A2, or A1 rebuts A2 and A2 does not undercut A1. • A1 strongly attacks A2 if A1 attacks A2 and A2 does not undercut A1. • A1 strongly undercuts A2 if A1 undercuts A2 and A2 does not undercut A1. A notion of attack is defined as a function x that assigns to each ELP P a bi- nary relation xP ⊆ ArgsP × ArgsP. Different notions of attack are partially ordered by defining: x ⊆ y iff ∀P : xP ⊆ yP. The inverse of a notion of attack x, denoted by x−1, is defined as: x−1 P = {(B,A)|(A,B) ∈ xP}. Thus, a = u ∪ r, d = u ∪ (r-u−1), sa = (u ∪ r)-u−1, and su = u-u−1. Since their framework does not consider priorities, undercuts play the prime role and notions of attack which are based on rebuttals, such as r or r-u−1, are not considered.
- 180. 166 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari Their definition of acceptability is parameterized on the notions of attack allowed for the proponent and the opponent. Basically, an argument is acceptable if it can be defended against any attack. Let x and y be notions of attack, A an argument, and S a set of arguments. Then A is x/y-acceptable with respect to S if for every argument B such that (B,A) ∈ x there exists an argument C ∈ S such that (C,B) ∈ y. A particular semantics can be obtained by choosing one notion of attack for the opponent, and another notion of attack as defense for the proponent. The uniformity of the definition makes it a convenient framework for comparing different argumentation semantics. Based on the notion of acceptability, they define a fixed point semantics for ar- guments: FP,x/y(S) = {A|A is x/y-acceptable with respect to S} and its least fixed point is denoted by JP,x/y. Thus, an argument A is x/y-justified if A ∈ JP,x/y, A is x/y- overruled if it is attacked by an x/y-justified argument, and A is x/y-defensible if it is neither x/y-justified nor x/y-overruled. For example, Dung’s grounded semantics is Ja/u, and Prakken and Sartor’s semantics without priorities or strict rules is Jd/su (see [26] for other examples). Consider, for instance, P1 = {(p ← not q), (q ← not p), (¬q ← r), (r ← not s), (¬s ← not s), (s)}. Here, all arguments (except [s]) are undercut by another argu- ment, and [¬s ← not s] rebuts (but does not defeat) [s]. Thus, [s] is identified as a justified argument in all semantics, except if a is allowed as an attack. We state several results (for other semantics see [26]). JP1,a/x= / 0 JP1,u/sa={[s],[¬q ← r], [p ← not q] } JP1,d/x={[s]} JP1,sa/su={[s],[p ← not q]} JP1,u/u={[s],[¬q ← r] } JP1,su/x={[s],[¬q ← r],[q ← not p ],[p ← not q]} Schweimeier and Schroeder proved a series of theorems, showing that some of the argumentation semantics defined above are subsumed by others. In fact some are actually equivalent. Thus, they established a hierarchy of argumentation semantics, where WFS and WFSXp also occur. One of their results states that WFS=u/u and WFSXp=u/a. They also showed which semantics satisfy the coherence principle and which ones generate a conflict free set of justified arguments. Finally, based on the dialectical proof theory of [24] they introduce a proof theory for x/y-justified arguments. 5 Dung’s approach In [10], Phan Minh Dung states that “there are extremely interesting relations be- tween argumentation and logic programming”. For example, he shows that logic programming can be shown to be a particular form of argumentation. He considers a logic program P as a finite set of clauses of the form B0 ← B1,...,Bm,¬Bm+1,..., ¬Bm+n (where the Bi’s are atoms). To capture the semantics of negation as finite failure (naff), he proposes that P can be transformed in an argumentation frame- work AFnaff = AR, attacks as follows: AR = {(K,k)| there exists a ground clause of P with head k and body K} ∪ {({¬k},k)|k is a ground atom}, and (K,h) at- tacks (K′,h′) iff the complement of h belongs to K′. That is, each ground instance
- 181. 8 Argument-based Logic Programming 167 of a clause of P constitutes an argument for its head. Then, Dung gives a series of theorems that relate semantics of logic programs and semantics of argumenta- tion frameworks. Similarly, an argumentation framework AFnapif that captures the semantics of negation of possibly infinite failure (general loop checking, as in the stable semantics) is introduced. In [10], Dung also shows that argumentation can be “viewed” as logic program- ming by introducing a general method for generating an interpreter for argumenta- tion. This method consists of a very simple logic program consisting of the following two clauses: C1 = acceptable(A) ← not defeated(A) C2 = defeated(A) ← attacks(B,A),acceptable(B) Consider an argumentation framework AF = (AR,attacks) where AR is a set of arguments and attack a relation over AR × AR. Let PAF be the logic pro- gram, PAF = {C1,C2} ∪ {attacks(B,A)|(B,A) ∈ attacks}. For each extension E of AF, let m(E) = {attacks(B,A)|(B,A) ∈ attacks} ∪ {acceptable(A)|A ∈ E} ∪{defeated(B)|B is attacked by some A ∈ E}. Then, Dung shows that 1. E is a stable extension of AF iff m(E) is stable model of PAF . 2. E is grounded extension of AF iff m(E) ∪ {not defeated(A)|A ∈ E} is the well- founded model of PAF . 6 Nute’s Defeasible Logic In [19, 20], Donald Nute introduced Logic for Defeasible Reasoning (LDR), a for- malism that provides defeasible reasoning with a simple representational language. Although LDR is not a defeasible argumentation formalism in itself, its implemen- tation, d-Prolog, defined as an extension of PROLOG, was the first language that introduced defeasible reasoning programming with specificity as a comparison cri- terion between rules. In LDR there are three types of rules: strict rules (e.g., emus → birds that rep- resents “emus are birds”), defeasible rules (e.g., birds ⇒ fly “birds usually fly”) and defeater rules (e.g., heavy ❀ ¬ fly “heavy animals may not fly”). The purpose of a defeater rule is to account for the exceptions to defeasible rules. Therefore, in contrast with the other two types of rules, defeater rules cannot be used to derive formulas. They can only be used to block an application of a defeasible rule. As shown below, the derivation of contradictory literals is prevented by the definition of a defeasible derivation. The original definitions of strict and defeasible derivation, introduced in [19], have evolved in successive articles [20, 4, 3]. For strict derivations only facts and strict rules can be used. However, for defeasible derivations a more sophisticated analysis is performed. A literal p has a defeasible derivation if p is the consequent of a rule (strict or defeasible) with antecedent A, such that (1) for every a ∈ A, a
- 182. 168 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari has a defeasible derivation, (2) the complement of p (p) has no strict derivation, and (3) for every rule R1 (strict, defeasible or defeater) with consequent p and antecedent B, it holds that (3a) there is some b ∈ B that has no defeasible derivation or (3b) there exists a (strict or defeasible) rule R2 with consequent p and antecedent C such that every c ∈ C has a defeasible derivation and R2 R1. Observe that only rules with complementary consequents can be compared. Thus, if a literal has a defeasible derivation, then no further analysis is performed. Therefore, a defeasible derivation cannot be considered as a single argument. A defeasible derivation is related to a tree of arguments because it encodes the analysis of all possible attacks and, by condition (3b), also counter-attacks. A comparison of d-Prolog with defeasible argumentation formalisms can be found in [25] and [15]. 7 Recent Developments In this section, we will briefly comment some recent developments that extend the formalisms mentioned above and provide new proposals. We also refer the inter- ested reader to [5], where an extensive review of current defeasible reasoning im- plementations is given. In [13], an argumentative reasoning service for multi-agent systems called DeLP-server is proposed. A DeLP-server is a stand-alone program that can interact with multiple client agents. A common (or public) DeLP-program can be stored in a server, and client agents (that can be distributed in remote hosts) may send queries to the server and receive the corresponding answer together with the explanation for that answer [14]. A DeLP-server can be consulted by several agents, and one par- ticular agent can consult several DeLP-servers, each of them providing a different shared knowledge base. To answer queries, a DeLP-server will use the common knowledge stored in it, together with individual knowledge that clients can send attached to a query, creating a particular context for that query (see [13] for details). This context is private knowledge that the server will use for answering the query and will not affect other future queries. That is, a client agent cannot make permanent changes to the public DeLP-program stored in a server. The temporal scope of the context sent in a query [Context, Q] is limited and disappears once the query Q has been answered. For instance, consider a DeLP-server that has the set Δ8.1 of Example 8.1 stored. An agent may send the contextual query [{∼electricity(d), emergency lights(d)} , lights on(d)] where the context {∼electricity(d), emergency lights(d)} will pro- vide to the server information about a particular room d. The server will use this context and Δ8.1 in order to return the answer for the query lights on(d). Since contextual information can be in contradiction with the information stored in the server, different types of contextual queries were defined [13]: prioritized, non-prioritized and restrictive. For instance, a prioritized contextual assigns more
- 183. 8 Argument-based Logic Programming 169 importance to the information sent by the agent and overrides the information stored in the server that produces the contradiction. Agents are not restricted to consult a unique DeLP-server, they may perform the same contextual query to different servers, and they may share different knowledge with other agents through different servers. Thus, several configurations of agents and servers can be established (statically or dynamically). For example, special- purpose DeLP-servers can be used, each of them representing particular shared knowledge of a specific domain. Argue tuProlog (AtuP) [6] is a prototype implementation of an argumentation engine based on Vreeswijk’s Argumentation System (AS) that can be used to im- plement a non-monotonic reasoning component in the Internet or agent-based ap- plications. The following description is taken from [5]. AtuP accepts formulae in an extended first-order language and returns answers to queries of acceptability on the basis of the semantics of credulously preferred sets. The language can be con- sidered to be a conservative extension of the basic language of PROLOG, enriched with numbers that quantify rule strength and degree of belief. Strict and defeasible rules can be entered into the system and typically exactly one query is supplied. The core prover of the implementation attempts to find an argument with a conclusion for the query. On the macro level arguments are constructed as nodes in a digraph, and AtuP tries to build an admissible set around an argument for the main claim. AtuP is currently implemented in Java. Finally, the issue of studying which properties are satisfied by the set of conse- quences of an argument-based reasoning system has been gaining attention in the community [8, 7]. A similar effort was conducted in the general area of nonmono- tonic reasoning formalisms since very early [12, 16, 18]. In [7] three postulates are advanced: Closure, Direct Consistency, and Indirect Consistency. The properties are studied in the context of the possible extensions that could be obtained from the argumentation system via different semantics such as the ones proposed in [10]. Closure is a form of completeness, i.e., all possible consequences of the knowledge base are obtained. Consistency (direct and indirect) requires that the set of conclu- sions be non contradictory in a logical sense. The idea of introducing postulates such as these is to guide the design of the reasoners in a manner that their consequences would satisfy the intuitions of the users of these systems. Even though the importance of this issue cannot be over- stated, the discussion of the problems of properly characterizing the behavior of an argument-based consequence operator is out of the scope of this chapter. 8 Conclusions In this chapter we have described several formalisms that integrate Logic Program- ming and Argumentation. In particular, we have described Defeasible Logic Pro- gramming (DeLP), which provides an extension of a logic programming language with a defeasible argumentation formalism. In DeLP a query succeeds if it is war-
- 184. 170 Alejandro J. Garcı́a, Jürgen Dix and Guillermo R. Simari ranted, i.e., if it is possible to build an argument that supports the query and this argument is found to be undefeated by a warrant procedure. We have described an argument-based extended logic programming formalism with defeasible priorities developed by Prakken and Sartor. In their formalism, arguments are expressed in a logic programming language with both strong and default negation. Conflicts be- tween arguments are decided with the help of priorities on the rules. These priorities can be defeasibly derived as conclusions within the system. An overview of the ar- gumentation semantics for extended logic programming developed by Schweimeier and Schroeder was given. Their proposal introduces a variety of notions of attack for extended logic programs from combinations of undercutting and rebuttal. As Phan Minh Dung emphasized in [10] “there are extremely interesting re- lations between argumentation and logic programming”. In this chapter we have covered some of them. Acknowledgements This research was funded by CONICET Argentina Project PIP 5050, and SGCyT, Universidad Nacional del Sur, Argentina. References 1. T. Alsinet, C. Chesñevar, L. Godo, S. Sandri, and G. Simari. Formalizing argumentative reasoning in a possibilistic logic programming setting with fuzzy unification. International Journal of Approximate Reasoning, 48(3):711–729, 2008. 2. T. Alsinet, C. Chesñevar, L. Godo, and G. Simari. A logic programming framework for possibilistic argumentation: Formalization and logical properties. Fuzzy Sets and Systems, 159(10):208–1228, 2008. 3. G. Antoniou, D. Billington, G. Governatori, M. J. Maher, and A. Rock. A family of defeasible reasoning logics and its implementation. In Proceedings of European Conference on Artificial Intelligence (ECAI), pages 459–463, 2000. 4. G. Antoniou, D. Billington, and M. J. Maher. Normal forms for defeasible logic. In Pro- ceedings of International Joint Conference and Symposium on Logic Programming, pages 160–174. MIT Press, 1998. 5. D. Bryant and P. J. Krause. A review of current defeasible reasoning implementations. The Knowledge Engineering Review, 23:1–34, 2008. 6. D. Bryant, P. J. Krause, and G. Vreeswijk. Argue tuProlog: A Lightweight Argumentation Engine for Agent Applications. In Proc. of 1st Int. Conference on Computational Models of Argument (COMMA06), pages 27–32. IOS Press, 2006. 7. M. Caminada and L. Amgoud. On the evaluation of argumentation formalisms. Artificial Intelligence, 171(5-6):286–310, 2007. 8. C. Chesñevar and G. Simari. Modelling inference in argumentation through labelled deduc- tion: Formalization and logical properties. Logica Universalis, 1(1):93–124, 2007. 9. J. Dix, C. Chesñevar, F. Stolzenburg, and G. Simari. Relating Defeasible and Normal Logic Programming through Transformation Properties. Theoretical Computer Science, 290(1):499– 529, 2002. 10. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning and logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995. 11. E. Ferretti, M. Errecalde, A. Garcı́a, and G. Simari. Decision rules and arguments in de- feasible decision making. In P. Besnard, S. Doutre, and A. Hunter, editors, Proc. 2nd Int.
- 185. 8 Argument-based Logic Programming 171 Conference on Computational Models of Arguments (COMMA), volume 172 of Frontiers in Artificial Intelligence and Applications, pages 171–182. IOS Press, 2008. 12. D. Gabbay. Theoretical foundations for non-monotonic reasoning in expert systems. In K. Apt, editor, Logics and Models of Concurrent Systems, pages 439–459. Springer-Verlag, 1985. 13. A. Garcı́a, N. Rotstein, M. Tucat, and G. Simari. An argumentative reasoning service for deliberative agents. In Z. Zhang and J. Siekmann, editors, LNAI 4798 Proceedings of the 2nd. International Conference on Knowledge Science, Engineering and Management (KSEM 2007), pages 128–139. Springer-Verlag, 2007. 14. A. J. Garcı́a, N. D. Rotstein, and G. R. Simari. Dialectical explanations in defeasible argu- mentation. In K. Mellouli, editor, ECSQARU, volume 4724 of Lecture Notes in Computer Science, pages 295–307. Springer, 2007. 15. A. J. Garcı́a and G. R. Simari. Defeasible logic programming: An argumentative approach. TPLP, 4(1-2):95–138, 2004. 16. S. Kraus, D. Lehmann, and M. Magidor. Nonmonotonic reasoning, preferential models and cumulative logics. Artificial Intelligence, 44:167–207, 1990. 17. V. Lifschitz. Foundations of logic programs. In G. Brewka, editor, Principles of Knowledge Representation, pages 69–128. CSLI Pub., 1996. 18. D. Makinson. General patterns in nonmonotonic reasoning. In D. Gabbay, editor, Handbook of Logic in Artificial Intelligence and Logic Programming (vol 3): Nonmonotonic and Uncertain Reasoning, pages 35–110. Oxford University Press, 1994. 19. D. Nute. Defeasible reasoning. In Proc. of the 20th annual Hawaii Int. Conf. on System Sciences, 1987. 20. D. Nute. Defeasible logic. In D. Gabbay, C. Hogger, and J.A.Robinson, editors, Handbook of Logic in Artificial Intelligence and Logic Programming, Vol 3, pages 355–395. Oxford University Press, 1994. 21. J. Pollock. Cognitive Carpentry: A Blueprint for How to Build a Person. MIT Press, 1995. 22. H. Prakken. Logical Tools for Modelling Legal Argument. A Study of Defeasible Reasoning in Law. Kluwer Law and Philosophy Library, 1997. 23. H. Prakken and G. Sartor. A dialectical model of assessing conflicting arguments in legal reasoning. Artificial Intelligence and Law, 4(331-368), 1996. 24. H. Prakken and G. Sartor. Argument-based logic programming with defeasible priorities. J. of Applied Non-classical Logics, 7(25-75), 1997. 25. H. Prakken and G. Vreeswijk. Logical systems for defeasible argumentation. In D.Gabbay, editor, Handbook of Philosophical Logic, 2nd ed. Kluwer, 2000. 26. R. Schweimeier and M. Schroeder. A parameterised hierarchy of argumentation semantics for extended logic programming and its application to the well-founded semantics. TPLP, 5(1-2):207–242, 2005. 27. G. R. Simari, C. I. Chesñevar, and A. J. Garcı́a. The role of dialectics in defeasible argu- mentation. In Proc. of the XIV Int. Conf. of the Chilenean Computer Science Society, pages 335–344, 1994. 28. G. R. Simari and R. P. Loui. A Mathematical Treatment of Defeasible Reasoning and its Implementation. Artificial Intelligence, 53:125–157, 1992. 29. F. Stolzenburg, A. J. Garcı́a, C. I. Chesñevar, and G. R. Simari. Computing generalized speci- ficity. Journal of Applied Non-Classical Logics, 13(1):87–113, 2003.
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- 187. Chapter 9 A Recursive Semantics for Defeasible Reasoning John L. Pollock 1 Reasoning in the Face of Pervasive Ignorance One of the most striking characteristics of human beings is their ability to function successfully in complex environments about which they know very little. Reflect on how little you really know about all the individual matters of fact that characterize the world. What, other than vague generalizations, do you know about the apples on the trees of China, individual grains of sand, or even the residents of Cincinnati? But that does not prevent you from eating an apple while visiting China, lying on the beach in Hawaii, or giving a lecture in Cincinnati. Our ignorance of individual matters of fact is many orders of magnitude greater than our knowledge. And the situation does not improve when we turn to knowledge of general facts. Modern sci- ence apprises us of some generalizations, and our experience teaches us numerous higher-level although less precise general truths, but surely we are ignorant of most general truths. In light of our pervasive ignorance, we cannot get around in the world just rea- soning deductively from our prior beliefs together with new perceptual input. This is obvious when we look at the varieties of reasoning we actually employ. We tend to trust perception, assuming that things are the way they appear to us, even though we know that sometimes they are not. And we tend to assume that facts we have learned perceptually will remain true, at least for awhile, when we are no longer perceiving them, but of course, they might not. And, importantly, we combine our individ- ual observations inductively to form beliefs about both statistical and exceptionless generalizations. None of this reasoning is deductively valid. On the other hand, we cannot be criticized for drawing conclusions on the basis of such non-conclusive ev- idence, because there is no feasible alternative. Our non-deductive reasoning makes our conclusions reasonable, but does not guarantee their truth. As our conclusions John L. Pollock Department of Philosophy, University of Arizona, Tucson, Arizona 85721. This work was supported by NSF grant no. IIS- 0412791 I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 173 DOI 10.1007/978-0-387-98197-0 9, c Springer Science+Business Media, LLC 2009
- 188. 174 John L. Pollock are not guaranteed to be true, we must countenance the possibility that new infor- mation will lead us to change our minds, withdrawing previously adopted beliefs. In this sense, our reasoning is “defeasible”. That is, it makes it reasonable for us to form beliefs, but it can be “defeated” by considerations that make it unreasonable to maintain the previously reasonable beliefs. If we are to understand how rational cognition works, we must know how de- feasible reasoning works, or ought to work. This chapter attempts to answer that question. 2 The Structure of Defeasible Reasoning 2.1 Inference Graphs I assume that much of our reasoning proceeds by stringing together individual in- ferences into more complex arguments. In philosophy it is customary to think of arguments as linear sequences of propositions, with each member of the sequence being either a premise or the conclusion of an inference (in accordance with some inference scheme) from earlier propositions in the sequence. However, this repre- sentation of arguments is an artifact of the way we write them. In many cases the ordering of the elements of the sequence is irrelevant to the structure of the argu- ment. For instance, consider an argument that proceeds by giving a subargument for P and an unrelated subargument for (P→Q), and then finishes by inferring Q by modus ponens. We might diagram this argument as in figure 9.1. The ordering of the elements of the two subarguments with respect to each other is irrelevant. If we write the argument for Q as a linear sequence of propositions, we must order the elements of the subarguments with respect to each other, thus introducing arti- ficial structure in the representation. For many purposes it is better to represent the argument graphically, as as in figure 9.1. Such a graph is an inference graph. The compound arrows linking elements of the inference graph represent the application of inference schemes. Fig. 9.1 An inference graph
- 189. 9 A Recursive Semantics for Defeasible Reasoning 175 More generally, we can take the elements of arguments to be Fitch-style sequents – ordered pairs of propositions and suppositions (sets of propositions), and inference rules like conditionalization can take advantage of that. However, for the purposes of this chapter, I will ignore that sophistication. In deductive reasoning, the inference schemes employed are deductive inference rules. What distinguishes deductive rea- soning from reasoning more generally is that the reasoning is not defeasible. More precisely, given a deductive argument for a conclusion, you cannot rationally deny the conclusion without denying one or more of the premises. In contrast, consider an inductive argument. Suppose we observe a number of swans and they are all white. This gives us a reason for thinking that all swans are white. If we subsequently journey to Australia and observe a black swan, we must retract that conclusion. But notice that this does not give us a reason for retracting any of the premises. It is still reasonable to believe that each of the initially observed swans is white. What distinguishes defeasible arguments from deductive arguments is that the addition of information can mandate the retraction of the conclusion of a defeasible argument without mandating the retraction of any of the premises or conclusions from which the retracted conclusion was inferred. 2.2 Rebutting defeaters Information that can mandate the retraction of the conclusion of a defeasible argu- ment constitutes a defeater for the argument. There are two kinds of defeaters. The simplest are rebutting defeaters, which attack an argument by attacking its conclu- sion. In the inductive example concerning white swans, what defeated the argument was the discovery of a black swan, and the reason that was a defeater is that it entails the negation of the conclusion, i.e., it entails that not all swans are white. More generally, a rebutting defeater could be any reason for denying the conclusion (deductive or defeasible). For instance, I might be informed by Herbert, an ornithol- ogist, that not all swans are white. People do not always speak truly, so the fact that he tells me this does not entail that it is true that not all swans are white. Neverthe- less, because Herbert is an ornithologist, his telling me that gives me a defeasible reason for thinking that not all swans are white, so it is a rebutting defeater. 2.3 Undercutting defeaters Not all defeaters are rebutting defeaters. Suppose Simon, whom I regard as very reli- able, tells me, “Don’t believe Herbert. He is incompetent.” That Herbert told me that not all swans are white gives me a reason for believing that not all swans are white, but Simon’s remarks about Herbert give me a reason for withdrawing my belief, and they do so without either (1) making me doubt that Herbert said what I took him to say or (2) giving me a reason for thinking it false that not all swans are white. Even if
- 190. 176 John L. Pollock Herbert is incompetent, he might have accidentally gotten it right that not all swans are white. Thus Simon’s remarks constitute a defeater, but not a rebutting defeater. This is an example of an undercutting defeater. The difference between rebutting de- featers and undercutting defeaters is that rebutting defeaters attack the conclusion of a defeasible inference, while undercutting defeaters attack the defeasible inference itself, without doing so by giving us a reason for thinking it has a false conclusion. We can think of an undercutting defeater as a reason for thinking that it is false that the premises of the inference would not be true unless the conclusion were true. More simply, we can think of it as giving us a reason for believing that (under the present circumstances) the truth of the premises does not guarantee the truth of the conclusion. It will be convenient to symbolize this as “premises⊗conclusion”. It is useful to expand our graphical representation of reasoning by including defeat rela- tions. Thus we might represent the preceding example as in figure 9.2. Here I have drawn the defeat relations using thick grey arrows. Note that the rebutting defeat is symmetrical, but undercutting defeat is not. Fig. 9.2 Inference graph with defeat 2.4 Computing Defeat-statuses We can combine all of a cognizer’s reasoning into a single inference graph and re- gard that as a representation of those aspects of his cognitive state that pertain to reasoning. The hardest problem in a theory of defeasible reasoning is to give a pre- cise account of how the structure of the cognizer’s inference graph determines what he should believe. Such an account is called a “semantics” for defeasible reasoning,
- 191. 9 A Recursive Semantics for Defeasible Reasoning 177 although it is not a semantics in the same sense as, for example, a semantics for first-order logic. If a cognizer reasoned only deductively, it would be easy to pro- vide an account of what he should believe. In that case, a cognizer should believe all and only the conclusions of his arguments (assuming that the premises are somehow initially justified). However, if an agent reasons defeasibly, then the conclusions of some of his arguments may be defeaters for other arguments, and so he should not believe the conclusions of all of them. For example, in figure 9.2, the cognizer first concludes “All swans are white”. Then he constructs an argument for a defeater for the first argument, at which point it would no longer be reasonable to believe its con- clusion. But then he constructs a third argument supporting a defeater for the second (defeating) argument, and that should reinstate the first argument. Obviously, the re- lationships between interacting arguments can be very complex. We want a general account of how it is determined which conclusions should be believed, or to use philosophical parlance, which conclusions are “justified” and which are not. This distinction enforces a further distinction between beliefs and conclusions. When a cognizer constructs an argument, he entertains the conclusion and he entertains the propositions comprising the intervening steps, but he need not believe them. Con- structing arguments is one thing. Deciding which conclusions to accept is another. What we want is a criterion which, when applied to the inference graph, determines which conclusions are defeated and which are not, i.e., a criterion that determines the defeat-statuses of the conclusions. The conclusions that ought to be believed are those that are undefeated. The remainder of the chapter will be devoted to proposing such a criterion. 3 The Multiple-Assignment Semantics Let us collect all of an agent’s arguments into an inference-graph, where the nodes are labeled by the conclusions of arguments, support-links tie nodes to the nodes from which they are inferred, and defeat-links indicate defeat relations between nodes. These links relate their roots to their targets. The root of a defeat-link is a sin- gle node, and the root of a support-link is a set of nodes. The analysis is somewhat simpler if we construct the inference-graph in such a way that when the same con- clusion is supported by two or more arguments, it is represented by a separate node for each argument. For example, consider the inference-graph diagrammed in fig- ure three, which represents two different arguments for (P Q) given the premises, P, Q, A, and (A → (P Q)). The nodes of such an inference-graph represent argu- ments rather than just representing their conclusions. In such an inference-graph, a node has at most one support-link. When it is unambiguous to do so, I will refer to the nodes in terms of the conclusions they encode. Because a conclusion can be supported by multiple arguments, it is the arguments themselves to which we must first attach defeat-statuses. Then a conclusion is un- defeated iff it is supported by at least one undefeated argument. The only exception to this rule is “initial nodes”, which (from the perspective of the inference graph)
- 192. 178 John L. Pollock Fig. 9.3 An inference graph are simply “given” as premises. Initial nodes are unsupported by arguments, but are taken to be undefeated. Ultimately, we want to use this machinery to model ratio- nal cognition. In that case, all that can be regarded as “given” is perceptual input (construed broadly to include such modes of perception as proprioception, intro- spection, etc.), in which case it may be inaccurate to take the initial nodes to encode propositions. It is probably better to regard them as encoding percepts.1 The node-basis of a node is the set of roots of its support-link (if it has one), i.e., the set of nodes from which the node is inferred in a single step. If a node has no support-link (i.e., it is initial) then the node-basis is empty. The node-defeaters are the roots of the defeat-links having the node as their target. Given an inference- graph, a semantics must determine which nodes encode (the conclusions of) argu- ments that ought to be accepted, i.e., that are not defeated. This is the defeat-status computation, and nodes are marked “defeated” or “undefeated”. The defeat-status computation is made more complex by the fact that some arguments support their conclusions more strongly than other arguments. For instance, if Jones tells me it is raining, and Smith denies it, and I regard them as equally reliable, then I have equally strong arguments both for believing that it is raining and for believing that it is not raining. In that case, I should withhold belief, not accepting either conclu- sion. On the other hand, if I regard Jones as much more reliable than Smith, then I have a stronger argument for believing that it is raining, and if the difference is great enough, that is the conclusion I should draw. So argument-strengths make a differ- ence. However, most semantics for defeasible reasoning ignore argument strengths, pretending that all initial nodes are equally well justified and all inference schemes equally strong. I will make this same simplifying assumption in this chapter. What can we say about the semantics in this simplified case? Let us define: A node of the inference-graph is initial iff its node-basis and list of node-defeaters are empty. It is initially tempting to try to characterize defeat-statuses recursively using the following two rules: (D1) Initial nodes are undefeated. (D2) A non-initial node is undefeated iff all the members of its node-basis 1 See [11] and [12] for a fuller discussion of this.
- 193. 9 A Recursive Semantics for Defeasible Reasoning 179 are undefeated and all node-defeaters are defeated. However, this recursion turns out to be ungrounded because we can have nodes of an inference-graph that defeat each other, as in inference-graph (4), where dashed ar- rows indicate defeasible inferences and heavy arrows indicate defeat-links. In com- puting defeat-statuses in inference-graph (4), we cannot proceed recursively using rules (D1) and (D2), because that would require us to know the defeat-status of Q before computing that of ∼Q, and also to know the defeat-status of ∼Q before com- puting that of Q. The general problem is that a node Q can have an inference/defeat- descendant that is a defeater of Q, where an inference/defeat-descendant of a node is any node that can be reached from the first node by following support-links and defeat-links. I will say that a node is Q-dependent iff it is an inference/defeat- descendant of a node Q. So the recursion is blocked in inference-graph (4) by there being Q-dependent defeaters of Q and ∼Q-dependent defeaters of ∼Q. Inference-graph (4) is a case of “collective defeat”. For example, let P be “Jones says that it is raining”, R be “Smith says that it is not raining”, and Q be “It is rain- ing”. Given P and Q, and supposing you regard Smith and Jones as equally reliable, what should you believe about the weather? It seems clear that you should withhold belief, accepting neither Q nor ∼Q. In other words, both Q and ∼Q should be de- feated. This constitutes a counter-example to rule (D2). So not only do rules (D1) and (D2) not provide a recursive characterization of defeat-statuses – they are not even true. The failure of these rules to provide a recursive characterization of defeat- statuses suggests that no such characterization is possible, and that in turn suggested to me (see [9, 10]) that rules (D1) and (D2) might be used to characterize defeat- statuses in another way. Reiter’s default logic [13] proceeded in terms of multiple “extensions”, and “skeptical default logic” characterizes a conclusion as following nonmonotonically from a set of premises and defeasible inference-schemes iff it is true in every extension. There are simple examples showing that this semantics is inadequate for the general defeasible reasoning of epistemic agents (see below), but the idea of having multiple extensions suggested to me that rules (D1) and (D2) might be used to characterize multiple “status assignments”. On this approach, a partial status assignment is an assignment of defeat-statuses to a subset of the nodes of the inference-graph in accordance with (D1) and (D2):
- 194. 180 John L. Pollock An assignment σ of “defeated” and “undefeated” to a subset of the nodes of an inference-graph is a partial status assignment iff: 1. σ assigns “undefeated” to any initial node; 2. σ assigns “undefeated” to a non-initial node α iff σ assigns “undefeated” to all the members of the node-basis of α and all node-defeaters of α are assigned “defeated”. My (1995) semantics defined: σ is a status assignment iff σ is a partial status assignment and σ is not properly contained in any other partial status assignment. My proposal was then: A node is undefeated iff every status assignment assigns “undefeated” to it; oth- erwise it is defeated. Belief in P is justified for an agent iff P is encoded by an undefeated node of the inference-graph representing the agent’s current epistemological state. I will refer to this semantics as the multiple-assignment semantics. To illustrate, consider inference-graph (4) again. There are two status assignments for this infer- ence graph: assignment 1: P “undefeated” R “undefeated” Q “undefeated” ∼Q “defeated” assignment 2: P “undefeated” R “undefeate” Q “defeated” ∼Q “undefeated” P and R are undefeated, but neither Q nor ∼Q is assigned “undefeated” in every assignment, so both are defeated.
- 195. 9 A Recursive Semantics for Defeasible Reasoning 181 The reason for making status assignments “partial” is that there are inference graphs for which it is impossible to construct status assignments assigning statuses to every node. One case in which this happens is when we have “self-defeating ar- guments”, i.e., arguments whose conclusions defeat some of the inferences leading to those conclusions. A simple example is inference-graph (5). A partial status as- signment must assign “undefeated” to P. If it assigned “undefeated” to Q then it would assign “undefeated” to R and P⊗Q, in which case it would have to assign “defeated” to Q. So it cannot assign “undefeated” to Q. If it assigned “defeated” to Q it would have to assign “defeated” to R and P⊗Q, in which case it would have to assign “undefeated” to Q. So that is not possible either. Thus a partial status as- signment cannot assign anything to Q, R, and P⊗Q. Hence there is only one status assignment (i.e., maximal partial status assignment), and it assigns “undefeated” to P and nothing to the other nodes. Accordingly, P is undefeated and the other nodes are defeated. An intuitive example having approximately the same form is shown in inference-graph (6). Here we suppose that people generally tell the truth, and this gives us a reason for believing what they tell us. However, some people suffer from a malady known as “pink-elephant phobia”. In the presence of pink elephants, they become strangely disoriented so that their statements about their surroundings cease to be reliable. Now imagine Robert, who tells us that the elephant beside him looks pink. In ordinary circumstances, we would infer that the elephant beside Robert does look pink, and hence probably is pink. However, Robert suffers from pink-elephant phobia. So if it were true that the elephant beside Robert is pink, we could not rely upon his report to conclude that it is. So we should not conclude that it is pink. We may be left wondering why he would say that it is, but we cannot explain his utterance by supposing that the elephant really is pink. So this gives us no reason at all for a judgment about the color of the elephant. On the other hand, it gives us no reason to doubt that Robert did say that the elephant is pink, or that Robert has pink-elephant phobia. Those are perfectly justified beliefs.
- 196. 182 John L. Pollock Inference-graphs (5) and (6) constitute intuitive counterexamples to default logic [13] and the stable model semantics [2] because there are no extensions. Hence on those semantics, P has the same status as Q, R, and P⊗Q. It is perhaps more ob- vious that this is a problem for those semantics if we imagine this self-defeating argument being embedded in a larger inference-graph containing a number of oth- erwise perfectly ordinary arguments. On these semantics, all of the nodes in all of the arguments would have to have the same status, because there would still be no extensions. But surely the presence of the self-defeating argument should not have the effect of defeating all other (unrelated) arguments. 4 A Problem Case The multiple-assignment semantics produces the intuitively correct answer for many complicated inference-graphs. For a number of years, I thought that, given the sim- plifying assumption that all arguments are equally strong, this semantics was cor- rect. But I no longer think so. Here is the problem. Contrast inference-graph (4) with inference-graph (7). Inference-graph (7) involves “odd-length defeat cycles”. For an example of inference-graph (7), let A = “Jones says that Smith is unreliable”, B = “Smith is unreliable”, C = “Smith says that Robinson is unreliable”, D = “Robinson is unreliable”, E = “Robinson says that Jones is unreliable”, F = “Jones is unreli- able”. Intuitively, this should be another case of collective defeat, with A, C, and E being undefeated and B, D, and F being defeated. The multiple-assignment se-
- 197. 9 A Recursive Semantics for Defeasible Reasoning 183 mantics does yield this result, but it does it in a peculiar way. A, C, and E must be assigned “undefeated”, but there is no consistent way to assign defeat-statuses to B, D, and F. Accordingly, there is only one status assignment (maximal partial status assignment), and it leaves B, D, and F unassigned. We get the right answer, but it seems puzzling that we get it in a different way than we do for even-length defeat cycles like that in inference-graph (4). This difference has always bothered me. That we get the right answer in a different way does not show that the semantics is incorrect. As long as otherwise equivalent inference-graphs containing odd-length and even-length defeat cycles always produce the same defeat-statuses throughout the graphs, there is no problem. However, they do not. Contrast inference-graphs (8) and (9). In inference-graph (8), there are two status assignments, one assigning “defeated” to B and “undefeated” to D, and the other assigning “undefeated” to B and “defeated” to D. On either status assignment, P has an undefeated defeater, so it is defeated on both status assignments, with the result that Q is undefeated on both status assignments. Hence Q is undefeated simpliciter. However, in inference-graph (9), there is only one status-assignment, and it assigns no status to any of B, D, F, P, or Q. Thus Q is defeated in inference-graph (9), but undefeated in inference-graph (8). This, I take it, is a problem. Although it might not be clear which inference- graph is producing the right answer, the right answer ought to be the same for both inference-graphs. Thus the semantics is getting one of them wrong. It is worth noting in passing that, as far as I know, no currently available semantics for defeasible reasoning handles (8) and (9) correctly. I take this to show that we need a different semantics. 5 A Recursive Semantics The multiple-assignment semantics is based upon the two rules:
- 198. 184 John L. Pollock (D1) Initial nodes are undefeated. (D2) A non-initial node is undefeated if all the members of its node-basis are undefeated and all node-defeaters are defeated. We have seen that these rules are not true as stated. For example, inference-graph (4) is a counterexample to rule (D2). Both Q and ∼Q should be defeated, but then both have undefeated node-bases but no undefeated defeaters. I tried to avoid this problem by imposing these rules instead on partial-status assignments. But perhaps we should take seriously the fact that these rules are simply wrong. In inference- graph (4), in computing the defeat-status of Q, what is crucial is that (a) its node- basis is undefeated, (b) the node-basis of its defeater is undefeated, and (c) there is no other defeater for ∼Q besides Q itself. We can capture this by asking whether ∼Q would be defeated if it were not defeated by Q. We can test this by removing the mutual defeat-links between Q and ∼Q, producing inference-graph (4*). In (4*), ∼Q is undefeated. The proposal is that this should make Q defeated in (4). Note that the defeaters we are removing in constructing inference-graph (4*) are those that are Q-dependent, i.e., those that can be reached by following paths from Q consisting of inference-links and defeat-links. Consider another example – inference-graph (10). In computing the defeat-status of Q, we note that its node-basis is undefeated, and its defeater P⊗Q is defeated only by the Q-dependent defeat-link from R⊗S. If we remove the Q-dependent defeat- links from inference-graph (10) we get inference-graph (10*). In inference-graph (10*), P⊗Q is undefeated, so again, the proposal is that this makes Q defeated in inference-graph (10). These examples suggest that we might replace rule (D2) by a rule that computes the defeat-statuses of defeat-links in a modified inference-graph from which we have removed those defeat-links that make the computation circular. Recall that a defeat- link or support-link extends from its root to its target. The root of a defeat-link is a single node, and the root of a support-link is a set of nodes. Let us define:
- 199. 9 A Recursive Semantics for Defeasible Reasoning 185 Definition 9.1. An inference/defeat-path from a node ϕ to a node θ is a sequence of support-links and defeat-links such that (1) ϕ is or is a member of the root of the first link in the path; (2) θ is the target of the last link in the path; (3) a member of the root of each link after the first member of the path is the target of the preceding link; (4) the path does not contain an internal loop, i.e., no two links in the path have the same target. Definition 9.2. θ is ϕ-dependent iff there is an inference/defeat-path from ϕ to ϕ. Definition 9.3. A circular inference/defeat-path from a node ϕ to itself is an inference/defeat-path from ϕ to a defeater for ϕ. Definition 9.4. A defeat-link is ϕ-critical iff it is a member of some minimal set of defeat-links such that removing all the defeat-links in the set suffices to cut all the circular inference/defeat-paths from ϕ to ϕ. It will be convenient to modify our understanding of initial nodes. Previously, I took them to be automatically undefeated, and we can still regard that as the default value, but it will also be useful to be able to stipulate that some of the initial nodes in a newly-constructed inference-graph are defeated. The construction I am go- ing to propose builds new inference-graphs as subgraphs of pre-existing inference- graphs by (1) deleting ϕ-critical links, and (2) making ϕ-independent nodes initial, i.e., deleting the arguments for them. The latter nodes, being ϕ-independent, have defeat-statuses that were computable in the original inference-graph without first having to compute a defeat-status for ϕ. I want to be able to simply stipulate that these newly-initial nodes have the same defeat-statuses in the new inference-graph as they had in the original. This allows us to define: Definition 9.5. If ϕ is a node of an inference-graph G, let Gϕ be the inference-graph that results from deleting all ϕ-critical defeat-links from G and making all mem- bers of the node-basis of ϕ and all ϕ-independent nodes initial-nodes (i.e., deleting their support-links and defeat-links) with stipulated defeat-statuses the same as their defeat-statuses in G. My proposed semantics now consists of two rules: (CL1) Initial nodes are undefeated unless they are stipulated to be defeated. (CL2) A non-initial node ϕ is undefeated in an inference-graph G iff all members of the node-basis of ϕ are undefeated in G and any defeater for ϕ is defeated in Gϕ. On the assumption that arguments cannot be circular, this pair of rules can be applied recursively to compute the defeat-status of any node in a finite inference- graph. The recursion simply steps through arguments, computing the defeat-status of each node ϕ after the defeat-statuses of the nodes in ϕ’s node-basis are com- puted. The problem of circular inference/defeat-paths is avoided by removing the ϕ-critical defeat-links and evaluating node-defeaters in Gϕ. I will refer to this new semantics as the critical-link semantics, and contrast it with the multiple-assignment semantics.
- 200. 186 John L. Pollock I believe that the critical-link semantics gets everything right that the multiple- assignment semantics got right. Consider a complex example. Inference-graph (11) illustrates the so called “lottery paradox” [3]. Here P reports a description (e.g., a newspaper report) of a fair lottery with one million tickets. P constitutes a defeasible reason for R, which is the description. That is, the newspaper report gives us a defeasible reason for believing the lottery is fair and has a million tickets. In such a lottery, each ticket has a probability of one in a million of being drawn, so for each i, the statistical syllogism gives us a reason for believing ∼Ti (“ticket i will not be drawn”). The supposed paradox is that although we thusly have a reason for believing of each ticket that it will not be drawn, we can also infer on the basis of R that some ticket will be drawn. Of course, this is not really a paradox, because the inferences are defeasible and this is a case of collective defeat. This results from the fact that for each i, we can infer Ti from (i) the description R (which entails that some ticket will be drawn) and (ii) the conclusions that none of the other tickets will be drawn. This gives us a defeating argument for the defeasible argument to the conclusion that ∼Ti, as diagrammed in inference-graph (11). The result is that for each i, there is a status assignment on which ∼Ti is assigned “defeated” and the other ∼T j’s are all assigned “undefeated”, and hence none of them are assigned “undefeated” in every status assignment. I believe that all (skeptical) semantics for defeasible reasoning get the lottery paradox right. A more interesting example is the “lottery paradox paradox”, dia- grammed in inference-graph (12). This results from the observation that because R entails that some ticket will be drawn, from the collection of conclusions of the form ∼Ti we can infer ∼R, and that is a defeater for the defeasible inference from ∼P to ∼R. This is a self-defeating argument. Clearly, the inferences in the lottery paradox should not lead us to disbelieve the newspaper’s description of the lottery, so R should be undefeated. Circumscription [5], in its simple non-prioritized form, gets this example wrong, because one way of minimizing abnormalities would be to block the inference from P to R. My own early analysis [8] also gets this wrong. This was the example that led me to the multiple-assignment semantics. The multiple- assignment semantics gets this right. We still have the same status assignments as in
- 201. 9 A Recursive Semantics for Defeasible Reasoning 187 inference-graph (11), and ∼R is defeated in all of them because it is inferred from the entire set of ∼Ti’s, and one of those is defeated in every status assignment. It will be convenient to have a simpler example of an inference-graph with the same general structure as the lottery paradox paradox. For that purpose we can use inference-graph (13). Here P and R should be undefeated, but T1, T2, and ∼R should be defeated. In the critical link semantics, to compute the defeat-status of R in inference-graph (13), we construct (13*) by removing the only defeat-link whose removal results in R no longer having an R-dependent defeater. In (13*), the triangle consisting of R, T1 and T2 is analogous to inference-graph (4), with the result that T1 and T2 are both defeated in inference-graph (13*). They constitute the node-basis for ∼R, so ∼R is also defeated in inference-graph (13*). Thus by (CL2), R is unde- feated in inference-graph (13). Turning to T1 and T2 in inference-graph (13), both have R as their node-basis, and R is undefeated. Thus to compute the defeat-status of T1 or T2, we construct inference-graph (13**), and observe that T1 and T2 are undefeated there. It then follows by (CL2) that T1 and T2 are defeated in inference- graph (13). Then because T1 and T2 are defeated, ∼R is defeated in inference-graph (13). So we get the intuitively correct answers throughout.
- 202. 188 John L. Pollock Inference-graph (13) also illustrates why, in constructing Gϕ, we remove only the ϕ-critical defeat-links, and not all of the ϕ-dependent defeat-links. All of the defeat- links in inference-graph (13) are R-dependent, and if we remove them all we get inference-graph (13***). But in inference-graph (13***), ∼R is undefeated. This would result in R being defeated in inference-graph (13) rather than undefeated. Thus it is crucial to remove only the ϕ-critical defeat-links rather than all the ϕ- dependent defeat-links. 6 The Problem Cases Now let us turn to some cases that the multiple-assignment semantics does not or may not get right. First, consider the pair of inference-graphs that motivated the search for a new semantics. These are inference-graphs (8) and (9). In these inference-graphs, not everyone agrees whether Q should come out defeated or un- defeated, but it does seem clear that whatever the right answer is, it should be the same for both inference-graphs. Unfortunately, on the multiple-assignment seman- tics, Q is undefeated in inference-graph (8) and defeated in inference-graph (9). On the critical-link semantics, we compute the defeat-statuses of B and D in inference-graph (8) by constructing inference-graph (8*). B and D are undefeated in inference-graph (8*), so each defeats the other in inference-graph (8), with the result that B and D are defeated in inference-graph (8). There are no P-critical defeat-links in (8), so removing P-critical defeat-links leaves inference-graph (8)
- 203. 9 A Recursive Semantics for Defeasible Reasoning 189 unchanged. B and D are defeated in inference-graph (8), so it follows that P is de- feated in inference-graph (8). Then because there are no Q-dependent defeat-links in inference-graph (8), Q is undefeated. The computation of defeat-statuses in inference-graph (9) works in exactly the same way, via inference-graph (9*), again producing the result that Q is undefeated. So on the critical-link semantics, we do not get a divergence between inference- graphs (8) and (9). Still, we can ask whether the answer we get for inference-graphs (8) and (9) is the correct answer. There is some intuitive reason for thinking so. In inference-graph (8), B and D are defeated, so they should not have the power to defeat P, and hence P should defeat Q. Similarly, in inference-graph (9), all three of B, D, and F are defeated, and so again, D should not have the power to defeat P, and hence P should defeat Q. However, not everyone agrees that this intuitive reasoning is correct. This issue is closely connected with a question that has puzzled theorists since the earliest work on the semantics of defeasible reasoning. The multiple-assignment semantics, as well as default logic, the stable model semantics, circumscription, and almost every familiar semantics for defeasible reasoning and nonmonotonic logic, supports what I have called [8] “presumptive defeat”.2 For example, consider inference-graph (14). On the multiple-assignment semantics, a defeated conclusion like Q that is as- signed “defeated” in some status assignment and “undefeated” in another retains the ability to defeat. That is because, in the assignment in which it is undefeated, the defeatee is defeated, and hence not undefeated in all status-assignments. In the case of inference-graph (14) this has the consequence that S is assigned “defeated” in those status-assignments in which Q is assigned “defeated”, but S is assigned “un- defeated” and ∼S is assigned “defeated” in those status-assignments in which Q is assigned “undefeated”. Touretzky, Horty, and Thomason [14] called this “ambigu- ity propagation”, and Makinson and Schlechta [4] called such arguments “Zombie arguments” (they are dead, but they can still get you). However, the critical-link se- mantics precludes presumptive defeat. It entails that Q, ∼Q, and hence ∼S, are all defeated, and S is undefeated. Is this the right answer? 2 The only semantics I know about that does not support presumptive defeat are certain versions of Nute’s [6] defeasible logic. See also Covington, Nute, and Vellino [1], and Nute [7].
- 204. 190 John L. Pollock Consider an example. You are sitting with Keith and Alvin, and the following conversation ensues: Keith: I heard on the news this morning that it is going to rain this afternoon. Alvin: Nonsense! I was sitting right beside you listening to the same weather report, and the announcer clearly said that it is going to be a sunny day in Tucson. Keith: You idiot, you must have cotton in your ears! It was perfectly clear that he said it is going ro rain. Alvin: You never pay attention. No one in his right mind could have thought he said it was going to rain. He said it would be sunny. ... At that point, you wander off shaking your head, still wondering what the weather is going to be. Then it occurs to you that it is about time for the noon News, so you turn on the radio and hear the announcer say, “This just in from the National Weather Service. It is going to rain in Tucson this afternoon.” Surely, that settles the matter. You will believe, with complete justification, that it is going to rain. The earlier conversation between Keith and Alvin does not defeat your judgment on the basis of the noon broadcast. This example has the form of inference-graph (14) if we let: S = “It is going to rain in Tucson this afternoon” Q = “The morning news said that S” P = “Alvin says that Q” R = “Keith says that ∼Q” A = “The noon news says that S” This seems to me to be a fairly compelling example of the failure of presumptive defeat. Formally, presumptive defeat arises for the multiple-assignment semantics from the fact that if a node P is defeated in one assignment and undefeated in an- other, then P-dependent nodes will also have different defeat-statuses in the different assignments unless one of their inference-ancestors is defeated absolutely (i.e., in all status assignments). A similar problem arises for inference-nodes P that cannot be assigned defeat-statuses in any assignments. This occurs, for example, in cases of self-defeat or when there are odd-length defeat cycles. In this case, no P-dependent node can be assigned a defeat-status either unless one of its inference-ancestors is defeated absolutely. For example, consider once more the sad case of Robert, the pink-elephant-phobic (inference-graph (6)). We observed that Robert’s statement that the elephant beside him is pink does not give us a good reason for believing that
- 205. 9 A Recursive Semantics for Defeasible Reasoning 191 it really is pink. Now suppose that Robert is accompanied by Herbert, who is also standing beside the elephant. While Robert is blathering about pink-elephants, Her- bert turns to you and says, “I read in the newspaper this morning that the President is going to visit China.” From this you infer that he did read that in the newspaper, and hence the President is probably going to visit China. Suppose, however, that Herbert also suffers from pink-elephant phobia. Does that make any difference? It does not seem so, because as we observed, Robert’s statement gives us no reason to think the elephant is pink, and so no reason to distrust Herbert’s statement. This sce- nario is diagrammed in inference-graph (15). However, on the multiple-assignment semantics, The elephant beside Robert and Herbert is pink has no status assignment, and hence neither does (People generally tell the truth and Herbert says that he read in the newspaper this morning that the President is going to visit China)⊗ Herbert read in the newspaper this morning that the President is going to visit China or Herbert read in the newspaper this morning that the President is going to visit China or The president is going to visit China. This seems clearly wrong. On the other hand, on the critical-link semantics, The elephant beside Robert and Herbert looks pink is defeated, and hence so is The elephant beside Robert and Herbert is pink and so is (People generally tell the truth and Herbert says that he read in the newspaper this morning that the President is going to visit China) ⊗ Herbert read in the newspaper this morning that the President is going to visit China. Accordingly, Herbert read in the newspaper this morning that the President is going to visit China and The president is going to visit China are undefeated, which is the intuitively correct result.
- 206. 192 John L. Pollock The upshot is that the critical-link semantics agrees with the multiple-assignment semantics on simple cases in which the latter seems to give the right answer, but the critical-link semantics also seems to get right a number of cases that the multiple- assignment semantics gets wrong. The test of a semantics for defeasible reasoning is that it agrees with our intuitions about clear cases. So we have reasonably strong inductive reasons for thinking that the critical-link semantics properly characterizes the semantics of defeasible reasoning. 7 Computing Defeat-Statuses Principles (CL1) and (CL2) provide a recursive characterization of defeat-status relative to an inference-graph. However, this characterization does not lend itself well to implementation because it requires the construction of modified inference- graphs, which would be computationally expensive. The objective of this section is to produce an equivalent recursive characterization that appeals only to the given inference-graph. A defeat-link is ϕ-critical iff it is a member of a minimal set such that removing all the defeat-links in the set suffices to cut all the circular inference/defeat-paths from ϕ to ϕ. A necessary condition for a defeat-link L to be ϕ-critical is that it lie on such a circular path. In general, there can be diverging and reconverging paths with several “parallel” defeat-links, as in figure 16. In figure 16, removing the defeat-link D3 suffices to cut both circular paths. But the set D1,D2 of parallel defeat-links is also a minimal set of defeat-links such that the removal of all the links in the set
- 207. 9 A Recursive Semantics for Defeasible Reasoning 193 suffices to cut all the circular inference/defeat-paths from ϕ to ϕ. Thus in figure 16, all of the defeat-links are ϕ-critical. However, lying on a circular inference/defeat- path is not a sufficient condition for being ϕ-critical. A defeat-link on a circular inference/defeat-path from ϕ to ϕ fails to be ϕ-critical when there is a path around it consisting entirely of support-links, as diagrammed in figure 17. In this case, you must remove D3 to cut both paths, but once you have done that, removing D1 is a gratuitous additional deletion. So D1 is not contained in a minimal set of deletions sufficient for cutting all the circular inference/defeat-paths from ϕ to ϕ, and hence D1 is not ϕ-critical. This phenomenon is also illustrated by inference-graph (13), and we saw that it is crucial to the computation of degrees of justification in that inference-graph that such defeat-links not be regarded as ϕ-critical. It turns out that this is the only way a defeat-link on a circular inference/defeat-path can fail to be ϕ-critical, as will now be proven. Let us say that a node α precedes a node β on Fig. 9.16 Parallel ϕ-critical defeat-links Fig. 9.17 Defeat link that is not ϕ-critical an inference/defeat-path iff α and β both lie on the path and either α = β or the path contains a subpath originating on α and terminating on β. Node-ancestors of a node are nodes that can be reached by following support-links backwards. It will be convenient to define: Definition 9.6. A defeat-link L is bypassed on an inference/defeat-path µ in G iff there is a node α preceding the root of L on µ and a node β preceded by the target of L on µ such that α = β or α is a node-ancestor of β in G.
- 208. 194 John L. Pollock Definition 9.7. µ is a ϕ-circular-path in G iff µ is a circular inference/defeat-path in G from ϕ to ϕ and no defeat-link in G is bypassed on µ. Lemma 9.1. If µ1 and µ2 are ϕ-circular-paths and every defeat-link in µ1 occurs in µ2, then µ1 and µ2 contain the same defeat-links and they occur in the same order. Proof. Proof: Suppose the defeat-links in µ1 are δ1,...,δn, occurring in that order. Suppose µ1 and µ2 differ first at the ith defeat-link. Then µ1 and µ2 look as in figure 18. But every defeat-link in µ1 occurs in µ2, so δi must occur later in µ2. But then the path from δi−1 to δi in µ1 is a bypass around δ∗ i in µ2, which is impossible if it is a ϕ-circular-path.✷ Fig. 9.18 Paths must agree Lemma 9.2. Every defeat-link in a ϕ-circular-path is ϕ-critical. Proof. Suppose δ is a defeat-link on the ϕ-circular-path µ. Let D be the set of all defeaters in the inference-graph other than those on µ. If deleting all members of D is sufficient to cut all ϕ-circular-paths not containing δ, then select a minimal subset D0 of D whose deletion is sufficient to cut all ϕ-circular-paths not containing δ. Adding δ to D0 gives us a set of defeat-links whose deletion is sufficient to cut all ϕ-circular-paths. Furthermore, it is minimal, because adding δ cannot cut any paths not containing δ, and all members of D0 are required to cut those paths. Thus δ is a member of a minimal set of defeat-links the deletion of which is sufficient to to cut all ϕ-circular-paths, i.e., δ is ϕ-critical. Thus if δ is not ϕ-critical, there is a ϕ-circular-path ν not containing δ and not cut by cutting all defeat-links not in µ. That is only possible if every defeat-link in ν is in µ. But then by the previous lemma, µ and ν must contain the same defeat-links, so contrary to supposition, δ is in ν. Thus the supposition that δ is not ϕ-critical is inconsistent with the supposition that it lies on a ϕ-circular-path. ✷ Lemma 9.3. If a defeat-link does not occur on any ϕ-circular-path then it is not ϕ-critical. Proof. For every circular inference/defeat-path µ from ϕ to ϕ there is a ϕ-circular- path ν such that every defeat-link in ν is in µ. ν results from removing bypassed defeat-links and support-links in µ and replacing them by their bypasses. It follows that any set of deletions of defeat-links that will cut all ϕ-circular-paths will also cut every circular inference/defeat-path from ϕ to ϕ. Conversely, ϕ-circular-paths are also circular-paths from ϕ to ϕ, so any set of deletions that cuts all circular- paths from ϕ to ϕ will also cut all ϕ-circular-paths. So the ϕ-circular-paths and the
- 209. 9 A Recursive Semantics for Defeasible Reasoning 195 circular-paths from ϕ to ϕ have the same sets of deletions of defeat-links sufficient to cut them, and hence the same minimal sets of deletions. If a defeat-link δ does not occur on any ϕ-circular-path, then it is irrelevant to cutting all the ϕ-circular-paths, and hence it is not in any minimal set of deletions sufficient to cut all circular-paths from ϕ to ϕ, i.e., it is not ϕ-critical. ✷ Theorem 4 follows immediately from lemmas 2 and 3: Theorem 9.4. A defeat-link is ϕ-critical in G iff it lies on a ϕ-circular-path in G. A further simplification results from observing that, for the purpose of deciding whether a defeat-link is ϕ-critical, all we have to know about ϕ-circular-paths is what defeat-links occur in them. It makes no difference what support-links they contain. So let us define: Definition 9.8. A ϕ-defeat-loop is a sequence µ of defeat-links for which there is a ϕ-circular-path ν such that the same defeat-links occur in µ and ν and in the same order. In other words, to construct a ϕ-defeat-loop from a ϕ-circular-path we simply remove all the support-links. We have the following very simple characterization of ϕ-defeat-loops: Theorem 9.5. A sequence δ1,...,δn of defeat-links is a ϕ-defeat-loop iff (1) ϕ is a node-ancestor of the root of δ1, but not of the root of any δk for k 1, (2) ϕ is the target of δn, and (3) for each k n, the target of δk is equal to or an ancestor of the root of δk+1, but not of the root of δk+j for j 1. The significance of ϕ-defeat-loops is that by omitting the support-links we make them easier to process, but we still have the simple theorem: Theorem 9.6. A defeat-link is ϕ-critical in G iff it lies on a ϕ-defeat-loop in G. In simple cases, Gϕ will be an inference-graph in which no node ψ has a ψ- critical defeat-link. But in more complex cases, like inference-graph (13), we have to repeat the construction, constructing first Gϕ, and then (Gϕ)ψ. Let us define re- cursively: Definition 9.9. Gϕ1,...,ϕn = Gϕ2,...,ϕn ϕ1 As formulated, the recursive semantics requires us to construct the inference- graphs Gϕ1,...,ϕn. To reformulate the semantics so as to avoid this, let us define recursively: Definition 9.10. A defeat-link δ of G is ϕ1,...,ϕn-critical in G iff (1) δ lies on a ϕ1-defeat- loop µ in G containing no ϕ2,...,ϕn-critical defeat-links. A defeat-link δ of G is hereditarily-ϕ1,...,ϕn-critical in G iff either δ is ϕ1,...,ϕn-critical in G or δ is hereditarily-ϕ2,...,ϕn in G. A defeater (i.e., a node) of G is hereditarily-ϕ1,...,ϕn-critical in G iff it is the root of a hereditarily-ϕ1,...,ϕn-critical defeat-link in G.
- 210. 196 John L. Pollock Obviously: Theorem 9.7. δ is hereditarily-ϕ1,...,ϕn-critical in G iff δ is ϕ1-critical in Gϕ2,...,ϕn or ϕ2-critical in Gϕ3,...,ϕn or ... or ϕn-critical in G. Note that a defeat-link that is ϕi-critical in Gϕi+1,...,ϕn does not exist in Gϕj+1,...,ϕn for j i, so: Theorem 9.8. δ is ϕ1-critical in Gϕ2,...,ϕn iff δ is ϕ1,...,ϕn-critical in G. Furthermore, a defeat-link still exists in Gϕ3,...,ϕn (i.e., has not been removed) iff it is not ϕ1,...,ϕn-critical in G. Where θ,ϕ2,...,ϕn are nodes of an inference-graph G, define: Definition 9.11. θ is ϕ-independent of ψ in G iff there is no inference/defeat-path in G from ϕ to θ. θ is ϕ1,...,ϕn-independent in G iff every inference/defeat-path in G from ϕ1 to θ contains a hereditarily-ϕ2,...,ϕn-critical defeat-link. Theorem 9.9. θ is ϕ1,...,ϕn-independent in G iff θ is ϕ1-independent in Gϕ2,...,ϕn. Let us define recursively: Definition 9.12. (a) If ψ is initial in G then ψ is ϕ1,...,ϕn-undefeated in G iff ψ is undefeated in G; (b) If ψ is ϕ1,...,ϕn-independent in G then ψ is ϕ1,...,ϕn-undefeated in G iff ψ is ϕ2,...,ϕn-undefeated in G; (c) Otherwise, ψ is ϕ1,...,ϕn-undefeated in G iff (1) all members of the node- basis of ψ are ϕ1,...,ϕn-undefeated in G, (2) all defeaters for ψ that are ϕ1,...,ϕn-independent of ψ in G and are not hereditarily-ϕ1,...,ϕn-critical in G (i.e., still exist in Gϕ3,...,ϕn) are ϕ1,...,ϕn-defeated in G, and (3) all de- featers for ψ that are ϕ1,...,ϕn-dependent of ψ in G and are not hereditarily- ϕ1,...,ϕn-critical in G (i.e., still exist in Gϕ3,...,ϕn) are ψ,ϕ1,...,ϕn-defeated in G, The reason this is a recursive definition is that we always reach an n at which there are no more ϕ1,...,ϕn-dependent defeaters, and then the values of all nodes are computed recursively in terms of the values assigned to initial nodes. It is now trivial to prove by induction on n that: Theorem 9.10. ψ is undefeated in Gϕ1,...,ϕn iff ψ is ϕ1,...,ϕn-undefeated in G. Thus we have a recursive definition of the defeat-status of a node that computes defeat-statuses entirely by reference to the given inference-graph rather than by building a sequence of modified inference-graphs in accordance with the original analysis. This is easily implemented with two pages of LISP code.
- 211. 9 A Recursive Semantics for Defeasible Reasoning 197 8 Conclusions In an environment of real-world complexity, it is impossible to know enough about the world to confine one’s reasoning to deductively valid inferences. One has to reason defeasibly, drawing conclusions that are made reasonable by one’s evidence, but be prepared to change one’s mind in the face of new evidence. The question then arises how defeasible reasoning ought to work. In particular, given a set of defeasible arguments some of which support defeaters for others, how is it determined which conclusions ought to be believed? Most semantics for defeasible reasoning agree with regard to simple cases, and produce intuitively congenial answers. But there are some complex cases that all existing semantics seem to get wrong. This chapter proposes a new semantics, based on the concept of a critical link, that arguably gets those cases right. Furthermore, the semantics is recursive and easily implemented. References 1. M. A. Covington, D. Nute, and A. Vellino. Prolog Programming in Depth. Prentice-Hall, Englewood Cliffs, New Jersey, second edition, 1997. 2. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming, and n-person games. Artificial Intelligence, 77(2):321–357, 1995. 3. H. E. Kyburg, Jr. Probability and the Logic of Rational Belief. Wesleyan University Press, Middletown, Conneticut, 1961. 4. D. Makinson and K. Schlechta. Floating conclusions and zombie paths: Two deep difficulties in the “directly skeptical” approach to inheritance nets. Artificial Intelligence, 48(2):199–209, 1991. 5. J. McCarthy. Applications of circumscription to formalizing common sense knowledge. Arti- ficial Intelligence, 28(1):89–116, 1986. 6. D. Nute. Basic defeasible logic. In L. F. del Cerro and M. Penttonen, editors, Intensional Logics for Programming, pages 125–154. Oxford University Press, USA, 1992. 7. D. Nute. Norms, priorities, and defeasibility. In P. McNamara and H. Prakken, editors, Norms, Logics and Information Systems, pages 201–218. IOS Press, Amsterdam, 1999. 8. J. Pollock. Defeasible reasoning. Cognitive Science, 11(4):481–518, 1987. 9. J. Pollock. Justification and defeat. Artificial Intelligence, 67(2):377–408, 1994. 10. J. Pollock. Cognitive Carpentry: A Blueprint for How to Build a Person. MIT Press, 1995. 11. J. Pollock. Perceiving and reasoning about a changing world. Computational Intelligence, 14(4):498–562, 1998. 12. J. Pollock and I. Oved. Vision, knowledge, and the mystery link. Philosophical Perspectives 19, pages 309–351, 2005. 13. R. Reiter. A Logic for Default Reasoning. Artificial Intelligence, 13(1,2):81–132, 1980. 14. D. S. Touretzky, J. F. Horty, and R. H. Thomason. A clash of intuitions: the current state of nonmonotonic multiple inheritance systems. In Proceedings IJCAI, pages 476–482, 1987.
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- 213. Chapter 10 Assumption-Based Argumentation Phan Minh Dung, Robert A. Kowalski and Francesca Toni 1 Introduction Assumption-Based Argumentation (ABA) [4, 3, 27, 11, 12, 20, 22] was developed, starting in the 90s, as a computational framework to reconcile and generalise most existing approaches to default reasoning [24, 25, 4, 3, 27, 26]. ABA was inspired by Dung’s preferred extension semantics for logic programming [9, 7], with its dialec- tical interpretation of the acceptability of negation-as-failure assumptions based on the notion of “no-evidence-to-the-contrary” [9, 7], by the Kakas, Kowalski and Toni interpretation of the preferred extension semantics in argumentation-theoretic terms [24, 25], and by Dung’s abstract argumentation (AA) [6, 8]. Because ABA is an instance of AA, all semantic notions for determining the “acceptability” of arguments in AA also apply to arguments in ABA. Moreover, like AA, ABA is a general-purpose argumentation framework that can be instan- tiated to support various applications and specialised frameworks, including: most default reasoning frameworks [4, 3, 27, 26] and problems in legal reasoning [27, 13], game-theory [8], practical reasoning and decision-theory [33, 29, 15, 28, 14]. How- ever, whereas in AA arguments and attacks between arguments are abstract and primitive, in ABA arguments are deductions (using inference rules in an underly- ing logic) supported by assumptions. An attack by one argument against another is a deduction by the first argument of the contrary of an assumption supporting the second argument. Differently from a number of existing approaches to non-abstract argumentation (e.g. argumentation based on classical logic [2] and DeLP [23]) ABA does not have explicit rebuttals and does not impose the restriction that arguments have consistent and minimal supports. However, to a large extent, rebuttals can be obtained “for Phan Minh Dung Asian Institute of Technology, Thailand, e-mail: dung@cs.ait.ac.th Robert A. Kowalski, Francesca Toni Imperial College London, UK, e-mail: {rak,ft}@doc.ic.ac.uk I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 199 DOI 10.1007/978-0-387-98197-0 10, c Springer Science+Business Media, LLC 2009
- 214. 200 Phan Minh Dung, Robert A. Kowalski and Francesca Toni free” [27, 11, 33]. Moreover, ABA arguments are guaranteed to be “relevant” and largely consistent [34]. ABA is equipped with a computational machinery (in the form of dispute deriva- tions [11, 12, 19, 20, 22]) to determine the acceptability of claims by building and exploring a dialectical structure of a proponent’s argument for a claim, an oppo- nent’s counterarguments attacking the argument, the proponent’s arguments attack- ing all the opponents’ counterarguments, and so on. This computation style, which has its roots in logic programming, has several advantages over other computational mechanisms for argumentation. The advantages are due mainly to the fine level of granularity afforded by interleaving the construction of arguments and determining their “acceptability”. The chapter is organised as follows. In Sections 2 and 3 we define the ABA notions of argument and attack (respectively). In Section 4 we define “acceptability” of sets of arguments, focusing on admissible and grounded sets of arguments. In Section 5 we present the computational machinery for ABA. In Section 6 we outline some applications of ABA. In Section 7 we conclude. 2 Arguments in ABA ABA frameworks [3, 11, 12] can be defined for any logic specified by means of inference rules, by identifying sentences in the underlying language that can be treated as assumptions (see Section 3 for a formal definition of ABA frameworks). Intuitively, arguments are “deductions” of a conclusion (or claim) supported by a set of assumptions. The inference rules may be domain-specific or domain-independent, and may represent, for example, causal information, argument schemes, or laws and regu- lations. Assumptions are sentences in the language that are open to challenge, for example uncertain beliefs (“it will rain”), unsupported beliefs (“I believe X”), or decisions (“perform action A”). Typically, assumptions can occur as premises of in- ference rules, but not as conclusions. ABA frameworks, such as logic programming and default logic, that have this feature are said to be flat [3]. We will focus solely on flat ABA frameworks. Examples of non-flat frameworks can be found in [3]. As an example, consider the following simplification of the argument scheme from expert opinion [38]: Major premise: Source E is an expert about A. Minor premise: E asserts that A is true. Conclusion: A may plausibly be taken as true. This can be represented in ABA by a (domain-independent) inference rule: 1 1 In this chapter, we use inference rule schemata, with variables starting with capital letters, to stand for the set of all instances obtained by instantiating the variables so that the resulting premises and conclusions are sentences of the underlying language. For simplicity, we omit the formal defi- nition of the language underlying our examples.
- 215. 10 Assumption-Based Argumentation 201 h(A) ← e(E,A),a(E,A),arguably(A) with conclusion h(A) (“A holds”) and premises e(E,A) (“E is an expert about A”), a(E,A) (“E asserts A”), and an assumption arguably(A). This assumption can be read in several ways, as “there is no reason to doubt that A holds” or “ the com- plement of A cannot be shown to hold” or “the defeasible rule – that a conclusion A holds if a person E who is an expert in A asserts that A is the case – should not apply”. The inference rule can be understood as the representation of this de- feasible rule as a strict (unchallangable) rule with an extra, defeasible condition – arguably(A) – that is open to challenge. This transformation of defeasible rules into strict rules with defeasible conditions is borrowed from Theorist [31]. Within ABA, defeasible conditions are always assumptions. Different representations of assump- tions correspond to different frameworks for defeasible reasoning. For example, in logic programming arguably(A) could be replaced by not ¬h(A) (here not stands for negation as failure), and in default logic it could become M h(A). Note that Verheij [37] also uses assumptions to represent defeasibility of rules. However, his approach amounts to treating every defeasible rule as an assumption. In ABA, attacks are always directed at the assumptions in inference rules. The transformation of defeasible rules into strict rules with defeasible conditions is also used to reduce rebuttal attacks to undercutting attacks, as we will see in Section 3. Note that, here and in all the examples given in this chapter, we represent condi- tionals as inference rules. However, as discussed in [11], this is equivalent to repre- senting them as object language implications together with modus ponens and and- introduction as more general inference rules. Representing conditionals as inference rules is useful for default reasoning because it inhibits the automatic application of modus tollens to object language implications. However, the ABA approach applies to any logic specified by means of inference rules, and is not restricted in the way illustrated in our examples in this chapter. Suppose we wish to apply the inference rule above to the concrete situation in which a professor of computer science (cs), say jo, advises that a software product sw meets a customer’s requirements (reqs) for speed (s) and usability (u). Suppose, moreover, that professors are normally regarded as experts within (w) their field. This situation can be represented by the additional inference rules: reqs(sw) ← h(ok(sw,s)),h(ok(sw,u)); a(jo,ok(sw,s)) ←; a(jo,ok(sw,u)) ←; prof(jo,cs) ←; w(cs,ok(sw,s)) ←; w(cs,ok(sw,u)) ←; e(X,A) ← prof(X,S),w(S,A),c prof(X,S) Note that all these inference rules except the last are problem-dependent. Note also that, in general, inference rules may have empty premises. The potential assumptions in the language underlying all these inference rules are (instances of) the formulae arguably(A) and c prof(X,S) (“X is a credible profes- sor in S”). Given these inference rules and pool of assumptions, there is an argument with assumptions {arguably(ok(sw,s)), arguably(ok(sw,u)), c prof(jo,cs)} sup- porting the conclusion (claim) reqs(sw). In the remainder, for simplicity we drop the assumptions arguably(A) and re- place the inference rule representing the scheme from expert opinion simply by
- 216. 202 Phan Minh Dung, Robert A. Kowalski and Francesca Toni h(A) ← e(E,A),a(E,A). With this simplification, there is an argument for reqs(sw) supported by {c prof(jo,cs)}. Informally, an argument is a deduction of a conclusion (claim) c from a set of as- sumptions S represented as a tree, with c at the root and S at the leaves. Nodes in this tree are connected by the inference rules, with sentences matching the conclusion of an inference rule connected as parent nodes to sentences matching the premises of the inference rule as children nodes. The leaves are either assumptions or the special extra-logical symbol τ, standing for an empty set of premises. Formally: Definition 10.1. Given a deductive system (L,R), with language L and set of infer- ence rules R, and a set of assumptions A ⊆ L, an argument for c ∈ L (the conclusion or claim) supported by S ⊆ A is a tree with nodes labelled by sentences in L or by the symbol τ, such that • the root is labelled by c • for every node N – if N is a leaf then N is labelled either by an assumption or by τ; – if N is not a leaf and lN is the label of N, then there is an inference rule lN ← b1,...,bm (m ≥ 0) and either m = 0 and the child of N is τ or m 0 and N has m children, labelled by b1,...,bm (respectively) • S is the set of all assumptions labelling the leaves. Throughout this chapter, we will often use the following notation • an argument for (claim) c supported by (set of assumptions) S is denoted by S ⊢ c in situations where we focus only on the claim c and support S of an argument. Note that our definition of argument allows for one-node arguments. These arguments consist solely of a single assumption, say α, and are denoted by {α} ⊢ α. A portion of the argument {c prof(jo,cs)} ⊢ reqs(sw) is given in Fig. 10.1. Here, for simplicity, we omit the right-most sub-tree with root h(ok(sw,u)), as this is a copy of the left-most sub-tree with root h(ok(sw,s)) but with s replaced by u throughout. Arguments, represented as trees, display the structural relationships between claims and assumptions, justified by the inference rules. The generation of argu- ments can be performed by means of a proof procedure, which searches the space of applications of inference rules. This search can be performed in the forward direc- tion, from assumptions to conclusions, in the backward direction, from conclusions to assumptions, or even “middle-out”. Our definition of tight arguments in [11] cor- responds to the backward generation of arguments represented as trees. Backward generation of arguments is an important feature of dispute derivations, presented in Section 5.2. Unlike several other authors, e.g. those of [2] (see also Chapter 7) and [23] (see also Chapter 8), we do not impose the restriction that the support of an argument be minimal. For example, consider the ABA representation of the scheme from expert
- 217. 10 Assumption-Based Argumentation 203 opinion, and suppose that our professor of computer science, jo, is also an engineer (eng). Suppose, moreover, that engineers are normally regarded as experts in com- puter science. These additional “suppositions” can be represented by the inference rules eng(jo) ←; e(X,A) ← eng(X),w(cs,A),c eng(X) with (instances of) the formula c eng(X) (“X is a credible engineer”) as additional assumptions. There are now three, different arguments for the claim reqs(sw): {c prof(jo,cs)} ⊢ reqs(sw), {c eng(jo)} ⊢ reqs(sw), and {c prof(jo,cs),c eng(jo)} ⊢ reqs(sw). Only the first two arguments have minimal support. However, all three arguments, including the third, “non-minimal” argument, are relevant, in the sense that their assumptions contribute to deducing the conclusion. Minimality is one way to ensure relevance, but comes at a computational cost. ABA arguments are guaranteed to be relevant without insisting on minimality. Note that the arguments of Chapter 9, defined as inference graphs, are also constructed to ensure relevance. Some authors (e.g. again [2] and [23]) impose the restriction that arguments have consistent support 2. We will see later, in Sections 3 and 4, that the problems arising for logics including a notion of (in)consistency can be dealt in ABA by reducing (in)consistency to the notion of attack and by employing a semantics that insists that sets of acceptable arguments do not attack themselves. reqs(sw) ❅ ❅ ❅ ❅ h(ok(sw,s)) h(ok(sw,u)) ❅ ❅ ❅ ❅ ............ e(jo,ok(sw,s)) a(jo,ok(sw,s)) ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ prof( jo,cs)) w(cs,ok(sw,s)) c prof(jo,cs)) τ τ τ Fig. 10.1 An example argument represented as a tree 2 Note that these authors define arguments with respect to an underlying logic with an explicit negation and hence a notion of consistency, such that inconsistency implies every sentence in the language. The logic underlying an ABA framework need not have an explicit negation and notion of inconsistency.
- 218. 204 Phan Minh Dung, Robert A. Kowalski and Francesca Toni 3 Attacks in ABA In ABA, the notion of attack between arguments is defined in terms of the contrary of assumptions: one argument S1 ⊢ c1 attacks another (or the same) argument S2 ⊢ c2 if and only if c1 is the contrary of an assumption in S2. In general, the contrary of an assumption is a sentence representing a challenge against the assumption. For example, the contrary of the assumption “it will rain” might be “the sky is clear”. The contrary of the assumption “perform action A” might be “perform action B” (where the actions A and B are mutually exclusive). The contrary of the assumption “I believe X” might be “there is evidence against X”. The contrary of an assumption can also represent critical questions addressed to an argument scheme. For example, the argument scheme from expert opinion in Section 2 can be challenged by such critical questions as [38]: CQ1: How credible is E as an expert source? CQ2: Is E an expert in the field that A is in? CQ3: Does E’s testimony imply A? CQ4: Is E reliable? CQ5: Is A consistent with the testimony of other experts? CQ6: Is A supported by evidence? For simplicity, we focus here solely on CQ1, because modelling the other ques- tions would require introducing additional assumptions to our earlier representation of the scheme. 3 Providing negative answers to CQ1 can be understood as prov- ing the contrary ¬c prof(jo,cs), ¬c eng(jo,cs) of the assumptions c prof(jo,cs), c eng(jo,cs) respectively. Contraries may be other assumptions or may be defined by inference rules, e.g. ¬c eng(E,cs) ← ¬prog(E); ¬prog(E) ← theo(E) where prog stands for “programmer” and theo stands for “theoretician”. The first rule can be used to challenge the assumption that an engineer is a credible expert in computer science by arguing that the engineer is not a programmer. The second rule can be used to show that an engineer is not a programmer by assuming that he/she is a theoretician (here theo(E) is an additional assumption). Given this representation, the argument {c eng(jo,cs)} ⊢ reqs(sw) is attacked by the argument {theo(jo)} ⊢ ¬c eng(jo,cs). Definition 10.2. Given a notion of contrary of assumptions 4, • an argument S1 ⊢ c1 attacks an argument S2 ⊢ c2 if and only if the conclusion c1 of the first argument is the contrary of one of the assumptions in the support S2 of the second argument; • a set of arguments Arg1 attacks a set of arguments Arg2 if an argument in Arg1 attacks an argument in Arg2. 3 For example, providing negative answers to CQ5 and CQ6 for A can understood as proving the contrary of the assumption arguably(A) introduced at the beginning of Section 2 but ignored afterwards. 4 See definition 10.3 for the formal notion of contrary.
- 219. 10 Assumption-Based Argumentation 205 This notion of attack between arguments depends only on attacking (“undercutting”) assumptions. In many other approaches, however, such as those of Pollock [30] and Prakken and Sartor [32], an argument can attack (“rebut”) another argument by deducing the negation of its conclusion. We reduce such “rebuttal” attacks to “undercutting” attacks, as described in [27, 11, 33, 34]. For example, consider the inference rules prog(X) ← works for(X,micro),nor(X); works for(jo,micro) ← where micro is the name of some company, nor(X) stands for “X is normal”, and the first inference rule represents the defeasible rule that “normally individuals working at micro are programmers”. From these and the earlier inference rule for ¬prog, we can construct both an argument for prog(jo) supported by {nor(jo)} and an argu- ment for ¬prog(jo) supported by {theo(jo)}. These arguments “rebut” one another but neither one undercuts the other. However, let us set the contrary of assumption theo(X) to prog(X) and the contrary of assumption nor(X) to ¬prog(X). Then, the effect of the rebuttals is obtained by undercutting the assumptions (supporting the arguments for prog( jo) and ¬prog(jo)). Note that an alternative approach to accommodate rebuttals could be to introduce an explicit additional notion of rebuttal attack as done in [10] for logic programming with two kinds of negation. To complete our earlier definition of argument and attack we need a definition of ABA framework: Definition 10.3. An ABA framework is a tuple L, R, A, where • (L,R) is a deductive system, with a language L and a set of inference rules R, • A ⊆ L is a (non-empty) set, whose elements are referred to as assumptions, • is a total mapping from A into L, where α is the contrary of α. 4 Acceptability of arguments in ABA ABA can be used to determine whether a given claim is to be “accepted” by a ra- tional agent. The claim could be, for example, a potential belief to be justified, or a goal to be achieved, represented as a sentence in L. In order to determine the “ac- ceptability” of the claim, the agent needs to find an argument for the claim that can be defended against attacks from other arguments. To defend an argument, other ar- guments may need to be found and they may need to be defended in turn. As in AA, this informal definition of “acceptability” can be formalised in many ways, using the notion of attack between arguments. In this chapter we focus on the following notions of “acceptable” sets of arguments: • a set of arguments is admissible if and only if it does not attack itself and it attacks every argument that attacks it; • an admissible set of arguments is complete if it contains all arguments that it defends, where a set of arguments Arg defends an argument arg if Arg attacks all arguments that attack {arg};
- 220. 206 Phan Minh Dung, Robert A. Kowalski and Francesca Toni • the least (with respect to set inclusion) complete set of arguments is grounded. As for AA (see [8] and Chapter 2), in ABA, given a proposed conclusion c, there always exists a grounded set of arguments, and this can be constructed bottom-up [3, 12]. Consider again our formulation of the scheme for expert opinion. The set con- sisting of the two arguments arg1={c eng(jo,cs)} ⊢ reqs(sw) arg2={nor(jo)} ⊢ prog(jo) is admissible, and as a consequence so is the claim reqs(sw). Indeed, this set does not attack itself and it attacks the argument arg3={theo(jo)} ⊢ ¬prog(jo). However, the set {arg1,arg2} is not (a subset of) the grounded set of arguments. But the set {arg4} is grounded, where arg4={c prof(jo,cs)} ⊢ reqs(sw). The notion of admissibility is credulous, in that there can be alternative, conflicting admissible sets. In the example above, {arg3} is also admissible, but in conflict with the admissible {arg1,arg2}. In some applications, it is more appropriate to adopt a sceptical notion of “ac- ceptability”. The notion of grounded set of arguments is sceptical in the sense that no argument in the grounded set is attacked by an admissible set of arguments. Other notions of credulous and sceptical “acceptable” set of arguments can be employed within ABA [3, 27, 12]. Note that the notions of “acceptable” sets of arguments given here are more struc- tured than the corresponding notions of “acceptable” sets of assumptions given in [3, 27, 11, 12]. The correspondence between “acceptability” of arguments and “ac- ceptability” of assumptions, given in [12], is as follows: • If a set of assumptions S is admissible/grounded then the union of all arguments supported by any subset of S is admissible/grounded; • If a set of arguments S is admissible/grounded then the union of all sets of as- sumptions supporting the arguments in S is admissible/grounded. Note that, if the underlying logic has explicit negation and inconsistency, and we ap- ply the transformation outlined in Section 3 (to reduce rebuttals to our undercutting attacks), then an argument has an inconsistent support if and only if it attacks itself. Thus, in such a case, no argument belonging to an “acceptable” set may possibly contain an argument with an inconsistent support [34]. 5 Computation of “acceptability” The notion of “acceptability” of sets of arguments provides a non-constructive spec- ification. In this section we show how to turn the specification into a constructive proof procedure. As argued in [6, 8], at a conceptual level, a proof procedure for
- 221. 10 Assumption-Based Argumentation 207 argumentation consists of two tasks, one for generating arguments and one for de- termining the “acceptability” of the generated arguments. We have already briefly discussed the computation of arguments in Section 2. Below, we first demonstrate how to determine the “acceptability” of arguments that are already constructed, in the spirit of AA, by means of dispute trees [11, 12]. Then, we discuss how dispute derivations [11, 12, 20, 22], which interleave constructing arguments and determin- ing their “acceptability”, can be viewed as generating “approximations” to “accept- able” dispute trees. Dispute derivations are inspired by SLDNF, the “EK” procedure of [17, 9, 7], and the “KT” procedure of [36, 26] for logic programming with negation as failure. Like SLDNF and EK, dispute derivations interleave two kinds of derivations (one for “proving” and one for “disproving”). Like EK, they accumulate defence assump- tions and use them for filtering. Like KT, they accumulate culprit assumptions and use them for filtering. 5.1 Dispute trees Dispute trees can be seen as a way of representing a winning strategy for a propo- nent to win a dispute against an opponent. The proponent starts by putting forward an initial argument (supporting a claim whose “acceptability” is under dispute), and then the proponent and the opponent alternate in attacking each other’s previously presented arguments. The proponent wins if he/she has a counter-attack against ev- ery attacking argument by the opponent. Definition 10.4. A dispute tree for an initial argument a is a (possibly infinite) tree T such that 1. Every node of T is labelled by an argument and is assigned the status of propo- nent node or opponent node, but not both. 2. The root is a proponent node labelled by a. 3. For every proponent node N labelled by an argument b, and for every argument c attacking b, there exists a child of N, which is an opponent node labelled by c. 4. For every opponent node N labelled by an argument b, there exists exactly one child of N which is a proponent node labelled by an argument attacking b. 5. There are no other nodes in T except those given by 1-4 above. The set of all arguments belonging to the proponent nodes in T is called the defence set of T. Note that a branch in a dispute tree may be finite or infinite. A finite branch repre- sents a winning sequence of arguments (within the overall dispute) that ends with an argument by the proponent that the opponent is unable to attack. An infinite branch represents a winning sequence of arguments in which the proponent counter-attacks every attack of the opponent, ad infinitum. Note that our notion of dispute tree intu- itively corresponds to the notion of winning strategy in Chapter 6.
- 222. 208 Phan Minh Dung, Robert A. Kowalski and Francesca Toni Fig. 10.2 illustrates (our notion of) dispute tree for an extension of our running example, augmented with the following rules ¬c prof(X,S) ← ret(X),inact(X); ¬c prof(X,S) ← admin(X),inact(X); act(X) ← pub(X); ret(jo) ←; admin( jo) ←; pub(jo) ← Here inact(X) is an assumption with inact(X) = act(X). These additions express that professors cannot be assumed to be credible (¬c prof) if they are retired (ret) or cover administrative roles (admin) and can be assumed to be inactive (inact). inact cannot be assumed if its contrary (act) can be shown, and this is so for pro- fessors with recent publications (pub). The resulting, overall ABA framework is summarised in Fig. 10.3. Note that the tree in Fig. 10.2 has an infinite (left-most) branch with {nor(jo)} ⊢ prog(jo) child of {theo(jo)} ⊢ ¬prog(jo) and {theo(jo)} ⊢ ¬prog(jo) child of {nor(jo)} ⊢ prog(jo) ad infinitum. Note also that our intention is to label the opponent nodes in the middle and right-most branches by two different arguments, but both denoted by {inact(jo)} ⊢ ¬c prof(jo,cs). The two arguments differ with respect to the infer- ence rules used to obtain them (the first and second inference rule for ¬c prof in Fig. 10.3, respectively) and thus have different representations as argument trees (as in Definition 10.1). The definition of dispute tree incorporates the requirement that the proponent must counter-attack every attack, but it does not incorporate the requirement that the proponent does not attack itself. This further requirement is incorporated in the definition of admissible and grounded dispute trees: Definition 10.5. A dispute tree T is ❅ ❅ ❅ ❅ P: {c eng(jo),c prof(jo,cs)} ⊢ reqs(sw) O: {theo(jo)} ⊢ ¬c eng(jo) O: {inact(jo)} ⊢ ¬c prof(jo,cs) O: {inact(jo)} ⊢ ¬c prof(jo,cs) P: {nor( jo)} ⊢ prog(jo) O: {theo(jo)} ⊢ ¬prog( jo) . . . . . . P: {} ⊢ act(jo) P: {} ⊢ act(jo) Fig. 10.2 A dispute tree for argument {c eng(jo),c prof(jo,cs)} ⊢ reqs(sw), with respect to the ABA framework in Fig. 10.3.
- 223. 10 Assumption-Based Argumentation 209 • admissible if and only if no argument labels both a proponent and an opponent node. 5 • grounded if and only if it is finite. Note that, by theorem 3.1 in [12], any grounded dispute tree is admissible. The re- lationship between admissible/grounded dispute tree and admissible/grounded sets of arguments is as follows: 1. the defence set of an admissible dispute tree is admissible; 2. the defence set of a grounded dispute tree is a subset of the grounded set of arguments; 3. if an argument a belongs to an admissible set of arguments A then there exists an admissible dispute tree for a with defence set A′ such that A′ ⊆ A and A′ is admissible; 4. if an argument a belongs to the grounded set of arguments A (and the set of all arguments supported by assumptions for the given ABA framework is finite) then there exists a grounded dispute tree for a with defence set A′ such that A′ ⊆ A and A′ is admissible. Results 1. and 3. are proven in [12] (theorem 3.2). Results 2. and 4. follow directly from theorem 3.7 in [26]. Note that the dispute tree in Fig. 10.2 is admissible but not grounded (since it has an infinite branch). However, the tree with root {c prof(jo,cs)} ⊢ reqs(sw) (arg4 in Section 4) and the two right-most subtrees in Fig. 10.2 is grounded. We can obtain finite trees from infinite admissible dispute trees by using “filter- ing” to avoid re-defending assumptions that are in the process of being “defended” or that have already successfully been “defended”. For example, for the tree in Fig. 10.2, only the (proponent) child of argument {theo(jo)} ⊢ ¬prog(jo) needs to be constructed. Indeed, since this argument is already being “defended”, the re- mainder of the (infinite) branch can be ignored. R: reqs(sw) ← h(ok(sw,s)),h(ok(sw,u)); h(A) ← e(E,A),a(E,A); e(X,A) ← eng(X),w(cs,A),c eng(X); e(X,A) ← prof(X,S),w(S,A),c prof(X,S); a( jo,ok(sw,s)) ←; a( jo,ok(sw,u)); eng(jo) ←; prof(jo,cs) ←; w(cs,ok(sw,s)) ←; w(cs,ok(sw,u)) ←; ¬c eng(E,cs) ← ¬prog(E); ¬prog(X) ← theo(X); prog(X) ← works for(X,micro),nor(X); works for(bob,micro) ←; ¬c prof(X,S) ← ret(X),inact(X); ret(jo) ←; ¬c prof(X,S) ← admin(X),inact(X); admin( jo) ←; act(X) ← pub(X); pub(jo) ← A: c eng(X); c prof(X,S); theo(X); nor(X); inact(X) : c eng(X) = ¬c eng(X); c prof(X,S) = ¬c prof(X,S); theo(X) = prog(X); nor(X) = ¬prog(X); inact(X) = act(X) Fig. 10.3 ABA framework for the running example. 5 Note that admissible dispute trees are similar to the complete argument trees of [2]. We use the term “argument tree” for arguments.
- 224. 210 Phan Minh Dung, Robert A. Kowalski and Francesca Toni 5.2 Dispute derivations Dispute derivations compute (“approximations of) dispute trees top-down, starting by constructing an argument supporting a given claim. While doing so, they per- form several kinds of “filtering” exploiting the fact that different arguments may share the same supporting assumptions. Assumptions that are already under attack in the dispute are saved in appropriate data structures (the defence assumptions and culprits), in order to avoid re-computation. The assumptions used by the propo- nent (defence assumptions) are kept separate from the assumptions used by the opponent and attacked by the proponent (culprits). The defence assumptions and culprits for the dispute tree in Fig. 10.2 are {c eng(jo),c prof(jo),nor(jo)} and {theo(jo),inact(jo)} respectively. Dispute derivations employ the following forms of filtering: 1. of defence assumptions by defence assumptions, e.g. performed on the defence assumption theo(jo) in the left-most branch of the dispute tree in Fig. 10.2 (this is analogous to the filtering of arguments we discussed earlier); 2. of culprits by defence assumptions and of defence assumptions by culprits, to guarantee that no argument labels both a proponent and opponent node in the tree and thus attacks itself (see Definition 10.5); 3. of culprits by culprits, for reasons of efficiency; e.g., if the dispute tree in Fig. 10.2 is generated left-to-right, the leaf in the right-most branch does not need to be generated, as the culprit inact(jo) has already been attacked in the middle branch. The first form of filtering is employed only for computing admissible sets, whereas the other two forms are employed for computing grounded as well as admissible sets. Dispute derivations are defined in such a way that, by suitably tuning parameters, they can interleave the construction of arguments and determing “acceptability”. This interleaving may be very beneficial, in general, as it allows • abandoning, during their construction, “potential arguments” that cannot be ex- panded into an actual argument in an “acceptable” set of proponent’s arguments, • avoiding the expansion of the opponent’s “potential arguments” into actual argu- ments when a culprit has already been identified and defeated. Informally speaking, a potential argument of a conclusion c from a set of premises P can be represented as a tree, with c at the root and P at the leaves. As in the case of “actual” arguments (as in Definition 10.1), nodes in the tree are connected by inference rules. However, whereas the leaves of an argument tree are only assump- tions or τ, the leaves of a potential argument can also be non-assumption sentences in L−A. Dispute derivations successively expand potential arguments, using infer- ence rules backwards to replace a non-assumption premise, e.g. p, that matches the conclusion of an inference rule, e.g. p ← Q, by the premises of the rule, Q. In this case, we also say that p is expanded to Q.
- 225. 10 Assumption-Based Argumentation 211 Fig. 10.1 without the dots is an example of a potential argument for reqs(sw), supported by assumption c prof(jo,cs) and non-assumption h(ok(sw,u)). Fig. 10.4 shows another example of a potential argument. In general there may be one, many or no actual arguments that can be obtained from a potential argument. For example, the potential argument in Fig. 10.1 may give rise to two actual arguments (supported by {c prof(jo,cs)} and {c prof(jo,cs), c eng(jo)} respectively), whereas the po- tential argument in Fig. 10.4 gives rise to exactly one actual argument (supported by {inact(jo)}). However, if no inference rules were given for ret, then no actual argument could be generated from the potential argument in Fig. 10.4. The benefits of interleaving mentioned earlier can be illustrated as follows: • A proponent’s potential argument is abandoned if it is supported by assumptions that would make the proponent’s defence of the claim unacceptable. For example, in Fig. 10.1, if the assumption c prof(jo,cs) is “defeated” before h(ok(sw,u)) is expanded, then the entire potential argument can be abandoned. • An opponent’s potential argument does not need to be expanded into an actual argument if a culprit can be selected and defeated already in this potential ar- gument. For example, by choosing as culprit and “defeating” the assumption inact(jo) in the potential argument in Fig. 10.4, the proponent “defeats” any argument that can be obtained by expanding the non-assumption ret(jo). However, when a potential argument cannot be expanded into an actual argument, defeating a culprit in the potential argument is wasteful. Nonetheless, dispute deriva- tions employ a selection function which, given a potential or actual argument, se- lects an assumption to attack or a non-assumption to expand. As a special case, the selection function can be patient [11, 20], always selecting non-assumptions in preference to assumptions, in which case arguments will be fully constructed before they are attacked. Even in such a case, dispute derivations still benefit from filtering. Informally, a dispute derivation is a sequence of transitions steps from one state of a dispute to another. In each such state, the proponent maintains a set P of (sen- tences supporting) potential arguments, representing a single way to defend the ini- tial claim, and the opponent maintains a set O of potential arguments, representing all ways to attack the assumptions in P. In addition, the state of the dispute contains the set D of all defence assumptions and the set C of all culprits already encoun- tered in the dispute. The sets D and C are used to filter potential arguments, as we discussed earlier. A step in a dispute derivation represents either a move by the proponent or a move by the opponent. ❅ ❅ ❅ ❅ ¬c prof(jo,cs) ret( jo) inact( jo) Fig. 10.4 A potential argument for ¬c prof(jo,cs), supported by assumption inact(jo) and non- assumption ret( jo).
- 226. 212 Phan Minh Dung, Robert A. Kowalski and Francesca Toni A move by the proponent either expands one of his/her potential arguments in P, or it selects an assumption in one of the opponent’s potential arguments in O and decides whether or not to attack it. In the first case, it expands the potential argument in only one way, and adds any new assumptions resulting from the expansion to D, checking that they are distinct from any assumptions in the culprit set C (filtering defence assumptions by culprits). In the second case, either the proponent ignores the assumption, as a non-culprit, or he/she adds the assumption to C (bearing in mind that, in order to defeat the opponent, he/she needs to counter-attack only one assumption in each of the opponent’s attacking arguments). In this latter case, he/she checks that the assumption is distinct from any assumptions in D (filtering culprits by defence assumptions) and checks that it is distinct from any culprit already in C (filtering culprits by culprits). If the selected culprit is not already in C, the pro- ponent adds the contrary of the assumption as the conclusion of a new, one-node potential argument to P (to construct a counter-attack). A move by the opponent, similarly, either expands one of his/her potential argu- ments in O, or it selects an assumption to attack in one of the proponent’s potential arguments in P. In the first case, it expands a non-assumption premise of the selected potential argument in all possible ways, replacing the selected potential argument in O by all the new potential arguments. In the second case, the opponent does not have the proponent’s dilemma of deciding whether or not to attack the assumption, be- cause the opponent needs to generate all attacks against the proponent. Thus, he/she adds the contrary of the assumption as the conclusion of a new, one-node potential argument to O. 6 A successful dispute derivation represents a single way for the proponent to sup- port and defend a claim, but all the ways that the opponent can try to attack the proponent’s arguments. Thus, although the proponent and opponent can attack one another before their arguments are fully completed, for a dispute derivation to be successful, all of proponent’s arguments must be actual arguments. In contrast, the opponent’s defeated arguments may be only potential. In this sense, dispute deriva- tions compute only “approximations” of dispute trees. However, for every success- ful dispute derivation, there exists a dispute tree that can be obtained by expanding the opponent’s potential arguments and dropping the potential arguments that cannot be expanded, as well as any of the proponent’s unnecessary counter-attacks [22]. We give an informal dispute derivation for the running example: P: I want to determine the “acceptability” of claim reqs(sw) (D = {} and C = {} initially). P: I generate a potential argument for reqs(sw) supported by {h(ok(sw,s)), h(ok(sw,u))}, and then expand it (through several steps) to one supported by {c prof(jo,cs), h(ok(sw,u))} (c prof(jo,cs) is added to D). O: I attack the assumption c prof(jo,cs) in this potential argument by looking for arguments for its contrary ¬c prof(jo,cs). 6 If computing admissibility, however, the opponent would not attack assumptions that already belong to D (filtering defence assumptions by defence assumptions).
- 227. 10 Assumption-Based Argumentation 213 O: I generate two potential arguments for ¬c prof(jo,cs), supported by {ret(jo), inact(jo)} and {admin(jo), inact(jo)} respectively. P: I choose inact(jo) as culprit in the first potential argument by O (inact(jo) is added to C), and generate (through several steps) an argument for its contrary act(jo), supported by the empty set. O: There is no point for me to expand this potential argument then. But I still have the attacking argument for ¬c prof(jo,cs), supported by {admin(jo), inact(jo)}. P: I again choose inact(jo) as culprit, which I have already defeated (inact(jo) ∈ C). P: There is no attacking argument that I still need to deal with: let me go back to expand the argument for reqs(sw). P: I need to expand h(ok(sw,u)), I can do so and generate (through several steps) an argument for reqs(sw) supported by {c prof(jo,cs),c eng(jo)} (c eng(jo) is added to D). O: I can attack this argument by generating (through several steps) an argument for the contrary ¬c eng(jo) of c eng(jo): this argument is supported by assump- tion theo(jo). P: I can attack this argument by generating (through several steps) a potential argument for the contrary of theo(jo). . . . This dispute corresponds to the top-down and right-to-left construction of (an ap- proximation of) the dispute tree in Fig. 10.2. The dispute ends successfully for com- puting admissible sets of arguments, but does not terminate for computing grounded sets of arguments. Note that dispute derivations are defined in terms of several pa- rameters: the selection function, the choice of “player” at any specific step in the derivation, the choice of potential arguments for the proponent/opponent to expand etc (see [20, 22]). Concrete choices for some of these parameters (e.g. the choice of the proponent’s arguments) correspond to concrete search strategies for finding dispute derivations and computing dispute trees. Concrete choices for other param- eters (e.g. the choice of “player”) determine how the dispute tree is constructed in a linear manner. Several notions of dispute derivations have been proposed, differing in the notion of “acceptability” and in the presentation of the computed set of “acceptable” argu- ments. More specifically, the dispute derivations of [11, 20, 22] compute admissible sets of arguments whereas the dispute derivations of [12] compute grounded and ideal (another notion of sceptical “acceptability”) sets of arguments. Moreover, the dispute derivations of [11, 12] compute the union of all sets of assumptions sup- porting the “acceptable” sets of arguments for the given claim, whereas the dispute derivations of [20, 22] also return explicitly the computed set of “acceptable” argu- ments and the attacking (potential) arguments, as well as an indication of the attack relationships amongst these arguments.
- 228. 214 Phan Minh Dung, Robert A. Kowalski and Francesca Toni 6 Applications In this section, we illustrate recent applications of ABA for dispute resolution (Sec- tion 6.1, adapted from [20]) and decision-making (Section 6.2, adapted from [15]). For simplicity, the description of these applications is kept short here. For more detail see [13] (for dispute resolution applied to contracts) and [33, 16, 29] (for decision-making in service-oriented architectures). We also show how ABA can be used to model the stable marriage problem (Section 6.3, building upon [8]). 6.1 ABA for dispute resolution Consider the following situation, inspired by a real-life court case on contract dis- pute. A judge is tasked with resolving a disagreement between a software house and a customer refusing to pay for a product developed by the software house. Sup- pose that this product is the software sw in the running example of Fig. 10.3. The judge uses information agreed upon by both parties, and evaluates the claim by the software house that payment should be made to them. All parties agree that payment should be made if the product is delivered on time (del) and is a good product (goodProd). They also agree that a product is not good (badProd) if it is late (lateProd) or does not meet its requirements. As before, we assume that these requirements are speed and usability. There is also the indisputable fact that the software was indeed delivered (del(sw)). This situation can be modelled by extending the framework of Fig. 10.3 with inference rules payment(sw) ← del(sw),goodProd(sw); badProd(sw) ← lateProd(sw); badProd(sw) ← ¬reqs(sw); del(sw) ← and assumptions goodProd(sw), ¬reqs(sw), with contraries: goodProduct(sw) = badProduct(sw); ¬reqs(sw) = reqs(sw) Given the expert opinions of jo (see Section 2), the claim that payment should be made is grounded (and thus admissible). 6.2 ABA for decision-making We use a concrete “home-buying” example, in which a buyer is looking for a prop- erty and has a number of goals including “structural” features of the property, such as its location and the number of rooms, and “contractual” features, such as the price, the completion date for the sale, etc. The buyer needs to decide both on a property, taking into account the features of the properties (Ri below), and general “norms” about structural properties (Rn below). The buyer also needs to decide and agree on a contract, taking into account norms about contractual issues (Rc below). A simple example of buyer is given by the ABA framework L, R, A, with:
- 229. 10 Assumption-Based Argumentation 215 • R = Ri ∪Rn ∪Rc and Ri : number of rooms = 5 ← house1; number of rooms = 4 ← house2; price = £400K ← house2 Rn : safe ← council approval,a1; ¬safe ← weak foundations,a2; council approval ← completion certi ficate,a3 Rc : seller moves abroad ← house2; quick completion ← seller moves abroad • A = Ad ∪Au ∪Ac and Ad = {house1,house2}; Ac = {a1,a2,a3}; Au = {¬council approval } • house1 = house2, house2 = house1, a1 = ¬safe, a2 = safe, a3 = ¬council approval, ¬council approval = council approval. Here, there are two properties for sale, house1 and house2. The first has 5 rooms, the second has 4 rooms and costs £400K (Ri). The buyer believes that a property ap- proved by the council is normally safe, a completion certificate normally indicates council approval, and a property with weak foundations is normally unsafe (Rn). The buyer also believes that the seller of the second property is moving overseas, and this means that the seller aims at a quick completion of the sale (Rc). Some of the assumptions in the example have a defeasible nature (Ac), others amount to mutually exclusive decisions (Ad) and finally others correspond to genuine uncer- tainties of the buyer (Au). This example combines default reasoning, epistemic reasoning and practical rea- soning. An example of “pure” practical reasoning in ABA applied to a problem described in [1] can be found in [35]. 6.3 ABA for the stable marriage problem Given two sets M, W of n men and n women respectively, the stable marriage prob- lem (SMP) is the problem of pairing the men and women in M and W in such a way that no two people of opposite sex prefer to be with each other rather than with the person they are paired with. The SMP can be viewed as a problem of finding stable extensions in an abstract argumentation framework [8]. Here we show that the problem can be naturally represented in an ABA frame- work L, R, A, with: • A = {pair(A,B) | A ∈ M,B ∈ W} • pair(A,B) = contrary pair(A,B) • R consists of inference rules contrary pair(A,B) ← prefers(A,D,B), pair(A,D); contrary pair(A,B) ← prefers(B,E,A), pair(E,B) together with a set P of inference rules of the form prefer(X,Y,Z) ← such that for each person A and for each two different people of the opposite sex
- 230. 216 Phan Minh Dung, Robert A. Kowalski and Francesca Toni B and C, either prefer(A,B,C) ← or prefer(A,C,B) ← belongs to P, where prefer(X,Y,Z) stands for X prefers Y to Z. The standard formulation of the stable marriage problem combines the rules, as- sumptions and contraries of this framework with the notion that a set of argu- ments/assumptions is “acceptable” if and only if it is stable, where • A set of arguments/assumptions is stable if and only if it does not attack itself, but attacks all arguments/assumptions not in the set. In SMP, the semantics of stable sets forces solutions to be total, pairing all men and women in the sets M and W. The semantics of admissible sets is more flexible. It does not impose totality, and it can be used when the sets M and W have different cardinalities. Consider for example the situation in which two men, a and b, and two women, c and d have the following preferences: prefer(a,d,c) ←; prefer(b,c,d) ←; prefer(c,a,b) ←; prefer(d,b,a) ← Although there is no stable solution in which all people are paired with their highest preference, there exist two alternative stable solutions: {pair(a,c), pair(b,d)} and {pair(a,d), pair(b,c)}. However, suppose a third woman, e, enters the scene, turns the heads of a and b, and expresses a preference for a over b, in effect adding to P: prefer(a,e,d) ←; prefer(a,e,c) ←; prefer(b,e,c) ←; prefer(b,e,d) ←; prefer(e,a,b) ← This destroys both stable solutions of the earlier SMP, but has a single maximally admissible solution: {pair(a,e), pair(b,c)}. Thus, by using admissibility we can drop the requirement that the number of men and women is the same. Similarly, we can drop the requirement that preferences are total, namely all preferences between pairs of the opposite sex are given. 7 Conclusions In this chapter, we have reviewed assumption-based argumentation (ABA), focusing on relationships with other approaches to argumentation, computation, and applica- tions. In contrast with a number of other approaches, ABA makes use of under- cutting as the only way in which one argument can attack another. The effect of rebuttal attacks is obtained in ABA by adding appropriate assumptions to rules and by attacking those assumptions instead. The extent to which such undercutting is an adequate replacement for rebuttals has been explored elsewhere [34], but merits further investigation. Also, again in contrast with some other approaches, we do not insist that the support of an argument in ABA be minimal and consistent. Instead of insisting on minimality, we guarantee that the support of an argument is relevant, as a side-effect of representing arguments as deduction trees. Instead of insisting that the support of an argument is consistent, we obtain a similar effect by imposing the restriction that “acceptable” sets of arguments do not attack themselves. ABA is an instance of abstract argumentation (AA), and consequently it inherits its various notions of “acceptable” sets of arguments. ABA also shares with AA the
- 231. 10 Assumption-Based Argumentation 217 computational machinery of dispute trees, in which a proponent and an opponent alternate in attacking each other’s arguments. However, ABA also admits the com- putation of dispute derivations, in which the proponent and opponent can attack and defeat each other’s potential arguments before they are completed. We believe that this feature of dispute derivations is both computationally attractive and psycholog- ically plausible when viewed as a model of human argumentation. The computational complexity of ABA has been investigated for several of its in- stances [5] (see also Chapter 5). The computational machinery of dispute derivations and dispute trees is the basis of the CaSAPI argumentation system 7 [19, 20, 22]. Although ABA was originally developed for default reasoning, it has recently been used for several other applications, including dispute resolution and decision- making. ABA is currently being used in the ARGUGRID project 8 to support ser- vice selection and composition of services in the Grid and Service-Oriented Ar- chitectures. ABA is also being used for several applications in multi-agent systems [21, 18] and e-procurement [29]. Acknowledgement This work was partially funded by the Sixth Framework IST programme of the EC, under the 035200 ARGUGRID project. References 1. T. Bench-Capon and H. Prakken. Justifying actions by accruing arguments. In Proc. COMMA’06, pages 247–258. IOS Press, 2006. 2. P. Besnard and A. Hunter. Elements of Argumentation. MIT Press, 2008. 3. A. Bondarenko, P. Dung, R. Kowalski, and F. Toni. An abstract, argumentation-theoretic approach to default reasoning. Artificial Intelligence, 93(1-2):63–101, 1997. 4. A. Bondarenko, F. Toni, and R. Kowalski. An assumption-based framework for non- monotonic reasoning. In Proc. LPRNR’93, pages 171–189. MIT Press, 1993. 5. Y. Dimopoulos, B. Nebel, and F. Toni. On the computational complexity of assumption-based argumentation for default reasoning. Artificial Intelligence, 141:57–78, 2002. 6. P. M. Dung. On the acceptability of arguments and its fundamental role in non-monotonic reasoning and logic programming. In Proc. IJCAI’93, pages 852–859. Morgan Kaufmann, 1993. 7. P. M. Dung. An argumentation theoretic foundation of logic programming. Journal of Logic Programming, 22:151–177, 1995. 8. P. M. Dung. On the acceptability of arguments and its fundamental role in non-monotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995. 9. P. M. Dung. Negations as hypotheses: An abductive foundation for logic programming. In Proc. ICLP, pages 3–17. MIT Press, 1991. 10. P. M. Dung. An argumentation semantics for logic programming with explicit negation. In Proc. ICLP, pages 616–630. MIT Press, 1993. 11. P. M. Dung., R. Kowalski, and F. Toni. Dialectic proof procedures for assumption-based, admissible argumentation. Artificial Intelligence, 170:114–159, 2006. 12. P. M. Dung, P. Mancarella, and F. Toni. Computing ideal sceptical argumentation. Artificial Intelligence, 171(10-15):642–674, 2007. 13. P. M. Dung and P. M. Thang. Towards an argument-based model of legal doctrines in common law of contracts. In Proc. CLIMA IX, 2008. 7 http://www.doc.ic.ac.uk/˜dg00/casapi.html 8 www.argugrid.eu
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- 233. Chapter 11 The Toulmin Argument Model in Artificial Intelligence Or: how semi-formal, defeasible argumentation schemes creep into logic Bart Verheij 1 Toulmin’s ‘The Uses of Argument’ In 1958, Toulmin published The Uses of Argument. Although this anti-formalistic monograph initially received mixed reviews (see section 2 of [20] for Toulmin’s own recounting of the reception of his book), it has become a classical text on argumen- tation, and the number of references to the book (when writing these words1 — by a nice numerological coincidence — 1958) continues to grow (see [7] and the special issue of Argumentation 2005; Vol. 19, No. 3). Also the field of Artificial Intelligence has discovered Toulmin’s work. Especially four of Toulmin’s themes have found follow-up in Artificial Intelligence. First, argument analysis involves half a dozen distinct elements, not just two. Second, many, if not most, arguments are substan- tial, even defeasible. Third, standards of good reasoning and argument assessment are non-universal. Fourth, logic is to be regarded as generalised jurisprudence. Us- ing these central themes as a starting point, this chapter provides an introduction to Toulmin’s argument model and its connections with Artificial Intelligence research. No attempt is made to give a comprehensive history of the reception of Toulmin’s ideas in Artificial Intelligence; instead a personal choice is made of representative steps in AI-oriented argumentation research. When Toulmin wrote his book, he was worried. He saw the influence of the successes of formal logic on the philosophical academia of the time, and was afraid that as a consequence seeing formal logic’s limitations would be inhibited. He wrote The Uses of Argument to fight the — in his opinion mistaken — idea of formal logic as a universal science of good reasoning. In the updated edition of The Uses of Argument [19], he describes his original aim as follows: to criticize the assumption, made by most Anglo-American academic philosophers, that any significant argument can be put in formal terms: not just as a syllogism, since for Aristotle himself any inference can be called a ‘syllogism’ or ‘linking of statements’, but a rigidly demonstrative deduction of the kind to be found in Euclidean geometry. ([19], vii) Bart Verheij Artificial Intelligence, University of Groningen 1 Source: Google Scholar citation count, April 1, 2008. I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 219 DOI 10.1007/978-0-387-98197-0 11, c Springer Science+Business Media, LLC 2009
- 234. 220 Bart Verheij Fig. 11.1 Toulmin’s layout of arguments with an example ([18], 104–5) In short: Toulmin wanted to argue that there are other arguments than formal ones. It is also clear from this quote that Toulmin’s goals were first and foremost aimed at his fellow philosophers. In the Preface to the 2003 edition, Toulmin says it thus: In no way had I set out to expound a theory of rhetoric or argumentation: my concern was with twentieth-century epistemology, not informal logic. ([19], vii) Let us look closer at some of Toulmin’s points. 1.1 Argument analysis involves half a dozen distinct elements, not just two Toulmin is perhaps most often read because of his argument diagram (Figure 1). Whereas a formal logical analysis uses the dichotomy of premises and conclusions when analyzing arguments, Toulmin distinguishes six different kinds of elements: Data, Claim, Qualifier, Warrant, Backing and Rebuttal. Before explaining the roles of these elements, let us look at Toulmin’s famous example of Harry, who may or may not be a British subject. When someone claims that Harry is a British subject, it is natural to ask, so says Toulmin: What have you got to go on? An answer to that question can provide the data on which the claim rests, here: Harry was born in Bermuda. But having datum and claim is not enough. A further important question needs to be answered. Toul- min phrases it thus: How do you get there? In other words, why do you think that the datum gives support for your claim? An answer to this question must take the form of a rule-like general statement, the warrant underlying the step from datum to claim. In the example, the warrant is that a man born in Bermuda will generally be a British subject. As the example shows, warrants need not express universal generalizations. Here the warrant is not that each man born in Bermuda is a British subject, but merely that a man born in Bermuda will generally be a British sub- ject. As a result, on the basis of datum and warrant a claim needs to be qualified. Here the claim becomes that presumably Harry is a British subject. When datum, qualified claim and warrant have been made explicit, a further question needs to be asked: Why do you think that the warrant holds? An answer will be provided by the backing of the warrant. In the example, Toulmin refers to the existence of statutes and other legal provisions (without specifying them) that can provide the backing
- 235. 11 The Toulmin Argument Model in Artificial Intelligence 221 for the warrant that who is born in Bermuda will generally be British subjects. The sixth and final kind of element to be distinguished is that of conditions of exception or rebuttal (101/93)2. Conditions of rebuttal indicate ‘circumstances in which the general authority of the warrant would have to be set aside’ or ‘exceptional circum- stances which might be capable of defeating or rebutting the warranted conclusion’ (101/94). In the example, Harry’s parents could be aliens or he could have become a naturalized American. Toulmin refers to Hart and Ross as predecessors for his discussion of rebuttal. Hart coined the term ‘defeasibility’ (see also [9]) and used it in legal and philosophical settings (contract, free will, responsibility), while Ross emphasized that moral rules must have exceptions (142/131–2). Here is Toulmin’s defence of the difference between a datum and the negation of a rebuttal, which predates discussions about the relation between rule conditions and exceptions: [T]he fact that Harry was born in Bermuda and the fact that his parents were not aliens are both of them directly relevant to the question of his present nationality; but they are relevant in different ways. The one fact is a datum, which by itself establishes a presumption of British nationality; the other fact, by setting aside one possible rebuttal, tends to confirm the presumption thereby created. (102/95) Summarizing, Toulmin distinguishes six kinds of elements in arguments: Claim: The Claim is the original assertion that we are committed to and must justify when challenged (97/90). It is the starting point of the argument. Datum: The Datum provides the basis of the claim in response to the question: What have you got to go on? (97–8/90) Warrant: The Warrant provides the connection between datum and claim. A warrant ex- presses that ‘[d]ata such as D entitle[s] one to draw conclusions, or make claims, such as C’. Warrants are ‘general, hypothetical statements, which can act as bridges, and autho- rise the sort of step to which our particular argument commits’. They are ‘rules, principles, inference-licences or what you will, instead of additional items of information’. (98/91) Qualifier: The Qualifier indicates the strength of the step from datum to claim, as conferred by the warrant (101/94) Backing: The Backing shows why a warrant holds. Backing occurs when not a particular claim is challenged, but the range of arguments legitimized by a warrant (103–4/95–6). Rebuttal: A Rebuttal can indicate ‘circumstances in which the general authority of the war- rant would have to be set aside’ or ‘exceptional circumstances which might be capable of defeating or rebutting the warranted conclusion’ (101/94). 1.2 Many, if not most, arguments are substantial, even defeasible Let us consider another of Toulmin’s example arguments: (1) Anne is one of Jack’s sisters; All Jack’s sisters have red hair; So, Anne has red hair. (123/115) This example is a variant of the paradigmatic example of a syllogism: ‘Socrates is a man. All men are mortal. So, Socrates is mortal’. Anyone accustomed to the 2 Page numbers before the slash refer to the original 1958 edition of The Uses of Argument [18], those after the slash to the updated 2003 edition [19]
- 236. 222 Bart Verheij standard logical treatment of syllogisms will recognize the following logical form underlying this type of syllogistic argument: (2) P(t) (∀x) (P(x) → Q(x)) ———————– Q(t) In this logical analysis, argument (1) has two premises and one conclusion. More- over, the two premises have clearly distinctive roles, one often referred to as the minor premise (P(t)), the other the major premise ((∀x) (P(x) → Q(x)). If we look at example (1) and its logical analysis (2) — an analysis to which Toul- min does not object for this type of argument3 — one may ask: Why the richness of primitives in his scheme? Shouldn’t we apply Occam’s razor and be satisfied with the good-old dichotomy of premises and conclusions instead of Toulmin’s six-fold scheme? Toulmin’s answer is: no, we shouldn’t be satisfied. A central place in the defence of his position is the claim that syllogistic arguments of the logical form in (2) are atypical, even rare (147–150/136–139; also: 125–6/116–7). They have special characteristics that do not hold for other kinds of arguments. Toulmin discusses the following five characteristics of (2)-fitting arguments: 1. They are unequivocal in their consequences. However, there are also arguments (e.g., the Harry example; Figure 1) that only allow draw- ing a conclusion tentatively. Hence the need for qualifiers. 2. They are formally valid. Toulmin speaks of a formally valid argument when the argument’s conclusion can be achieved by ‘shuffling’ the premises and their constituent parts (118/110). Arguments of the form ‘D; W. So C’ can in this way be phrased as formally valid, but arguments of the form ‘D; B. So C’ normally cannot. Toulmin refers to the Harry example (Figure 1) to make his point (123/114). 3. They are expressed in terms of ‘logical words’. Toulmin says it thus: ‘The acceptable, logical words include ‘all’, ‘some’, ‘or’, and a few others: these are firmly herded away from the non-logical goats, i.e. the generality of nouns, adjectives and the like, and unruly connectives and quantifiers such as ‘most’, ‘few’, ‘but’.’ (149/138) 4. They are warrant-using. But, says Toulmin, there are also warrant-establishing arguments, as they for instance occur in scientific papers (120–1/112–3). Toulmin refers to Ryle, who contrasted warrant-using and warrant-establishing arguments by the analogy of taking a journey along a railway al- ready built and the building of a fresh railway. Toulmin connects warrant-using arguments to the term ‘deduction’, and warrant-establishing arguments to ‘induction’. 5. They are analytic. Toulmin calls an argument analytic if and only if the backing for the warrant authorising it includes, explicitly or implicitly, the information conveyed in the conclusion itself. For instance, the universal statement that all of Jack’s sisters have red hair, in a way includes that Anne, who is one of Jack’s sisters, has red hair (123–127/114–118). Arguments that are not analytic are substantial. 3 The class of analytic arguments, for which both ‘D. W. So, C’ and ‘D. B. So, C’ can be expressed in the formally valid way (2) (123/114).
- 237. 11 The Toulmin Argument Model in Artificial Intelligence 223 It is because of the accidental concurrence of these five properties in (2)-arguments that the idea has come about that all arguments have them, and must have them; and this is an unfortunate fact of history, says Toulmin. In the connection of analytic versus substantial arguments, Toulmin distinguishes two variants of the Anne argument (1) (124/115): (1, backing version) Anne is one of Jack’s sisters; Each one of Jack’s sisters has (been checked individually to have) red hair; So, Anne has red hair. (1, warrant version) Anne is one of Jack’s sisters; Any sister of Jack’s will (i.e. may be taken to) have red hair; So, Anne has red hair. Note the different phrasing of the general statement used in the first and the second variant. In the former, it is formulated as a backing, here taking the form of an empirical fact about Jack’s sisters, thereby encompassing the instance of sister Anne having red hair (at a certain moment). In the latter, it is formulated as a warrant, i.e., an inference-licensing general statement (‘may be taken to’). The former can be used as a backing for the second. Under which circumstances is (1, backing version) a genuinely analytic argument defending the claim that Anne has red hair? Well, says Toulmin, ‘only if at this very moment I have all of Jack’s sisters in sight. .... The thing to do now is use one’s eyes, not hunt up a chain of reasoning’ (126/117). The ‘So’ in the argument could be just as well replaced by ‘In other words’ or ‘That is to say’. In all other situations (which is: most), the conclusion will not be given with da- tum and backing, hence the argument will be a substantial one. Toulmin continues: ‘If the purpose of an argument is to establish conclusions about which we are not entirely confident by relating them back to other information about which we have greater assurance, it begins to be a little doubtful whether any genuine, practical argument could ever be properly analytic.’ Here is how Toulmin extends the Anne example, making it fit his own format: Datum: Anne is one of Jack’s sisters. Claim: Anne has red hair. Warrant: Any sister of Jack’s will (i.e. may be taken to) have red hair. Backing: All his sisters have previously been observed to have red hair. Qualifier: Presumably Rebuttal: Anne has dyed/gone white/lost her hair ... Note how Toulmin has added a qualifier and rebuttals, even though the backing assumes that all sisters have been checked. But Toulmin says rightly that normally an argument like this occurs later than at the time of establishing the warrant by the backing; hence making it non-demonstrative/subject to exceptions/defeasible/..., hence substantial, and not analytic (in Toulmin’s sense). Checking hair colour today is not a guarantee for hair colour tomorrow. Toulmin mentions one field in which arguments seem to be safe: mathematics. But then again: ‘As a model argument for formal logicians to analyse, it [i.e., a
- 238. 224 Bart Verheij solution to a mathematical problem] may be seducingly elegant, but it could hardly be less representative’ (127/118). 1.3 Standards of good reasoning and argument assessment are not universal, but context-dependent According to Toulmin, our standards for the assessment of real arguments are not universal, but depend on a context. In a section, where he discusses this issue, he uses the term ‘possibility’ as an illustration (36/34): whereas in mathematics ‘pos- sibility’ has to do with the absence of demonstrable contradiction, in most cases ‘possibility’ is based on a stronger standard. His example statement is ‘Dwight D. Eisenhower will be selected to represent the U.S.A. in the Davis Cup match against Australia’. This statement involves no contradiction, while still (now former, then actual) President Eisenhower will not be considered a possible team member. In other words, ‘possibility’ is judged using different standards, some more for- mal (‘absence of contradiction’), others more substantial (‘being a top-level tennis player’). The example is however an example of different standards for the possible, not of different standards for the assessment of arguments, his ultimate aim. Here is a succinct phrasing of his position: It is unnecessary, we argued, to freeze statements into timeless propositions before admit- ting them into logic: utterances are made at particular times and in particular situations, and they have to be understood and assessed with one eye on this context. The same, we can now argue, is true of the relations holding between statements, at any rate in the majority of practical arguments. The exercise of the rational judgement is itself an activity carried out in a particular context and essentially dependent on it: the arguments we encounter are set out at a given time and in a given situation, and when we come to assess them they have to be judged against this background. So the practical critic of arguments, as of morals, is in no position to adopt the mathematician’s Olympian posture. (182–3/168–9; the quote appears in a section entitled “Logic as a System of Eternal Truths”) According to Toulmin, the differences between standards of reasoning are reflected in the backings that are accepted to establish warrants. For instance, he considers the following three warrants (103–4/96): A whale will be a mammal. A Bermudan will be a Briton. A Saudi Arabian will be a Muslim. Each of these warrants gives in a similar way the inferential connection between certain types of data and certain kinds of claims. The first allows inferring that a particular whale is a mammal, the second that a particular Bermudan is a Briton, the third that a particular Saudi Arabian is a Muslim (all these inferences, of course, subject to qualification and rebuttal). The different standards become visible when information about the corresponding backings is inserted: A whale will be (i.e. is classifiable as) a mammal A Bermudan will be (in the eyes of the law) a Briton A Saudi Arabian will be (found to be) a Muslim
- 239. 11 The Toulmin Argument Model in Artificial Intelligence 225 Toulmin explains (104/96): One warrant is defended by relating it to a system of taxonomical classification, another by appealing to the statutes governing the nationality of people born in the British colonies, the third by referring to the statistics which record how religious beliefs are distributed among people of different nationalities. For Toulmin, the establishment of standards of argument assessment, hence of good reasoning, is an empirical question, cf. the following excerpt: Accepting the need to begin by collecting for study the actual forms of argument current in any field, our starting-point will be confessedly empirical: we shall study ray-tracing techniques because they are used to make optical inferences, presumptive conclusions and ‘defeasibility’ as an essential feature of many legal arguments, axiomatic systems because they reflect the pattern of our arguments in geometry, dynamics and elsewhere. (257/237)4 Toulmin goes one step further. Our standards of good reasoning are not only to be established empirically, they are also to be considered historically: they change over time and can be improved upon: To think up new and better methods of arguing in any field is to make a major advance, not just in logic, but in the substantive field itself: great logical innovations are part and parcel of great scientific, moral, political or legal innovations. [...] We must study the ways of arguing which have established themselves in any sphere, accepting them as historical facts; knowing that they may be superseded, but only as the result of a revolutionary advance in our methods of thought. (257/237) Because of his views that standards of good reasoning and argument assessment are non-universal and depend on field, even context, Toulmin has been said to revive Aristotle’s Topics (Toulmin 2003, viii). 1.4 Logic is generalised jurisprudence Toulmin discusses the relation of logic with a number of research areas (3–8/3–8). When logic is regarded as psychology, it deals with the laws of thought, distinguish- ing between what is normal and abnormal, thereby perhaps even allowing a kind of “psychopathology of cognition” (5/5). In logic as psychology, the goal is at heart descriptive: to formulate generalisations about thinkers thinking. But logic can also be seen as a kind of sociology. Then it is not individual thinkers that are at issue, but the focus is on general habits and practices. Here Toulmin refers to Dewey, who explains the passage from the customary to the mandatory: inferential habits can turn into inferential norms. Logic can also be regarded as a kind of technology, i.e., as providing a set of recipes for rationality or the rules of a craft. Here he speaks of logic as an art, like medicine. In this analogy, logic aims at the formulation of maxims, ‘tips’, that remind thinkers how they should think. And then there is logic as mathematics. There the goal of logic becomes to find truths about logical rela- tions. There is no connection with thinking and logic becomes an objective science. 4 Toulmin here considers the study of defeasibility an empirical question, to be performed by looking at the law! Toulmin has predicted history, by foreseeing what actually has happened and still is happening in the field of AI law.
- 240. 226 Bart Verheij Finally Toulmin comes to the metaphor that he prefers and uses as the basis for his work: to view logic as jurisprudence: Logic is concerned with the soundness of the claims we make-with the solidity of the grounds we produce to support them, the firmness of the backing we provide for them-or, to change the metaphor, with the sort of case we present in defence of our claims. (7/7) The jurisprudence metaphor emphasises the critical, procedural function of logic, thereby fundamentally changing the perspective on logic. It helps to change logic from an ‘idealised logic’ to a ‘working logic’ (cf. the title of the fourth essay in The Uses of Argument). At the end of his book he says that jurisprudence should not be seen as merely an analogy, but, more strongly, as providing an example to follow, as being a kind of ‘best practice’: Jurisprudence is one subject which has always embraced a part of logic within its scope, and what we called to begin with ‘the jurisprudential analogy’ can be seen in retrospect to amount to something more than a mere analogy. If the same as has long been done for legal arguments were done for arguments of other types, logic would make great strides forward. (255/235) 2 The reception and refinement of Toulmin’s ideas in AI The reception of Toulmin’s ideas is marked by historical happenstance. It was al- ready mentioned that his original audience, primarily the positivist, logic-oriented philosophers of knowledge of the time, was on the whole critical. For Toulmin’s main messages to be appreciated a fresh crowd was needed. It was found in a radi- cal movement in academic research and education refocusing on the analysis and as- sessment of real-life argument. This movement, referred to by names such as speech communication, informal logic and argumentation theory, started to blossom from the 1970s, continuing so to the present day (see [22]). One thing that Toulmin and this movement shared was the relativising, at times antagonistic, attitude towards logic as a formal science. The swing had swung back to exploring the possibilities of more formal approaches in the 1990s, when Toulmin’s project of treating logic as a generalised jurisprudence was almost literally taken up in the field of Artificial In- telligence and Law (see Feteris’ [4] for a related development in argumentation the- ory). Successful attempts were made to formalize styles of legal reasoning in a way that respected actual legal reasoning. The approach taken in this field was rooted in an independent development in Artificial Intelligence, where so-called nonmono- tonic logics were studied from the 1980s. In that line of research, formal logical systems were studied that allowed for the retraction of conclusions when new in- formation, indicating exceptional or contradictory circumstances, became available. Also in the 1990s, the study of nonmonotonic logics evolved towards what might be called argumentation logics. More generally, attention was reallocated to imple- mented systems and an agent-oriented perspective. The following does not give a fully representative, historical account of AI work taking up Toulmin’s ideas. A personal choice of relevant research has been made in
- 241. 11 The Toulmin Argument Model in Artificial Intelligence 227 order to highlight how Toulmin’s points of view have been adopted and refined in Artificial Intelligence. 2.1 Reiter’s default rules An early strand of research in Artificial Intelligence, in which a number of Toulmin’s key positions are visible, is Reiter’s work on the logic of default reasoning [15]. Re- iter’s formalism is built around the concept of a default: an expression α : Mβ1, ..., Mβn / γ, in which α, β1, ..., βn, and γ are sentences of first-order logic. Defaults are a kind of generalized rules of inference. The sentence α is the default’s prereq- uisite, playing the role of what Toulmin refers to as the datum. The sentence γ is the default’s consequent, comparable to Toulmin’s notion of a claim. The sentences βi are called the default’s justifications. A default expresses that its consequent follows given its prerequisite, but only when its justifications can consistently be assumed. Reiter does not refer to Toulmin in his highly influential 1980 paper, nor in his other work. Being thoroughly embedded in the fertile logic-based AI community of the time, Reiter does not refer to less formal work. Still, in Reiter’s work two important ideas defended by Toulmin recur in a formal version. The first is the idea of defeasibility. As said, Reiter’s defaults are a kind of generalized rules of inference, but of a defeasible kind. For instance, the default p : M¬e / q expresses that q follows from p unless ¬e cannot be assumed consistently. Reiter’s formal definitions are such that, given only p, it follows that q, while if both p and e are given q does not follow. Reiter’s justifications are hence closely related to Toulmin’s rebuttals, but as opposites: in our example the opposite e of the default’s justification ¬e is a kind of rebuttal in Toulmin’s sense. This holds more generally: opposites of justifications can be thought of as formal versions of Toulmin’s rebuttals. There is a second way in which Reiter’s work formally explicates one of Toul- min’s prime concerns: defaults are contingent rules of inference, in the sense that they are not fixed in the logical system, as is the case for the natural deduction rules of first-order logic. Concretely, in Reiter’s approach, defaults are part of the the- ory from which consequences can be drawn, side by side with the other, factual, information. One can therefore say that Toulmin’s creed that standards of reason- ing are field-dependent has found a place in Reiter’s work. There is one important limitation however. Although Reiter’s defaults can be used to construct arguments — in what Toulmin refers to as warrant-using arguments —, they cannot be argued about. In other words, there is no counterpart of warrant-establishing arguments. A default can for instance not have a default as its conclusion. Since Reiter’s defaults are givens, it is not possible to give reasons for why they hold. Whereas in Toulmin’s model warrants do not stand by themselves, but can be given support by backings, this has no counterpart for Reiter’s defaults. (See section 2.7 for an approach to warrant-establishing arguments.) How did Reiter extend or refine Toulmin? The first way is obvious: Reiter has given a precise explication of a part of Toulmin’s notions, which is a direct conse- quence of the fact that Reiter’s approach is formally specified, whereas Toulmin’s only exists in the form of an informal philosophical essay. Reiter has shown that it is possible to give a formal elaboration of rebuttals and of warrants.
- 242. 228 Bart Verheij The other way is perhaps more important, as it concerns a genuine extension of what Toulmin had in mind: Reiter’s logical formalism proposes a way of determin- ing which consequences follow from given information. The key formal notion is that of an extension of a default theory (consisting of a set of factual assumptions and a set of defaults), which can be thought of as a possible set of consequences of the theory. Essentially, a set of sentences S is an extension of a default theory if S is equal to the set of consequences of the factual information that one obtains by ap- plying a subset of the defaults, namely those defaults the justifications of which are consistent with S. (Note that S occurs in the definiens and in the definiendum.) Let me show how and to what extent Reiter’s system can formalize Toulmin’s Harry- example. We will leave out the qualifier and the backing as these have no obvious counterpart in Reiter’s work. The core of a formalization of the Harry example is the default d(x) : M¬r1(x), M¬r2(x) / c(x) and its instance d(t) : M¬r1(t), M¬r2(t) / c(t). The following code is used: t Harry d(t) Harry was born in Bermuda c(t) Harry is a British subject r1(t) Both his parents were aliens r2(t) He has become a naturalized American The default expresses that it follows that Harry is a British subject given that he is born in Bermuda, as long as it can be consistently assumed that his parents are not aliens and he has not become a naturalized American. Note that the default is a kind of hybrid of the example’s warrant and rebuttals and that the default’s list of justifications is not open-ended (in contrast with Toulmin’s list of rebuttals). Now consider two sets of sentences: S1, the first-order closure of d(t), c(t) and S2, the closure of d(t), r1(t). Then S1 is the unique extension of the theory consisting of the default and the factual information d(t), while S2 is the unique extension of the theory consisting of the default and the factual information d(t) and r1(t). (Analo- gous facts hold when the other rebuttal r2(t) is used.) These facts can be interpreted as saying that, given the warrant encoded by the default, the claim follows from the data, but only when there is no rebuttal. Three ways in which this formal version refines Toulmin’s treatment seem note- worthy. First, here there is a distinction between ‘generic’ warrants and ‘specific’ warrants. The former is for Toulmin a pleonasm, while he does not consider the latter. Here the distinction is clear: on the one hand there is the generic inference license that a man born in Bermuda will generally be a British subject, on the other the specific inference license that if Harry was born in Bermuda, he is a British sub- ject. Second, whereas Toulmin only treats single, unstructured sentences, Reiter’s formal system inherits the elegant additional structuring of first-order sentences. For instance, disjunction and conjunction are directly inherited. Third and finally, Reiter’s version specifies what happens when there is more than one default. Es- pecially, his version incorporates naturally the situations that a sentential element (datum, rebuttal, ...) of one instance of Toulmin’s model can be the claim of another. It has sometimes been charged against Toulmin that his model does not allow such recursiveness.
- 243. 11 The Toulmin Argument Model in Artificial Intelligence 229 2.2 Pollock’s undercutting and rebutting defeaters Pollock’s work on the philosophy and AI of argumentation has rightly achieved recognition in today’s argumentation research. He can be regarded as being the first who combined theoretical, computational and practical considerations in his design of an ‘artificial person’, OSCAR (see, e.g., [12]). In this high ambition, he has had no followers. Pollock’s work started with roots close to Toulmin’s original audi- ence, namely philosophers of knowledge. Gradually he began using methods from the field of Artificial Intelligence, where his ideas have gained most attention. Pol- lock does not seem to have been directly influenced by Toulmin. In his [11], where Pollock connects philosophical approaches to defeasibility with AI approaches, he cites work on defeasibility by Chisholm (going back to 1957, hence a year before Toulmin’s The Uses of Argument) and himself (going back to 1967). Here are some of Pollock’s definitions [11]: P is a prima facie reason for S to believe Q if and only if P is a reason for S to believe Q and there is an R such that R is logically consistent with P but (P R) is not a reason for S to believe Q. R is a defeater for P as a prima facie reason for Q if and only if P is a reason for S to believe Q and R is logically consistent with P but (P R) is not a reason for S to believe Q. So prima facie reasons are reasons that sometimes lead to their conclusion, but not always, namely not when there is a defeater. There is a close connection with non- monotonic consequence relations: when P is a prima facie reason for Q, and R is a defeater for P as a reason, then Q follows from P, but not from P R. Pollock goes on to distinguish between two kinds of defeaters: R is a rebutting defeater for P as a prima facie reason for Q if and only if R is a defeater and R is a reason for believing ∼Q. R is an undercutting defeater for P as a prima facie reason for S to believe Q if and only if R is a defeater and R is a reason for denying that P wouldn’t be true unless Q were true. Undercutting defeaters only attack the inferential connection between reason and conclusion, whereas a defeater is rebutting if it is also a reason for the opposite of the conclusion. Pollock remarks that ‘P wouldn’t be true unless Q were true’ is a kind of conditional, different from the material conditional of logic, but having learnt from an initial analysis, which he no longer finds convincing, he maintains that it is otherwise not clear how to analyze this conditional ([11], 485).5 Pollock’s finding that there are different kinds of defeaters has been recognized as an important contribution both for the theory and for the practical analysis of arguments. Nothing of the sort can be found in Toulmin’s The Uses of Argument.6 (See section 2.7 for more on different conceptions of a rebuttal.) 5 As far as I know, Pollock’s later work (e.g., his [12]) does not contain a new analysis of this conditional. See section 2.7 for an approach addressing this. 6 Pollock’s work contributes significantly to several other aspects of argumentation (e.g., argu- ment evaluation, semi-formal rules of inference and software implementation). See also Verheij’s discussion [26], 104–110.
- 244. 230 Bart Verheij 2.3 Prakken, Sartor Hage on reasoning with legal rules Toulmin’s idea that logic should be regarded as a generalised form of jurisprudence (section 1.4), was taken up seriously in the 1990s in the field of Artificial Intel- ligence and Law. The work by Prakken, Sartor and Hage on reasoning with legal rules [13, 6] is representative.7 Influenced by logic-based knowledge representation (see, e.g., chapter 10 of [16]), Prakken Sartor and Hage use an adapted first-order language as the basis of their formalism. For instance, here is a formal version of the rule that someone has legal capacity unless he can be shown to be a minor ([13], 340): r1: ∼x is a minor ⇒ x has legal capacity Here r1 is the name of the rule, which can be used to refer to it, and ‘x is a minor’ and ‘x has legal capacity’ are unary predicates. The tilde represents so-called weak negation, which here means that the rule’s antecedent is fulfilled when it cannot be shown that x is a minor. If ordinary negation were used, the fulfilment of the antecedent would require something stronger, namely that it can be shown that x is not a minor. In the system of Prakken Sartor, arguments are built by applying Modus po- nens to rules. There are two ways in which arguments can attack each other. First, an argument can attack a weakly negated assumption in the antecedent of a rule used in the attacked argument. Second, two arguments can have opposite conclusions. Information about rule priorities (expressed using the rules names) is then used to compare the arguments. Argument evaluation is defined in terms of winning strate- gies in dialogue games: an argument is called justified when it can be successfully defended against an opponent’s counterarguments. Hage’s approach [6], in several ways similar to Prakken Sartor’s, is more am- bitious and philosophically radical.8 For Hage, rules are first-and-foremost to be thought of as things with properties. As a result, a rule is formalized as a structured term. A rule’s properties are then formalized using predicates. For instance, the fact that the rule that thieves are punishable, is valid is formalized as Valid(rule(theftl, thief(x), punishable(x))). Here ‘theft1’ is the name of the rule, ‘thief(x)’ the rule’s antecedent and ‘punish- able(x)’ its consequent. Hage’s work takes the possibilities of a knowledge repre- sentation approach to the modelling of legal reasoning to its limits. For instance, there are dedicated predicates to express reasons, rule validity, rule applicability and the weighing of reasons. How does the work by Prakken, Sartor Hage relate to Toulmin’s views? First, they have provided an operationalisation of Toulmin’s idea of law-inspired logic, by formalizing aspects of legal reasoning. Second, they have refined Toulmin’s treat- ment of argument. Notably, Prakken Sartor have modelled specific kinds of re- buttal, namely by the attack of weakly negated assumptions and on the basis of 7 Some other important AI Law work concerning argumentation is for instance [1, 2, 5, 10]. 8 Hage’s philosophical and formal theory of rules and reasons Reason-Based Logic was initiated by Hage and further developed in cooperation with Verheij.
- 245. 11 The Toulmin Argument Model in Artificial Intelligence 231 rule priorities, and embedded them in an argumentative dialogue. Hage has added a further kind of rebuttal, namely by the weighing of reasons. Also, Hage has distin- guished the validity of a rule from its applicability. The former can be regarded as an expression of a warrant in Toulmin’s sense, and since in Hage’s system rule validity can depend on other information, it is natural to model Toulmin’s backings as rea- sons for the validity of a rule. And perhaps most importantly: Prakken Sartor and Hage (and other AI law researchers) have worked on the embedding of defeasible argumentation in a genuine procedural, dialogical setting (see also section 2.5). A further refinement of Toulmin’s view is given by Verheij and colleagues [28], who show how two kinds of warrants (viz. legal rules and legal principles) with apparent logical differences, can be seen as extremes of a spectrum. 2.4 Dung’s admissible sets Dung’s paper [3] has supplied an abstract mathematical foundation for formal work on argumentation. Following earlier mathematically flavoured work (e.g., [17, 29, 30]), his abstraction of only looking at the attack relation between argu- ments has helped organize the field, e.g., by showing how several formal systems of nonmonotonic reasoning can be viewed from the perspective of argument attack. A set of (unstructured) arguments with an attack relation is called an argumentation framework. Dung has studied the mathematics of three types of subsets of the set of argu- ments of an argumentation framework: stable, preferred and grounded extensions. A set of arguments is a stable extension if it attacks all arguments not in the set. A set of arguments is a preferred extension if it is a maximal set of arguments without internal conflicts and attacking all arguments attacking the set. The grounded exten- sion (there is only one) is the result of an inductive process: starting from the empty set, consecutively arguments are added that are only attacked by arguments already defended against. Stable extensions can be regarded as an ‘ideal’ interpretation of an argumentation framework. When an extension is stable, all conflicts between arguments can be regarded as solved. It turns out that sometimes there are distinct ways of resolving the conflicts (e.g., when two arguments attack each other, each argument by itself is a stable extension) and that sometimes there is no way (e.g., when an argument is self-attacking). Preferred extensions are a generalization of stable extensions, as all stable extensions are also preferred. However, an argumentation framework always has a preferred extension (perhaps several). Preferred extensions can be regarded as showing how as many conflicts as possible can be resolved by counterattack. The grounded extension exists always and is a subset of all preferred and stable extensions. Dung’s work shows that the mathematics of argument attack is non-trivial and interesting. Thereby he has significantly extended our understanding of Toulmin’s concept of rebuttal.
- 246. 232 Bart Verheij 2.5 Walton’s argumentation schemes Toulmin’s proposal that the maxims provided by a standard formal logical system (such as first order predicate logic) are not the only criteria for good reasoning and argument assessment, posed a new problem: if there are other, more field- and context-dependent standards of reasoning, what are they? For, though many recog- nized the shortcomings of formal logic for practical argument assessment, few were happy with the possible relativistic implication that anything goes. A good way to avoid the trap of uncontrolled relativism is to provide a systematic specification of standards of good reasoning. One approach in this direction, which is especially close to Toulmin’s concep- tion of warrants, can be found in Walton’s work on argumentation schemes (e.g., [31]). Argumentation schemes can be thought of as a semi-formal generalization of the rules of inference found in formal logic. Argument from expert opinion is an example ([31], 65): E is an expert in domain D. E asserts that A is known to be true. A is within D. Therefore, A may (plausibly) be taken to be true. As Walton’s argumentation schemes are context-dependent, not universal; defeasi- ble, not strict; and concrete, not abstract, there is a strong analogy with Toulmin’s warrants. There are two important differences though. First, Walton’s argumentation schemes are structured, whereas Toulmin’s warrants are not. Walton’s argumenta- tion schemes have premises, consisting of one or more sentences (often with in- formal variables), and a conclusion;9 Toulmin’s warrants are expressed as rule-like statements, such as ‘A man born in Bermuda will generally be a British subject’.10 By giving generic inference licenses more structure, as in Walton’s work, the ques- tion arises whether they become formal enough to give rise to a kind of ‘concrete logic’. Verheij [24] argues that it is a matter of choice, perhaps: taste, whether one draws the border between form and content on either side of argumentation schemes. To indicate the somewhat ambiguous status of argumentation schemes, the term ‘semi-formal’ may be most appropriate. Second, Walton’s argumentation schemes have associated critical questions. Crit- ical questions help evaluating applications of an argumentation scheme. As a result, they play an important role in the evaluation of practical arguments. For instance, Walton lists the following critical questions for the scheme ‘Argument from expert opinion’ ([31], 65): 1. Is E a genuine expert in D? 2. Did E really assert A? 9 Sometimes Walton’s schemes take another form, e.g., small chains of argument steps or small dialogues; see Verheij’s [24] for a format for the systematic specification of argumentation schemes inspired by knowledge engineering technology. 10 Occasionally, a bit more structure is made explicit. For instance, when Toulmin phrases a warrant in an ‘if ... then ...’ form (e.g., ‘If anything is red, it will not also be black’, 98/91), thereby making an antecedent and consequent recognizable.
- 247. 11 The Toulmin Argument Model in Artificial Intelligence 233 3. Is A relevant to domain D? 4. Is A consistent with what other experts in D say? 5. Is A consistent with known evidence in D? Critical questions are related to argument attack, as they point to circumstances in which application of the scheme is problematic (e.g., [24]). For instance, the question ‘Is E a genuine expert in D?’ questions whether the element ‘E is an expert in domain D’ in the premises of the scheme really holds. Some critical questions are like Toulmin’s notion of rebuttal. For instance, the question ‘Is A consistent with what other experts in D say?’ points to a rebuttal ‘A is not consistent with what other experts in D say’, which, if accepted, can raise doubt whether the conclusion can justifiably be drawn. In general, four types of critical questions can be distinguished [24]: 1. Critical questions concerning the conclusion of an argumentation scheme. Are there other reasons, based on other argumentation schemes for or against the scheme’s conclusion? 2. Critical questions concerning the elements of the premises of an argumentation scheme. Is E an expert in domain D? Did E assert that A is known to be true? Is A within D? 3. Critical questions based on the exceptions of an argumentation scheme. Is A consistent with what other experts in D say? Is A consistent with known evidence in D? 4. Critical questions based on the conditions of use of an argumentation scheme. Do experts with respect to facts like A provide reliable information concerning the truth of A? The critical questions associated with an argumentation scheme point to the dia- logical setting of argumentation. Toulmin mentions the dialogical and procedural setting of argumentation (as, e.g., when discussing the jurisprudence metaphor for logic), but the discussion is not elaborate. Much work on the relation between argu- mentation and dialogue has been done. There is for instance the pragma-dialectical school (e.g., [21]), but also Walton’s conception of argumentation is embedded in a procedural, dialogical setting. For instance, Walton [32] expresses a view on how to determine the relevance of an argument in a dialogue. There are six issues to take into account: the dialogue type11, the stage the dialogue is in, the dialogue’s goal, the type of argument, which is determined by the argumentation scheme underlying the argument, the prior sequence of argumentation, and the institutional and social setting. In conclusion, Walton’s work has played a significant role in two developments in AI with respect to Toulmin’s main themes. First, the study of argumentation schemes by Walton and others has made a start with the systematic specification of context-dependent, defeasible, concrete standards of argument assessment, as sought for by Toulmin. And, second, the idea of considering argumentation from a procedural, dialogue perspective has been elaborated upon. 2.6 Reed Rowe’s argument analysis software Further steps towards the realization of Toulmin’s goals have been made by the recent advent of software-support of argumentative tasks, often using argument dia- grams [8, 26]. In this connection, Reed Rowe’s work on the Araucaria tool [14] is 11 See also Walton Krabbe’s [33], a treatment of dialogue types that is especially influential in research in AI and multi-agent systems.
- 248. 234 Bart Verheij especially relevant for the achievement of Toulmin’s goals, as they have presented Araucaria specifically as a software tool for argument analysis. Araucaria uses an argument diagramming format, in which the recursive tree-structure of reasons sup- porting conclusions is depicted. It is also possible to indicate statements that are in conflict. Araucaria’s standard diagramming format12 is different from Toulmin’s in several ways, but especially by not graphically distinguishing warrants from data. In an interestingly different way, however, Araucaria’s standard format does include the idea of context-dependent types of reasoning as argued for by Toulmin, namely by its incorporation of Walton-style argumentation schemes (cf. section 2.5). Ar- gumentation schemes can be used in Araucaria to label argumentative steps. For instance, a concrete argument ‘There is smoke. Therefore, there is fire’ could be la- belled as an instance of the scheme ‘Argument from sign’, thereby giving access two critical questions, such as ‘Are there other events that would more reliably account for the sign?’. By this possibility, Reed Rowe’s Araucaria is a useful step towards software-supported argument assessment. The tool provides a significant extension of Toulmin’s aim to change logic from an ‘idealised logic’ to a ‘working logic’. 2.7 Verheij’s formal reconstruction of Toulmin’s scheme Already the examples in this chapter show a wide variety of approaches to — what might be called — semi-formal defeasible argumentation; and this is just the tip of the iceberg. By this embarrassment of riches, the question arises whether there are fundamental differences, e.g., between explicitly Toulmin-oriented approaches and other; or is the similarity of subject matter strong enough to allow for a synthesis of approaches? Looking for answers, I have attempted to reconstruct Toulmin’s scheme using modern formal tools [25]. I used the abstract argumentation logic DefLog [23]. DefLog uses two connectives × and ❀: the first for expressing the defeat of a prima facie justified statement (‘negation-as-defeat’, the semantics of which falls outside the scope of this chapter), the second for expressing a conditional relation between statements (‘primitive implication’, validating Modus Ponens, but lacking a so-called introduction rule).13 Toulmin’s notion of a qualifier has been left out of the reconstruction. The key to the translation of Toulmin’s scheme into DefLog is to explicitly ex- press that a datum leads to a claim; in DefLog: D ❀ C. DefLog’s primitive im- plication can be thought of as expressing a specific inference license. It is an ex- plicit expression of what Toulmin refers to as a ‘logical gulf’ (9/9) that seems to exist between a reason and the state of affairs it supports. In this way, the licens- ing of concrete argument steps is removed from the logic, i.e., the fixed formalized background specifying general argument validity, and shifted to the contingent in- formation. In this way, it becomes possible to express substantial arguments about concrete inferential bridges. 12 In later versions, two alternative formats are provided: Toulmin’s and Wigmore’s. 13 DefLog is formally an extension of Dung’s abstract argumentation framework (section 2.4), as Dung’s attack between two arguments A and B can be expressed as A ❀ ×B. DefLog analogues of Dung’s stable and preferred semantics are defined and proven to coincide with Dung’s when DefLog’s language is restricted to Dung’s.
- 249. 11 The Toulmin Argument Model in Artificial Intelligence 235 In particular, the role of a warrant can now be expressed as a reason for such a conditional statement: W ❀ (D ❀ C). To formally show that a datum and claim are specific, whereas a warrant is to be thought of as a generic inference license, we can use variables and their instances: W ❀ (D(t) ❀ C(t)). This clarifies the distinctions between the following three: (1) A man born in Bermuda will generally be a British subject. (2) If Person was born in Bermuda, then generally Person is a British subject. (3) If Harry was born in Bermuda, then generally he is a British subject. The first is the ordinary language expression of a warrant as a rule-like statement (formally: W). The third is a conditional sentence (formally: D(t) ❀ C(t)), instan- tiating the second, which is a scheme of conditional sentences (formally: D(x) ❀ C(x), where x is a variable, that can be instantiated by the concrete term t). Note that only (1) and (3) can occur in actual texts, whereas (2) is — by its use of a variable Person — an abstraction. One could say however that (1) and (2) imply each other: a warrant corresponds to a scheme of argumentative steps from datum to claim. Given this analysis of warrants and their relation to datum and claim, backings are simply reasons for warrants: B ❀ W. Rebuttals are an ambiguous concept in Toulmin’s treatment. He associates rebut- tals with ‘circumstances in which the general authority of the warrant would have to be set aside’ (101/94), ‘exceptional circumstances which might be capable of defeat- ing or rebutting the warranted conclusion’ (101/94) and with the (non )applicability of a warrant (102/95). It turns out that these three can be distinguished, and, given the present analysis of the warrant-datum-claim part of Toulmin’s scheme, even ex- tended to five kinds of rebuttals, as there are five different statements that can be argued against: the datum D, the claim C, the warrant W, the conditional D(t) ❀ C(t), expressing the inferential bridge from datum to claim, and the conditional W ❀ (D(t) ❀ C(t)), which expresses the application of the warrant in the concrete situ- ation (see [25] for a more extensive explanation). A rebuttal of the latter conditional coincides conceptually with an undercutting defeater in the sense of Pollock’s. Note that the analysis suggests an answer to Pollock’s open issue of how to analyze the conditional ‘P wouldn’t be true unless Q were true’ (section 2.2). If U undercuts P as a reason for Q, we would write U ❀ ×(P ❀ Q). In this analysis, there is a natural extension of Toulmin’s concept of warrant: just as it is necessary to specify which data imply claims (by Toulmin’s warrants), it is necessary to specify which rebuttals block the application of warrants. Informally: it is a matter of substance, not logic, which statements are rebuttals. It must be shown by argument whether some statement is a rebuttal. In the law, for instance, not only legal rules (a kind of warrants) find backing in statutes, but also exceptions to rules. In the present analysis, dealing with ‘rebuttal warrants’ is a matter of course, since there is an explicit expression that R is a rebuttal.14 A side effect of the reconstruction is that arguments modelled according to Toul- min’s scheme can be formally evaluated. For instance, assuming that datum and 14 When R is a rebuttal in the sense of a Pollockian undercutter, this requires a sentence of the form R ❀ ×(W ❀ (D ❀ C)), expressing that, if R holds, the warrant W is not applicable.
- 250. 236 Bart Verheij Fig. 11.2 An entangled dialectical argument warrant hold, but not a rebuttal, the claim follows; when also a rebuttal is assumed, the claim does not follow.15 A rebuttal of a rebuttal can be shown to reinstate a claim. Verheij [24] extends the approach to include Walton’s argumentation schemes. The result of the formal reconstruction of Toulmin’s scheme showed some ex- tensions, while retaining the original flavour. Figure 2 (using a diagramming format used in [27]) illustrates the basic relations between statements as distinguished here: Claims can have reasons for and against them (Jim’s testimony supporting the as- sault by Jack, and Paul’s attacking it), and the inferential bridges (the conditionals connecting reasons with their conclusions, here drawn as arrows) can be argued about just like other statements. The resulting argument structures are dialectical, by their incorporation of pros and cons, and entangled by their allowing the support and attack of inferential bridges. 3 Concluding remarks It has been shown that central points of view argued for by Toulmin (1958), in par- ticular the defeasibility of argumentation, the substantial, instead of formal, nature of standards of argument assessment, and the richer set of building blocks for argu- ment analysis, are very much alive. Also Toulmin’s ‘research program’ of treating logic as generalised jurisprudence has been taken up (with or without reference to him) and proven to be fertile. There have also been refinements and extensions. It is now known that defeasible argumentation has interesting (and intricate) formal properties. There exist formal systems for the evaluation of defeasible arguments. Toulmin’s argument diagram and its associated set of building blocks for argument analysis have been made pre- cise and become refined, e.g., by distinguishing between kinds of argument attack. Not only is there now a wealth of studies of domain-bound, concrete forms of ar- 15 Formally: using W ❀ (D ❀ C), R ❀ ×(W ❀ (D ❀ C)), two sentences expressing that W is a warrant and R a rebuttal blocking its application, respectively, as background, and then assuming W, and D, one finds a unique dialectical interpretation, in which C holds, whereas adding R to the assumptions leads to a unique dialectical interpretation in which C does not hold.
- 251. 11 The Toulmin Argument Model in Artificial Intelligence 237 gumentation, also methods for their systematic investigation have been proposed. Defeasible argumentation has been embedded in procedural models of dialogue. Toulmin’s wish to develop logic into a practical tool has found a modern guise in the form of argumentation-support software, aiming at argument analysis and pro- duction. Notwithstanding recent progress, there is ample room for innovative research. Some possible directions of future research are the continuing systematisation and specification of argumentation schemes; the further organisation of the wealth of evaluation paradigms for defeasible argumentation (‘semantics’); the prolongation of research aiming at practically useful software tools, especially when supported by user studies or commercial success; the implementation of software agents capable of argumentative behaviour; and the coupling of empirical work on reasoning and argumentation to the findings in AI. Toulmin ends his introduction in a modest, but hopeful mood: The studies which follow are, as I have said, only essays. If our analysis of arguments is to be really effective and true-to-life it will need, very likely, to make use of notions and distinctions that are not even hinted at here. But of one thing I am confident: that by treating logic as generalised jurisprudence and testing our ideas against our actual practice of argument-assessment, rather than against a philosopher’s ideal, we shall eventually build up a picture very different from the traditional one. The most I can hope for is that some of the pieces whose shape I have here outlined will keep a place in the finished mosaic. (10/10) As evidenced by the research discussed in this chapter, the present state of the art in AI-inspired argumentation research shows that Toulmin’s hope has been fulfilled. Acknowledgements The author would like to thank David Hitchcock, Douglas Walton and James Freeman for comments on a prepublication version of this text. References 1. K. D. Ashley. Modeling legal argument. Reasoning with cases and hypotheticals. The MIT Press, Cambridge (Massachusetts), 1990. 2. T. J. M. Bench-Capon. Persuasion in practical argument using value-based argumentation frameworks. Journal of Logic and Computation, 13(3):429–448, 2003. 3. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77:321–357, 1995. 4. E. T. Feteris. Fundamentals of legal argumentation. A Survey of Theories on the Justification of Judicial Decisions. Kluwer Academic Publishers, Dordrecht, 1999. 5. T. F. Gordon. The Pleadings Game. An Artificial Intelligence Model of Procedural Justice. Kluwer Academic Publishers, Dordrecht, 1995. 6. J. C. Hage. A theory of legal reasoning and a logic to match. Artificial Intelligence and Law, 4:199–273, 1996. 7. D. L. Hitchcock and B. Verheij, editors. Arguing on the Toulmin Model. New Essays in Ar- gument Analysis and Evaluation (Argumentation Library, Volume 10). Springer-Verlag, Dor- drecht, 2006. 8. P. A. Kirschner, S. J. Buckingham Shum, and C. S. Carr. Visualizing Argumentation: Software Tools for Collaborative and Educational Sense-Making. Springer-Verlag, London, 2002.
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- 253. Chapter 12 Proof Burdens and Standards Thomas F. Gordon and Douglas Walton 1 Introduction This chapter explains the role of proof burdens and standards in argumentation, il- lustrates them using legal procedures, and surveys the history of research on compu- tational models of these concepts. It also presents an original computational model which aims to integrate the features of these prior systems. The ‘mainstream’ conception of argumentation in the field of artificial intelli- gence is monological [6] and relational [14]. Argumentation is viewed as taking place against the background of an inconsistent knowledge base, where the knowl- edge base is a set of propositions represented in some formal logic. Argumentation in this conception is a method for deducing warranted propositions from an incon- sistent knowledge base. Which statements are warranted depends on attack relations among the arguments [10] which can be constructed from the knowledge base. The notions of proof standards and burden of proof become relevant only when argumentation is viewed as a dialogical process for making justified decisions. The input to the process is an initial claim or issue. The goal of the process is to clarify and decide the issues, and produce a justification of the decision which can with- stand a critical evaluation by a particular audience. The role of the audience could be played by the respondent or a neutral-third party, depending on the type of di- alogue. The output of this process consists of: 1) a set of claims, 2) the decision to accept or reject each claim, 3) a theory of the generalizations of the domain and the facts of the particular case, and 4) a proof justifying the decision of each issue, showing how the decision is supported by the theory. Notice that a theory or knowledge-base is part of the output of argumentation dialogues, not, as in the relational conception, its input. This is because, as has been Thomas F. Gordon Fraunhofer FOKUS, Berlin, Germany, e-mail: thomas.gordon@fokus.fraunhofer.de Douglas Walton University of Windsor, Windsor, Canada, e-mail: dwalton@uwindsor.ca I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 239 DOI 10.1007/978-0-387-98197-0 12, c Springer Science+Business Media, LLC 2009
- 254. 240 Thomas F. Gordon and Douglas Walton repeatedly recognized [31, 33, 21], the generalizations (rules) of some domain and the particular facts of a problem or case are dependent on one another and need to be constructed together, in an iterative process. For example, one of the founders of the field of computer science and law, Jon Bing, wrote in 1982: Legal reasoning is not primarily deductive, but rather a modeling process of shaping an understanding of the facts, based on evidence, and an interpretation of the legal sources, to a construct a theory for some legal conclusion. [7] The concept of proof in argumentation is weaker than it is in mathematics. The proof need not demonstrate that a proposition is necessarily true, given a set of ax- ioms assumed to be true. Rather, as in law, a proof in argumentation is a structure which demonstrates to a particular audience that a proposition statisfies its applica- ble proof standard. Since expressive logics are undecidable or intractable, the theory constructed during the dialogue cannot usually serve as a proof. A burden of proof is not discharged if the audience must solve a hard problem to construct the proof for themselves from the theory. There are several kinds of proof burdens. The distinctions between them can only be understood with a deeper analysis of particular argumentation processes. There are many kinds of argumentation processes, each regulated by its own procedural rules, usually called ‘protocols’ in AI. Walton has developed a typology of dialogue types, classifying persuasion dialogues, negotiation, and deliberation, among other types [37]. For our purpose of illustrating different kinds of proof burdens, it is sufficient to use a simplified description of civil procedure, roughly based on the law of Califor- nia [34]. A civil case begins by the plaintiff filing a complaint, stating a claim against the defendant. The complaint is the first step in the pleadings phase of the case. It contains, in addition to the claim, assertions about the facts of the case which the plaintiff contends are sufficient, if true, to prove the defendant has breached some obligation legally entitling the plaintiff to some remedy or compensation. The de- fendant then has several options for responding to the complaint. For the sake of brevity we will mention just one, filing an answer in which the factual allegations are each denied or conceded and asserting additional facts, called an affirmative de- fense, which may be useful for defeating or undercutting later arguments put forward by the plaintiff. The final step in the pleadings phase gives the plaintiff an opportu- nity to file a reply in which he concedes or denies the additional facts alleged by the defendant in his answer. The next phase of the process provides the parties various methods to discover evidence, for example by interviewing witnesses under oath, called taking depositions. At the trial, this evidence is presented to the judge, and possibly a jury, and further evidence is produced by examining and cross-examining witnesses during the trial. At the end of the trial, the evidence is passed on to the trier-of-fact, either the judge or the jury, if there is one. If there is a jury, the judge first instructs the jury about the relevant law, since the jury is only responsible for finding the facts. After the jury has completed its deliberations, it reports its verdict to the judge, who then enters his judgment upon the verdict. The judgment may be appealed by the losing party, but we will end our exposition of legal procedure here.
- 255. 12 Proof Burdens and Standards 241 Our account of legal burdens of proof below is based in part on [30]. The first kind of burden of proof is called the burden of claiming. A person who feels he has a right to some legal remedy has the burden of initiating the proceeding by filing a complaint, which must allege facts sufficient to prove the operative facts of legal rules entitling him to some remedy. The second type of burden of proof is called the burden of questioning or contesting. During pleading, any allegations of fact by either party are implicitly conceded unless they are denied. The third type of burden is called the burden of production. It is the burden to discover and bring forward evidence supporting the contested factual allegations in the pleadings. The fourth type of burden of proof is the burden of persuasion. In a civil proceeding, this burden becomes operative only at the end of the trial, when the evidence and arguments are put to the jury to decided the factual issues. In a civil proceeding, the plaintiff has the burden of persuasion for all operative facts of his complaint and the defendant has the burden of persuasion for all affirmative defenses, i.e. exceptions. In criminal cases this is different. The prosecution has the burden of persuasion for all facts of the case, whether or not they are the operative facts of the elements of the alleged crime, or defenses, such as self-defense in a murder case. The fifth type of burden is called the tactical burden of proof. During the trial, arguments are put forward by both parties, pro and con the various claims at issue. At a finer level of granularity, the argumentation phase can be broken down conceptually into a sequence of stages, where each stage consists of all the arguments which have been put forward by both parties so far in the proceeding. The parties take turns putting forward arguments, by introducing new evidence. The next stage is constructed by adding the arguments put forward during this turn to all the previous arguments. The tactical burden arises from considering whether the arguments of a stage would be sufficient to meet the burden of persuasion with regard to some issue, if hypothetically the trial were to end at the stage and the issues where immediately put to the jury. The tactical burden of proof is the only burden of proof which, strictly speaking, can shift back and forth between the parties during the proceeding. How does the burden of persuasion operate? Essentially the jury has the task of weighing the arguments pro and con each proposition at issue. If the pro arguments are not deemed to sufficiently outweigh the con arguments, then the jury must reject the alleged fact by deciding that the alleged fact is not true. Because of the way the burden of persuasion is allocated, this amounts to accepting the default truth value of the proposition at issue. When do pro arguments ‘sufficiently’ outweigh con arguments to meet the bur- den of persuasion? This leads us to our final topic, proof standards. The question is how to aggregate or ‘accrue’ [24] arguments pro and con some claim. In the le- gal domain, four proof standards for factual issues exist, at least in common law jurisdictions. The scintilla of evidence proof standard is met if there is “any evi- dence at all in a case, even a scintilla, tending to support a material issue . . . ” [8, p. 1207] The preponderance of evidence proof standard is met by “evidence which as a whole shows that the fact sought to be proved . . . is more credible and convinc- ing to the mind.” [8, p. 1064]. The clear and convincing evidence proof standard is the “measure or degree of proof which will produce in mind of trier of facts a
- 256. 242 Thomas F. Gordon and Douglas Walton firm belief or conviction as to allegations sought to be established; it is intermedi- ate, being more than preponderance, but not to extent of such certainty as is required beyond reasonable doubt . . . ” [8, p. 227]. Finally, the beyond reasonable doubt stan- dard is the strongest legal proof standard, applicable in criminal cases. It requires evidence which leaves the trier of fact “fully satisfied, entirely convinced, . . . to a moral certainty” [8, p. 147]. In our view proof standards cannot and should not be interpreted probabilisti- cally. The first and most important reason is that probability theory is applicable only if statistical knowledge about prior and conditional probabilities is available. Presuming the existence of such statistical information would defeat the whole pur- pose of argumentation about factual issues, which is to provide methods for making justified decisions when knowledge of the domain is lacking. Another argument against interpreting proof standards probabilistically is more technical. Arguments for and against some proposition are rarely independent. What is needed is some way to accrue arguments which does not depend on the assumption that the argu- ments or evidence are independent. Prakken has identified three principles any formal account of accrual must satisfy [24]: 1) Combining several arguments pro or con some proposition can not only strengthen one’s position, but also weaken it. 2) Once several arguments have been accrued, the individual arguments, considered separately, should have no impact on the acceptability of the proposition at issue, and 3) Finally, any argument which is ‘flawed’ may not take part in the aggregation process. The models of proof standards presented in the section are designed to respect these principals. 2 Formal Model Our goal in this section is to define an abstract formal model of argumentation as a theory and proof construction process for making justified decisions. Inspired by Dung’s model of abstract argumentation frameworks, the model shall be as abstract and simple as possible while being sufficient for capturing the distinctions between the various types of proof burdens and proof standards identified in the introduction and meeting other known requirements. It is not intended to be a comprehensive formal model of argumentation. We will also take care to abstract from the details of the legal domain and, in particular, the law of civil procedure. We begin with the concept of an argument. Unlike Dung, we cannot leave this concept fully abstract, since our aim is to model burden of proof and proof stan- dards. The proponent of an argument has the burden of production for its ordinary premises; while the respondent has the burden of production for any exceptions. Moreover, since the task of proof standards is to aggregate arguments pro and con some proposition at issue, the model must represent not only the premises of argu- ments, and distinctions between types of premises, but also their conclusions. These considerations lead us to the following definition of argument. Definition 12.1 (argument). Let L be a propositional language. An argument is a tuple P,E,c where P ⊂ L are its premises, E ⊂ L are its exceptions and c ∈ L
- 257. 12 Proof Burdens and Standards 243 is its conclusion. For simplicity, c and all members of P and E must be literals, i.e. either an atomic proposition or a negated atomic proposition. Let p be a literal. If p is c, then the argument is an argument pro p. If p is the complement of c, then the argument is an argument con p. Since all conclusions of arguments are literals according to this definition, the axioms of the theory constructed during argumentation consist only of literals. Other propositions of the theory can be derived from these axioms using the inference rules of classical logic and the argumentation schemes of the domain. To model the distinctions between the various kinds of burden of proof, we must model argumentation as a process, consisting of several phases. It is sufficient to distinguish three phases, the opening, argumentation and closing phases of the pro- cess. Since typically argumentation takes place in dialogues, we will use the term ‘dialogue’ as the generic name for argumentation processes. Definition 12.2 (dialogue). A dialogue is a tuple O,A,C, where O, A and C, the opening, argumentation, and closing phases of the dialogue, respectively, are each sequences of stages. A stage is a tuple arguments,status, where arguments is a set of arguments and status is a function mapping the conclusions of the arguments in arguments to their dialectical status in the stage, where the status is a member of {claimed,questioned}. In every chain of arguments, a1,...an, constructable from arguments by linking the conclusion of an argument to a premise of another argu- ment, a conclusion of an argument ai may not be a premise of an argument aj, if j i. A set of arguments which violates this condition is said to contain a cycle and a set of arguments which complies with this condition is called cycle-free. Notice that the cycles defined here are not the same as cycles in a Dung argu- mentation framework. Whereas the links (arcs) between arguments in the directed graph induced by a Dung argumentation framework model the attack relation, the links in the directed graph induced by arguments in our system model the premise and conclusion relations. Notice that, in our system, arguments both pro and con some proposition can be included in a set of arguments without causing a cycle. Constraining the arguments of a stage to be cycle-free is intended to simplify the evaluation of arguments. The set of arguments is intended to model the current state of the proof being constructed by the parties in the dialogue, not a ‘pool of information’ for constructing proofs. Intuitively, proofs should not contain cycles. Next we need a structure for evaluating arguments, to assess the acceptability of propositions at issue. As in value-based argumentation frameworks [4, 5] arguments are evaluated with respect to an audience, such as the trier-of-fact (judge or jury) in legal trials. Definition 12.3 (audience). An audience is a structure assumptions,weight, where assumptions ⊂ L is a consistent set of literals assumed to be acceptable by the au- dience and weight is a partial function mapping arguments to real numbers in the range 0.guatda.com/cmx.p0...1.0, representing the relative weights assigned by the audience to the arguments.
- 258. 244 Thomas F. Gordon and Douglas Walton Whereas in value-based argumentation frameworks the audience is defined by a partial-order on a set of values, which is then used to constrain the attack relation on arguments in a Dung argumentation framework, the audience in our system models the relative strength of arguments for this audience. Intuitively, a stronger argument does not necessarily attack a weaker argument. Both arguments could be arguments pro the same proposition, for example. Thus, these two conceptions of an audience are not directly comparable. An argument evaluation structure associates an audience with a stage of dialogue and assigns proof standards to propositions, providing a basis for evaluating the acceptability of propositions to this audience. Definition 12.4 (argument evaluation structure). An argument evaluation struc- ture is a tuple stage,audience,standard, where stage is a stage in a dialogue, audience is an audience and standard is a total function mapping propositions in L to their applicable proof standards in the dialogue. A proof standard is a func- tion mapping tuples of the form issue,stage,audience to the Boolean values true and false, where issue is a proposition in L, stage is a stage and audience is an audience. Given an argument evaluation structure, the acceptability of a proposition can be defined as follows. Definition 12.5 (acceptability). A literal p is acceptable in an argument evalua- tion structure stage,audience,standard if and only if standard(p,stage,audience) is true. The argument evaluation structure is the component of this formal model which is most like a Dung abstract argumentation framework. The role of the attack re- lation in abstract argumentation frameworks is played by competing pro and con arguments aggregated by proof standards, using the relative weights assigned the arguments by an audience. Obviously much of the work of argument evaluation has been delegated to the proof standards. We cannot say anything about the computational properties of ac- ceptability in an argument evaluation structure until these standards have been de- fined. All the proof standards make use of the concept of argument applicability, so let us define this concept first. Definition 12.6 (argument applicability). Let stage,audience,standard be an ar- gument evaluation structure. An argument P,E,c is applicable in this argument evaluation structure if and only if • the argument is a member of the arguments of the stage, • every proposition p ∈ P, the premises, is an assumption of the audience or, if nei- ther p nor p is an assumption, is acceptable in the argument evaluation structure and • no proposition p ∈ E, the exceptions, is an assumption of the audience or, if nei- ther p nor p is an assumption, is acceptable in the argument evaluation structure.
- 259. 12 Proof Burdens and Standards 245 Now we are ready to define the proof standards, beginning with scintilla of the evidence. A proposition satisfies the scintilla standard, in our model, if it is sup- ported by at least one applicable pro argument. Definition 12.7 (scintilla of evidence). Let stage,audience,standard be an argu- ment evaluation structure and let p be a literal in L. scintilla(p,stage,audience) = true if and only if there is at least one applicable argument pro p in stage. Scintilla is the weakest of the proof standards we will define and is the only one which can be met by complementary literals in the same argument evaluation structure. That is, if p is an atomic proposition, both p and ¬p can be acceptable in an argument evaluation structure using the scintilla of evidence standard. This would be the case if p has an applicable con argument, as well as an applicable pro argument. Let us now turn our attention to the three most important legal proof standards: preponderance of the evidence, clear and convincing evidence and beyond reason- able doubt. Intuitively, preponderance is satisfied if the pro arguments outweigh the con arguments, by however much. The issue we have to face when formalizing pre- ponderance is how to aggregate the weights of a set of arguments for the purpose of this comparison. The clear and convincing evidence standard requires more proof than the preponderance standard: not only must the pro arguments outweigh the con arguments, the weight of the pro arguments and the difference in weight of the pro and con arguments both must exceed some thresholds. Finally, the beyond a reasonable doubt standard goes further. Not only must the arguments be clear and convincing, but, as the name of the standard suggests, the weight of the con argu- ments must be below the threshold of ‘reasonable doubt’. Definition 12.8 (preponderance of the evidence). Let stage,audience,standard be an argument evaluation structure and let p be a literal in L. preponderance (p,stage,audience) = true if and only if • there is at least one applicable argument pro p in stage and • the maximum weight assigned by the audience to the applicable arguments pro p is greater than the maximum weight of the applicable arguments con p. The preponderance of the evidence standard was called the best argument stan- dard in [18]. Definition 12.9 (clear and convincing evidence). Let stage,audience,standard be an argument evaluation structure and let p be a literal in L. clear-and-convincing (p,stage,audience) = true if and only if • the preponderance of the evidence standard is met, • the maximum weight of the applicable pro arguments exceeds some threshold α, and • the difference between the maximum weight of the applicable pro arguments and the maximum weight of the applicable con arguments exceeds some threshold β.
- 260. 246 Thomas F. Gordon and Douglas Walton Definition 12.10 (beyond reasonable doubt). Let stage,audience,standard be an argument evaluation structure stage,audience,standard and let p be a literal in L. beyond-reasonable-doubt(p,stage,audience) = true if and only if • the clear and convincing evidence standard is met and • the maximum weight of the applicable con arguments is less than some threshold γ. We assume the α, β and γ thresholds used by the clear and convincing evidence and beyond a reasonable doubt standards are set by the applicable protocol of the dialogue. At first glance, using maximum weights to aggregate pro and con arguments might seem unintuitive. One might be inclined to compare the sums of the weights of the applicable pro and con arguments. However, since arguments cannot be as- sumed to be independent, summing weights would risk taking the same information or reasons into account multiple times. When several weak arguments can be com- bined to make a stronger argument, this can be achieved by joining their premises together into a single argument, as discussed further next. We leave it up to the au- dience to judge the effect of any possible interdependencies among the premises on the weight of the argument. Both alternatives, summing weights or taking their maximum weight, have the property of taking all arguments into account. Assuming arguments are stated in their strongest form, aggregating arguments using their maximum weight satisfies all three of Prakken’s principles of accrual [24]: 1) Aggregated arguments can be evaluated by the audience to be stronger or weaker than the arguments considered separately; 2) Once several arguments have been accrued, the individual arguments, considered separately, have no effect on the acceptability of the proposition at issue; and 3) Any argument which is ‘flawed’ does not take part in the aggregation process. By ‘stating arguments in their strongest form’ we mean the following. Let p and q be two propositions which, when they are true, are evidence pro a third proposition, r. This can be expressed as either two convergent arguments or as a linked argument [38]. The convergent arguments would be: a1: r since p. a2: r since q. The linked form of this argument ‘accrues’ p and q into a single argument: a3: r since p and q. If the linked argument, a3, is more persuasive, that is if the party putting forward this argument estimates it would be given more weight by the audience than either a1 or a2, then we assume a3 will be put forward. To illustrate this more concretely, let’s return to Prakken’s example about jog- ging when it is both hot and rainy. The strongest arguments con jogging are the convergent arguments: a4: not jogging since hot.
- 261. 12 Proof Burdens and Standards 247 a5: not jogging since rainy. The strongest argument pro jogging is the following linked argument: a6: jogging since hot and rainy. Returning to Prakken’s three principles of accrual: 1) The audience can decide whether to give a6 greater or lesser weight than each of the arguments a4 and a5; 2) If the accrued argument, a6, is given greater weight, then a6 will defeat both a4 and a5, rendering them ineffective, using any of the proofs standards which ag- gregate arguments by weight; and 3) Inapplicable arguments are not be taken into consideration using any proof standard. In [18] one further proof standard was defined, called dialectical validity. For the sake of completeness we include its definitions here. Definition 12.11 (dialectical validity). Let stage,audience,standard be an argu- ment evaluation structure stage,audience,standard and let p be a literal in L. dialectical-validity(p,stage,audience,) = true if and only if there is at least one ap- plicable argument pro p in stage and no argument con p in stage is applicable. The dialectical validity standard is suitable for aggregating arguments from gen- eral rules and exceptions, where any applicable exception is enough to override the general rule. One of the requirements identified in the introduction is that checking proofs should be an easy task. In more computational terms, using our formal model, the is- sue is whether the acceptability of a proposition in an argument evaluation structure is tractably decidable. We conjecture that this is the case, but will not try to prove this formally here. The reasons for this conjecture are many. The argument eval- uation structure has been designed in part to achieve this goal, by making several restrictions: 1) The language is propositional, not first-order; 2) premises, excep- tions and conclusions of arguments must be literals; and 3) the set of arguments of a stage is finite and, by definition, acyclic. Of course the computational complexity of acceptability also depends on the complexity of the proof standards applied. Now we are ready to turn to modeling the various kinds of burden of proof. The burdens of claiming and questioning must be met during the opening phase of the dialogue. The burden of production and the tactical burden of proof are relevant only during the argumentation phase. Finally, the burden of persuasion comes into play in the closing phase, but is also used hypothetically to estimate the tactical burden of proof during the argumentation phase. The purpose of the opening stage of a dialogue is to frame the issues. Arguments put forward in the argumentation stage must be relevant to the issues raised in the opening stage. Depending on the protocol of the dialogue, a proposition claimed in the opening stage may be deemed conceded unless it is questioned, requiring the audience to assume it is true, following the principal of “silence implies consent.” Definition 12.12 (burdens of claiming and questioning). Let s1,...,sn be the stages of the opening phase of a dialogue. Let argumentsn,statusn be the last stage,
- 262. 248 Thomas F. Gordon and Douglas Walton sn, of the opening phase. A party has met the burden of claiming a proposition p if and only if statusn(p) ∈ {claimed,questioned}, that is, if and only if statusn(p) is defined. The burden of questioning a proposition p has been met if and only if statusn(p) = questioned. Notice that a questioned proposition satisfies the burden of claiming, since it is assumed that only propositions which have been claimed in an earlier stage are questioned. This simple model defines only minimal requirements for raising issues in the opening phase of a dialogue. The argumentation protocol of a dialogue may state additional requirements. For example, according to the law of civil procedure in Cal- ifornia, the plaintiff must state a cause of action: the facts claimed must be sufficient to give the plaintiff a right to judicial relief, as a matter of law. The burden of production is relevant only during the argumentation phase of a dialogue. The burden of production for some proposition is satisfied if it is accept- able at the end of the argumentation phase using the the weakest proof standard, scintilla of the evidence. The party who puts forward an argument has the burden of production for its premises. Similarly, the respondent to an argument has the burden of production for each exception. The audience used to assess the burden of production depends on the protocol of the particular dialogue. In civil proceedings in California, the judge is the audience during the argumentation phase, i.e. the trial. Definition 12.13 (burden of production). Let s1,...,sn be the stages of the argu- mentation phase of a dialogue. Let argumentsn,statusn be the last stage, sn, of the argumentation phase. Let audience be the relevant audience for assessing the burden of production, depending on the protocol of the dialogue. Let AES be the argument evaluation structure sn,audience,standard, where standard is a function mapping every proposition to the scintilla of evidence proof standard. The burden of production for a proposition p has been met if and only if p is acceptable in AES. Even though the weakest proof standard, scintillia of the evidence, is used to test whether the burden of production has been met, an arbitrary, or silly, argument would not be sufficient, since only applicable arguments are taken into consider- ation by all proof standards. A silly argument can be defeated by questioning or attacking its premises. Arguments can be undercut in our system by first revealing, if necessary, an implicit premise about the applicability of the warrant underlying the argument to this case and then attacking this premise [18]. Since arguments put forward to met the burden of production can be defeated by further arguments, the burden of production may be met at some stage, si, of a dialogue, but not met at some later stage, sj, where j i. If the burden of production is not met at the end of the argumentation phase, the audience in the closing phase may be required, depending on the dialogue type, to assume that the proposition is false. In this case, the burden of persuasion for this proposition becomes irrelevant.
- 263. 12 Proof Burdens and Standards 249 The burden of persuasion plays a role only in the closing phase of the dialogue. The burden of persuasion is met only if at the end of the closing phase the propo- sition at issue is acceptable to the audience. The way proof standards are assigned to propositions depends on the type of dialogue and is regulated by the argumen- tation protocol. In legal proceedings in California, the proof standards are assigned by the judge, since this is a question of law, not fact. In civil proceedings, the usual proof standard is preponderance of the evidence. In criminal proceedings, the proof standard is beyond reasonable doubt. Definition 12.14 (burden of persuasion). Let s1,...,sn be the stages of the closing phase of a dialogue. Let argumentsn,statusn be the last stage, sn, of the closing phase. Let audience be the relevant audience for assessing the burden of persuasion, depending on the dialogue type and its protocol. Let AES be the argument evalua- tion structure sn,audience,standard, where standard is a function mapping every proposition to its applicable proof standard for this type of dialogue. The burden of persuasion for a proposition p has been met if and only if p is acceptable in AES. In some cases, the party which has the burden of production in the argumenta- tion phase may not have the burden of persuasion in the closing phase. This is the case, for example, in criminal law proceedings. The defendant, as usual, has the burden of production for exceptions, such as self-defense in murder cases, but once this burden has been met, the burden of persuasion is passed to the prosecution. If any evidence of self-defense has been brought forward, satisfying the burden of production, the prosecution has the burden of persuading the trier of fact, beyond a reasonable doubt, that the defendant did not act in self-defense. This can be achieved in our model by making the exception an ordinary premise after the burden of production has been met in the argumentation phase. For exam- ple, let P,E,c be an argument and e be a proposition in E, meaning “self defense”. After the burden of production for e has been met, the other side can be given the burden of persuasion by removing e from E and adding ¬e to P. It may seem odd to modify the argument in this way, but keep in mind the arguments of a stage do not represent the speech acts of the parties, but rather the state of the proof being con- structed collaboratively by all parties, according the protocol of the dialogue type. The stage must be modified in some way to reflect this change, and modifying the arguments of the stage is one way to accomplish this. One kind of burden of proof remains to be defined formally, the tactical burden of proof. The tactical burden is the only one which can shift back and forth between the parties. It is relevant only during the argumentation phase of the dialogue. We defined the burden of persuasion first, even though it is applicable only in the later closing stage, because the tactical burden of proof requires the burden of persuasion to be estimated. At each stage of the argumentation phase, a party must decide whether stronger arguments might be necessary to persuade the audience. In some dialogue types, the audience may reveal its assumptions and evaluations (weight assignments) during the argumentation phase, at least provisionally. This will be the case, for example, in two-party dialogues where the audience to be persuaded is the same as the respondent. In legal proceedings this is not the case, since the
- 264. 250 Thomas F. Gordon and Douglas Walton respondent is the defendant and the audience is the judge or jury. In such cases it will be necessary to make assumptions about the audience. Definition 12.15 (tactical burden of proof). Let s1,...,sn be the stages of the ar- gumentation phase of a dialogue. Assume audience is the audience which will as- sess the burden of persuasion in the closing phase. Assume standard is the function which will be used in the closing phase to assign a proof standard to each propo- sition. For each stage si in s1,...,sn, let AESi be the argument evaluation structure si,audience,standard. The tactical burden of proof for a proposition p is met at stage si if and only if p is acceptable in AESi. Both sides in a dialogue can have a tactical burden. Intuitively, a party has a tactical burden of proof for a proposition p at some stage si only if p is not acceptable in si and the party has an interest in proving p, either because proving p is relevant for proving some claim of the party or disproving some claim of the other party, given the arguments which have been put forward. A fuller, more complete account of the tactical burden of proof would require the parties, their claims and the concept of relevance to be modeled. This completes our formal model of proof standards and burden of proof. Again, we do not claim this is a comprehensive dialogical model of argumentation. Many important elements of argumentation have been abstracted out for the sake of brevity, such as the parties, argumentation schemes and their critical questions, di- alogue types, argumentation protocols and commitment stores. Our aim here was to define the simplest, most abstract possible model which is sufficient for distin- guishing the various kinds of proof standards and burdens of proof which have been discussed in the computational models of argument literature. Of course, we cannot prove that we have achieved this goal. We leave it up to others in the field to try to develop a simpler model with this scope. In the introduction we formulated our view of argumentation as a kind of pro- cess for making justified decisions, where the input to the process is an initial claim or issue and the output consists of a set of claims, the decision to accept or reject each claim, a theory and a proof. Unlike assumption-based instantiations of Dung’s model of abstract argumentation frameworks [9], in which a theory or knowledge base is presumed as part of the input, in our model a theory and a proof are con- structed together during argumentation and are part of the output. The theory con- structed is the deductive closure, in classical propositional logic, of the set of all propositions which have been assumed by the audience or, if neither the proposi- tion or its complement has been assumed, are acceptable in the final stage of the closing phase of the dialogue. The proof constructed is represented by the argument evaluation structure of the final stage. 3 Survey of Prior Research Early work in the field of Artificial Intelligence and Law recognized the utility of defeasible rules, subject to exceptions, as a tool for allocating the burden of proof
- 265. 12 Proof Burdens and Standards 251 and developed nonmonotonic logics for reasoning with such rules [13, 35]. But different proof standards or kinds of proof burdens were not yet distinguished in these models. In the Pleadings Game [14], inspired by legal proceedings, argumentation was modeled as dialogical process consisting of several phases, in which theories and arguments are constructed dynamically by the parties during the process. The Plead- ings Game modeled the burdens of claiming, questioning and production, but it did not explicitly use this terminology. Proof standards and the burden of persuasion, were not modeled, as they do not play a role in the opening phase. To our knowledge, the first effort to develop a computational model of proof standards was by Freeman and Farley in 1996 [12]. They modeled argumentation as a dialectical process during which an acyclic argument graph is constructed by putting forward pro and con arguments constructed (‘invented’) from a propositional rule-base. The model of proof standards comes into play when evaluating the argu- ments in the graph. An argument is defendable if each premise satisfies its proof standard. Five proof standards were defined (scintilla of evidence, preponderance, dialectical validity, beyond a reasonable doubt and beyond a doubt). The relative weights of arguments were not assigned by an audience, but rather computed from certainty factors assigned to premises and the kind of argumentation scheme used to construct the argument, using the weakest premise principle [32]. The model of ar- gumentation developed in this paper has borrowed much from Freeman and Farley, but there are some important differences. By restricting the arguments which can be put forward to those which can be constructed from a static rule-base and model of the facts, provided as input to the dialectical process, Freeman and Farley’s model is more of a relational model of argumentation than a theory construction model. The relative strength of arguments is determined by the rules and facts, rather than by the intersubjective judgment of an audience. Only the burden of persuasion has been modeled by this work. Finally, no attempt was made to model proof burdens other than the burden of persuasion in this work. The Zeno system [17] was inspired by Freeman and Farley’s work. Zeno’s model of the structure of argument graphs was based on Kunz and Rittel’s [22] concept of issue-based information systems (IBIS). Zeno supported arguments about both fac- tual issues and issues about which action to take to solve some problem or achieve some goal (practical reasoning). For factual issues, three proof standards were mod- eled (scintilla of evidence, preponderance of the evidence and beyond reasonable doubt). For issues about actions, two further proof standards were provided (no bet- ter alternative and best choice). The relative strength of arguments in Zeno was computed from qualitative constraints (equations and inequalities) over proposi- tions. The qualitative constraints were issues which could be argued about. Only the burden of persuasion was modeled in Zeno. As in Freeman and Farley’s work, Zeno did not explicitly model an audience. On the contrary, in Zeno it was assumed the parties would argue about the evaluation of their own arguments, by putting forward and arguing about constraints. Prakken formulated the three principles of argument accrual [24] explained in the introduction, which our model has been designed to satisfy. He compared automatic
- 266. 252 Thomas F. Gordon and Douglas Walton and manual approaches to accrual. In the manual approach, arguments are accrued by changing their representation in the model. In the automatic approach, argu- ments are accrued by the argument evaluation process, without having to modify the representation of the arguments. Prakken illustrated this approach with a novel system, using a rule-based instantiation of Dung’s model of an abstract argumenta- tion framework, in which the conclusions of rules are labeled with the set of their premises. The attack relation of the argumentation framework is defined so that an argument A does not attack an argument B if the set of premises of the conclu- sion of the argument A is a subset of the (accrued) set of premises of the argument B. This leads to a bottom-up inference process which, Prakken notes, is similar to Reason-Based Logic [20]: first all arguments pro and con some leaf proposition are combined into two competing sets of accruals; next the conflict between these accruals is resolved; and finally the process iterates moving up the tree, with only the winning defeasible conclusion being used. The way arguments are evaluated in our model has much in common with the bottom-up inference approach taken by both Reason-Based Logic and Prakken’s system. Nonetheless, our approach to the accrual problem is manual, not automatic, as responsibility for accruing arguments is allocated to the parties who put them forward in dialogues. But unlike Bayesian Networks, which require a complete conditional probabilities table to be provided as input to the process, our approach does not require all possible arguments to be formulated. Accruing arguments when possible to strengthen one’s case is part of the burden of proof allocated to the parties. Prakken points out that there is a trade- off between automatic and manual approaches to accrual: in the former unwanted inferences must be expressly blocked while in the latter accruals must be expressly formulated. So neither approach is clearly superior. In 2006, Prakken and Sartor published the first formal account of the distinctions between the burden of production, the burden of persuasion and the tactical burden of proof [28]. Their model is based on an interpretation of presumptions as default rules, formalized using an extended version of their Inference System (IS) defeasi- ble logic [27], called the Litigation Inference System (LIS) [23], which includes as part of the input, together with the defeasible rules, an assignment of the burden of persuasion for literals to either the plaintiff or defendant in the proceeding. Prakken and Sartor’s model of the distinctions between these three proof burdens is more concrete than our model, as it is limited to arguments constructed from defeasible rules. Our model abstracts away the process of constructing arguments and can be instantiated with models of various argumentation schemes, for example for argu- ments from cases as well as defeasible rules.1 Prakken and Sartor did not attempt to model the burdens of claiming or questioning or support the use of various proof standards. Continuing work began with Reed and Walton in 2005 on dialogues about the burden of proof [26], Prakken and Sartor in 2007 presented a model of arguments about the allocation of the burden of persuasion [29]. They note that ‘the argumen- 1 We have done some research recently on constructing arguments automatically from formal mod- els of ontologies, rules, and cases, in a way which is well integrated with the model of argument presented here [16].
- 267. 12 Proof Burdens and Standards 253 tation games we define in this paper are not intended as a model of actual legal dialogue but as a proof theory for a nonmonotonic logic. . . . All we claim is that our games draw the correct defeasible inferences from a given body of information and an associated allocation of the burden of persuasion. It remains to be seen how the present logical model can be integrated with dialogical and procedural mod- els of legal argument”. Clearly, their work is a relational model of argumentation, not a theory construction model. In their conclusion, Prakken and Sartor express concern that their system “lacks an extension-based semantics in the style of [10]” and note, citing [23], that this “raises questions about the adequacy of ‘mainstream’ nonmonotonic logics for representing legal reasoning”.2 The idea of using audiences in our model was inspired by work by Bench-Capon, Doutre and Dunne [5]. Bench-Capon’s focus is on modeling practical reasoning, i.e. the process of making decisions about which action to take in order to achieve goals which promote an agent’s values. Since different agents can have different values, as well as different priority orderings on their set of values, arguments about action can only be evaluated against the values of a particular agent, or ‘audience’. Bench- Capon contrasted practical reasoning with argumentation about “what is true in a sit- uation”, i.e. the facts of a case, and appears to consider audiences to be relevant only for the former. As illustrated by legal trials, however, audiences can be important in argumentation dialogues about factual issues as well, since different persons will judge the probative weight of evidence, such as witness testimony, differently. Our model of an audience is more abstract, since it orders arguments by their strength, regardless of the kind of argumentation scheme which has been applied to construct the argument. Formally, Bench-Capon’s model is an extension of Dung’s concept of an abstract argumentation framework. Although a formal dialogue game is de- fined, the game serves as a proof theory for a relational model of argument. The input to the system is a Value-Based Argumentation Framework (VAF) consisting of a set of arguments, an attack relation over these arguments, a set of values, and a mapping from arguments to values. An audience is defined to be a binary rela- tion over these values, modeling the preference ordering over these values of the audience. The output of the dialogue game is the set of arguments which are objec- tively or subjectively acceptable. Arguments which are objectively acceptable are, or should be, acceptable by all (rational) audiences. Conversely, arguments which are subjectively acceptable are acceptable to a particular, given audience. Bench- Capon compares the distinction between objective and subjective acceptability with the concept of credulous and skeptical inference familiar from nonmonotonic logic and Dung argumentation frameworks. The distinction between objective and sub- jective acceptability, like the distinction between credulous and skeptical inference, possibly can be viewed as a simple two-level model of proof standards, but it is questionable that this distinction is of much use outside of the procedural context of dialogues in which the allocation of the burden of proof matters, and such dialogues were outside the scope of this work. 2 But see [36] for a defense of nonmonotonic logic for legal reasoning, even when the burden of persuasion is divided among the parties.
- 268. 254 Thomas F. Gordon and Douglas Walton Atkinson and Bench-Capon recently modeled a variety of proof standards us- ing Dung-style argumentation frameworks [2]. This paper also recognizes the need for audiences in dialogues about factual issues, as well as teleological issues about values promoted by alternative courses of action. The basic idea of this paper elab- orates on the idea discussed above, of using the distinction between various kinds of acceptability in Dung argumentation frameworks to define proof standards. The scintilla of evidence standard is modeled as membership in some preferred exten- sion (credulous acceptability). Preponderance of the evidence corresponds to mem- bership in all preferred extensions (skeptical acceptability). And, finally, beyond reasonable doubt corresponds to membership in the grounded extension, if there is one. Ways of modeling proof standards between preponderance of the evidence and beyond reasonable doubt, such as the clear and convincing evidence standard, are proposed which make use of an assignment of ‘probabilities’, i.e. weights, to arguments. However this idea is discarded with the argument that this information is not usually available. In our model this information is provided by the intersub- jective judgment of the audience. Later in their paper, Atkinson and Bench-Capon suggest the use of audiences to derive preferences over arguments, starting with Bench-Capon’s value-based argumentation frameworks for arguing about teleolog- ical issues. This idea is extended to arguments about factual issues by ordering evi- dence, for example by ordering witness testimony using information about witness credibility. An advantage of Bench-Capon and Atkinson’s line of research is that it elabo- rates rationality constraints on an audience’s assignment of strengths to arguments, using for example its value preferences or its assessment of the relative credibil- ity of witnesses. Although our model leaves the weights assigned by audiences to arguments completely unconstrained, it has the advantage of being more general, applying to arguments from any argumentation scheme. Perhaps it is possible to combine the benefits of these two approaches. This may be an interesting topic for future research. A disadvantage of modeling proof standards using Dung argumentation frame- works is the computational complexity of testing whether the proof standard has been met. We have argued that, intuitively, to satisfy a burden of proof, the party with the burden must present the proof in a form which is easy to check. The other party shouldn’t have to solve an undecidable or intractable problem to check the proof. As discussed previously, in Section 2, this criterion could be satisfied for the proposed scintilla of evidence proof standard, modeled as credulous acceptability, by requiring the proponent to produce an admissible set of arguments. The admissi- ble set could serve as a representation of the proof, since checking whether the set is admissible and whether an argument is a member of this set are both tractable problems. But it is not clear what structures could serve as proofs for the models proposed by Atkinson and Bench-Capon for the stricter proof standards. A related issue is whether or not such proofs could be presented in a form which is compre- hensible to human users, using for example some kind of argument mapping or other visualization technique. Existing argument mapping methods have been developed for presenting proofs (argument graphs) of the kind we have presented in this paper
- 269. 12 Proof Burdens and Standards 255 [3, 11, 1, 15]. While Dung argumentation frameworks are often displayed as graphs, these graphs do not represent proofs of the acceptability of any of the arguments in the graph. Prakken recently published a formal model which explicates the role of judges in deciding issues regarding the admissibility of evidence and the allocation of the burdens of production and persuasion [25]. Interestingly, dialogues are divided into two, rather than tree phases, in the model, called the pleadings phase and the de- cision phase. The pleadings phases encompasses the opening and argumentation phases of our model. The decision phase corresponds to our closing phase. The judge plays a role in the decision phase comparable to the audience in the closing phase of our model, but arguments are evaluated by having the judge put forward further arguments, which are then evaluated using Prakken’s LIS nonmonotonic logic [23], discussed above. Arguments are not weighed and proof standards are not part of the model. Prakken and Sartor’s chapter on a “Logical Analysis of Burdens of Proof” in [30] highly influenced our model of the distinctions among the various kinds of proof burdens. The main difference between our systems is that they use Dung’s abstract argumentation framework to evaluate the arguments at each stage of the dialogue. We have already expressed our reservations about this approach, which does not take into consideration that a burden of proof entails an obligation to put forward the proof in a form which can be tractably checked. In their conclusion they point out that their use of a nonmonotonic logic for evaluating arguments in a stage could be replaced by any formalism “which accepts as input a description of an ev- idential problem and produces as output a fallible assessment whether a claim has been proven”. This is one way looking at we have done here, by replacing their non- monotonic logic with a structure designed to make argument evaluation tractable for a variety of common proof standards. Proof standards are not given much attention in their chapter, except to suggest that they could be handled using some mecha- nism for ordering arguments by their strength. Finally, although they recognize the role of the finder-of-fact, they did not explictly model audiences or their impact on assessing proof burdens. For example, the evalution of the tactical burden of proof in their model does not require a party to estimate an audience’s assessment of the weight of arguments. Rather, this information appears to be assumed as input to the dialogue, available as common knowledge to both parties. The model of argumentation developed in this paper is most closely related to the Carneades system, which consists of both a mathematical model of argumen- tation and software tools for supporting argumentation tasks based on this model.3 Carneades is work in progress and thus prior publications about the system differ in their details. For example, the version of the model in [19] gave the term ‘presump- tion’ a technical meaning which is confusing for those familar with the concept of a presumption in the legal domain. This was pointed out by Prakken and Sartor [28] and corrected when Prakken joined us for the next version of Carneades [18]. Simi- larly, the use of weights to order arguments in [19] was replaced by a partial order in 3 http://carneades.berlios.de
- 270. 256 Thomas F. Gordon and Douglas Walton [18], but now we have come full circle, by again using numeric weights. The initial models of the preponderance of the evidence and beyond reasonable doubt proof standards of [19] were removed from the [18] version, since we had our doubts about their adequacy as models of these legal standards. We feel confident enough about the models of these legal standards in this paper to want to publish them in order to obtain critical feedback. Aside from improved models of various proof standards, the main contribution of the new version presented here is that it more clearly and explicitly models argumentation as a theory and argument construction process, consisting of a sequence of stages divided into opening, argumentation and closing phases. Although we had already suggested in [18] how to use Carneades to model the distinction between the burden of production and the burden of persua- sion, the version in this article models these distinctions more explicitly and extends the model to cover the burdens of claiming and questioning in the opening phase and the tactical burden of proof. This version also introduces an explicit model of audiences. Previous versions had used the concept of an argumentation context to model argument strengths or priorities, with no reference to an audience. 4 Concluding Remarks Viewing argumentation as dialogical process for making justified decisions raises a number of issues which have no place in relational, monological accounts of ar- gumentation, proof burdens and standards among them. Thus it should come as no surprise that the concept of proof has thus far received little attention in mainstream accounts of argumentation in artificial intelligence. An argument may be acceptable in a Dung-style argumentation framework, or a proposition may be warranted by a default theory in some nonmonotonic logic, but what mathematical structures are adequate as models of proofs of these or other inference relations? We have argued that checking a proof should be an easy, tractable problem. Argumentation frame- works and default theories do not, in general, meet this requirement. And other, less computational requirements could also be formulated, such as transparency and comprehensibility for the intended class of audiences. Another distinction between relational and dialogical conceptions of argumenta- tion concerns their input/out relations. Whereas in relational accounts an argumen- tation framework or default theory is provided as input and the task is to derive ac- ceptable arguments or warranted propositions, argumentation dialogues begin with a claim or issue and construct, as part of their output, theories and proofs. Argumen- tation dialogues includes synthetic as well as analytic tasks. Abstract argumentation frameworks, which focus on attack relations among ar- guments, are not well-suited to modeling proof standards, at least not the familiar proof standards from the legal domain. The intuitive idea of legal proof standards since Roman times, symbolized by the scales of the goddess Justitia, involves the weighing of arguments or evidence pro and con some claim. This simple idea can- not be directly represented using abstract argumentation frameworks. Attempts to
- 271. 12 Proof Burdens and Standards 257 model proof standards as variations of credulous and skeptical acceptability in an argumentation framework are not very promising, since they largely leave open the question of how to represent proofs in an understandable form which can be easily checked. Research to date on modeling proof standards and burdens calls into question the common research strategy in the field of computational models of argumentation which presumes that valid dialogical models of argumentation as a process can be constructed on the foundation of relational models of argument as a nonmonotonic inference relation. We recommend a research strategy which begins with a task and requirements analysis of argumentation dialogues in a variety of application do- mains. Acknowledgements We’d like to thank Trevor Bench-Capon and Henry Prakken for their helpful comments on various drafts of this article. References 1. T. Anderson, D. Schum, and W. Twining. Analysis of Evidence. Cambridge University Press, 2nd edition, 2005. 2. K. Atkinson and T. Bench-Capon. Argumentation and standards of proof. In ICAIL ’07: Proceedings of the 11th International Conference on Artificial Intelligence and Law, pages 107–116, New York, NY, USA, 2007. ACM. 3. M. C. Beardsley. Practical Logic. Prentice Hall, New York, 1950. 4. T. Bench-Capon. Persuasion in practical argument using value-based argumentation frame- works. Journal of Logic and Computation, 13(3):429–448, 2003. 5. T. J. Bench-Capon, S. Doutre, and P. E. Dunne. Audiences in argumentation frameworks. Artificial Intelligence, 171(42-71), 2007. 6. P. Besnard and A. Hunter. Elements of Argumentation. MIT Press, 2008. 7. J. Bing. Uncertainty, decisions and information systems. In C. Ciampi, editor, Artificial Intelligence and Legal Information Systems. North-Holland, 1982. 8. H. C. Black. Black’s Law Dictionary. West Publishing Co., 1979. 9. A. Bondarenko, P. M. Dung, R. A. Kowalski, and F. Toni. An abstract, argumentation-theoretic approach to default reasoning. Artificial Intelligence, 93(1-2):63–101, 1997. 10. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–357, 1995. 11. J. B. Freeman. Dialectics and the Macrostructure of Arguments: A Theory of Argument Struc- ture. Walter de Gruyter, Berlin / New York, 1991. 12. K. Freeman and A. M. Farley. A model of argumentation and its application to legal reasoning. Artificial Intelligence and Law, 4(3-4):163–197, 1996. 13. T. F. Gordon. Some problems with Prolog as a knowledge representation language for legal expert systems. In C. Arnold, editor, Yearbook of Law, Computers and Technology, pages 52–67. Leicester Polytechnic Press, Leicester, England, 1987. 14. T. F. Gordon. The Pleadings Game; An Artificial Intelligence Model of Procedural Justice. Springer, New York, 1995. Book version of 1993 Ph.D. Thesis; University of Darmstadt. 15. T. F. Gordon. Visualizing Carneades argument graphs. Law, Probability and Risk, 6(1-4):109– 117, 2007. 16. T. F. Gordon. Hybrid reasoning with argumentation schemes. In Proceedings of the 8th Workshop on Computational Models of Natural Argument (CMNA 08), pages 16–25, Patras, Greece, July 2008. The 18th European Conference on Artificial Intelligence (ECAI 2008).
- 272. 258 Thomas F. Gordon and Douglas Walton 17. T. F. Gordon and N. Karacapilidis. The Zeno argumentation framework. In Proceedings of the Sixth International Conference on Artificial Intelligence and Law, pages 10–18, Melbourne, Australia, 1997. ACM Press. 18. T. F. Gordon, H. Prakken, and D. Walton. The Carneades model of argument and burden of proof. Artificial Intelligence, 171(10-11):875–896, 2007. 19. T. F. Gordon and D. Walton. The Carneades argumentation framework — using presumptions and exceptions to model critical questions. In P. E. Dunne and T. J. Bench-Capon, editors, Computational Models of Argument. Proceedings of COMMA 2006, pages 195–207, Amster- dam, September 2006. IOS Press. 20. J. Hage. A theory of legal reasoning and a logic to match. Artificial Intelligence and Law, 4(3-4):199–273, 1996. 21. H. L. A. Hart. Essays in Jurisprudence and Philosophy. Oxford University Press, 1983. 22. W. Kunz and H. W. Rittel. Issues as elements of information systems. Technical report, Institut für Grundlagen der Planung, Universität Stuttgart, 1970. also: Center for Planning and Development Research, Institute of Urban and Regional Development Research. Working Paper 131, University of California, Berkeley. 23. H. Prakken. Modeling defeasibility in law: Logic or Procedure? Fundamenta Informaticae, 48:253–271, 2001. 24. H. Prakken. A study of accrual of arguments, with applications to evidential reasoning. In Proceedings of the Tenth International Conference on Artificial Intelligence and Law, pages 85–94, New York, 2005. ACM Press. 25. H. Prakken. A formal model of adjudication. In S. Rahman, editor, Argumentation, Logic and Law. Springer Verlag, Dordrecht, 2008. 26. H. Prakken, C. Reed, and D. Walton. Dialogues about the burden of proof. In Proceedings of the Tenth International Conference on Artificial Intelligence and Law, pages 85–94, Bologna, 2005. ACM Press. 27. H. Prakken and G. Sartor. A dialectical model of assessing conflicting argument in legal reasoning. Artificial Intelligence and Law, 4(3-4):331–368, 1996. 28. H. Prakken and G. Sartor. Presumptions and burden of proof. In T. van Engers, editor, Legal Knowledge and Information Systems. JURIX 2006: The Nineteenth Annual Conference, pages 21–30, Amsterdam, 2006. IOS Press. 29. H. Prakken and G. Sartor. Formalizing arguments about the burden of persuasion. In Proceed- ings of the 11th International Conference on Artificial Intelligence and Law, pages 97–106, New York, 2007. Stanford University, ACM Press. 30. H. Prakken and G. Sartor. A logical analysis of burdens of proof. In H. Kaptein, H. Prakken, and B. Verheij, editors, Legal Evidence and Proof: Statistics, Stories, Logic, Applied Legal Philosophy Series. Ashgate Publishing, 223–253, 2009. 31. J. Rawls. Outline of a decision procedure for ethics. Philosophical Review, 177–197, 1951. 32. N. Rescher. Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge. State University of New York Press, 1977. 33. H. W. Rittel and M. M. Webber. Dilemmas in a general theory of planning. Policy Science, 4:155–169, 1973. 34. M. Rosenberg, J. B. Weinstein, H. Smit, and H. L. Korn. Elements of Civil Procedure. Foun- dation Press, 1976. 35. G. Sartor. Defeasibility in legal reasoning. In Informatics and the Foundations of Legal Rea- soning, Law and philosophy library, pages 119–157. Kluwer Academic Publishers, Dordrecht, 1995. 36. K. Satoh, S. Tojo, and Y. Suzuki. Formalizing a switch of burden of proof by logic program- ming. In Proceedings of the First International Workshop on Juris-Informatics (JURISIN 2007), pages 76–85, Miyazaki, Japan, 2007. 37. D. Walton. The new dialectic: A method of evaluating an argument used for some purpose in a given case. ProtoSociology, 13:70–91, 1999. 38. D. Walton. Fundamentals of Critical Argumentation. Cambridge University Press, Cam- bridge, UK, 2006.
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- 275. Chapter 13 Dialogue Games for Agent Argumentation Peter McBurney and Simon Parsons 1 Introduction The rise of the Internet and the growth of distributed computing have led to a ma- jor paradigm shift in software engineering and computer science. Until recently, the notion of computation has been variously construed as numerical calculation, as information processing, or as intelligent symbol analysis, but increasingly, it is now viewed as distributed cognition and interaction between intelligent entities [60]. This new view has major implications for the conceptualization, design, engineer- ing and control of software systems, most profoundly expressed in the concept of systems of intelligent software agents, or multi-agent systems [99]. Agents are soft- ware entities with control over their own execution; the design of such agents, and of multi-agent systems of them, presents major research and software engineering challenges to computer scientists. One key challenge is the design of means of communication between intelli- gent agents. Considerable research effort has been expended on the design of ar- tificial languages for agent communications, such as DARPA’s Knowledge Query and Manipulation Language (KQML) [33] and the Foundation for Intelligent Phys- ical Agents’ (now IEEE FIPA) Agent Communications Language (FIPA ACL) [35]. These languages, and languages like them, have been designed to be widely ap- plicable. As well as being a strength, this feature can also be a weakness: agents participating in conversations have too many choices of what to utter at each turn, and thus agent dialogues may endure a state-space explosion. Allowing sufficient flexibility of expression while avoiding state-space explosion had led agent communications researchers to the study of formal dialogue games; these are rule-governed interactions between two or more players (or agents), where Department of Computer Science, University of Liverpool, UK e-mail: mcburney@liverpool.ac.uk Department of Computer and Information Science, Brooklyn College, New York, USA e-mail: parsons@sci.brooklyn.cuny.edu I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 261 DOI 10.1007/978-0-387-98197-0 13, c Springer Science+Business Media, LLC 2009
- 276. 262 McBurney and Parsons each player “moves” by making utterances, according to a defined set of rules. Al- though their study dates to at least the time of Aristotle [5], dialogue games have found recent application in philosophy, computational linguistics and Artificial In- telligence (AI). In philosophy, dialogue games have been used to study fallacious reasoning [41, 62] and to develop a game-theoretic semantics for various logics, e.g., intuitionistic and for classical logics [59]. In linguistics, they have been used to ex- plain sequences of human utterances [57], with subsequent application to machine- based natural language processing and generation [49], and to human-computer in- teraction [9]. Within computer science and AI, they have been applied to modeling complex human reasoning, for example in legal domains [81], and to requirements specification for complex software systems [34]. Dialogue games differ from the games of economic game theory in that payoffs for winning or losing a game are not considered, and, indeed, the notions of winning and losing are not always ap- plicable to dialogue games. They also differ from the abstract games recently used as a semantics for interactive computation [1], since these latter games do not share the rich rule structure of dialogue games, nor are these latter intended to themselves have a semantic interpretation involving the co-ordination of actions among a group of agents. This chapter considers the application of formal dialogue games for agent com- munication and interaction using argumentation. We begin, in the next subsection, with a brief overview of an influential typology of human dialogues, which have proven useful in classifying agent interactions. Because the design of artificial lan- guages for communication between software agents shares much with the study of natural human languages, we structure this chapter according to the standard divi- sion within linguistic theory between syntax, semantics and pragmatics; we do this despite this division being imprecise and contested within linguistics (e.g., [58]). Very broadly (following [58]), we may view: syntax as being concerned with the sur- face form and combinatorial properties of utterances, words and their components; semantics as being concerned with the truth or falsity of utterances; and pragmatics as being concerned with those aspects of the meaning of utterances other than their truth or falsity.1 Section 2 thus presents a model of a formal dialogue game protocol, focusing primarily on the syntax of such dialogues. We follow this in Section 3 with a discussion of the semantics and the pragmatics of agent dialogues. Section 4 then presents an illustrative example, taken from [68], while Section 5 considers protocol design and assessment. The chapter ends with a brief conclusion in Section 6. 1.1 Types of dialogues An influential model of human dialogues is the typology of primary dialogue types of argumentation theorists Douglas Walton and Erik Krabbe [96]. This categoriza- tion is based upon the information the participants have at the commencement of a 1 Note that the word semantics is used differently here than in the study of argumentation frame- works, as in Chapter 2 of this volume.
- 277. 13 Dialogue Games 263 dialogue (of relevance to the topic of discussion), their individual goals for the di- alogue, and the goals they share. Information-Seeking Dialogues are those where one participant seeks the answer to some question(s) from another participant, who is believed by the first to know the answer(s). In Inquiry Dialogues the participants collaborate to answer some question or questions whose answers are not known to any one participant. Persuasion Dialogues involve one participant seeking to persuade another to accept a proposition he or she does not currently endorse. In Negotiation Dialogues, the participants bargain over the division of some scarce resource. If a negotiation dialogue terminates with an agreement, then the resource has been divided in a manner acceptable to all participants. Participants of Delib- eration Dialogues collaborate to decide what action or course of action should be adopted in some situation. Here, participants share a responsibility to decide the course of action, or, at least, they share a willingness to discuss whether they have such a shared responsibility. Participants may have only partial or conflicting in- formation, and conflicting preferences. As with negotiation dialogues, if a deliber- ation dialogue terminates with an agreement, then the participants have decided on a mutually-acceptable course of action. In Eristic Dialogues, participants quarrel verbally as a substitute for physical fighting, aiming to vent perceived grievances. Several comments are important to make here. The first is that although Walton and Krabbe talk about the goal of a dialogue and the goal of a dialogue type,2 only participants can have goals since only they are sentient. Participants may believe that a dialogue interaction they enter has an ostensible purpose, but their own goals or the goals of the other participants may not be consistent with this purpose. For ex- ample, participants may enter a negotiation dialogue in order to reach an agreement (a deal) over the allocation of some resource; or they may enter it to prevent any such agreement being reached, or to delay agreement [24], or to prove that no such agreement is possible, or to gather information from the other participants, or even to signal something to some third party, not in the dialogue. Participants in dialogues may also seek to hide their true goals from the other participants [25, 64]. Instead of dialogue goals it makes sense only to speak of participant goals and dialogue outcomes [74]. Secondly, most actual dialogue occurrences — both human and agent — involve mixtures of these dialogue types. A purchase transaction, for example, may com- mence with a request from a potential buyer for information from a seller, proceed to a persuasion dialogue, where the seller seeks to persuade the potential buyer of the importance of some feature of the product, and then transition to a negotia- tion, where each party offers to give up something he or she desires in return for something else. The two parties may or may not be aware of the different nature of their discussions at each phase, or of the transitions between phases. Instances of individual dialogue types contained entirely within other dialogue types are said to be embedded [96]. Several formalisms have been suggested for computational rep- resentation of combinations of dialogue: first, the Dialogue Frames of Reed [84], which enable iterated, sequential and embedded dialogues to be represented; sec- 2 as do others, e.g., [80].
- 278. 264 McBurney and Parsons ond, the Agent Dialogue Frameworks of McBurney and Parsons, based on PDL [66], which permit iterated, sequential, parallel and embedded dialogues to be rep- resented; and third, the more abstract RASA frameworks of Miller and McBurney [73], which permit iterated, sequential, parallel and embedded combinations of any types of agent interaction protocols. All these formalisms are neutral with regard to the modeling of the primary dialogue types themselves, allowing the primary types to be represented in any convenient form, and allowing for types other than the six of the Walton and Krabbe typology to be included. Walton and Krabbe do not claim their typology is comprehensive, and some recent research has explored other types and sub-types, e.g., [14]. Researchers in multi-agent systems and in argumentation have articulated dia- logue game protocols for many of the types in the Walton and Krabbe typology. For example, the two-party protocol of Amgoud, Maudet and Parsons [3], which is based on MacKenzie’s philosophical dialogue game DC [62], supports persuasion, inquiry and information-seeking dialogues; a subsequent extension of this protocol with additional locutions supports negotiation dialogues [4]. Information-seeking dialogues have been considered by Hulstijn [49], and analyzed by Cogan, Parsons and McBurney [14]; indeed this latter work, which examines the pre- and post- conditions of dialogues over beliefs in fine detail, identifies several new types of dialogues not explicitly included in the Walton and Krabbe typology. A study of dif- ferent persuasion protocols can be found in the review paper by Prakken [80]; other protocols for persuasion dialogues include the PADUA protocol for arguments from experience by Wardeh, Bench-Capon and Coenen [97] and a protocol for arguments over access to information by Doutre and colleagues [22, 23, 78]. Protocols for multi-agent inquiry dialogues have been proposed and studied by McBurney and Parsons [65], who consider the circumstances under which an in- quiry dialogue may converge to the truth, and by Black and Hunter [11], whose agent reasoning architecture enables generative inquiry dialogues, i.e., those where new proposals may emerge for consideration and possible endorsement by the agents participating. For dialogues over beliefs (information-seeking, inquiry and persuasion dialogues), Parsons and Sklar consider the question of convergence of beliefs of agents engaged in repeated dialogues with one another [77]. In addition to [4] cited above, protocols for negotiation dialogues include those of Sadri, Toni and Torroni [87], McBurney, van Eijk, Parsons and Amgoud [63], and Karunatil- lake [54]. Regarding dialogues over action which are not negotiations: McBurney, Hitchcock and Parsons [64] and Tang and Parsons [92] have presented protocols for deliberation dialogues; Atkinson, Bench-Capon and McBurney have given a repre- sentation for proposals for actions and a dialogue game protocol to discuss these proposals [6]; and Atkinson, Girle, McBurney and Parsons have presented a dia- logue game protocol for dialogues over commands [7]. Finally, the dialogue-game protocols presented in the work of Dignum, Dunin-Kȩplicz and Verbrugge [20, 21] are intended to enable agents to form teams and to agree joint intentions, respec- tively.
- 279. 13 Dialogue Games 265 2 Syntax The syntax of a language concerns the surface form of words and phrases, and how these may be combined. Accordingly, defining the syntax of an agent dialogue game protocol usually involves the specification of the possible utterances which agents can make (the locutions) and the rules which govern the order in which utterances can be made. Since the work of Hamblin [41], it has become standard to talk of speakers in a dialogue incurring commitments: a speaker who asserts a statement as being true, for example, may be committed to justifying this assertion when chal- lenged by another participant, or else allowed (or even forced) to retract the asser- tion. Although such dialogical commitments may be viewed as aspects of the se- mantics (the meaning) of utterances, the rules regarding commitments are typically included in the specification of dialogue syntax because these rules often influence the order of utterances. The various commitments of the participants are usually tracked in a publicly-readable database, called a commitment store. Within the agents communications community, it has become standard to view utterances as composed of two layers: an inner layer comprising the topics of dis- cussion, and an outer (or wrapper) layer, comprising the locutions. An utterance can thus be seen as an instantiated locution, with one variable of instantiation being the topic. This structure, adopted for both KQML and FIPA ACL, provides great flexi- bility, since agents encoded appropriately may use the same wrappers to undertake dialogues over different topics. We now present a generic framework for specification of a dialogue game pro- tocol in terms of its key components, adapted from [66].3 We first assume that the topics of discussion between the agents (the inner layer) can be represented in some logical language, whose well-formed formulae are denoted by the lower-case Ro- man letters, p, q, r, etc. A dialogue game specification then comprises the following elements, each of which concern the wrapper layer of communications: Commencement Rules: Rules which define the circumstances under which the dialogue commences. Locutions: Rules which indicate what utterances are permitted. Typically, legal locutions permit participants to assert propositions, permit others to question or contest prior assertions, and permit those asserting propositions which are sub- sequently questioned or contested to justify their assertions. Justifications may involve the presentation of a proof of the proposition or an argument for it. The dialogue game rules may also permit participants to utter propositions to which they assign differing degrees of commitment, for example: one may merely pro- pose a proposition, a speech act which entails less commitment than would an assertion of the same proposition. Rules for Combination of Locutions: Rules which define the dialogical contexts under which particular locutions are permitted or not, or obligatory or not. For 3 We are also informed by [80]; note, however, that work defines a mathematical model for analyz- ing multi-party dialogues, rather than defining a framework for specification of dialogue protocols for agent communications.
- 280. 266 McBurney and Parsons instance, it may not be permitted for a participant to assert a proposition p and subsequently the proposition ¬p in the same dialogue, without in the interim having retracted the former assertion. Commitments: Rules which define the circumstances under which participants incur dialogical commitments by their utterances, and thus alter the contents of the participants’ associated commitment stores. For example, a question posed by one agent to another may impose a commitment on the second to provide a response; until provided, this commitment remains undischarged. Rules for Combination of Commitments: Rules which define how commitments are combined or manipulated when utterances incurring conflicting or comple- mentary commitments are made. For example, the rules may allow a speaker to assert the truth of a proposition and then to assert its negation, with the commit- ment store holding only the most recent asserted proposition, or the store may hold the earlier proposition until explicitly retracted. These rules become partic- ularly important when multiple dialogues are involved, as when one dialogue is embedded within another; in such a case, the commitments incurred in the inner dialogue may take priority over those of the outer dialogue, or vice versa [66]. Rules for Speaker Order: Rules which define the order in which speakers may make utterances. It may be that any speaker may speak at any time, as in FIPA ACL, or that there are rules regarding turn-taking. Termination Rules: Rules that define the circumstances under which the dialogue ends. It is worth noting here that more than one notion of commitment is present in the literature on dialogue games. For example, Hamblin treats commitments in a purely dialogical sense: “A speaker who is obliged to maintain consistency needs to keep a store of statements representing his previous commitments, and require of each new statement he makes that it may be added without inconsistency to this store. The store represents a kind of persona of beliefs; it need not correspond with his real beliefs . . .” [41, p. 257]. In contrast, Walton and Krabbe [96, Chapter 1] treat com- mitments as obligations to (execute, incur or maintain) a course of action, which they term action commitments. These actions may be utterances in a dialogue, as when a speaker is forced to defend a proposition he has asserted against attack from others; so Walton and Krabbe also consider propositional commitment as a special case of action commitment [96, p. 23]. As with Hamblin’s treatment, such dialogi- cal commitments to propositions may not necessarily represent a participant’s true beliefs. In contrast, Singh’s social semantics [90], requires participants in an interac- tion to express publicly their beliefs and intentions, and these expressions are called social commitments. These include both expressions of belief in some propositions and expressions of intent to execute or incur some future actions.4 Our primary mo- tivation is the use of dialogue games as the basis for interaction protocols between autonomous agents. Because such agents will typically enter into these interactions 4 It is worth noting that all these notions of commitment differ from that commonly used in discus- sion of agent’s internal states, namely the idea of the persistence of a belief or an intention [99, p. 205].
- 281. 13 Dialogue Games 267 in order to achieve some wider objectives, and not just for the enjoyment of the interaction itself, we believe it is reasonable to define commitments in terms of fu- ture actions or propositions external to the dialogue. In a commercial negotiation dialogue, for instance, the utterance of an offer may express a willingness by the speaker to undertake a subsequent transaction on the terms contained in the offer. For this reason, we can view commitments as semantic mappings between locutions and subsets of some set of statements expressing actions or beliefs external to the dialogue. 3 Semantics and Pragmatics 3.1 Purposes of Semantics We begin this section by discussing the concept of semantics for agent communica- tions languages and dialogue protocols. These languages and protocols are clearly media for communication (between software entities and/or their human principals) and so researchers have naturally looked to theories developed in human linguistics to understand them. But, unlike human languages, agent communications languages and dialogue protocols are also formal constructs, usually defined explicitly and of- ten computationally; thus, understanding their properties can also usefully draw on notions from logic and mathematics. Moreover, because these communications lan- guages and dialogue protocols are usually intended to be used by autonomous soft- ware entities, they are also programming languages, since software agents will use them to construct sequences of utterances — commands — with which to interact with one another. The theory of programming language semantics is therefore also relevant to their study. It is thus important to keep in mind the different functions which a semantics for an agent communications language or dialogue protocol may be required to serve: • To provide a shared understanding to participants in a communicative interac- tion of the meaning of individual utterances, of sequences of utterances, and of dialogues. • To provide a shared understanding to designers of agent protocols and to the (possibly distinct) designers of agents using those protocols of the meaning of individual utterances, of sequences of utterances, and of dialogues. • To provide a means by which the properties of individual agent communications languages and protocols may be studied formally and with rigor. • To provide a means by which different agent communications languages and protocols may be compared with one another formally and with rigor. • To provide a means by which languages and protocols may be readily imple- mented in production systems. • To help ensure that implementation of agent communications in open, distributed agent systems is undertaken uniformly.
- 282. 268 McBurney and Parsons Different types of semantics may serve these various purposes to varying degrees, and so it may be useful to develop more than one semantics for a communications language or protocol. 5 In addition, an articulation of semantics could be undertaken at one or more different levels: for each individual utterance, or speech act; for specified short sequences of utterances,6 such as a question-and-answer sequence; for complete sequences of utterances, or dialogues; and for dialogue protocols. Most current published work on agent dialogue protocols presents a semantics defined in terms of individual utterances. In the terms defined below, these semantics are most often axiomatic or operational, and are much less often denotational. 3.2 Types of Semantics We have inherited two conflicting notions of semantics, one deriving from linguis- tics and the other from mathematical logic. As linguists normally understand these terms, the syntax of a language is “the formal relation of signs to one another” and the semantics of the language “the relations of signs to the objects to which the signs are applicable” (Morris [75], cited in [58, p. 1]). Thus, it makes sense to speak of the truth of a sign (or of an utterance in a language using such signs), since this indicates that the sign has a relationship to external objects in the world. Within mathematics and mathematical logic, a different understanding of semantics has arisen, beginning with Pieri [79] and Hilbert [43] and first articulated formally by Tarski [94]. In this tradition, a semantics for a formal language is a relationship between that language and a space M of mathematical structures, called models. A statement S in the language specifies a subset M(S) of M. Such a statement is said to be true in a particular model M0 if M0 ∈ M(S). A statement is said to be logically true if it is true in every model, i.e., if M(S) = M.7 These two notions of semantics — one linguistic, one mathematical — collide.8 In particular, in the mathematical framework, benefit may be gained from defining different semantic mathematical structures for the one language; Tarski himself, for instance, defined topological [93] and discrete lattice [71] semantics for propositional logic. The ben- efits of this are that different semantic frameworks may enable different properties of the language to be studied and may provide different insights. Insight may also be gained by comparing the structures with each other, a subject known as model theory or metamathematics [48]. But, defining and comparing alternative structures 5 Traditional mathematical communications theory, due to Shannon and Weaver [89], explicitly ignores the semantics of messages, and so provides little guidance to designers, developers or users of agent communications languages and protocols. 6 These are known as conversations in the agent communications literature, e.g., [39]. 7 Note that Tarski only applied his framework to formal, or mathematical, languages, and was skeptical about its applicability to natural language [94, pp. 163–165]. 8 Their first skirmish was the argument between Hilbert and Frege over the meaning of Hilbert’s ax- ioms for geometry: Frege took what we are calling a linguistic approach, Hilbert a model-theoretic approach; see [95, pp. 408–412] and [46, pp. 7–10].
- 283. 13 Dialogue Games 269 in this way makes no sense in the linguistic understanding of semantics: how could a language admit more than one set of relationships to the truth? Agent communications languages and dialogue protocols straddle this divide. Because they are formal languages, insight into their properties can be gained by defining semantic relationships to mathematical structures, and studying these struc- tures. However, because they are also intended as media for communication, just as natural language is, each agent using a particular communications language or pro- tocol will wish to ensure that all users share a common understanding of utterances.9 To verify that agents have the same understanding — the same semantics — for a communications language ultimately requires some form of inspection of their in- ternal states or, equivalently, their program code. This is a challenging, and perhaps conceptually impossible, undertaking since a sufficiently-clever agent can always simulate insincerely any required internal state.10 Rather, in this chapter, our use of semantic frameworks differs from that in linguistics: first, as in model theory, semantic structures are a means to understand the properties of a formal agent com- munications language, and, second, because our focus is on computer systems, these structures are a means to support the engineering of multi-agent systems software and to aid uniformity of implementation when software engineering is undertaken by different development teams. It is therefore helpful to consider several different types of semantic frameworks for formal languages. In doing so, we draw on the summary of the literature on pro- gramming language semantics presented by van Eijk in [29, Section 1.2.2]; however, we make no claims that the typology is comprehensive. One type of semantics de- fines each locution of a communications language in terms of the pre-conditions which must exist before the locution can be uttered, and possibly also the post- conditions which apply following its utterance, in a STRIPS-like fashion [32]. This is called an axiomatic semantics [29, 72]. For agent communications languages and dialogue protocols we distinguish between public and private axiomatic approaches. In public axiomatic approaches, the pre-conditions and post-conditions all describe states or conditions of the dialogue which can be observed by all participants. In private axiomatic approaches, at least some of the pre- or post-conditions describe states or conditions which are internal to one or more of the participants, and thus not directly observable by the others. For example, the semantic language, SL, for the locutions of the Agent Communications Language, FIPA ACL, of the Founda- tion for Intelligent Physical Agents (FIPA), is a private axiomatic semantics of the speech acts of the language, defined in terms of the beliefs, desires and intentions of the participating agents [35]. For example, the inform locution in the FIPA ACL language, allows one agent, say agent A, to tell another agent, say B, some propo- sition p. The FIPA ACL semantics of inform only permits agent A to do this if [35, p. 10]: (a) agent A believes p to be true, (b) agent A intends that agent B believes p to be true, and (c) agent A believes that agent B does not already have a belief about 9 Friedrich Dürrenmatt’s novel, Die Panne, shows what tragic consequences may follow when participants assign very different meanings to the same conversation [28]. 10 For more on this, see [98].
- 284. 270 McBurney and Parsons the truth of p.11 Similarly, the semantics defined for many dialogue game protocols for agent interaction, e.g., [3], are also private axiomatic semantics. In contrast, the semantics provided for dialogue games used for modeling legal reasoning in [10] is a public axiomatic semantics. A second type of semantics, an operational semantics, considers the dialogue locutions as computational instructions which operate successively on the states of some abstract machine. Under this approach, the participating agents and their shared dialogue are viewed conceptually as parts of a large abstract or virtual com- puter, whose overall state may be altered by the utterance of valid locutions or by internal decision processes of the participants; it is as if these locutions or decisions were commands in some computer programme language acting on the virtual ma- chine.12 The utterances and agent decision-mechanisms are thus seen as state tran- sition operators, and the operational semantics defines these transitions precisely [29]. This approach to the semantics of agent communications languages makes ex- plicit any link between the internal decision mechanisms of the participating agents and their public utterances to one another. The semantics therefore enables the re- lationships between the mental states of the participants and the public state of the dialogue to be seen explicitly, and shows how these relationships change as a re- sult of utterances and internal agent decisions. Thus, an operational semantics will typically make assumptions about the internal decision-mechanisms of the agents participating in the interaction; the actual agents engaged in a communicative in- teraction may not necessarily use the decision-process or realize the mental states assumed. Operational semantics have recently been defined for some agent commu- nications languages, for example, in [30, 44] and for some dialogue protocols, e.g., information-passing interactions [19], negotiation dialogue protocols [54, 63], and a general argumentation protocol [68]. Third, in denotational semantics, each element of the language syntax is as- signed a relationship to an abstract mathematical entity, its denotation. The possi- ble worlds, or Kripkean, semantics defined for modal logic syntax is an example of such a semantics for a logical language [56]. However, two decades before Kripke’s work, a denotational semantics mapping logical formulae to subsets of a topologi- cal space was given for the modal logic system S4 [91]. For argumentation systems, three denotational semantics have been provided for the ICRF’s Logic of Argu- mentation LA [55]. In the first of these, Ambler [2] articulated a category-theoretic semantics [61] for LA, by extending to arguments the Curry-Howard isomorphism, which connects proofs in a deductive logic to the morphisms of a free cartesian closed category. In this semantics, propositions (i.e., premises or claims) correspond to objects in a particular enriched category, and arguments linking propositions to morphisms between the associated objects. A second denotational semantics for LA, due to Parsons [76], connects argumentation systems to qualitative probabilistic net- works (QPNs). In this semantics, propositions correspond to nodes in a QPN, and arguments linking propositions to edges between the associated nodes. Das [17] 11 Note that condition (a) enforces sincerity on the speaker, which is not necessarily desirable. Also, condition (c) precludes the use of inform in authentication dialogues. 12 This virtual machine is purely a conceptual construct and does not need to exist in reality.
- 285. 13 Dialogue Games 271 articulated a third denotational semantics for logics of argumentation, based on a Kripkean possible-worlds structure. In this semantics, different arguments are as- sumed classified according to the degree of support they provide for propositions; these differential degrees of support are translated into separate hyper-relations over the accessibility relations of the Kripke structure.13 Perhaps the first example of a denotational semantics for a dialogue protocol was the possible-worlds semantics for question-response interactions defined by Ham- blin in 1956 [40]. Although possible-worlds and category-theoretic denotational se- mantics have a long subsequent history in mathematical linguistics, only recently have denotational semantics been defined for agent dialogue protocols. In [67], McBurney and Parsons articulated a category-theoretic semantics, called a Trace Semantics, for a broad class of deliberation dialogue protocols. In this semantics, articulation of proposals for action by participants correspond to the creation of ob- jects in certain categories, while participant preferences between these proposals correspond to the existence of arrows (morphisms) between the corresponding ob- jects. Thus, the semantics is constructed jointly and incrementally by the dialogue participants as the dialogue proceeds, in a manner similar to the natural language se- mantics of Discourse Representation Theory [53] (which uses possible worlds), or the argumentation graph of Gordon’s Pleadings Game [38]. A similar denotational semantics, constructed jointly and incrementally by the participants, is outlined by Atkinson and colleagues in [6], for a dialogue protocol for arguments over propos- als for action. In this semantics, the mathematical entities constructed are topoi and maps between them, rather than simply categories.14 For the denotational semantics approach to be useful, we must be able to derive the semantic mapping of a compound statement in the language from the seman- tic mappings of its elements, a property called compositionality. This property is not always present; for example, it may be absent if the language contains com- pound statements with infinite combinations of elements or if compound statements have denotations which differ from the composition of those of their elements, as in Hintikka’s Independence-Friendly (IF) Logic [47]. In these cases, a specific type of denotational semantics, game-theoretic semantics, has sometimes proven use- ful [45]. In this semantics, each well-formed statement in the language is associated with a conceptual game between two players, a protagonist and an antagonist. A statement in the language is considered to be true when and only when a winning strategy exists for the protagonist in the associated game; a winning strategy for a player is a rule giving that player moves for the game such that executing these moves guarantees the player can win the game, no matter what moves are made by the opposing player. Game semantics have been articulated for propositional and predicate logics [59], linear logic [1], and for probability theory [18], among others. 13 This semantics may be viewed as a form of quantification over possible worlds, of which a more general formalism is that developed subsequently (and independently) by van Eijk and his colleagues to compare network topologies [31]. 14 Topoi are generalizations of the category of sets, and incorporate a categorial analogue of the notion of set membership [37].
- 286. 272 McBurney and Parsons What value do these different types of semantics have? Axiomatic semantics show the pre- and post-conditions of individual utterances in a communications in- teraction. They may also be used to show the pre- and post-conditions of sequences of utterances, or even entire dialogues [14, 74]. Thus, they provide a set of rules to regulate participation in rule-governed interactions. Operational semantics, by showing the state transitions effected by utterances, may be used to identify dia- logue states which are not reachable or from which no legal utterance may be made. These semantics can be used, therefore, to demonstrate that termination of dialogues between participants using a particular protocol is or is not possible. Operational semantics also identify which internal agent decision-mechanisms are needed by agents in order to issue and comprehend received utterances. Properties of dialogue protocols may also be demonstrated using denotational semantics. In [67], we used the Deliberation Trace Semantics to generalize a result of Harsanyi [42] regarding the pareto-optimality of deals achieved using Zeuthen’s Monotonic Concession Pro- tocol (MCP) [100]. Game semantics have also been used to study the properties of formal argumentation systems and dialogue protocols, such as their computational complexity [26], or the extent of truth-convergence under an inquiry dialogue pro- tocol [65], and to identify acceptable sets of arguments in argument frameworks [13, 51]. 3.3 Pragmatics Following Levinson [58], we view the study of language pragmatics as dealing with those aspects of linguistic meaning not covered by considerations of truth and fal- sity. Chief among these aspects are the desires and intentions of speakers, and these are usually communicated by means of speech acts, non-propositional utterances in- tended to or perceived to change the state of the world. Examples of speech acts are utterances in which a speaker proposes that some action be undertaken, or promises to undertake it, or commands another to perform it. Modern speech act theory was initially due to Austin [8] and Searle [88], who classified spoken utterances by their intended and actual effects on the world (including the internal mental states of those hearing the utterances), and developed pre-conditions for those effects to be realized. Drawing on this theory, Bretier, Cohen, Levesque, Perrault and Sadek were able to present pre- and post-conditions for agent utterances in terms of the mental states of the participants [12, 15, 16]. This work formed the basis for the axiomatic Semantic Language SL of the FIPA Agent Communications Language ACL mentioned above [35]. One of the criticisms made of FIPA ACL is that the language does not support argumentation [70]; accordingly, McBurney and Parsons [68] extended this language by defining five additional locutions to enable agents to assert, question, challenge, justify and retract statements with one another. A set of locution-combination rules are given (although any such rules are absent from the specification of FIPA ACL itself), along with an axiomatic semantics in the style of
- 287. 13 Dialogue Games 273 SL and an operational semantics. This protocol is discussed in the next Section, as an example of these ideas. Long before Austin and Searle, Reinach [85] had noted that speech acts typically require endorsement, or uptake, from the hearer before changing the state of the world; a speaker may promise a hearer to perform some action, but the speaker is only obligated to act once the promise is accepted by the hearer. Speech acts are thus essentially social activities and cannot, normally, be executed by a lone reasoner: their natural home is a multi-party dialogue. This observation is particularly true for those speech acts for which the speaker does not have power of retraction or revocation. In [69], McBurney and Parsons presented an analysis of the differences in meaning between, for instance, commands and promises. Once uptaken (i.e., once in effect), a command may only be revoked by the original speaker, whereas a promise may only be revoked by the agent to which the promise was made, not the speaker. However, capturing such differences in the syntax of utterances can be difficult. For example, the syntactical form of the two utterances: I command you to wash the car. I promise you to wash the car. is identical, but the illocutionary force, the effects on the world, the nature of the obligation incurred, the identity of the agent with revocation powers, and even the identity of the agent intended by the speaker to perform the action are differ- ent. Although formal agent communications languages should be less ambiguous than natural language, an interpretation of the syntax of utterances is required for elimination of any ambiguity in meaning. McBurney and Parsons [69] dealt with this problem by modifying the denotational trace semantics of [67] to map action- utterances to tuples in a partitioned tuple space [36]. The different dialogical powers that participating agents have of issuance, endorsement and revocation for particular types of utterance then correspond to different permissions to write, copy and delete (respectively) tuples from associated sub-spaces of the tuple space. 4 Example As an example of the ideas in this chapter, we present the Fatio protocol of [68]. This protocol comprises five locutions which may be added to the 22 locutions of FIPA ACL, in order to enable communicating agents to engage in rational argument: assert, question, challenge, justify and retract. These five locutions are subject to six locution-combination rules, which together encode a particular dialectical ar- gumentation theory. Because these locutions are intended to be complementary to FIPA ACL, there are no locutions for commencing or terminating a dialogue.15 For reasons of space, we only give examples of two of the five legal locutions of Fatio: 15 It would be easy to take these from another protocol, such as [63].
- 288. 274 McBurney and Parsons F1: assert(Pi,φ): A speaker Pi asserts a statement φ ∈ C (a belief, an intention, a social connection, an external commitment, etc). In doing so, Pi creates a di- alectical obligation within the dialogue to provide justification for φ if required subsequently by another participant. F3: challenge(Pj,Pi,φ): A speaker Pj challenges a prior utterance of assert(Pi,φ) by another participant Pi, and seeks a justification for φ. In contrast to a ques- tion, with this locution, Pj also creates a dialectical obligation on himself to pro- vide a justification for not asserting φ, for example an argument against φ, if questioned or challenged. Thus, challenge(Pj,Pi,φ) is a stronger utterance than question(Pj,Pi,φ). For illustration, we present two of the six Fatio locution-combination rules. Here, Φ ⊢+ φ indicates that Φ is an argument in support of φ. CR2: The utterances question(Pj,Pi,φ) and challenge(Pj,Pi,φ) may be made at any time following an utterance of assert(Pi,φ). Similarly, the utterances question(Pj,Pi,Φ) and challenge(Pj,Pi,Φ) may be made at any time following an utterance of justify(Pi,Φ ⊢+ φ). CR3: Immediately following an utterance of question(Pj,Pi,φ) or challenge(Pj, Pi,φ), the speaker Pi of assert(Pi,φ) must reply with justify(Pi,Φ ⊢+ φ), for some Φ ∈ A. In [68], both an axiomatic and an operational semantics for this protocol are ar- ticulated. The axiomatic semantics is defined in terms of the beliefs, desires and intentions of the participating agents consistent with the axiomatic semantics SL of FIPA ACL. For example, the semantics of the locution assert(.) is defined as fol- lows, with Biφ indicating that “Agent i believes that φ is true”, and Diφ that “Agent i desires that φ be true.” • assert(Pi,φ) Pre-conditions: A speaker Pi desires that each participant Pj(j = i), believes that Pi believes the proposition φ ∈ C. ((Pi,φ,+) ∈ DOS(Pi))∧(∀j = i)(DiBjBiφ). Post-conditions: Each participant Pk(k = i), believes that participant Pi desires that each participant Pj(j = i), believe that Pi believes φ. (Pi,φ,+) ∈ DOS(Pi) ∧(∀k = i)(∀j = i)(BkDiBjBiφ). Dialectical Obligations: (Pi,φ,+) is added to DOS(Pi), the Dialectical Obliga- tions Store of speaker Pi. Similarly, the operational semantics for Fatio defined in [68] articulates the state- transition effected by an assert(.) utterance (here labeled F1) on the mental states of, firstly, the agent who uttered it and, secondly, on any agent who heard it. TR2: Pi, D1, utter-assert(φ) F1 → Pi, D5, listen TR3: Pi, D1, utter-assert(φ) F1 → Pj, D5, do-mech(D2) These excerpts from the semantics of Fatio are intended simply for illustration. Full details are given with the protocol definition [68].
- 289. 13 Dialogue Games 275 5 Protocol design and assessment The science and software engineering of agent communications and interactions is still in its infancy. Accordingly, designers of multi-agent dialogue game protocols still have little guidance for design questions such as: How many locutions should there be? What types of locutions should be included, e.g., assertions, questions, etc? What are the appropriate rules for the combination of locutions? When should be- havior be forbidden, e.g., repeated utterance of one locution? Under what conditions should dialogues be made to terminate? When are dialogues conducted according to a particular protocol guaranteed to terminate? What are the properties of a proposed protocol? Similarly, the immaturity of the discipline means that software developers and their agents still lack answers to questions such as: How may different protocols be compared and differences measured? Which protocols are to be preferred under which circumstances? In other words, what are their advantages and disadvantages? How should a system developer (or an agent) choose between two protocols which both support the same type of dialogue, for example, two negotiation protocols? When are dialogue game protocols preferable to other forms of agent interaction, such as auction mechanisms or general agent communications languages, such as FIPA ACL? Some work has been undertaken which would assist with such questions. For ex- ample, McBurney, Parsons and Wooldridge [70], proposed thirteen desirable prop- erties of agent interaction protocols using dialogue games, and then applied these properties to assess several dialogue game protocols and FIPA ACL; all were found wanting, to a greater or lesser extent. From an empirical perspective, Karunatil- lake [54] undertook an evaluation of various negotiation protocols. This work used simulation studies to compare performance in negotiation interactions for agents using protocols with and using protocols without argumentation, in order to iden- tify the circumstances under which argumentation-based negotiation was beneficial. From a theoretical perspective, Johnson, McBurney and Parsons [52] defined var- ious measures of protocol equivalence, both syntactical and semantic, and showed the relationship of these measures to one another. Knowing that two protocols are equivalent allows inference about their properties (such as termination), and about their compliance with a given specification, such as that laid down as the standard for interacting within some electronic institution [86]. 6 Conclusion In this chapter we have given a brief introduction to the theory of dialogue game protocols for agent interaction and argument, a subject which has become impor- tant with the rise of multi-agent systems. We have focused on the syntax and the semantics of these protocols because these topics are important, not only for anal- ysis of protocols, but also for the software engineering specification, design and
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- 295. Chapter 14 Models of Persuasion Dialogue Henry Prakken 1 Introduction This chapter1 reviews formal dialogue systems for persuasion. In persuasion dia- logues two or more participants try to resolve a conflict of opinion, each trying to persuade the other participants to adopt their point of view. Dialogue systems for persuasion regulate how such dialogues can be conducted and what their outcome is. Good dialogue systems ensure that conflicts of view can be resolved in a fair and effective way [6]. The term ‘persuasion dialogue’ was coined by Walton [13] as part of his influential classification of dialogues into six types according to their goal. While persuasion aims to resolve a difference of opinion, negotiation tries to resolve a conflict of interest by reaching a deal, information seeking aims at trans- ferring information, deliberation wants to reach a decision on a course of action, inquiry is aimed at “growth of knowledge and agreement” and quarrel is the ver- bal substitute of a fight. This classification leaves room for shifts of dialogues of one type to another. In particular, other types of dialogues can shift to persuasion when a conflict of opinion arises. For example, in information-seeking a conflict of opinion could arise on the credibility of a source of information, in deliberation the participants may disagree about likely effects of plans or actions and in negotiation they may disagree about the reasons why a proposal is in one’s interest. The formal study of dialogue systems for persuasion was initiated by Hamblin [5]. Initially, the topic was studied only within philosophical logic and argumenta- tion theory [15, 7], but later several fields of computer science also became inter- ested in this topic. In general AI the embedding of nonmonotonic logic in models of persuasion dialogue was seen as a way to deal with resource-bounded reasoning [6, 2], while in AI Law persuasion was seen as an appropriate model of legal Henry Prakken Department of Information and Computing Sciences, Utrecht University, and Faculty of Law, Uni- versity of Groningen, e-mail: henry@cs.uu.nl 1 This chapter is a revised and updated version of [11]. I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 281 DOI 10.1007/978-0-387-98197-0 14, c Springer Science+Business Media, LLC 2009
- 296. 282 Henry Prakken procedures [4]. In intelligent tutoring, systems for teaching argumentation skills have been founded on models of persuasion dialogue [16]. Finally, in the field of multi-agent systems dialogue systems have been incorporated into models of ratio- nal agent interaction [8]. To delineate the scope of this chapter, it is useful to discuss what is the subject matter of dialogue systems. According to Carlson [3] dialogue systems define the principles of coherent dialogue, that is, the conditions under which an utterance is appropriate. The leading principle here is that an utterance is appropriate if it furthers the goal of the dialogue. For persuasion this means that an utterance should contribute to the resolution of the conflict of opinion that triggered the persuasion. Thus according to Carlson the principles governing the use of utterances should not be defined at the level of individual speech acts but at the level of the dialogue in which the utterance is made. Carlson therefore proposes a game-theoretic approach to dialogues, in which speech acts are viewed as moves in a game and rules for their appropriateness are formulated as rules of the game. Most work on formal dialogue systems for persuasion follows this approach and therefore this chapter will assume a game format of dialogue systems. It should be noted that the term dialogue system as used in this chapter only covers the rules of the game, i.e., which moves are allowed; it does not cover principles for playing the game well, i.e., strategies and heuristics for the individual players. The latter are instead aspects of agent models. Below in Section 2 an example persuasion dialogue will be presented, which will be used for illustration throughout the paper. Then in Section 3 a formal framework for specifying dialogue game systems is proposed, which in Section 4 is instantiated for persuasion dialogues and in Section 5 is used for discussing and comparing three systems proposed in the literature. 2 An example persuasion dialogue The following example persuasion dialogue exhibits some typical features of per- suasion and will be used in this chapter to illustrate different degrees of expressive- ness and strictness of the various persuasion systems. Paul: My car is safe. (making a claim) Olga: Why is your car safe? (asking grounds for a claim) Paul: Since it has an airbag, (offering grounds for a claim) Olga: That is true, (conceding a claim) but this does not make your car safe. (stat- ing a counterclaim) Paul: Why does that not make my care safe? (asking grounds for a claim) Olga: Since the newspapers recently reported on airbags expanding without cause. (stating a counterargument by providing grounds for the counterclaim) Paul: Yes, that is what the newspapers say (conceding a claim) but that does not prove anything, since newspaper reports are very unreliable sources of technologi- cal information. (undercutting a counterargument)
- 297. 14 Models of Persuasion Dialogue 283 Olga: Still your car is still not safe, since its maximum speed is very high. (alter- native counterargument) Paul: OK, I was wrong that my car is safe. This dialogue illustrates several features of persuasion dialogues. • Participants in a persuasion dialogue not only exchange arguments and coun- terarguments but also express various propositional attitudes, such as claiming, challenging, conceding or retracting a proposition. • As for arguments and counterarguments it illustrates the following features. – An argument is sometimes attacked by constructing an argument for the oppo- site conclusion (as in Olga’s two counterarguments) but sometimes by saying that in the given circumstances the premises of the argument do not support its conclusion (as in Paul’s counterargument). This is Pollock’s well-known distinction between rebutting and undercutting counterarguments [9]. – Counterarguments are sometimes stated at once (as in Paul’s undercutter and Olga’s last move) and are sometimes introduced by making a counterclaim (as in Olga’s second and third move). – Natural-language arguments sometimes leave elements implicit. For example, Paul’s second move arguably leaves a commonsense generalisation ‘Cars with airbags usually are safe’ implicit. • As for the structure of dialogues, the example illustrates the following features. – The participants may return to earlier choices and move alternative replies: in her last move Olga states an alternative counterargument after she sees that Paul had a strong counterattack on her first counterargument. Note that she could also have moved the alternative counterargument immediately after her first, to leave Paul with two attacks to counter. – The participants may postpone their replies, sometimes even indefinitely: with her second argument why Paul’s car is not safe, Olga postpones her reply to Paul’s counterattack on her first argument for this claim; if Paul fails to suc- cessfully attack her second argument, such a reply might become superfluous. 3 Elements of dialogue systems In this section a formal framework for specifying dialogue systems is proposed. To summarise, dialogue systems have a dialogue goal and at least two participants, who can have various roles. Dialogue systems have two languages, a a communica- tion language wrapped around a topic language. Sometimes, dialogues take place in a context of fixed and undisputable knowledge, such as the relevant laws in a le- gal dispute. The heart of a dialogue system is formed by a protocol, specifying the allowed moves at each point in a dialogue, the effect rules, specifying the effects of utterances on the participants’ commitments, and the outcome rules, defining the
- 298. 284 Henry Prakken outcome of a dialogue. Two kinds of protocol rules are sometimes separately de- fined, viz. turntaking and termination rules. Let us now specify these elements more formally. The definitions below of dia- logues, protocols and strategies are based on Chapter 12 of [1] as adapted in [10]. As for notation, the complement ϕ of a formula ϕ is ¬ϕ if ϕ is a positive formula and ψ if ϕ is a negative formula ¬ψ. Definition 14.1. (Dialogue systems) A dialogue system consists of the following elements. • A topic language Lt, closed under classical negation. • A communication language Lc, consisting of a set of speech acts with a content. The set of dialogues, denoted by M≤∞, is the set of all sequences from Lc, and the set of finite dialogues, denoted by M∞, is the set of all finite sequences from Lc. For any dialogue d = m1,...,mn,..., the subsequence m1,...,mi is denoted with di. • A dialogue purpose. • A set A of participants (or ‘players’) and a set R of roles, defined as disjoint subsets of A. A participant a may or may not have a, possibly inconsistent, belief base Σa ⊆ Pow(Lt), which may or may not change during a dialogue. Further- more, each participant has a, possibly empty set of commitments Ca ⊆ Lt, which usually changes during a dialogue. • A context K ⊆ Lt, containing the knowledge that is presupposed and must be respected during a dialogue. The context is assumed consistent and remains the same throughout a dialogue. • A logic L for Lt, which may or may not be monotonic and which may or may not be argument-based. • A set of effect rules C for Lc, specifying for each utterance ϕ ∈ Lc its effects on the commitments of the participants. These rules are specified as functions – Ca : M∞ −→ Pow(Lt) Changes in commitments are completely determined by the last move in a dia- logue and the commitments just before making that move: – If d = d′ then Ca(d,m) = Ca(d′,m) • A protocol Pr for Lc, specifying the allowed (or ‘legal’) moves at each stage of a dialogue. Formally, A protocol on Lc is a function Pr with domain the context plus a nonempty subset D of M∞ taking subsets of Lc as values. That is: – Pr : Pow(Lt)×D −→ Pow(Lc) such that D ⊆ M∞. The elements of D are called the legal finite dialogues. The elements of Pr(d) are called the moves allowed after d. If d is a legal dialogue and Pr(d) = ∅, then d is said to be a terminated dialogue. Pr must satisfy the following condition: for all finite dialogues d and moves m, d ∈ D and m ∈ Pr(d) iff d,m ∈ D.
- 299. 14 Models of Persuasion Dialogue 285 It is useful (although not strictly necessary) to explicitly distinguish elements of a protocol that regulate turntaking and termination: – A turntaking function is a function T : D×Pow(Lt) −→ Pow(A). A turn of a dialogue is defined as a maximal sequence of moves in the dialogue in which the same player is to move. Note that T can designate more than one player as to-move next. – Termination is above defined as the case where no move is legal. Accordingly, an explicit definition of termination should specify the conditions under which Pr returns the empty set. • Outcome rules OK, defining the outcome of a dialogue given a context. For in- stance, in negotiation the outcome is an allocation of resources, in deliberation it is a decision on a course of action, and in persuasion dialogue it is a winner and a loser of the persuasion dialogue. The outcome must be defined for termi- nated dialogues and may be defined for nonterminated ones; in the latter case the outcome rules capture an ‘anytime’ outcome notion. Note that no relations are assumed between a participant’s commitments and be- lief base. Commitments are an agent’s publicly declared points of view about a proposition, which need not coincide with the agent’s internal beliefs. Definition 14.2. (Some protocol types) • A protocol has a public semantics if the set of legal moves is always independent from the agents’ belief bases. • A protocol is context-independent if the set of legal moves and the outcome is always independent of the context, so if Pr(K,d) = Pr(∅,d) and OK(d) = O∅(d) for all K and d. • A protocol Pr is fully deterministic if Pr always returns a singleton or the empty set. It is deterministic in Lc if the set of moves returned by Pr at most differ in their content but not in their speech act type. • A protocol is unique-move if the turn shifts after each move; it is multiple-move otherwise. Paul and Olga (ct’d): The protocol in our running example is multiple-move. Dialogue participants can have strategies and heuristics for playing the dialogue game in ways that promote their individual dialogue goal. The notion of a strategy for a participant a can be defined in the game-theoretical sense, as a function from the set of all finite legal dialogues in which a is to move into Lc. A strategy for a is a winning strategy if in every dialogue played in accord with the strategy a realises his dialogue goal (for instance, winning in persuasion). Heuristics generalise strategies in two ways: they may leave the choice for some dialogues undefined and they may specify more than one move as a choice option. More formally: Definition 14.3. (strategies and heuristics) Let Da, a subset of D, be the set of all dialogues where a is to move, and let D′ a be a subset of Da. Then a strategy and a heuristic for a are defined as functions sa and ha as follows.
- 300. 286 Henry Prakken • sa : Da −→ Lc • ha : D′ a −→ Pow(Lc) 4 Persuasion Let us now become more precise about persuasion. Walton Krabbe [14] define persuasion dialogues as dialogues with as goal to resolve a conflict of points of view between at least two participants. A point of view with respect to a proposition can be positive (for), negative (against) or doubtful. The participants aim to per- suade the other participant(s) to accept their point of view. According to Walton Krabbe a conflict is resolved if all parties share the same point of view on the propo- sition that is at issue. They distinguish disputes as a subtype of persuasion dialogues where two parties disagree about a single proposition ϕ, such that at the start of the dialogue one party has a positive (ϕ) and the other party a negative (¬ϕ) point of view towards the proposition. Dialogue systems for persuasion can be formally defined as a particular class of instantiations of the general framework. Definition 14.4. (dialogue systems for persuasion) A dialogue system for persua- sion is a dialogue system with at least the following instantiations of Definition 14.1. • The dialogue purpose is resolution of a conflict of opinion about one or more propositions, called the topics T ⊆ Lt. This dialogue purpose gives rise to the following participant roles and outcome rules. • The participants can have the following roles. To start with, prop(t) ⊆ A, the proponents of topic t, is the (nonempty) set of all participants with a positive point of view towards t. Likewise, opp(t) ⊆ A, the opponents of t, is the (nonempty) set of all participants with a doubtful point of view toward a topic t. Together, the proponents and opponents of t are called the adversaries with respect to t. For any t, the sets prop(t) and opp(t) are disjoint but do not necessarily jointly exhaust A. The remaining participants, if any, are the third parties with respect to t, assumed to be neutral towards t. Note that this allows that a participant is a proponent of both t and ¬t or has a positive attitude towards t and a doubtful attitude towards a topic t′ that is logically equivalent to t. Since protocols can deal with such situations in various ways, this should not be excluded by definition. • The Outcome rules of systems for persuasion dialogues define for a dialogue d, context K and topic t the winners and losers of d with respect to topic d. More precisely, O consists of two partial functions w and l: – w : D×Pow(Lt)×Lt −→ Pow(A) – l : D×Pow(Lt)×Lt −→ Pow(A) such that they are defined at least for all terminated dialogues but only for those t that are a topic of d. These functions will be written as wK t (d) and lK t (d) or,
- 301. 14 Models of Persuasion Dialogue 287 if there is no danger for confusion, as wt(d) and lt(d). They further satisfy the following conditions for arbitrary but fixed context K: – wt(d)∩lt(d) = ∅ – wt(d) = ∅ iff lt(d) = ∅ – if | A | = 2, then wt(d) and lt(d) are at most singletons • Next, to make sense of the notions of proponent and opponent, their commit- ments at the start of a dialogue should not conflict with their points of view. – If a ∈ prop(t) then t ∈ Ca(∅) – If a ∈ opp(t) then t ∈ Ca(∅) • Finally, in persuasion at most one side in a dialogue gives up, i.e., – wt(d) ⊆ prop(t) or wt(d) ⊆ opp(t) ; and – If a ∈ wt(d) then · if a ∈ prop(t) then t ∈ Ca(d) · if a ∈ opp(t) then t ∈ Ca(d) These conditions ensure that a winner did not change its point of view. Note that they make that two-person persuasion dialogues are zero-sum games. Per- haps this is the main feature that sets persuasion apart from information seeking, deliberation and inquiry. Note that the two last winning conditions of the last bullet lack their only-if part. This is to allow for a distinction between so-called pure persuasion and conflict resolution. The outcome of pure persuasion dialogues is fully determined by the participants’ points of view and commitments: Definition 14.5. (types of persuasion systems) • A dialogue system is for pure persuasion iff for any terminated dialogue d it holds that a ∈ wt(d) iff – either a ∈ prop(t) and t ∈ Ca′ (d) for all a′ ∈ prop(d)∪opp(d) – or a ∈ opp(t) and t ∈ Ca′ (d) for all a′ ∈ prop(d)∪opp(d) • Otherwise, it is for conflict resolution. In addition, pure persuasion dialogues are assumed to terminate as soon as the right-hand-side conjuncts of one of these two winning conditions hold. Paul and Olga (ct’d): In our running example, if the dialogue is regulated by a protocol for pure persuasion, it terminates after Paul’s retraction. In conflict resolution dialogues the outcome is not fully determined by the par- ticipant’s points of view and commitments. In other words, in such dialogues it is possible that, for instance, a proponent of ϕ loses the dialogue about ϕ even if at termination he is still committed to ϕ. A typical example is legal procedure, where a third party can determine the outcome of the case. For instance, a crime suspect can be convicted even if he maintains his innocence throughout the case.
- 302. 288 Henry Prakken If the system has an anytime outcome notion, then another distinction can be made [6]: a protocol is immediate-response if the turn shifts just in case the speaker is the ‘current’ winner and if it then shifts to a ‘current’ loser. As for the communication language and effect rules, some common elements can be found throughout the literature. Below are the most common speech acts, with their informal meaning and the various names they have been given in the literature.2 • claim ϕ (assert, statement, ...). The speaker asserts that ϕ is the case. • why ϕ (challenge, deny, question, ...) The speaker challenges that ϕ is the case and asks for reasons why it would be the case. • concede ϕ (accept, admit, ...). The speaker admits that ϕ is the case. • retract ϕ (withdraw, no commitment, ..) The speaker declares that he is not com- mitted (any more) to ϕ. Retractions are ‘really’ retractions if the speaker is com- mitted to the retracted proposition, otherwise it is a mere declaration of non- commitment (for example, in reply to a question). • ϕ since S (argue, argument, ...) The speaker provides reasons why ϕ is the case. Some protocols do not have this move but instead require that reasons be pro- vided by a claim ϕ or claim S move in reply to a why ψ move (where S is a set of propositions). Also, in some systems the reasons provided for ϕ can have structure, for example, of a proof three or a deduction. • question ϕ The speaker asks the hearers’ opinion on whether ϕ is the case. Paul and Olga (ct’d): In this communication language our example from Section 2 can be more formally displayed as follows: P1: claim safe O2: why safe P3: safe since airbag O4: concede airbag O5: claim ¬ safe P6: why ¬ safe O7: ¬ safe since newspaper P8: concede newspaper P9: so what since ¬ newspapers reliable O10: ¬ safe since high max. speed P11: retract safe Most dialogue systems have a notion of typical replies to certain speech acts, although usually this is left implicit in the replies that are allowed by the protocol rules. In most systems these typical replies are as displayed in Table 14.1. Paul and Olga (ct’d): With this table our running example can be displayed as in Figure 14.1, where the boxes stand for moves and the links for reply relations. The reply notion induces another distinction between dialogue protocols. Definition 14.6. A dialogue protocol is unique-reply if at most one reply to a move is allowed throughout a dialogue; otherwise it is multiple-reply. Paul and Olga (ct’d): The protocol governing our running example is multiple- reply, as illustrated by the various branches in Figure 14.1. 2 To make this chapter more uniform, the present terminology will be used even if the original publication of a system uses different terms.
- 303. 14 Models of Persuasion Dialogue 289 Table 14.1 Locutions and typical replies Locutions Replies claim ϕ why ϕ, claim ϕ, concede ϕ why ϕ ϕ since S (alternatively: claim S), retract ϕ concede ϕ retract ϕ ϕ since S why ψ (ψ ∈ S), concede ψ (ψ ∈ S), ϕ′ since S′ question ϕ claim ϕ, claim ϕ, retract ϕ Fig. 14.1 Reply structure of the example dialogue As for the commitment rules, the following ones are generally accepted in the literature. (Below pl denotes the speaker of the move; effects on the other parties’ commitments are only specified when a change is effected.) • If pl(m) = claim(ϕ) then Cpl(d,m) = Cpl(d)∪{ϕ} • If pl(m) = why(ϕ) then Cpl(d,m) = Cpl(d) • If pl(m) = concede(ϕ) then Cpl(d,m) = Cpl(d)∪{ϕ} • If pl(m) = retract(ϕ) then Cpl(d,m) = Cpl(d) −{ϕ} • If pl(m) = ϕ since S then Cpl(d,m) ⊇ Cpl(d)∪prem(A) The rule for since uses ⊇ since such a move may commit to more than just the premises of the moved argument. For instance, in [10] the move also commits to ϕ, since arguments can also be moved as counterarguments instead of as replies to
- 304. 290 Henry Prakken challenges of a claim. And in some systems that allow incomplete arguments, such as [14], the move also commits the speaker to the material implication S → ϕ. Paul and Olga (ct’d): According to these rules, the commitment sets of Paul and Olga at the end of the example dialogue are - CP(d11) ⊇ {airbag, newspaper, ¬ newspapers reliable} - CO(d11) ⊇ {¬ safe, airbag, newspaper, high max. speed} 5 Three systems Now three persuasion protocols from the literature will be discussed. The first is pri- marily based on commitments, the second defines protocols as finite state machines, while the third exploits an explicit reply structure on the communication language. 5.1 Walton and Krabbe (1995) The first system to be discussed is Walton Krabbe’s dialogue system PPD for “per- missive persuasion dialogues” [14]. In PPD, dialogues have no context. The players are called White (W) and Black (B). They are assumed to declare zero or more “as- sertions” and “concessions” in an implicit preparatory phase of a dialogue. Each participant is proponent of his own and opponent of the other participant’s initial assertions. B must have declared at least one assertion, and W starts a dialogue. The communication language consists of challenges, (tree-structured) arguments, con- cessions, questions, resolution demands (“resolve”), and two retraction locutions, one for assertion-type and one for concession-type commitments. It has no explicit reply structure but the protocol reflects the reply structure of Table 14.1 above. The logical language is that of propositional logic and the logic consists of an incomplete set of deductively valid inference rules: they are incomplete to reflect that for natural language no complete logic exists. Although an argument may thus be incomplete, its mover becomes committed to the material implication premises → conclusion, which is then open for discussion. The commitment rules are standard but Walton Krabbe distinguish between several kinds of commitments for each participant, viz. assertions, concessions and dark-side commitments. Initial assertions and premises of arguments are placed in the assertions while conceded propositions are placed in the concessions. Only as- sertions can be challenged. Dark-side commitments are hidden or veiled commit- ments of an agent, of which they are often unaware. This makes them hard to model computationally, for which reason they will be ignored below. The protocol is driven by two main factors: the contents of the commitment sets and the content of the last turn. W starts and in their first turn both W and B either concede or challenge each initial assertion of the other party. Then each turn must
- 305. 14 Models of Persuasion Dialogue 291 reply to all moves in the other player’s last turn except concessions and retractions; in particular, for since moves each premise must be conceded or challenged, includ- ing the hidden premise of incomplete arguments. Multiple replies are allowed, such as alternative arguments for the same assertion. Counterarguments are not allowed. In sum, the PPD protocol is nondeterministic, multi-move and multi-reply but post- ponement of replies is not allowed. Dark-side commitments prevent the protocol from having a public semantics. Most protocol rules refer to the participants’ commitments. To start with, chal- lenges, concessions and retractions always concern commitments. Second, a speaker cannot challenge or concede his own commitments, and question ϕ and ϕ since S may not be used if the listener is committed to ϕ. Furthermore, if a participant has inconsistent commitments, the other participant can demand resolution of the inconsistency by using the resolve speech act. Also, if a participant’s commitments logically imply an assertion of the other participant but do not contain that assertion, then the initial participant must either concede the assertion or retract one of the im- plying commitments. Retractions must be successful in that the retracted proposition is not still implied by the speaker’s commitments. Finally, the commitments deter- mine the outcome of a dialogue: dialogues terminate after a predetermined number of turns, and the outcome of terminated dialogues is defined as for pure persuasion. Table 14.2 contains an example dialogue.3 The first column numbers the turns, and the second contains the moves made in each turn. The other columns contain the assertions and concessions of W and B: the first row contains the initial commit- ments and the other rows indicate changes in these sets: +ϕ means that ϕ is added and −ϕ that it is deleted. If the dialogue terminates here, there is no winner, since neither player has conceded any of the other player’s assertions or retracted any of his own. Several points are worth noting about this example. Firstly, B in his first turn moves a complex argument, where the second argument supports a premise of the first: for this reason i is not added to B’s assertions. Next, in his second turn, W first concedes j and then asserts j as a premise of an argument; only after the second move has W incurred a burden to defend j if challenged. However, B in his second turn cannot challenge j since B is itself committed to j: if B wants to challenge j, he must first retract j. Note further that after B concedes f ∧ j → a in his second turn, his commitments logically imply a, which is an assertion of W. Therefore B must in the same turn either concede a or retract one of the implying commitments. B opts for the latter, retracting f. Next consider B’s second move of his second turn: remarkably, B becomes committed to a tautology but W still has the right to challenge it at his third turn. Finally, the example illustrates that the protocol only partly enforces relevance of moves. For instance, at any point a participant could have moved question ϕ for any ϕ not in the commitments of the listener. Paul and Olga (ct’d): Let us finally reconstruct our running example in PPD. To start with, Paul’s initial claim must now be modelled as an initial assertion in the 3 In this section the dialogue participants will be denoted with W and B, except if they have propo- nent/opponent roles throughout the dialogues, in which case they are called P and O.
- 306. 292 Henry Prakken Table 14.2 An example PPD dialogue Turn Moves AW CW AB CB {a} {b,c} {d,e} {f,g} W1 why d concede e +e B1 why a d since h,i, i since j,k +h,h∧i → d, j,k, j ∧k → i W2 concede j + j concede k +k why j ∧k → i concede h∧i → d +h∧i → d why h a since f, j + f, j, f ∧ j → a B2 h since l,l → h +l,l → k, l ∧(l → k) → k j ∧k → i since m +m,m → (j ∧k → i) concede f ∧ j → a + f ∧ j → a retractC f − f preparatory phase. Since arguments can be incomplete, they can be modelled as in the example’s original version. Two features of PPD make a straightforward mod- elling of the example impossible. The first is that PPD requires that every claim or argument is replied to in the next turn and the second is that explicit counterargu- ments are not allowed. To deal with the latter, it must be assumed that Olga has also declared an initial assertion, viz. that Paul’s car is not safe. (P0: claim safe O0: claim ¬ safe) O1: why safe P2: safe since airbag P3: why ¬ safe O4: concede airbag O5: ¬ safe since newspaper Here a problem arises, since Olga now has to either concede or challenge Paul’s hidden premise airbag → safe. If Olga concedes it, she is forced to also concede Paul’s initial claim, since it is now implied by Olga’s commitments. If, on the other hand, Olga challenges the hidden premise, then at his next turn Paul must provide an argument for it, which he does not do in our original example. Similar problems arise with the rest of the example. Let us now, to proceed with the example, ignore this ‘completeness’ requirement of turns. P6: concede newspaper Here another problem arises, since PPD does not allow Paul to move his undercut- ting counterargument against O5. The only way to attack O5 is by challenging its unstated premise (newspaper → ¬ safe).
- 307. 14 Models of Persuasion Dialogue 293 In sum, two features of PPD prevent a fully natural modelling of our example: the monotonic nature of the underlying logic and the requirement to reply to each claim or argument of the other participant. 5.2 Parsons, Wooldridge Amgoud (2003) In a series of papers Parsons, Wooldridge Amgoud have developed an approach to specifying dialogue systems for various types of dialogues. Here the persuasion system of [8] will be discussed. The system is for dialogues on a single topic between two players called White (W) and Black (B). Dialogues have no context but each participant has a, possibly inconsistent, belief base Σ. The communication language consists of claims, chal- lenges, and concessions; it has no explicit reply structure but the protocol largely conforms to Table 14.1. Claims can concern both individual propositions and sets of propositions. The logical language is propositional. Its logic is an argument- based nonmonotonic logic in which arguments are classical proofs from consistent premises and counterarguments negate a premise of their target. Conflict relations between arguments are resolved with a preference relation on the premises such that arguments are as good as their least preferred premises. Argument acceptability is defined with grounded semantics. In dialogues, arguments cannot be moved as such but only implicitly as claim S replies to challenges of another claim ϕ, such that S is consistent and S ⊢ ϕ. Finally, the commitment rules are standard and commitments are only used to enlarge the player’s belief base with the other player’s commit- ments; they do not constrain move legality nor determine the dialogue’s outcome. An important feature of the system is that the players are assumed to adopt an assertion and an acceptance attitude, which they must respect throughout the dia- logue. The attitudes are defined relative to their internal belief base (which remains constant throughout a dialogue) plus both players’ commitment sets (which may vary during a dialogue). The following assertion attitudes are distinguished: a con- fident agent can assert any proposition for which he can construct an argument, a careful agent can do so only if he can construct such an argument and cannot con- struct a stronger counterargument, and a thoughtful agent can do so only if he can construct an acceptable argument for the proposition. The corresponding acceptance attitudes also exist: a credulous agent accepts a proposition if he can construct an argument for it, a cautious agent does so only if in addition he cannot construct a stronger counterargument and a skeptical agent does so only if he can construct an acceptable argument for the proposition. It can be debated whether such the requirement to respect these attitudes must be part of a protocol or of a participant’s heuristics. According to one approach, a dialogue protocol should only enforce coherence of dialogues [14, 10]; according to another approach, it should also enforce rationality and trustworthiness of the agents engaged in a dialogue [8]. The second approach allows protocol rules to refer to an agent’s internal belief base and therefore such protocols do not have a
- 308. 294 Henry Prakken public semantics. The first approach does not allow such protocol rules and instead regards assertion and acceptance attitudes as an aspect of agent design. The formal definition of the persuasion protocol is as follows. Definition 14.7. (PWA persuasion protocol) A move is legal iff it does not repeat a move of the same player, and satisfies the following procedure: 1. W claims ϕ. 2. B concedes ϕ if its acceptance attitude allows, if not B asserts ¬ϕ if its assertion attitude allows it, or otherwise challenges ϕ. 3. If B claims ¬ϕ, then goto 2 with the roles of the players reversed and ¬ϕ in place of ϕ. 4. If B has challenged, then: a. W claims S, an argument for ϕ; b. Goto 2 for each s ∈ S in turn. 5. B concedes ϕ if its acceptance attitude allows, or the dialogue terminates. Dialogues terminate as specified in condition 5, or when the move required by the procedure cannot be made, or when the player-to-move has conceded all claims made by the hearer. No win and loss functions are defined, but the possible outcomes are defined in terms of the propositions claimed by one player and conceded by the other. This protocol is unique-move except that if one element of a claim S move is conceded, another element may be replied-to next. Also, it is unique-reply except that each element of a claim S move can be separately challenged or conceded. The protocol is deterministic in Lc but not fully deterministic, since if a player can construct more than one argument for a challenged claim, he has a choice. Let us first consider some simple dialogues that fit this protocol. Example 14.1. First, let ΣW = {p} and ΣB = ∅. Then the only legal dialogue is: W1: claim p, B1: concede p B1 is B’s only legal move, whatever its acceptance attitude, since after W1, B must reason from ΣB ∪CW (d1) = {p} so that B can construct the trivial argument ({p}, p). Here the dialogue terminates. This example illustrates that since the players must reason with the commitments of the other player, they can learn from each other. However, the next example illus- trates that the same feature sometimes makes them learn too easily. Example 14.2. Assume ΣW = {q,q → p} and ΣB = {¬q}, where all formulas are of the same preference level. W1: claim p Now whatever her acceptance attitude, B has to concede p since she can construct the trivial argument ({p}, p) for p while she can construct no argument for ¬p. Yet B has an attacker for W’s only argument for p, namely, ({¬q},¬q), which attacks ({q,q → p}, p) and is not weaker than its target. So even though p is not acceptable on the basis of the agents’ joint knowledge, W1 can win a dialogue about p.
- 309. 14 Models of Persuasion Dialogue 295 This example thus illustrates that if the players must reason with the other player’s commitments, one player can sometimes ‘force’ an opinion onto the other player by simply making a claim. A simple solution to this problem is to restrict the in- formation with which agent reason to their own beliefs and commitments. A more refined option is to assume that the agents have knowledge about the reliability of information sources and to let them use it in the acceptance policies. Paul and Olga (ct’d): Finally, our running example can be modelled in this ap- proach as follows. Let us give Paul and Olga the following beliefs: ΣW = {airbag, airbag → safe, ¬(newspaper → ¬ safe)} ΣB = {newspaper, newspaper → ¬ safe} (Note that Paul’s undercutter must now be formalised as the negation of Olga’s material implication.) Assume that all these propositions are equally preferred. We must also make some assumptions on the players’ assertion and acceptance atti- tudes. Let us first assume that Paul is thoughtful and skeptical while Olga is careful and cautious, and that they only reason with their own beliefs and commitments. P1: claim safe O2: claim ¬ safe Olga could not challenge Paul’s main claim as in the example’s original version, since she can construct an argument for ‘¬safe’, while she cannot construct an ar- gument for ‘safe’. So she had to make a counterclaim. Now since players may not repeat moves, Paul cannot make the remove required by the protocol and his asser- tion attitude, namely, claiming ‘safe’, so the dialogue terminates without agreement. Let us now assume that the players must also reason with each others commit- ments. Then the dialogue evolves as follows: P1: claim safe O2: concede safe Olga has to concede, since she can use Paul’s commitment to construct the trivial argument ({safe}, safe), while her own argument for ‘¬ safe’ is not stronger. So here the dialogue terminates with agreement on ‘safe’, even though this proposition is not acceptable on the basis of the players’ joint beliefs. So far, neither of the players could develop their arguments. To change this, as- sume now that Olga is also thoughtful and skeptical, and that the players reason with each others commitments. Then: P1: claim safe O2: why safe Olga could not concede, nor could she state her argument for ¬ safe since it is not preferred over its attacker ({safe},safe). So she had to challenge. P3: claim {airbag, airbag → safe} Now Olga can create a (trivial) argument for ‘airbag’ by using Paul’s commitments, but she can also create an argument for its negation by using her own beliefs. Neither is acceptable, so she must challenge. Likewise for the second premise, so: O4: why airbag P5: claim {airbag} O6: why airbag → safe P7: claim {airbag → safe}
- 310. 296 Henry Prakken Here the nonrepetition rule makes the dialogue terminate without agreement. Note that only Paul could develop his arguments. To give Olga a chance to develop her arguments, let us make her careful and skeptical while the players still reason with each others commitments. Then: P1: claim safe O2: claim ¬ safe In the new dialogue state Paul’s argument for ‘safe’ is not acceptable any more, since it is not preferred over its attacker ({¬ safe}, ¬ safe). So he must challenge. P3: why ¬ safe O4: claim {newspaper, newspaper → ¬ safe } Although Paul can construct an argument for Olga’s first premise, namely, ({¬(newspaper → ¬ safe’}, safe), it is not acceptable since it is not preferred over its attacker based on Olga’s second premise. So he must challenge. P5: why newspaper O6: claim {newspaper} Olga had to reply with a (trivial) argument for her first premise, after which Paul cannot repeat his challenge, so he has to go to the second premise of O4. Based on his beliefs and Olga’s commitments he can construct (trivial) arguments both for and against it and neither of these is acceptable. So he must again challenge. P7: why newspaper → ¬ safe O8: claim {newspaper → ¬ safe} Here the nonrepetition rule again makes the dialogue terminate without agreement. In this dialogue only Olga could develop her arguments (although she could not state her second counterargument). In conclusion, the PWA persuasion protocol leaves little room for choice and ex- ploring alternatives, and induces one-sided dialogues in that at most one side can develop their arguments for a certain issue. Also, the examples suggest that if a claim is accepted, it is accepted in the first ‘round’ of moves (but this should be formally verified). On the other hand, the strictness of the protocol induces short di- alogues which are guaranteed to terminate, which promotes efficiency. Also, thanks to the strong assumptions on the logic and the participants’ beliefs and reasoning behaviour, PWA have been able to prove several interesting properties of their pro- tocols. Finally, without the requirement to respect the assertion and acceptance atti- tudes the protocol would be much more liberal while still enforcing some coherence. 5.3 Prakken (2005) In [10] I proposed a formal framework for systems for two-party persuasion dia- logues and instantiated it with some example protocols. The participants have pro- ponent and opponent role, and their beliefs are irrelevant to the protocols, so that these have a public semantics. Dialogues have no context. The framework abstracts from the communication language except for an explicit reply structure. It also ab- stracts from the logical language and the logic, except that the logic is assumed to
- 311. 14 Models of Persuasion Dialogue 297 be argument-based and to conform to grounded semantics and that arguments are trees of deductive and/or defeasible inferences, as in e.g. [9]. A main motivation of the framework is to ensure focus of dialogues while yet allowing for freedom to move alternative replies and to postpone replies. This is achieved with two main features of the framework. Firstly, Lc has an explicit reply structure, where each move either attacks or surrenders to its target. An example Lc of this format is displayed in Table 14.3. Secondly, winning is defined for each dia- Table 14.3 An example Lc in Prakken’s framework Acts Attacks Surrenders claim ϕ why ϕ concede ϕ ϕ since S why ψ(ψ ∈ S) concede ψ (ψ ∈ S) ϕ′ since S′ concede ϕ (ϕ′ since S′ defeats ϕ since S) why ϕ ϕ since S retract ϕ concede ϕ retract ϕ logue, whether terminated or not, and it is defined in terms of a notion of dialogical status of moves. The dialogical status of a move is recursively defined as follows, exploiting the tree structure of dialogues generated by the reply structure on Lc. A move is in if it is surrendered or else if all its attacking replies are out. (This implies that a move without replies is in). And a move is out if it has a reply that is in. Then a dialogue is (currently) won by the proponent if its initial move is in while it is (currently) won by the opponent otherwise. Together, these two features of the framework support a notion of relevance that ensures focus while yet leaving a degree of freedom: a move is relevant just in case making its target out would make the speaker the current winner. Termination is defined as the situation that a player is to move but has no legal moves. Various results are proven about the relation between being the current winner of a dialogue and what is defeasibly implied by the arguments exchanged during the dialogue. As for dialogue structure, the framework allows for all kinds of protocols. The instantiations of [10] are all multi-move and multi-reply; one of them has the com- munication language of Table 14.3 and is constrained by the requirement that each move be relevant. This makes the protocol immediate-response, which implies that each turn consists of zero or more surrenders followed by one attacker. Within these limits postponement of replies is allowed, sometimes even indefinitely. Let us next discuss some examples, assuming that the protocol is further in- stantiated with Prakken Sartor’s argument-based version of prioritised extended logic programming [12]. This logic uses grounded semantics and supports argu- ments about rule priorities. (The examples below should speak for themselves so no formal definitions about the logic will be given. Note that since the rules are logic-programming rules, they do not satisfy contraposition or modus tollens. Rule
- 312. 298 Henry Prakken connectives are tagged with a rule name, which is needed to express rule priorities in the object language). Consider two agents with the following belief bases: ΣP = {p, p ⇒r1 q, q ⇒r2 r, p∧s ⇒r3 r2 r4} ΣO = {t,t ⇒r4 ¬r}. Then the following is legal in [10]’s so-called relevant protocol (with each move its target is indicated between square brackets): P1[−]: claim r O2[P1]: why r P3[O2]: r since q,q ⇒ r O4[P3]: why q P5[O4]: q since p, p ⇒ q O6[P5]: concede p ⇒ q O7[P5]: why p (Note that unlike in [8] but like in [14], arguments can be stepwise built in sev- eral moves.) Here P has several allowed moves, viz. retracting any of his argument premises or his claim, or giving an argument for p. All these moves are relevant but if P makes any retraction then an argument for p ceases to be relevant, since it cannot make P the current winner. Moreover, if P retracts r as a reply to P1 then the dialogue terminates with a win for O. O could at all points after P3 have moved her argument against r. For instance: O7[P3]: ¬r since t,t ⇒ ¬r P8[O7]: r2 r4 since p,s, p∧s ⇒ r1 r4 P8 is a priority argument which makes P3 strictly defeat O7 (note that the fact that s is not in P’s own belief base does not make the move illegal). At this point, P1 is in; O has various allowed moves, viz. challenging or conceding any (further) premise of P’s arguments, moving a counterargument to P5 or a second counterargument to P3, and conceding P’s initial claim. This example shows that the participants have much more freedom in this system than in the one of [8], since they are not bound by assertion and acceptance attitudes and the protocol is structurally less strict. The downside of this is that dialogues can be much longer, that the participants can lie and that they can prevent losing by simply continuing to attack the other participant. Another drawback of the present approach is that not all natural-language dia- logues have an explicit reply structure. For example, often one player tries to extract seemingly irrelevant concessions from the other player with the aim to lure her into a contradiction, as in as in the following witness cross-examination dialogue: Witness: Suspect was at home with me that day. Prosecutor: Are you a student? Witness: Yes. Prosecutor: Was that day during summer holiday? Witness: Yes. Prosecutor: Aren’t all students away during summer holiday? In [14] such dialogues can be modelled with the question locution but at the price of decreased coherence and focus.
- 313. 14 Models of Persuasion Dialogue 299 Paul and Olga (ct’d): Let us finally model our running example in this protocol. Figure 14.2 displays the dialogue tree, where moves within solid boxes are in and moves within dotted boxes are out. Fig. 14.2 The example dialogue in Prakken’s approach As can be easily checked, this formalisation captures all aspects of the exam- ple’s original version, except that arguments have to be complete and that counter- arguments cannot be introduced by a counterclaim. (But other instantiations of the framework may be possible without these limitations.) 6 Conclusion In this chapter a formal framework for dialogue systems for persuasion was pro- posed, which was then used to critically discuss three systems from the literature. Concluding, we can say that the formal study of persuasion dialogue has resulted in a number of interesting dialogue systems, some of which have been applied in insightful case studies or applications. On the other hand, there is still much room for refining or extending the various sytems with, for example, more refined com- munication languages or with different modes of reasoning, such as probabilistic, case-based or coherence-based reasoning. Also, the integration of persuasion with other types of dialogues should be studied. Another important research issue is the study of strategies and heuristics for individual participants and how these interact with the protocols to yield certain properties of dialogues. One aspect of such stud- ies is the development of quality measures for dialogues as to how well they satisfy certain desirable properties. More generally, a formal metatheory of systems, their interrelations and their combinations with agent models is still in its early stages.
- 314. 300 Henry Prakken Perhaps the main challenge in tackling all these issues is how to reconcile the need for flexibility and expressiveness with the aim to enforce coherent dialogues. The answer to this challenge may well vary with the nature of the context and ap- plication domain, and a precise description of the grounds for such variations would provide important insights in how dialogue systems for persuasion can be applied. References 1. J. Barwise and L. Moss. Vicious Circles. Number 60 in CSLI Lecture Notes. CSLI Publica- tions, Stanford, CA, 1996. 2. G. Brewka. Dynamic argument systems: a formal model of argumentation processes based on situation calculus. Journal of Logic and Computation, 11:257–282, 2001. 3. L. Carlson. Dialogue Games: an Approach to Discourse Analysis. Reidel Publishing Com- pany, Dordrecht, 1983. 4. T. Gordon. The Pleadings Game: an exercise in computational dialectics. Artificial Intelli- gence and Law, 2:239–292, 1994. 5. C. Hamblin. Fallacies. Methuen, London, 1970. 6. R. Loui. Process and policy: resource-bounded non-demonstrative reasoning. Computational Intelligence, 14:1–38, 1998. 7. J. Mackenzie. Question-begging in non-cumulative systems. Journal of Philosophical Logic, 8:117–133, 1979. 8. S. Parsons, M. Wooldridge, and L. Amgoud. Properties and complexity of some formal inter- agent dialogues. Journal of Logic and Computation, 13, 2003. 347-376. 9. J. Pollock. Cognitive Carpentry. A Blueprint for How to Build a Person. MIT Press, Cam- bridge, MA, 1995. 10. H. Prakken. Coherence and flexibility in dialogue games for argumentation. Journal of Logic and Computation, 15:1009–1040, 2005. 11. H. Prakken. Formal systems for persuasion dialogue. The Knowledge Engineering Review, 21:163–188, 2006. 12. H. Prakken and G. Sartor. Argument-based extended logic programming with defeasible pri- orities. Journal of Applied Non-classical Logics, 7:25–75, 1997. 13. D. Walton. Logical dialogue-games and fallacies. University Press of America, Inc., Lanham, MD., 1984. 14. D. Walton and E. Krabbe. Commitment in Dialogue. Basic Concepts of Interpersonal Rea- soning. State University of New York Press, Albany, NY, 1995. 15. J. Woods and D. Walton. Arresting circles in formal dialogues. Journal of Philosophical Logic, 7:73–90, 1978. 16. T. Yuan, D. Moore, and A. Grierson. A human-computer dialogue system for educational debate: A computational dialectics approach. International Journal of Artificial Intelligence in Education, 18:3–26, 2008.
- 315. Chapter 15 Argumentation for Decision Making Leila Amgoud 1 Introduction Decision making, often viewed as a form of reasoning toward action, has raised the interest of many scholars including economists, psychologists, and computer scien- tists for a long time. Any decision problem amounts to selecting the “best” or suf- ficiently “good” action(s) that are feasible among different alternatives, given some available information about the current state of the world and the consequences of potential actions. Available information may be incomplete or pervaded with un- certainty. Besides, the goodness of an action is judged by estimating how much its possible consequences fit the preferences of the decision maker. This agent is assumed to behave in a rational way [29], at least in the sense that his decisions should be as much as possible consistent with his preferences. Classical decision theory, as developed mainly by economists, has focused on mak- ing clear what is a rational decision maker. Thus, they have looked for principles for comparing different alternatives. The inputs of this approach are a set of can- didate actions, and a function that assesses the value of their consequences when the actions are performed in a given state, together with complete or partial infor- mation about the current state of the world. In other words, such an approach dis- tinguishes between knowledge and preferences, which are respectively encoded in practice by a distribution function assessing the plausibility of the different states of the world, and by a utility function encoding preferences by estimating how good a consequence is. The output is a preference relation between actions encoded by the associated principle. Note that such an approach aims at rank-ordering a group of candidate actions rather than focusing on a candidate action individually. Moreover, the candidate actions are supposed to be feasible. What is worth noticing is that in such an approach, the principles that are defined for comparing pairs of alterna- tives are given in terms of analytical expressions that summarize the whole decision Institut de Recherche en Informatique de Toulouse, IRIT-UPS 118 route de Narbonne, 31062 Toulouse, Cedex, France, e-mail: amgoud@irit.fr I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 301 DOI 10.1007/978-0-387-98197-0 15, c Springer Science+Business Media, LLC 2009
- 316. 302 L. Amgoud process. It is then hard for a person who is not familiar with the abstract decision methodology, to understand why a proposed alternative is good, or better than an- other. It is thus important to have an approach in which one can better understand the underpinnings of the evaluation. Argumentation is the most appropriate way to advocate a choice thanks to its explanatory power. Argumentation has been introduced in decision making analysis by several re- searchers only in the last few years (e.g. [15, 20, 23]). Indeed, in everyday life, deci- sion is often based on arguments and counter-arguments. Argumentation can be also useful for explaining a choice already made. Recently, in [1], a decision model in which some decision criteria were articulated in terms of a two-steps argumentation process has been proposed. At the first step, called inference step, the model uses a Dung style system in which arguments in favor/against each option are built, then evaluated using a given acceptability semantics. At the second step, called compar- ison step, pairs of alternatives are compared using a given criterion. This criterion is generally based on the “accepted” arguments computed at the inference step. The model returns thus, an ordering on the set of options, which may be either partial or total depending on the decision criterion that is encoded. This approach presents a great advantage since not only the best alternative is provided to the user but also the reasons justifying this recommendation. In what follows, we will develop that argument-based model for decision making. 2 A general framework for argumentative decision making Solving a decision problem amounts to defining a pre-ordering, usually a complete one, on a set D of possible options (or candidate decisions), on the basis of the different consequences of each decision. Let us illustrate this problem through a simple example borrowed from [20]. Example 15.1 (Having or not a surgery). The example is about having a surgery (sg) or not (¬sg), knowing that the patient has colonic polyps. The knowledge base contains the following information: • having a surgery has side-effects, • not having surgery avoids having side-effects, • when having a cancer, having a surgery avoids loss of life, • if a patient has cancer and has no surgery, the patient would lose his life, • the patient has colonic polyps, • having colonic polyps may lead to cancer. In addition to the above knowledge, the patient has also some goals like: “no side effects” and “to not lose his life”. Obviously it is more important for him to not lose his life than to not have side effects. Let L denote a logical language. From L, a finite set D = {d1,...,dn} of n op- tions is identified. An option di may be a conjunction of other options in D. Let us,
- 317. 15 Argumentation for Decision Making 303 for instance, assume that an agent wants a drink and has to choose between tea, milk or both. Thus, there are three options: d1 : tea, d2 : milk and d3 : tea and milk. In Example 15.1, the set D contains two options: d1 : sg and d2 : ¬sg. Argumentation is used in this chapter for ordering the set D. An argumentation- based decision process can be decomposed into the following steps: 1. Constructing arguments in favor/against statements (beliefs or decisions) 2. Evaluating the strength of each argument 3. Determining the different conflicts among arguments 4. Evaluating the acceptability of arguments 5. Comparing decisions on the basis of relevant “accepted” arguments Note that the first four steps globally correspond to an “inference problem” in which one looks for accepted arguments, and consequently warranted beliefs. At this step, one only knows what is the quality of arguments in favor/against candidate deci- sions, but the “best” candidate decision is not determined yet. The last step answers this question once a decision principle is chosen. 2.1 Types of arguments As shown in Example 15.1, decisions are made on the basis of available knowledge and the preferences of the decision maker. Thus, two categories of arguments are distinguished: i) epistemic arguments justifying beliefs and are themselves based only on beliefs, and ii) practical arguments justifying options and are built from both beliefs and preferences/goals. Note that a practical argument may highlight either a positive feature of a candidate decision, supporting thus that decision, or a negative one, attacking thus the decision. Example 15.2 (Example 15.1 cont.). In this example, α = [“the patient has colonic polyps”, and “having colonic polyps may lead to cancer”] is considered as an ar- gument for believing that the patient may have cancer. This epistemic argument involves only beliefs. While δ1 = [“the patient may have a cancer”, “when having a cancer, having a surgery avoids loss of life”] is an argument for having a surgery. This is a practical argument since it supports the option “having a surgery”. Note that such argument involves both beliefs and preferences. Similarly, δ2 = [“not hav- ing surgery avoids having side-effects”] is a practical argument in favor of “not having a surgery”. However, the two practical arguments δ3 = [“having a surgery has side-effects”] and δ4 = [“the patient has colonic polyps”, and “having colonic polyps may lead to cancer”, “if a patient has cancer and has no surgery, the patient would lose his life”] are respectively against surgery and no surgery since they point out negative consequences of the two options. In what follows, Ae denotes a set of epistemic arguments, and Ap denotes a set of practical arguments such that Ae ∩ Ap = ∅. Let A = Ae ∪ Ap (i.e. A will
- 318. 304 L. Amgoud contain all those arguments). The structure and origin of the arguments are assumed to be unknown. Epistemic arguments will be denoted by variables α1,α2,..., while practical arguments will be referred to by variables δ1,δ2,... When no distinction is necessary between arguments, we will use the variables a,b,c,... Example 15.3 (Example 15.1 cont.). Ae = {α} while Ap = {δ1,δ2,δ3,δ4}. Let us now define two functions that relate each option to the arguments support- ing it and to the arguments against it. • Fp : D → 2Ap is a function that returns the arguments in favor of a candidate decision. Such arguments are said pro the option. • Fc : D → 2Ap is a function that returns the arguments against a candidate deci- sion. Such arguments are said cons the option. The two functions satisfy the following requirements: • ∀d ∈ D, ∄δ ∈ Ap s.t. δ ∈ Fp(d) and δ ∈ Fc(d). This means that an argument is either in favor of an option or against that option. It cannot be both. • If δ ∈ Fp(d) and δ ∈ Fp(d′) (resp. if δ ∈ Fc(d) and δ ∈ Fc(d′)), then d = d′. This means that an argument refers only to one option. • Let D = {d1,...,dn}. Ap = ( Fp(di)) ∪ ( Fc(di)), with i = 1,...,n. This means that the available practical arguments concern options of the set D. When δ ∈ Fx(d) with x ∈ {p,c}, we say that d is the conclusion of δ, and we write Conc(δ) = d. Example 15.4 (Example 15.1 cont.). The two options of the set D = {sg,¬sg} are supported/attacked by the following arguments: Fp(sg) = {δ1}, Fc(sg) = {δ3}, Fp(¬sg) = {δ2}, and Fc(¬sg) = {δ4}. 2.2 Comparing arguments As pointed out by several researchers (e.g. [14, 18]), arguments may have forces of various strengths. These forces play two key roles: i) they may be used in order to refine the notion of acceptability of epistemic or practical arguments, ii) they allow the comparison of practical arguments in order to rank-order candidate decisions. Generally, the strength of an epistemic argument reflects the quality, such as the certainty level, of the pieces of information involved in it. Whereas the strength of a practical argument reflects both the quality of knowledge used in the argument, as well as how important it is to fulfill the preferences to which the argument refers. In our particular application, three preference relations between arguments are defined. The first one, denoted by ≥e, is a (partial or total) preorder1 on the set Ae. The second relation, denoted by ≥p, is a (partial or total) preorder on the set Ap. 1 A preorder is a binary relation that is reflexive and transitive
- 319. 15 Argumentation for Decision Making 305 Finally, a third relation, denoted by ≥m (m stands for mixed relation), captures the idea that any epistemic argument is stronger than any practical argument. The role of epistemic arguments in a decision problem is to validate or to undermine the beliefs on which practical arguments are built. Indeed, decisions should be made under “certain” information. Thus, ∀α ∈ Ae, ∀δ ∈ Ap, (α,δ) ∈≥m and (δ,α) / ∈≥m. Note that (a,b) ∈≥x, with x ∈ {e, p,m}, means that a is at least as good as b. At some places, we will also write a ≥x b. In what follows, x denotes the strict relation associated with ≥x. It is defined as follows: (a,b) ∈x iff (a,b) ∈≥x and (b,a) / ∈≥x. When (a,b) ∈≥x and (b,a) ∈≥x, we say that a and b are indifferent, and we write a ≈x b. When (a,b) / ∈≥x and (b,a) / ∈≥x, the two arguments are said incomparable. Example 15.5 (Example 15.1 cont.). ≥e = {(α,α)} and ≥m = {(α,δ1),(α,δ2)}. Now, regarding ≥p, one may, for instance, assume that δ1 is stronger than δ2 since the goal satisfied by δ1 (namely, not loss of life) is more important than the one satisfied by δ2 (not having side effects). Thus, ≥p = {(δ1,δ1), (δ2,δ2), (δ1,δ2)}. This example will be detailed in a next section. 2.3 Attacks among arguments Since knowledge may be inconsistent, the arguments may be conflicting too. In- deed, epistemic arguments may attack each others. Such conflicts are captured by the binary relation Re ⊆ Ae ×Ae. This relation is assumed abstract and its origin is not specified. Epistemic arguments may also attack practical arguments when they challenge their knowledge part. The idea is that an epistemic argument may un- dermine the beliefs part of a practical argument. However, practical arguments are not allowed to attack epistemic ones. This avoids wishful thinking. This relation, denoted by Rm, contains pairs (α,δ) where α ∈ Ae and δ ∈ Ap. We assume that practical arguments do not conflict. The idea is that each practical argument points out some advantage or some weakness of a candidate decision, and it is crucial in a decision problem to list all those arguments for each candidate deci- sion, provided that they are accepted w.r.t. the current epistemic state, i.e built from warranted beliefs. According to the attitude of the decision maker in face of uncer- tain or inconsistent knowledge, these lists associated with the candidate decisions may be taken into account in different manners, thus leading to different orderings of the decisions. This is why all accepted arguments should be kept, whatever their strengths, for preserving all relevant information in the decision process. Otherwise, getting rid of some of those accepted arguments (w.r.t. knowledge), for instance be- cause they would be weaker than others, may prevent us to have a complete view of the decision problem and then may even lead us to recommend decisions that would be wrong w.r.t. some decision principles (agreeing with the presumed de- cision maker’s attitude). This point will be made more concrete in a next section. Thus, the relation Rp ⊆ Ap ×Ap is equal to the empty set (Rp = ∅).
- 320. 306 L. Amgoud Each preference relation ≥x (with x ∈ {e, p,m}) is combined with the conflict relation Rx into a unique relation between arguments, denoted by Defx and called defeat relation, in the same way as in ([4], Definition 3.3, page 204). Definition 15.1 (Defeat relation). Let A be a set of arguments, and a, b ∈ A. (a,b) ∈ Defx iff (a,b) ∈ Rx, and (b,a) / ∈x. Let Defe, Defp and Defm denote the three defeat relations corresponding to the three attack relations. In case of Defm, the second bullet of Definition 15.1 is always true since epistemic arguments are strictly preferred (in the sense of ≥m) to any practical arguments. Thus, Defm = Rm (i.e. the defeat relation is exactly the attack relation Rm). The relation Defp is the same as Rp, thus it is empty. However, the relation Defe coincides with its corresponding attack relation Re in case all the arguments of the set Ae are incomparable. 2.4 Extensions of arguments Now that the sets of arguments and the defeat relations are identified, we can define the decision system. Definition 15.2 (Decision system). Let D be a set of options. A decision system for ordering D is a triple AF = (D,A,Def) where A = Ae ∪Ap 2 and Def = Defe ∪ Defp ∪Defm 3. Note that a Dung style argumentation system is associated to a decision system AF = (D,A,Def), namely the system (A,Def). This latter can be seen as the union of two distinct argumentation systems: AFe = (Ae,Defe), called epistemic system, and AFp = (Ap,Defp), called practical system. The two systems are related to each other by the defeat relation Defm. Due to Dung’s acceptability semantics defined in [17] or their extensions defined in [8], it is possible to identify among all the conflicting arguments, which ones will be kept for ordering the options. Recall that an acceptability semantics amounts to define sets of arguments that satisfy a consistency requirement and must defend all their elements. In what follows, E1,...,Ex denote the different extensions of the system (A,Def) under a given semantics. Using these extensions, a status is assigned to each argument of AF as follows. Definition 15.3 (Argument status). Let AF = (D,A,Def) be a decision system, and E1,...,Ex its extensions under a given semantics. Let a ∈ A. • a is skeptically accepted iff a ∈ Ei, ∀Ei=1,...,x, Ei = ∅. • a is credulously accepted iff ∃Ei such that a ∈ Ei and ∃Ej such that a / ∈ Ej. 2 Recall that options are related to their supporting and attacking arguments by the functions Fp and Fc respectively. 3 Since the relation Defp is empty, then Def = Defe ∪Defm.
- 321. 15 Argumentation for Decision Making 307 • a is rejected iff ∄Ei such that a ∈ Ei. A consequence of Definition 15.3 is the following one. Property 15.1. Each argument has exactly one status. Let Acc(x,y) be a function that returns the skeptically accepted arguments of de- cision system x under semantics y (y ∈ {ad,st, pr} with ad (resp. st and pr) stands for admissible (resp. stable and preferred) semantics). This set may contain both epistemic and practical arguments. Such arguments are very important in argumen- tation process since they support the conclusions to be inferred from a knowledge base or the options that will be chosen. Indeed, for ordering the candidate decisions, only skeptically accepted practical arguments are used. The following result shows the links between the sets of accepted arguments under different semantics. Property 15.2. Let AF = (D,A,Def) be a decision system. • Acc(AF,ad) = ∅. • If AF has no stable extensions, then Acc(AF,st)=∅ and Acc(AF,st)⊆Acc(AF, pr). • If AF has stable extensions, then Acc(AF, pr) ⊆ Acc(AF,st). From the above property, one concludes that in a decision problem, it is not interesting to use admissible semantics. The reason is that no argument is accepted. Consequently, argumentation will not help at all for ordering the different candidate decisions. Let us illustrate this issue through the following simple example. Example 15.6. Let us consider the decision system AF = (D,Ae ∪ Ap,Def) where D = {d1,d2}, Ae = {α1,α2,α3}, Ap = {δ} and Def = {(α1,α2), (α2,α1), (α1,α3), (α2,α3), (α3,δ)}. We assume that Fp(d1) = δ whereas Fp(d2) = Fc(d2) = ∅. The admissible extensions of this system are: E1 = {}, E2 = {α1}, E3 = {α2}, E4 = {α1,δ} and E5 = {α2,δ}. Under admissible semantics, the practical argument δ is not skeptically accepted. Thus, the two options d1 and d2 may be equally preferred since the first one has an argument but not an accepted one, and the second has no argument at all. However, the same decision system has two preferred extensions: E4 and E5. Under preferred semantics, the set Acc(AF, pr) contains the argument δ (i.e. Acc(AF, pr) = {δ}). Thus, it is natural to prefer option d1 to d2. Consequently, in the following, we will use stable semantics if the system has stable extensions, otherwise preferred semantics will be considered for computing the set Acc(AF,y). Since the defeat relation Defp is empty, it is trivial that the practical system AFp has exactly one preferred/stable extension which is the set Ap itself. Property 15.3. The practical system AFp = (Ap,Defp) has a unique preferred/stable extension, which is the set Ap. It is important to notice that the epistemic system AFe in its side is very general and does not necessarily present particular properties like for instance the existence
- 322. 308 L. Amgoud of stable/preferred extensions. In what follows, we will show that the result of the decision system depends broadly on the outcome of this epistemic system. The first result states that the epistemic arguments of each admissible extension of AF consti- tute an admissible extension of the epistemic system AFe. Theorem 15.1. Let AF = (D,Ae ∪ Ap,Defe ∪ Defp ∪ Defm) be a decision system, E1,...,En its admissible extensions, and AFe = (Ae,Defe) its epistemic system. • ∀Ei, the set Ei ∩Ae is an admissible extension of AFe. • ∀E′ such that E′ is an admissible extension of AFe, ∃Ei such that E′ ⊆ Ei ∩Ae. It is easy to show that when Defm is empty, i.e. no epistemic argument defeats a practical one, then the extensions of AF (under a given semantics) are exactly the different extensions of AFe (under the same semantics) augmented by the set AFp. Theorem 15.2. Let AF = (D,Ae ∪ Ap,Defe ∪ Defp ∪ Defm) be a decision system. Let E1,...,En be the extensions of AFe under a given semantics. If Defm = ∅ then ∀Ei with i = 1,...,n, then the set Ei ∪Ap is an extension of AF. Finally, it can be shown that if the empty set is the only admissible extension of the decision system AF, then the empty set is also the only admissible extension of the corresponding epistemic system AFe. Moreover, each practical argument is attacked by at least one epistemic argument. Theorem 15.3. Let AF = (D,Ae ∪ Ap,Defe ∪ Defp ∪ Defm) be a decision system. The only admissible extension of AF is the empty set iff: 1. The only admissible extension of AFe is the empty set, and 2. ∀δ ∈ Ap, ∃α ∈ Ae such that (α,δ) ∈ Defm. At this step, we have only defined the accepted arguments among all the existing ones. However, nothing is yet said about which option to prefer. In the next section, we will study different ways of comparing pairs of options on the basis of skeptically accepted practical arguments. 2.5 Ordering options Comparing candidate decisions, i.e. defining a preference relation ' on the set D of options, is a key step in a decision process. In an argumentation-based approach, the definition of this relation is based on the sets of “accepted” arguments pros or cons associated with candidate decisions. Thus, the input of this relation is no longer Ad, but the set Acc(AF,y)∩Ad, where Acc(AF,y) is the set of skeptically accepted arguments of the decision system (D,A,Def) under stable or preferred semantics. In what follows, we will use the notation Acc(AF) for short. Note that in a decision system, when the defeat relation Defm is empty, the epistemic arguments become useless for the decision problem, i.e. for ordering options. Thus, only the practical
- 323. 15 Argumentation for Decision Making 309 system AFp is needed. Depending on what sets are considered and how they are handled, one can roughly distinguish between three categories of principles: Unipolar principles: are those that only refer to either the arguments pros or the arguments cons. Bipolar principles: are those that take into account both types of arguments at the same time. Non-polar principles: are those where arguments pros and arguments cons a given choice are aggregated into a unique meta-argument. It results that the negative and positive polarities disappear in the aggregation. Whatever the category is, a relation ' should suitably satisfy the following min- imal requirements: 1. Transitivity: The relation should be transitive (as usually required in decision theory). 2. Completeness: Since one looks for the “best” candidate decision, it should then be possible to compare any pair of choices. Thus, the relation should be complete. 2.5.1 Unipolar principles In this section we present basic principles for comparing decisions on the basis of only arguments pros. Similar ideas apply to arguments cons. We start by presenting those principles that do not involve the strength of arguments, then their respective refinements when strength is taken into account. A first natural criterion consists of preferring the decision that has more arguments pros. Definition 15.4 (Counting arguments pros). Let AF = (D,A,Def) be a decision system and Acc(AF) its accepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff |Fp(d1)∩Acc(AF)| ≥ |Fp(d2)∩Acc(AF)|. Property 15.4. This relation is a complete preorder. Note that when the decision system has no accepted arguments (i.e. Acc(AF) = ∅), all the options in D are equally preferred w.r.t. the relation '. It is also worth men- tioning that with such a principle, one may prefer a decision d, which has three arguments pointing all to the same positive consequence, to decision d′, which is supported by two arguments pointing to different consequences. When the strength of arguments is taken into account in the decision process, one may think of preferring a choice that has a dominant argument, i.e. an argument pros that is preferred w.r.t. the relation ≥p⊆ Ap ×Ap to any argument pros the other choices. This principle is called promotion focus principle in [2].
- 324. 310 L. Amgoud Definition 15.5. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff ∃δ ∈ Fp(d1)∩Acc(AF) such that ∀δ′ ∈ Fp(d2)∩Acc(AF),δ ≥p δ′ . With this criterion, if the decision system has no accepted arguments, then all the options in D are equally preferred. The above definition relies heavily on the relation ≥p that compares practical arguments. Thus, the properties of this criterion depends on those of ≥p. Namely, it can be checked that the above criterion works properly only if ≥p is a complete preorder. Property 15.5. If the relation ≥p is a complete preorder, then ' is also a complete preorder. Note that the above relation may be found to be too restrictive, since when the strongest arguments in favor of d1 and d2 have equivalent strengths (i.e. are indif- ferent), d1 and d2 are also seen as equivalent. However, we can refine the above definition by ignoring the strongest arguments with equal strengths, by means of the following strict preorder. Definition 15.6. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D, and ≥p be a complete preorder. Let (δ1, ..., δr), (δ′ 1, ..., δ′ s) such that ∀δi=1,...,r, δi ∈ Fp(d1) ∩ Acc(AF), and ∀δ′ j=1,...,s, δ′ j ∈ Fp(d2)∩Acc(AF). Each of these vectors is assumed to be decreasingly ordered w.r.t ≥p (e.g. δ1 ≥p ... ≥p δr). Let v = min(r, s). d1 ' d2 iff: • δ1 p δ′ 1, or • ∃ k ≤ v such that δk p δ′ k and ∀ j k, δj ≈p δ′ j, or • r v and ∀ j ≤ v, δj ≈p δ′ j. Till now, we have only discussed decision principles based on arguments pros. However, the counterpart principles when arguments cons are considered can also be defined. Thus, the counterpart principle of the one defined in Definition 15.4 is the following complete preorder: Definition 15.7 (Counting arguments cons). Let AF = (D,A,Def) be a decision system and Acc(AF) its accepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff |Fc(d1)∩Acc(AF)| ≤ |Fc(d2)∩Acc(AF)|. The principles that take into account the strengths of arguments have also their coun- terparts when handling arguments cons. The prevention focus principle prefers a decision when all its cons are weaker than at least one argument against the other decision. Formally: Definition 15.8. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff ∃δ ∈ Fc(d2)∩Acc(AF) such that ∀δ′ ∈ Fc(d1)∩Acc(AF),δ ≥p δ′ .
- 325. 15 Argumentation for Decision Making 311 As in the case of arguments pros, when the relation ≥p is a complete preorder, the above relation is also a complete preorder, and can be refined into the following strict one. Definition 15.9. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. Let (δ1, ..., δr), (δ′ 1, ..., δ′ s) such that ∀δi=1,...,r, δi ∈ Fc(d1) ∩ Acc(AF), and ∀δ′ j=1,...,s, δ′ j ∈ Fc(d2) ∩ Acc(AF). Each of these vectors is assumed to be decreas- ingly ordered w.r.t ≥p (e.g. δ1 ≥p ... ≥p δr). Let v = min(r, s). d1 ≻ d2 iff: • δ′ 1 p δ1, or • ∃ k ≤ v such that δ′ k p δk and ∀ j k, δj ≈p δ′ j, or • v s and ∀ j ≤ v, δj ≈p δ′ j. 2.5.2 Bipolar principles Let’s now define some principles where both types of arguments (pros and cons) are taken into account when comparing decisions. Generally speaking, we can conjunc- tively combine the principles dealing with arguments pros with their counterpart handling arguments cons. For instance, the principles given in Definition 15.4 and Definition 15.7 can be combined as follows: Definition 15.10. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff • |Fp(d1)∩Acc(AF)| ≥ |Fp(d2)∩Acc(AF)|, and • |Fc(d1)∩Acc(AF)| ≤ |Fc(d2)∩Acc(AF)|. However, note that unfortunately this is no longer a complete preorder. Similarly, the principles given respectively in Definition 15.5 and Definition 15.8 can be combined into the following one: Definition 15.11. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff: • ∃ δ ∈ Fp(d1)∩Acc(AF) such that ∀ δ′ ∈ Fp(d2)∩Acc(AF), δ ≥p δ′, and • ∄ δ ∈ Fc(d1)∩Acc(AF) such that ∀ δ′ ∈ Fc(d2)∩Acc(AF), δ ≥p δ′. This means that one prefers a decision that has at least one supporting argument which is better than any supporting argument of the other decision, and also that has not a very strong argument against it. Note that the above definition can be also refined in the same spirit as Definitions 15.6 and 15.9. Another family of bipolar decision principles applies the Franklin principle which is a natural extension to the bipolar case of the idea underlying Definition 15.6. This principle consists, when comparing pros and cons a decision, of ignoring pairs of arguments pros and cons which have the same strength. After such a simplification, one can apply any of the above bipolar principles. In what follows, we will define formally the Franklin simplification.
- 326. 312 L. Amgoud Definition 15.12 (Franklin simplification). Let AF = (D,A,Def) be a decision system and Acc(AF) its accepted arguments. Let d ∈ D. Let P = (δ1, ..., δr), C = (δ′ 1, ..., δ′ m) such that ∀δi,δi ∈ Fp(d) ∩ Acc(AF) and ∀δ′ j,δ′ j ∈ Fc(d) ∩ Acc(AF). Each of these vectors is assumed to be decreasingly ordered w.r.t ≥p (e.g. δ1 ≥p ... ≥p δr). The result of the simplification is P′ = (δj+1, ..., δr), C′ = (δ′ j+1, ..., δ′ m) s.t. • ∀ 1 ≤ i ≤ j, δi ≈p δ′ i and (δj+1 p δ′ j+1 or δ′ j+1 p δj+1) • If j = r (resp. j = m), then P′ = ∅ (resp. C′ = ∅). 2.5.3 Non-polar principles In some applications, the arguments in favor of and against a decision are aggre- gated into a unique meta-argument having a unique strength. Thus, comparing two decisions amounts to compare the resulting meta-arguments. Such a view is well in agreement with current practice in multiple criteria decision making, where each decision is evaluated according to different criteria using the same scale (with a positive and a negative part), and an aggregation function is used to obtain a global evaluation of each decision. Definition 15.13 (Aggregation criterion). Let AF = (D,A,Def) be a decision sys- tem and Acc(AF) its accepted arguments. Let d1, d2 ∈ D. Let (δ1, ..., δn)4 and (δ′ 1, ..., δ′ m)5 (resp. (γ1,...,γl)6 and (γ′ 1,...,γ′ k)7) the vectors of the arguments pros and cons the decision d1 (resp. d2). d1 ' d2 iff h(δ1, ..., δn, δ′ 1, ..., δ′ m) ≥p h(γ1, ..., γl, γ′ 1, ..., γ′ k), where h is an aggregation function. A simple example of this aggregation attitude is computing the difference of the number of arguments pros and cons. Definition 15.14. Let AF = (D,A,Def) be a decision system and Acc(AF) its ac- cepted arguments. Let d1, d2 ∈ D. d1 ' d2 iff |Fp(d1) ∩ Acc(AF)| − |Fc(d1) ∩ Acc(AF)| ≥ |Fp(d2)∩Acc(AF)|−|Fc(d2)∩Acc(AF)|. This has the advantage to be again a complete preorder, while taking into account both pros and cons arguments. 3 A typology of formal practical arguments This section presents a systematic study of practical arguments. Epistemic argu- ments will not be discussed here because they have been much studied in the litera- 4 Each δi ∈ Fp(d1)∩Acc(AF). 5 Each δ′ i ∈ Fc(d1)∩Acc(AF). 6 Each γi ∈ Fp(d2)∩Acc(AF). 7 Each γ′ i ∈ Fc(d2)∩Acc(AF).
- 327. 15 Argumentation for Decision Making 313 ture (eg. [3, 9, 26]), and their handling does not make new problems in the general setting of Section 2, even in the decision process perspective of this chapter. More- over, they only play a role when the knowledge base is inconsistent. Before pre- senting the different types of practical arguments, we start first by introducing the logical language as well as the different bases needed in a decision making problem. 3.1 Logical representation of knowledge and preference This section introduces the representation setting of knowledge and preference which are here distinct, as it is in classical decision theory. Moreover, preferences are supposed to be handled in a bipolar way meaning that what the decision maker is really looking for may be more restrictive than what it is just willing to avoid. In what follows, a vocabulary P of propositional variables contains two kinds of vari- ables: decision variables, denoted by v1,...,vn, and state variables. Decision vari- able are controllable, that is their value can be fixed by the decision maker. Making a decision then amounts to fixing the truth value of every decision variable. On the contrary, state variables are fixed by nature, and their value is a matter of knowl- edge by the decision maker. He has no control on them (although he may express preferences about their values). 1. D is a set of formulas built from decision variables. Elements of D represent the different alternatives, or candidate decisions. Let us consider the example of an agent who wants to know whether she should take her umbrella, her raincoat or both. In this case, there are two decision variables: umb (for umbrella) and rac (for raincoat). Assume that this agent hesitates between the three following options: i) d1 : umb (i.e. to take only her umbrella), ii) d2 : rac (i.e. to take only her raincoat), or iii) d3 : umb ∧ rac (i.e. to take both). Thus, D = {d1,d2,d3}. Note that elements of D are not necessarily mutually exclusive. In the example, if the agent chooses the option d3 then the two other options are satisfied. 2. G is a set of propositional formulas built from state variables. It gathers the goals of an agent (the decision maker). A goal represents what the agent wants to achieve, and has thus a positive flavor. This means that if g ∈ G, the decision maker wants that the chosen decision leads to a state of affairs where g is true. This base may be inconsistent. In this case it would be for sure impossible to satisfy all the goals, which would induce the simultaneous existence of practical arguments pros and cons. In general G contains several goals. Clearly, an agent should try to satisfy all goals in its goal base G if possible. This means that G may be thought as a conjunction. However, the two goal bases G = {g1,g2} and G′ = {g1 ∧g2} although they are logically equivalent, will not be handled in the same way in an argumentative perspective, since in the second case there is no way to consider intermediary objectives such as here satisfying g1, or satisfying g2 only, in case it turns out that it is impossible to satisfy g1 ∧g2. This means that our approach is syntax-dependent.
- 328. 314 L. Amgoud 3. The set R contains propositional formulas built from state variables. It gathers the rejections of an agent. A rejection represents what the agent wants to avoid. Clearly rejections express negative preferences. The set {¬r|r ∈ R} describing what is acceptable for the agent is assumed to be consistent, since acceptable alternatives should satisfy ¬r due to the rejection of r, and at least there should remain some possible worlds that are not rejected. There are at least two rea- sons for separately considering a set of goals and a set of rejections. First, since agents naturally express themselves in terms of what they are looking for (i.e. their goals), and in terms of what they want to avoid (i.e. their rejections), it is better to consider goals and rejections separately in order to articulate arguments referring to them in a way easily understandable for the agents. Moreover, recent cognitive psychology studies [13] have confirmed the cognitive validity of this distinction between goals and rejections. Second, if r is a rejection, this does not necessarily mean that ¬r is a goal, and thus rejections cannot be equivalently restated as goals. For instance, in case of choosing a medical drug, one may have as a goal the immediate availability of the drug, and as a rejection its availability only after at least two days. In such a case, if the candidate decision guarantees the availability only after one day, this decision will for sure avoid the rejection without satisfying the goal. Another simple example is the case of an agent who wants to get a cup of either coffee or tea, and wants to avoid getting no drink. If the agent obtains a glass of water, again he would avoid its rejection, without be- ing completely satisfied. We can imagine different forms of consistency between the goals and the rejections. A minimal requirement is to have G∩R = ∅. 4. The set K represents the background knowledge that is not necessarily assumed to be consistent. The argumentation framework for inference presented in Section 2 will handle such inconsistency, namely with the epistemic system. Elements of K are propositional formulas built from the alphabet P, and assumed to be put in a clausal form. The base K contains basically two kinds of clauses: i) those not involving any element from D which encode pieces of knowledge or factual information (possibly involving goals) about how the world is; ii) those involving one negation of a formula d of the set D, and which states what follows when decision d is applied. Thus, the decision problem we consider will be always encoded with the four above sets of formulas (with the restrictions stated above). Moreover, we may suppose that each of the three bases K, G, and R are stratified. Having K stratified would mean that we consider that some pieces of knowledge are fully certain, while others are less certain (maybe distinguishing between several levels of partial certainty such as “almost certain”, “rather certain”, etc.). Clearly, formulas that are not certain at all cannot be in K. Similarly, having G (resp. R) stratified means that some goals (resp. rejections) are imperative, while some others are less important (one may have more than two levels of importance). Completely unimportant goals (resp. rejections) do not appear in any stratum of G (resp. R). It is worth pointing out that we assume that candidate decisions are all considered as a priori equally potentially suitable, and thus there is no need to have D stratified.
- 329. 15 Argumentation for Decision Making 315 Definition 15.15 (Decision theory). A decision theory (or a theory for short) is a tuple T = D, K, G, R. 3.2 A typology of formal practical arguments Each candidate decision may have arguments in its favor (called pros), and argu- ments against it (called cons). In the following, an argument is associated with an alternative, and always either refers to a goal or to a rejection. Arguments pros point out the “existence of good consequences” or the “absence of bad consequences” for a candidate decision. A good consequence means that applying decision d will lead to the satisfaction of a goal, or to the avoidance of a rejection. Similarly, a bad con- sequence means that the application of d leads for sure to miss a goal, or to reach a rejected situation. We can distinguish between practical arguments referring to a goal, and those arguments referring to rejections. When focusing on the base G, an argument pro corresponds to the guaranteed satisfaction of a goal when there exists a consistent subset S of K such that S∪{d} ⊢ g. Definition 15.16 (Positive arguments pro). Let T be a theory. A positively ex- pressed argument in favor of an option d is a tuple δ = S,d,g s.t: 1. S ⊆ K, d ∈ D, g ∈ G, S∪{d} is consistent 2. S∪{d} ⊢ g, and S is minimal for set inclusion among subsets of K satisfying the above criteria (arguments of Type PP). S is called the support of the argument, and d is its conclusion. In what follows, Supp denotes a function that returns the support S of an argument, Conc denotes a function that returns the conclusion d of the argument, and Result denotes a function that returns the consequence of the decision. The consequence may be either a goal as in the previous definition, or a rejection as we can see in the next definitions of argument types. The above definition deserves several comments: 1) The consistency of S ∪ {d} means that d is applicable in the context S, in other words that we cannot prove from S that d is impossible. This means that impossible alternatives w.r.t. K have been already taken out when defining the set D. In the par- ticular case where the base K would be consistent, then condition 1, namely S∪{d} is consistent, is equivalent to K∪{d} is consistent. But, in the case where K is in- consistent, independently from the existence of a PP argument, it may happen that for another consistent subset S′ of K, S′ ⊢ ¬d. This would mean that there is some doubt about the feasibility of d, and then constitute an epistemic argument against d. In the general framework proposed in section 2, such an argument will overrule decision d since epistemic arguments take precedence over any practical argument (provided that this epistemic argument is not itself killed by another epistemic argu- ment). 2) Note that argument of type PP are reminiscent of the practical syllogism recalled in the introduction. Indeed, it emphasizes that a candidate decision might be chosen
- 330. 316 L. Amgoud if it leads to the satisfaction of a goal. However, this is only a clue for choosing the decision since this last may have arguments against, which would weaken it, or there may exist other candidate decisions with stronger arguments. Moreover, due to the nature of the practical syllogism, it is worth noticing that practical arguments have an abductive form, contrarily to epistemic arguments that are defined in a deductive way, as revealed by their formal respective definitions. Another type of arguments pros refers to rejections. It amounts to avoid a rejec- tion for sure, i.e. S∪{d} ⊢ ¬r (where S is a consistent subset of K). Definition 15.17 (Negative arguments pros). Let T be a theory. A negatively ex- pressed argument in favor of an option is a tuple δ = S,d,r s.t: 1. S ⊆ K, d ∈ D, r ∈ R, S∪{d} is consistent 2. S ∪ {d} ⊢ ¬r and S is minimal for set inclusion among subsets of K satisfying the above criteria (arguments of Type NP). Arguments cons highlight the existence of bad consequences for a given can- didate decision. Negatively expressed arguments cons are defined by exhibiting a rejection that is necessarily satisfied. Formally: Definition 15.18 (Negative arguments cons). Let T be a theory. A negatively ex- pressed argument against an option d is a tuple δ = S,d,r s.t: 1. S ⊆ K, d ∈ D, r ∈ R, S∪{d} is consistent, 2. S∪{d} ⊢ r and S is minimal for set inclusion among subsets of K satisfying the above criteria (arguments of Type NC). Lastly, the absence of positive consequences can also be seen as an argument against (cons) an alternative. Definition 15.19 (Positive arguments cons). Let T be a theory. A positively ex- pressed argument against an option d is a tuple δ = S,d,g s.t: 1. S ⊆ K, d ∈ D, g ∈ G, S∪{d} is consistent, 2. S ∪ {d} ⊢ ¬g and S is minimal for set inclusion among subsets of K satisfying the above criteria (arguments of Type PC). Let us illustrate the previous definitions on an example. Example 15.7. Two decisions are possible, organizing a show (d), or not (¬d). Thus D = {d,¬d}. The knowledge base K contains the following pieces of knowledge: if a show is organized and it rains then small money loss (¬d ∨¬r ∨sml); if a show is organized and it does not rain then benefit (¬d ∨r ∨b); small money loss entails money loss (¬sml ∨ml); if benefit there is no money loss (¬b∨¬ml); small money loss is not large money loss (¬sml ∨¬lml); large money loss is money loss (¬lml ∨ ml); there are clouds (c); if there are clouds then it may rain (¬c∨r). All these pieces of knowledge are in the stratum of level n, except the last one which is in a stratum with a lower level due to uncertainty. Consider now the cases of two organizers (O1 and O2) having different preferences. O1 does not want any loss R = {ml}, and
- 331. 15 Argumentation for Decision Making 317 would like benefit G = {b}. O2 does not want large money loss R = {lml}, and would like benefit G = {b}. In such case, it is expected that O1 prefers ¬d to d, since there is a NC argument against d and no argument for ¬d. For O2, there is no longer any NC argument against d. He might even prefer d to ¬d, if he is optimistic and he considers that there is a possibility that it does not rain (leading to a potential PP argument under the hypothesis to have ¬r in K. Due to the asymmetry in the human mind between what is rejected and what is desired, the former being usually considered as stronger than the latter, one may assume that NC arguments are stronger than PC arguments, and conversely PP argu- ments are stronger than NP arguments. 4 Related work Some works have been done on arguing for decision. Quite early, in [22] Brewka and Gordon have outlined a logical approach to decision (for negotiation purposes), which suggests the use of defeasible consequence relation for handling prioritized rules, and which also exhibits arguments for each choice. However, arguments are not formally defined. In the framework proposed by Fox and Parsons in [20], no ex- plicit distinction is made between knowledge and goals. However, in their examples, values (belonging to a linearly ordered scale) are assigned to formulas which repre- sent goals. These values provide an empirical basis for comparing arguments using a symbolic combination of strengths of beliefs and goals values. This symbolic com- bination is performed through dictionaries corresponding to different kinds of scales that may be used. In this work, only one type of arguments is considered in the style of arguments in favor of beliefs. In [10], Bonet and Geffner have also proposed an approach to qualitative decision, inspired from Tan and Pearl [27], based on “action rules” that link a situation and an action with the satisfaction of a positive or a nega- tive goal. However in contrast with the previous two works and the work presented in this paper, this approach does not refer to any model in argumentative inference. Other researchers in AI, working on practical reasoning, starting with the generic question “what is the right thing to do for an agent in a given situation” [24, 25], have proposed a two steps process to answer this question. The first step, often called de- liberation [29], consists of identifying the goals of the agent. In the second step, they look for ways of achieving those goals, i.e. for plans, and thus for intermediary goals and sub-plans. Such an approach raises issues such as: how are goals generated ? are actions feasible ? do actions have undesirable consequences ? are sub-plans compat- ible ? are there alternative plans for achieving a given goal, etc. In [12], it has been argued that this can be done by representing the cognitive states, namely agent’s beliefs, desires and intentions (thus the so-called BDI architecture). This requires a rich knowledge/preference representation setting, which contrasts with the classical decision setting that directly uses an uncertainty distribution (a probability distri- bution in the case of expected utility), and a utility (value) function. Besides, the
- 332. 318 L. Amgoud deliberation step is merely an inference problem since it amounts to finding a set of desires that are justified on the basis of the current state of the world and of condi- tional desires. Checking if a plan is feasible and does not lead to bad consequences is still a matter of inference. A decision problem only occurs when several plans or sub-plans are possible, and one of them has to be chosen. This latter issue may be viewed as a classical decision problem. What is worth noticing in most works on practical reasoning is the use of argument schemes for providing reasons for choos- ing or discarding an action (e.g. [19, 21]). For instance, an action may be considered as potentially useful on the basis of the practical syllogism [28]: • G is a goal for agent X • Doing action A is sufficient for agent X to carry out goal G • Then, agent X ought to do action A The above syllogism is in essence already an argument in favor of doing action A. However, this does not mean that the action is warranted, since other argu- ments (called counter-arguments) may be built or provided against the action. Those counter-arguments refer to critical questions identified in [28] for the above syllo- gism. In particular, relevant questions are “Are there alternative ways of realizing G?”, “Is doing A feasible?”, “Has agent X other goals than G?”, “Are there other consequences of doing A which should be taken into account?”. Recently in [6, 7], the above syllogism has been extended to explicitly take into account the reference to ethical values in arguments. 5 Conclusion The chapter has proposed an abstract argumentation-based framework for decision making. The main idea behind this work is how to define a complete preorder on a set of candidate decisions on the basis of arguments. The framework distinguishes between two types of arguments: epistemic arguments that support beliefs and prac- tical arguments that justify candidate decisions. Each practical argument concerns only one candidate decision, and may be either in favor of that decision or against it. The framework follows two main steps: i) an inference step in which arguments are evaluated using acceptability semantics. This step amounts to return among the practical arguments, those which are warranted in the current state of information, i.e. the “accepted” arguments. ii) A pure decision step in which candidate decisions are compared on the basis of accepted practical arguments. For the second step of the process, we have proposed three families of principles for comparing pairs of choices. An axiomatic study and a cognitive validation of these principles are worth developing, in particular in connection with [11, 16]. The proposed approach is very general and includes as particular cases already studied argumentation-based deci- sion systems. Moreover it has been shown in [5] that the approach is suitable for multiple criteria decision making as well as decision making under uncertainty. In particular, the approach has been shown to fully agree with qualitative decision mak-
- 333. 15 Argumentation for Decision Making 319 ing under uncertainty, and to distinguish between pessimistic and optimistic attitude of the decision maker. Although our model is quite general, it may be still worth extending along dif- ferent lines. First, the use of default knowledge could be developed. Second, our approach does not take into account rules that recommend or disqualify decisions in given contexts. Such rules should incorporate modalities for distinguishing between strong and weak recommendations. Moreover, they are fired by classical argumen- tative inference. This contrasts with our approach where the only arguments per- taining to decisions have an abductive structure. Recommendation rules may also turn to be inconsistent with other pieces of knowledge in practical arguments pros or cons w.r.t. a decision. Lastly, agents may base their decision on two types of in- formation, namely generic knowledge and a repertory of concrete reported cases. Then, past observations recorded in the repertory may be the basis of a new form of arguments by exemplification of cases where a decision has succeeded or failed. This would amount to relate argumentation and case-based decision. References 1. L. Amgoud. A general argumentation framework for inference and decision making. In 21st Conference on Uncertainty in Artificial Intelligence, UAI’2005, pages 26–33, 2005. 2. L. Amgoud, J-F. Bonnefon, and H. Prade. An argumentation-based approach to multiple cri- teria decision. In Proceedings of the 8th European Conference on Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU’05), pages 269–280, 2005. 3. L. Amgoud and C. Cayrol. Inferring from inconsistency in preference-based argumentation frameworks. In International Journal of Automated Reasoning, 29, N2:125–169, 2002. 4. L. Amgoud and C. Cayrol. A reasoning model based on the production of acceptable argu- ments. In Annals of Mathematics and Artificial Intelligence, 34:197–216, 2002. 5. L. Amgoud and H. Prade. Explaining qualitative decision under uncertainty by argumentation. In Proceedings of the 21st National Conference on Artificial Intelligence (AAAI’06), pages 219–224, 2006. 6. K. Atkinson. Value-based argumentation for democratic decision support. In Proceed- ings of the First International Conference on Computational Models of Natural Argument (COMMA’06), pages 47–58, 2006. 7. K. Atkinson, T. Bench-Capon, and P. McBurney. Justifying practical reasoning. In Pro- ceedings of the Fourth Workshop on Computational Models of Natural Argument (CMNA’04), pages 87–90, 2004. 8. P. Baroni, M. Giacomin, and G. Guida. Scc-recursiveness: a general schema for argumentation semantics. In Artificial Intelligence, 168 (1-2):162–210, 2005. 9. Ph. Besnard and A. Hunter. A logic-based theory of deductive arguments. In Artificial Intel- ligence, 128:203–235, 2001. 10. B. Bonet and H. Geffner. Arguing for decisions: A qualitative model of decision making. In Proceedings of the 12th Conference on Uncertainty in Artificial Intelligence (UAI’96), pages 98–105, 1996. 11. J.-F. Bonnefon and H. Fargier. Comparing sets of positive and negative arguments: Empirical assessment of seven qualitative rules. In Proceedings of the 17th European Conference on Artificial Intelligence (ECAI’06), pages 16–20, 2006. 12. M. Bratman. Intentions, plans, and practical reason. Harvard University Press, Mas- sachusetts, 1987.
- 334. 320 L. Amgoud 13. J.T. Cacioppo, W.L. Gardner, and G.G. Bernston. Beyond bipolar conceptualizations and measures: The case of attitudes and evaluative space. In Personality and Social Psychology Review, 1:3–25, 1997. 14. C. Cayrol, V. Royer, and C. Saurel. Management of preferences in assumption-based reason- ing. In Lecture Notes in Computer Science, 682:13–22, 1993. 15. Y. Dimopoulos, P. Moraitis, and A. Tsoukias. Argumentation based modeling of decision aiding for autonomous agents. In IEEE-WIC-ACM International Conference on Intelligent Agent Technology, pages 99–105, 2004. 16. D. Dubois and H. Fargier. Qualitative decision making with bipolar information. In Proceed- ings of the 10th International Conference on Principles of Knowledge Representation and Reasoning (KR’06), pages 175–186, 2006. 17. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. In Artificial Intelligence, 77:321–357, 1995. 18. M. Elvang-Goransson, J. Fox, and P. Krause. Dialectic reasoning with inconsistent informa- tion. In Proceedings of 9th Conference on Uncertainty in Artificial Intelligence (UAI’93), pages 114 – 121, 1993. 19. J. Fox and S. Das. Safe and Sound. Artificial Intelligence in Hazardous Applications. AAAI Press, The MIT Press, 2000. 20. J. Fox and S. Parsons. On using arguments for reasoning about actions and values. In Proceed- ings of the AAAI Spring Symposium on Qualitative Preferences in Deliberation and Practical Reasoning, Stanford, 1997. 21. R. Girle, D. Hitchcock, P. McBurney, and B. Verheij. Decision support for practical reasoning. C. Reed and T. Norman (Editors): Argumentation Machines: New Frontiers in Argument and Computation. Argumentation Library. Dordrecht, The Netherlands: Kluwer Academic, 2003. 22. T. Gordon and G. Brewka. How to buy a porsche: An approach to defeasible decision making (preliminary report). In In the workshop of Comutational Dialectics, 1994. 23. T. F. Gordon and N. I. Karacapilidis. The Zeno Argumentation Framework. Kunstliche Intel- ligenz, 13(3):20–29, 1999. 24. J. Pollock. The logical foundations of goal-regression planning in autonomous agents. In Artificial Intelligence, 106(2):267–334, 1998. 25. J. Raz. Practical reasoning. Oxford, Oxford University Press, 1978. 26. G. R. Simari and R. P. Loui. A mathematical treatment of defeasible reasoning and its imple- mentation. In Artificial Intelligence and Law, 53:125–157, 1992. 27. S. W. Tan and J. Pearl. Qualitative decision theory. In Proceedings of the 11th National Conference on Artificial Intelligence (AAAI’94), pages 928–933, 1994. 28. D. Walton. Argument schemes for presumptive reasoning, volume 29. Lawrence Erlbaum Associates, Mahwah, NJ, USA, 1996. 29. M. J. Wooldridge. Reasoning about rational agents. MIT Press, Cambridge Massachusetts, London England, 2000.
- 335. Chapter 16 Argumentation and Game Theory Iyad Rahwan and Kate Larson 1 What Game Theory Can Do for Argumentation In a large class of multi-agent systems, agents are self-interested in the sense that each agent is interested only in furthering its individual goals, which may or may not coincide with others’ goals. When such agents engage in argument, they would be expected to argue strategically in such a way that makes it more likely for their argumentative goals to be achieved. What we mean by arguing strategically is that instead of making arbitrary arguments, an agent would carefully choose its argu- mentative moves in order to further its own objectives. The mathematical study of strategic interaction is Game Theory, which was pi- oneered by von Neuman and Morgenstern [13]. A setting of strategic interaction is modelled as a game, which consists of a set of players, a set of actions available to them, and a rule that determines the outcome given players’ chosen actions. In an argumentation scenario, the set of actions are typically the set of argumentative moves (e.g. asserting a claim or challenging a claim), and the outcome rule is the criterion by which arguments are evaluated (e.g. a judge’s attitude or a social norm). Generally, game theory can be used to achieve two goals: 1. undertake precise analysis of interaction in particular strategic settings, with a view to predicting the outcome; 2. design rules of the game in such a way that self-interested agents behave in some desirable manner (e.g. tell the truth); this is called mechanism design; Both these approaches are quite useful for the study of argumentation in multi- agent systems. On one hand, an agent may use game theory to analyse a given argumentative situation in order to choose the best strategy. On the other hand, we Iyad Rahwan British University in Dubai, UAE University of Edinburgh, UK, e-mail: irahwan@acm.org Kate Larson University of Waterloo, Canada e-mail: klarson@cs.uwaterloo.ca I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 321 DOI 10.1007/978-0-387-98197-0 16, c Springer Science+Business Media, LLC 2009
- 336. 322 Iyad Rahwan and Kate Larson may use mechanism design to design the rules (e.g. argumentation protocol) in such a way as to promote good argumentative behaviour. In this chapter, we will discuss some early developments in these directions. In the next section, we motivate the usefulness of game theory in argumentation using a novel game. After providing a brief background on game theory in Section 3, we introduce our Argumentation Mechanism Design approach in Section 4 and present some preliminary results in Section 5. Finally, we discuss related work in Section 6 and conclude in Section 7 2 The “Argumentative Battle of the Sexes” Game Consider the following situation involving the couple Alice (A) and Brian (B), who want to decide on an activity for the day.1 Brian thinks they should go to a soc- cer match (argument α1) while Alice thinks they should attend the ballet (argument α2). There is time for only one activity, however (hence α1 and α2 defeat one an- other). Moreover, while Alice prefers the ballet to the soccer, she would still rather go to a soccer match than stay at home. Likewise, Brian prefers the soccer match to the ballet, but also prefers the ballet to staying home. Formally, we can write uA(ballet) uA(soccer) uA(home) and uB(soccer) uB(ballet) uB(home). Alice has a strong argument which she may use against going to the soccer, namely by claiming that she is too sick to be outdoors (argument α3). Brian simply cannot attack this argument (without compromising his marriage at least). Likewise, Brian has an irrefutable argument against the ballet; he could claim that his ex-wife will be there too (argument α4). Alice cannot stand her! Using Dung’s abstract ar- gumentation model [1], which is described in detail in Chapter 2, the argumentative structure of this situation can be modelled as shown in Figure 16.1(a). Alice can choose to say nothing, utter argument α2 or α3 or both. Similarly, Brian can choose to say nothing, utter argument α1 or α4 or both. For the sake of the example, we will suppose that Alice and Brian use the grounded semantics as the argumentative foundation of their marriage! The question we are interested in here is: What will Alice and Bob say? or at least: What are they likely to say? The strategic encounter, on the other hand, can be modelled as shown in the table in Figure 16.1(b). Each cell corresponds to a strategy profile in which Alice and Brian reveal a particular set of arguments. The numbers in the cells correspond to the utilities they obtain once the grounded extension is calculated on their revealed arguments. For example, if Alice utters {α2} while Brian utters {α1,α4}, we end up with a sub-graph of Figure 16.1(a) in which α3 is missing. The grounded extension of this argument graph admits arguments {α1,α4}. This corresponds to a situation where Brian wins and the couple head to the soccer. Thus, he gets the highest utility of 2, while Alice gets her second-preferred outcome with utility 1. This representa- tion, shown in Figure 16.1(b), is known as a normal form game. 1 We call this the argumentative battle of the sexes game. It is similar, but not identical to the well-known “Battle of the Sexes” game.
- 337. 16 Argumentation and Game Theory 323 α4 α3 α1: They should go to the soccer. α2: They should go to the ballet. α3: Alice too sick for the outdoors. α4: Brian’s ex-wife at the ballet. α1 α2 (a) (b) {α1} {α2} {} 0 0 2 1 {} 1 2 0 0 Brian Alice {α3} 0 0 0 0 2 1 {α2, α3} 2 1 {α4} 0 0 0 0 0 0 0 0 {α1, α4} 1 2 1 2 0 0 0 0 Nash Equilibria: ({α2}, {α1, α4}) ({α2, α3}, {α1}) ({}, {α1, α4}) ({α2, α3}, {}) Fig. 16.1 Simple argumentative scenario and its normal form game representation. The outcome is decided using grounded semantics. The normal-form game can be used to deduce a number of things about this particular scenario. First, it is never in either Alice or Brian’s best interest to utter their irrefutable argument (α3 or α4) without also stating their preferred activity (α1 or α2), since by announcing their preferred activity then they may possibly attend some event (either ballet or soccer) while if one of them only announces their irrefutable argument then both agents are certain to stay at home (the least preferred outcome for both Alice and Brian). That is, {α3} is weakly dominated by {α2,α3} (and {α2}) and α4 is weakly dominated by {α1,α4} (and {α1}). Given a game, we are interested in finding the Nash equilibria of the game. A Nash equilibrium is a strategy profile (a listing of one strategy for each agent) where no agent wants to change its strategy, assuming that the other agents do not change. The Nash equilibria are the stable outcomes of the game. Consider the strategy profile in which Alice says that she is sick and suggests the ballet (i.e. she utters {α2,α3}) and Brian simply suggests the soccer match (i.e. he utters {α1}). This outcome is a Nash equilibrium. On one hand, given that Alice states {α2,α3}, Brian has no incentive to deviate to any other strategy. If he mentions his ex-wife’s atten- dance at the ballet (uttering {α4} or {α1,α4}), he shoots himself in the foot and ends up spending the day at home. And if he stays quiet (uttering {}), he cannot influence the outcome anyway. On the other hand, assuming that Brian announces {α1} then Alice is best-off stating {α2,α3} since by doing so, she gets the outcome that she prefers (the ballet). In fact, we list four Nash equilibria in Figure 16.1(b).2 The analysis that we just concluded does not allow us to identify a single outcome for the example. However, it does identify some interesting strategic phenomena. In particular, it shows that it is never in Alice and Brian’s interest for them both to use their irrefutable arguments. For example, if Brian is confident that Alice will state 2 A reader familiar with game theory will note that we only list the pure strategy Nash equilibria. In addition to these four equilibria, there are three mixed equilibria in which players randomize their strategies.
- 338. 324 Iyad Rahwan and Kate Larson Fig. 16.2 A (pruned) game tree for the argumentative battle of the sexes game. that she is too sick (α3) then Brian should not bring up his argument against the ballet. While the above analysis did not allow us to identify a single outcome of the scenario, at least we were able to rule out so many unstable outcomes. Indeed, in some situations, there is a single Nash equilibrium, which makes predicting the outcome easier. So far, we used a normal-form representation to model the argument game. While this representation is useful for many purposes due to its simplicity, it fails to capture the dynamic aspect of argumentation: the fact that argumenta- tive moves are normally made interactively over multiple time steps. The appro- priate tool to model such dynamics are extensive-form games, which we discuss next. An extensive-form game with perfect information explicitly captures the fact that agents may take turns when choosing actions (for example, declaring arguments). A game tree is used to represent the game. Each node in the tree is associated with an agent whose turn it is to take an action. A path in the tree represents the sequence of actions taken, and leafs nodes are the final outcomes, given the actions on the path to the leaf node. In these games we assume that the actions that each agent takes are fully observable by all the other agents. We can model the interaction between Alice and Brian using an extensive-form game, if we assume that Alice and Brian take turns uttering arguments. We will
- 339. 16 Argumentation and Game Theory 325 assume that i) an agent can only make one argument at a time, ii) agents can not repeat arguments, and iii) if at some step an agent decides not to make an argument, then they are not allowed to make any more arguments in the future. We will also assume, for the sake of the example, that Alice gets to make the first argument. Figure 16.2 shows the game tree for the argumentative battle of the sexes game. Most of the paths which result in an outcome where both agents will get zero have been pruned from the tree. These paths are not played in equilibrium, and their removal allows us to focus on the relevant parts of the tree. Since Alice gets to make the first move, she has to decide whether to offer no arguments at all {}, suggest that they go to the ballet (α2), or present her counter- argument to the soccer match before the soccer match is even brought up (α3). Based on the argument uttered by Alice, Brian then gets to make a decision. If Alice had made no argument, then as long as Brian announces α1 (and possibly also α4) then they will go to the soccer match. Brian will receive utility of two and Alice will receive utility of one (the subtree on the left). If Alice suggested going to the ballet (α2) then Brian is best off immediately raising his counter-argument to the ballet (α4). This is because Alice’s best counter-argument is then to say nothing, which allows Brian to then present soccer (α1) as an alternative. This results in Brian getting his favorite outcome (soccer) since Alice’s only other option would be to raise argument α3, resulting in them both staying at home (the least preferred outcome). If instead, Brian had not made an argument or had uttered α1, then Alice would have been able to to raise her counter-argument, resulting in them both going to the ballet. Finally, if Alice first announces her counter-argument to soccer (α3) then Brian will end up announcing, at most, argument α1 since raising the counter- argument to the ballet (α4) will result in them both staying at home. This means that the outcome will be that both Alice and Brian will go to the ballet. Alice uses this reasoning in order to determine what her initial action should be. She will realize that if she makes no argument, or initially suggests the ballet (α2) then Brian will be able to take actions so that they end up going to the soccer match. However, if Alice starts with her counter-argument to the soccer match (α3) then she can force Brian into a situation where he is best off not making his counter-argument to the ballet, and so they will both end up going to the ballet, Alice’s preferred outcome. Therefore, in equilibrium, Alice will state her objection to soccer (α3) first, which will force Brian to either make no argument or make (already defeated) argument α1, which then allows Alice to counter with the ballet proposal (α2). This equilibrium is called a subgame perfect equilibrium and is a refinement of the Nash equilibria. We note that by going first, Alice had an advantage over Brian since by carefully choosing her first argument she could force the outcome that she wanted. If Brian had gone first, then he would have been best off first announcing α4, his counter- argument to the ballet. This would have allowed him to get the outcome that he preferred, that is, the soccer match. Thus in the extensive-form game analysis of argumentation, the order in which agents make arguments is critical in the analysis and in the outcome.
- 340. 326 Iyad Rahwan and Kate Larson A number of researchers have proposed using extensive-form games of perfect information to model argumentation. For example, Procaccia and Rosenschein [10] proposed a game-based argumentation framework where they extend Dung’s abstract argumentation framework by mapping argumentation frameworks into extensive-form games of perfect information. A similar approach has recently been proposed by Riveret et al [12], giving an extensive-form game characterisation of Prakken’s dialectical framework [8]. In both cases, the authors show how standard backward induction techniques can be used to eliminate dominated strategies and characterise Nash equilibrium strategies. 3 Technical Background Before we present a precise formal mapping of abstract argumentation into game theory, in this section, we give a brief background on key game-theoretic concepts. Readers who lack background in game theory may consult a more comprehensive introduction to the field, such as [5]. 3.1 Game Theory The field of game theory studies strategic interactions of self-interested agents. We assume that there is a set of self-interested agents, denoted by I. We let θi ∈ Θi denote the type of agent i which is drawn from some set of possible types Θi. The type represents the private information and preferences of the agent. An agent’s preferences are over outcomes o ∈ O, where O is the set of all possible outcomes. We assume that an agent’s preferences can be expressed by a utility function ui(o,θi) which depends on both the outcome, o, and the agent’s type, θi. Agent i prefers outcome o1 to o2 when ui(o1,θi) ui(o2,θi). When agents interact, we say that they are playing strategies. A strategy for agent i, si(θi), is a plan that describes what actions the agent will take for every decision that the agent might be called upon to make, for each possible piece of information that the agent may have at each time it is called to act. That is, a strategy can be thought as a complete contingency plan for an agent. We let Σi denote the set of all possible strategies for agent i, and thus si(θi) ∈ Σi. When it is clear from the context, we will drop the θi in order to simplify the notation. We let strategy profile s = (s1(θ1),...,sI(θI)) denote the outcome that results when each agent i is playing strategy si(θi). As a notational convenience we define s−i(θ−i) = (s1(θi),...,si−1(θi−1),si+1(θi+1),...,sI(θI)) and thus s = (si,s−i). We then interpret ui((si,s−i),θi) to be the utility of agent i with type θi when all agents play strategies specified by strategy profile (si(θi),s−i(θ−i)).
- 341. 16 Argumentation and Game Theory 327 Similarly, we also define: θ−i = (θ1,...,θi−1,θi+1,...,θI) Since the agents are all self-interested, they will try to choose strategies which maximize their own utility. Since the strategies of other agents also play a role in de- termining the outcome, the agents must take this into account. The solution concepts in game theory determine the outcomes that will arise if all agents are rational and strategic. The most well known solution concept is the Nash equilibrium. A Nash equilibrium is a strategy profile in which each agent is following a strategy which maximizes its own utility, given its type and the strategies of the other agents. Definition 16.1 (Nash Equilibrium). A strategy profile s∗ = (s∗ 1,...,s∗ I ) is a Nash equilibrium if no agent has incentive to change its strategy, given that no other agent changes. Formally, ∀i,∀s′ i,ui(s∗ i ,s∗ −i,θi) ≥ ui(s′ i,s∗ −i,θi). Although the Nash equilibrium is a fundamental concept in game theory, it does have several weaknesses. First, there may be multiple Nash equilibria and so agents may be uncertain as to which equilibrium they should play. Second, the Nash equi- librium implicitly assumes that agents have perfect information about all other agents, including the other agents’ preferences. A stronger solution concept in game theory is the dominant-strategy equilibrium. A strategy si is said to be dominant if by playing it, the utility of agent i is maximized no matter what strategies the other agents play. Definition 16.2 (Dominant Strategy). A strategy s∗ i is dominant if ∀s−i, ∀s′ i, ui(s∗ i ,s−i,θi) ≥ ui(s′ i,s−i,θi). Sometimes, we will refer to a strategy satisfying the above definition as weakly dominant. If the inequality is strict (i.e. instead of ≥), we say that the strategy is strictly dominant. A dominant-strategy equilibrium is a strategy profile where each agent is play- ing a dominant strategy. This is a very robust solution concept since it makes no assumptions about what information the agents have available to them, nor does it assume that all agents know that all other agents are being rational (i.e. trying to maximize their own utility). However, there are many strategic settings where no agent has a dominant strategy. A third solution concept is the Bayes-Nash equilibrium. We include it for the sake of completeness. In the Bayes-Nash equilibrium the assumption made for the Nash equilibrium, that all agents know the preferences of others, is relaxed. Instead, we assume that there is some common prior F((Θ1,...,ΘI)), such that the agents’ types are distributed according to F. Then, in equilibrium, each agent chooses the strategy that maximizes it’s expected utility given the strategies other agents are playing and the prior F.
- 342. 328 Iyad Rahwan and Kate Larson Definition 16.3 (Bayes-Nash Equilibrium). A strategy profile s∗ = (s∗ i ,s∗ −i) is a Bayes-Nash equilibrium if Eθ−i [ui((s∗ i (θi),s∗ −i(·)),θi)] ≥ Eθ−i [ui((s′ i(θi),s∗ −i(·)),θi)] ∀θi,∀s′ i. 3.2 Mechanism Design The problem that mechanism design studies is how to ensure that a desirable system- wide outcome or decision is made when there is a group of self-interested agents who have preferences over the outcomes. In particular, we often want the outcome to depend on the preferences of the agents. This is captured by a social choice function. Definition 16.4 (Social Choice Function). A social choice function is a rule f : Θ1 × ... ×ΘI → O, that selects some outcome f(θ) ∈ O, given agent types θ = (θ1,...,θI). The challenge, however, is that the types of the agents (the θ′ i s) are private and known only to the agents themselves. Thus, in order to select an outcome with the social choice function, one has to rely on the agents to reveal their types. However, for a given social choice function, an agent may find that it is better off if it does not reveal its type truthfully, since by lying it may be able to cause the social choice function to choose an outcome that it prefers. Instead of trusting the agents to be truthful, we use a mechanism to try to reach the correct outcome. A mechanism M = (Σ,g(·)) defines the set of allowable strategies that agents can chose, with Σ = Σ1 × ··· × ΣI where Σi is the strategy set for agent i, and an outcome function g(s) which specifies an outcome o for each possible strategy pro- file s = (s1,...,sI) ∈ Σ. This defines a game in which agent i is free to select any strategy in Σi, and, in particular, will try to select a strategy which will lead to an outcome that maximizes its own utility. We say that a mechanism implements social choice function f if the outcome induced by the mechanism is the same outcome that the social choice function would have returned if the true types of the agents were known. Definition 16.5 (Implementation). A mechanism M = (Σ,g(·)) implements social choice function f if there exists an equilibrium s∗ such that ∀θ ∈ Θ, g(s∗ (θ)) = f(θ). While the definition of a mechanism puts no restrictions on the strategy spaces of the agents, an important class of mechanisms are the direct-revelation mechanisms (or simply direct mechanisms). Definition 16.6 (Direct-Revelation Mechanism). A direct-revelation mechanism is a mechanism in which Σi = Θi for all i, and g(θ) = f(θ) for all θ ∈ Θ. In words, a direct mechanism is one where the strategies of the agents are to an- nounce a type, θ′ i to the mechanism. While it is not necessary that θ′ i = θi, the
- 343. 16 Argumentation and Game Theory 329 important Revelation Principle (see below for more details) states that if a social choice function, f(·), can be implemented, then it can be implemented by a direct mechanism where every agent reveals its true type [5]. In such a situation, we say that the social choice function is incentive compatible. Definition 16.7 (Incentive Compatible). The social choice function f(·) is incen- tive compatible (or truthfully implementable) if the direct mechanism M = (Θ,g(·)) has an equilibrium s∗ such that s∗ i (θi) = θi. If the equilibrium concept is the dominant-strategy equilibrium, then the social choice function is strategy-proof. In this chapter we will on occasion call a mecha- nism incentive-compatible or strategy-proof. This means that the social choice func- tion that the mechanism implements is incentive-compatible or strategy-proof. 3.3 The Revelation Principle Determining whether a particular social choice function can be implemented, and in particular, finding a mechanism which implements a social choice function appears to be a daunting task. In the definition of a mechanism, the strategy spaces of the agents are unrestricted, leading to an infinitely large space of possible mechanisms. However, the Revelation Principle states that we can limit our search to a special class of mechanisms [5, Ch 14]. Theorem 16.1 (Revelation Principle). If there exists some mechanism that imple- ments social choice function f in dominant strategies, then there exists a direct mechanism that implements f in dominant strategies and is truthful. The intuitive idea behind the Revelation Principle is fairly straightforward. Sup- pose that you have a, possibly very complex, mechanism, M, which implements some social choice function, f. That is, given agent types θ = (θ1,...,θI) there ex- ists an equilibrium s∗(θ) such that g(s∗(θ)) = f(θ). Then, the Revelation Principle states that it is possible to create a new mechanism, M′, which, when given θ, will then execute s∗(θ) on behalf of the agents and then select outcome g(s∗(θ)). Thus, each agent is best off revealing θi, resulting in M′ being a truthful, direct mechanism for implementing social choice function f. The Revelation Principle is a powerful tool when it comes to studying imple- mentation. Instead of searching through the entire space of mechanisms to check whether one implements a particular social choice function, the Revelation Prin- ciple states that we can restrict our search to the class of truthful, direct mecha- nisms. If we can not find a mechanism in this space which implements the social choice function of interest, then there does not exist any mechanism which will do so. It should be noted that while the Revelation Principle is a powerful analysis tool, it does not imply we should only design direct mechanisms. Some reasons why one rarely sees direct mechanisms in the “real world” include (among others);
- 344. 330 Iyad Rahwan and Kate Larson • they can place a high computational burden on the mechanism since it is required to execute agents’ strategies, • agents’ strategies may be computationally difficult to determine, and • agents may not be willing to reveal their true types because of privacy concerns. 4 Argumentation Mechanism Design Mechanism design (MD) is a sub-field of game theory concerned with the follow- ing question: what game rules guarantee a desirable social outcome when each self-interested agent selects the best strategy for itself? In other words, while game theory is concerned with a given strategic situation modelled as a game, mechanism design is concerned with designing the game itself. As such, one might actually call it reverse game theory. In this section we define the mechanism design problem for abstract argumenta- tion. We dub this new approach ‘Argumentation Mechanism Design’ (ArgMD). Let AF = A,R be an argumentation framework with a set of arguments A and a binary defeat relation R. We define a mechanism with respect to AF and semantics S, and we assume that there is a set of I self-interested agents. We define an agent’s type to be its set of arguments. Definition 16.8 (Agent Type). Given an argumentation framework A,R, the type of agent i, Ai ⊆ A, is the set of arguments that the agent is capable of putting for- ward. There are two things to note about this definition. Firstly, an agent’s type can be seen as a reflection of its expertise or domain knowledge. For example, medical experts may only be able to comment on certain aspects of forensics in a legal case, while a defendant’s family and friends may be able to comment on his/her character. Also, such expertise may overlap, so agent types are not necessarily disjoint. For example, two medical doctors might have some identical argument, and so on. The second thing to note about the definition is that agent types do not include the defeat relation. In other words, we implicitly assume that the notion of defeat is common to all agents. That is, given two arguments, no agent would dispute whether one attacks another. This is a reasonable assumption in systems where agents use the same logic to express arguments or at least multiple logics for which the notion of defeat is accepted by everyone (e.g. conflict between a proposition and its nega- tion). Disagreement over the defeat relation itself requires a form of hierarchical (meta) argumentation [7], which is a powerful concept, but is beyond the scope of the present chapter. Given the agents’ types (argument sets) a social choice function f maps a type profile into a subset of arguments; f : 2A ×...×2A → 2A
- 345. 16 Argumentation and Game Theory 331 While our definition of an argumentation mechanism will allow for generic social choice functions which map type profiles into subsets of arguments, we will be particularly interested in argument acceptability social choice functions. We denote by Acc(A,R,S) ⊆ A the set of acceptable arguments according to semantics S.3 Definition 16.9 (Argument Acceptability Social Choice Functions). Given an ar- gumentation framework A,R with semantics S, and given an agent type profile (A1,...,AI), the argument acceptability social choice function f is defined as the set of acceptable arguments given the semantics S. That is, f(A1,...,AI) = Acc(A1 ∪...∪AI,R,S). As is standard in the mechanism design literature, we assume that agents have preferences over the outcomes o ∈ 2A , and we represent these preferences using utility functions where ui(o,Ai) denotes agent i’s utility for outcome o when its type is argument set Ai. Agents may not have incentive to reveal their true type because they may be able to influence the final argument status assignment by lying, and thus obtain higher utility. There are two ways that an agent can lie in our model. On one hand, an agent might create new arguments that it does not have in its argument set. In the rest of the chapter we will assume that there is an external verifier that is capable of checking whether it is possible for a particular agent to actually make a particular argument. Informally, this means that presented arguments, while still possibly de- feasible, must at least be based on some sort of demonstrable ‘plausible evidence.’ If an agent is caught making up arguments then it will be removed from the mech- anism. For example, in a court of law, any act of perjury by a witness is punished, at the very least, by completely discrediting all evidence produced by the witness. Moreover, in a court of law, arguments presented without any plausible evidence are normally discarded (e.g. “I did not kill him, since I was abducted by aliens at the time of the crime!”). For all intents and purposes this assumption (also made by Glazer and Rubinstein [2]) removes the incentive for an agent to make up facts. A more insidious form of manipulation occurs when an agent decides to hide some of its arguments. By refusing to reveal certain arguments, an agent might be able to break defeat chains in the argument framework, thus changing the final set of acceptable arguments. For example, a witness may hide evidence that implicates the defendant if the evidence also undermines the witness’s own character. This type of lie is almost impossible to detect in practice, and it is this form of strategic behaviour that we will be the most interested in. As mentioned earlier, a strategy of an agent specifies a complete plan that de- scribes what action the agent takes for every decision that a player might be called upon to take, for every piece of information that the player might have at each time that it is called upon to act. In our model, the actions available to an agent involve announcing sets of arguments. Thus a strategy si ∈ Σi for agent i would specify for 3 Here, we assume that S specifies both the classical semantics used (e.g. grounded, preferred, stable) as well as the acceptance attitude used (e.g. sceptical or credulous).
- 346. 332 Iyad Rahwan and Kate Larson each possible subset of arguments that could define its type, what set of arguments to reveal. For example, a strategy might specify that an agent should reveal only half of its arguments without waiting to see what other agents are going to do, while another strategy might specify that an agent should wait and see what arguments are revealed by others, before deciding how to respond. In particular, beyond specifying that agents are not allowed to make up arguments, we place no restrictions on the allowable strategy spaces, when we initially define an argumentation mechanism. Later, when we talk about direct argumentation mechanisms we will further restrict the strategy space. We are now ready to define our argumentation mechanism. We first define a generic mechanism, and then specify a direct argumentation mechanism, which due to the Revelation Principle, is the type of mechanism we will study in the rest of the chapter. Definition 16.10 (Argumentation Mechanism). Given an argumentation frame- work AF = A,R and semantics S, an argumentation mechanism is defined as MS AF = (Σ1,...,ΣI,g(·)) where Σi is an argumentation strategy space of agent i and g : Σ1 ×...×ΣI → 2A . Note that in the above definition, the notion of dialogue strategy is broadly con- strued and would depend on the protocol used. In a direct mechanism, however, the strategy spaces of the agents are restricted so that they can only reveal a subset of arguments. Definition 16.11 (Direct Argumentation Mechanism). Given an argumentation framework AF = A,R and semantics S, a direct argumentation mechanism is de- fined as MS AF = (Σ1,...,ΣI,g(·)) where Σi = 2A and g : Σ1 ×...ΣI → 2A . In Table 16.1, we summarise the mapping of multi-agent abstract argumentation as an instance of a mechanism design problem. MD Concept ArgMD Instantiation Agent type θi ∈ Θi Agent’s arguments θi = Ai ⊆ A Outcome o ∈ O Accepted arguments Acc(.) ⊆ A Utility ui(o,θi) Preferences over 2A (what arguments end up being accepted) Social choice function f : Θ1 ×...×ΘI → O f(A1,...,AI ) = Acc(A1 ∪...∪AI,R,S). by some argument acceptability criterion Mechanism M = (Σ,g(·)) where Σ = Σ1 ×···×ΣI and g : Σ → O Σi is an argumentation strategy, g : Σ → 2A Direct mechanism: Σi = Θi Σi = 2A (every agent reveals a set of arguments) Truth revelation Revealing Ai Table 16.1 Abstract argumentation as a mechanism
- 347. 16 Argumentation and Game Theory 333 5 Case Study: Implementing the Grounded Semantics In this section, we demonstrate the power of our ArgMD approach by showing how it can be used to systematically analyse the strategic incentives imposed by a well-established argument evaluation criterion. In particular, we specify a direct- revelation argumentation mechanism, in which agents’ strategies are to reveal sets of arguments, and where the mechanism calculates the outcome using sceptical (grounded) semantics.4 That is, we look at the grounded semantics as if it was designed as a mechanism and analyse it from that perspective. We show that, in general, this mechanism gives rise to strategic manipulation. We prove, however, that under various conditions, this mechanism turns out to be strategy-proof. In a direct argumentation mechanism, each agent i’s available actions are Σi = 2A . We will refer to a specific action (i.e. set of declared arguments) as A◦ i ∈ Σi. We now present a direct mechanism for argumentation based on a sceptical ar- gument evaluation criteria. The mechanism calculates the grounded extension given the union of all arguments revealed by agents. Definition 16.12 (Grounded Direct Argumentation Mechanism). A grounded di- rect argumentation mechanism for argumentation framework A,R is Mgrnd AF = (Σ1,...,ΣI,g(.)) where: – Σi ∈ 2A is the set of strategies available to each agent; – g : Σ1 ×···×ΣI → 2A is an outcome rule defined as: g(A◦ 1,...,A◦ I ) = Acc(A◦ 1 ∪ ···∪A◦ I ,R,Sgrnd) where Sgrnd denotes sceptical grounded acceptability seman- tics. To simplify our analysis, we will assume below that agents can only lie by hiding arguments, and not by making up arguments. Formally, this means that ∀i, Σi ∈ 2Ai . For the sake of illustration, we will consider a particular family of preferences that agents may have. According to these preferences, every agent attempts to max- imise the number of arguments in Ai that end up being accepted. We call this pref- erence criteria the individual acceptability maximising preference. Definition 16.13 (Acceptability maximising preferences). An agent i has indi- vidual acceptability maximising preferences if and only if ∀o1,o2 ∈ O such that |o1 ∩Ai| ≥ |o2 ∩Ai|, we have ui(o1,Ai) ≥ ui(o2,Ai). Let us now consider aspects of incentives using mechanism Mgrnd AF through an example. Example 16.1. Consider grounded direct argumentation mechanism with three agents x, y and z with types Ax = {α1,α4,α5}, Ay = {α2} and Az = {α3} respectively. And suppose that the defeat relation is defined as follows: R = {(α1,α2), (α2,α3), (α3,α4), (α3,α5)}. If each agent reveals its true type (i.e. A◦ x = Ax; A◦ y = Ay; and 4 In the remainder of the chapter, we will use the term sceptical to refer to sceptical grounded, since the chapter focuses on the grounded semantics.
- 348. 334 Iyad Rahwan and Kate Larson α1 α2 α3 α4 α5 (a) Argument graph in case of full revelation α4 α3 α2 α5 (b) Argument graph with α1 withheld Fig. 16.3 Hiding an argument is beneficial (case of acceptability maximisers) A◦ z = Az), then we get the argument graph depicted in Figure 16.3(a). The mecha- nism outcome rule produces the outcome o = {α1,α3}. If agents have individual ac- ceptability maximising preferences, with utilities equal to the number of arguments accepted, then: ux(o,{α1,α4,α5}) = 1; uy(o,{α3}) = 1; and uz(o,{α2}) = 0. It turns out that the mechanism is susceptible to strategic manipulation, even if we suppose that agents do not lie by making up arguments (i.e., they may only withhold some arguments). In this case, for both agents y and z, revealing their true types weakly dominates revealing nothing at all. However, it turns out that agent x is better off revealing {α4,α5}. By withholding α1, the resulting argument network becomes as depicted in Figure 16.3(b), for which the output rule produces the outcome o′ = {α2,α4,α5}. This outcome yields utility 2 to agent x, which is better than the truth-revealing strategy. Remark 16.1. Given an arbitrary argumentation framework AF and agents with ac- ceptability maximising preferences, mechanism Mgrnd AF is not strategy-proof. The following theorem provides a full characterisation of strategy-proof mech- anisms for sceptical argumentation frameworks for agents with acceptability max- imising preferences. Theorem 16.2. Let AF be an arbitrary argumentation framework, and let EGR(AF) denote its grounded extension. Mechanism Mgrnd AF is strategy-proof for agents with acceptability maximising preferences if and only if AF satisfies the following con- dition: ∀i ∈ I,∀S ⊆ Ai and ∀A−i, we have |Ai ∩ EGR(Ai ∪ A−i,R)| ≥ |Ai ∩ EGR((AiS)∪A−i,R)|. Although the above theorem gives us a full characterisation, it is difficult to apply in practice. In particular, the theorem does not give us an indication of how agents (or the mechanism designer) can identify whether the mechanism is strategy-proof for a class of argumentation frameworks by appealing to their graph-theoretic prop- erties. Below, we provide an intuitive, graph-theoretic condition that is sufficient to ensure that Mgrnd AF is strategy-proof when agents have focal arguments. Let α,β ∈ A. We say that α indirectly defeats β, written α ֒→ β, if and only if there is an odd-length path from α to β in the argument graph.
- 349. 16 Argumentation and Game Theory 335 Theorem 16.3. Suppose agents have individual acceptability maximising prefer- ences. If each agent’s type corresponds to a conflict-free set of arguments which does not include (in)direct defeats (formally ∀i∄α1,α2 ∈ Ai such that α1 ֒→ α2), then Mgrnd AF is strategy-proof. Note that in the theorem, ֒→ is over all arguments in A. Intuitively, the condition in the theorem states that all arguments of every agent must be conflict-free (i.e. consistent), both explicitly and implicitly. Explicit consistency implies that no argu- ment defeats another. Implicit consistency implies that other agents cannot possibly present a set of arguments that reveal an indirect defeat among one’s own argu- ments. More concretely, in Example 16.1 and Figure 16.3, while agent x’s argument set Ax = {α1,α4,α5} is conflict-free, when agents y and z presented their own argu- ments α2 and α3, they revealed an implicit conflict in x’s arguments. In other words, they showed that x contradicts himself (i.e. committed a fallacy of some kind). In addition to characterising a sufficient graph-theoretic condition for strategy- proofness, Theorem 16.3 is useful for individual agents. As long as the agent knows that it is not possible for a path to be created which causes an (in)direct defeat among its arguments (i.e., a fallacy to be revealed), then the agent is best off revealing all its arguments. The agent only needs to know that no argument imaginable can reveal conflicts among its own arguments. We now ask whether the sufficient condition in Theorem 16.3 is also necessary for agents to reveal all their arguments truthfully. Example 16.2 shows that this is not the case. In particular, for certain argumentation frameworks, an agent may have truthtelling as a dominant strategy despite the presence of indirect defeats among its own arguments. α1 α6 α2 α3 α4 α5 Fig. 16.4 Strategy-proofness despite indirect self-defeat Example 16.2. Consider the variant of Example 16.1 with the additional argument α6 and defeat (α6,α3). Let the agent types be Ax = {α1,α4,α5,α6}, Ay = {α2} and Az = {α3} respectively. The full argument graph is depicted in Figure 16.4. Under full revelation, the mechanism outcome rule produces the outcome o = {α1,α4,α5,α6}. Note that in Example 16.2, truth revelation is now a dominant strategy for x (since it gets all its arguments accepted) despite the fact that α1 ֒→ α4 and α1 ֒→ α5. This hinges on the presence of an argument (namely α5) that cancels out the negative effect of the (in)direct self-defeat among x’s own arguments.
- 350. 336 Iyad Rahwan and Kate Larson 6 Related Work 6.1 Pareto Optimality of Outcomes A well-known property of the grounded semantics is that it is extremely sceptical, accepting only undefeated arguments and arguments defended by undefeated argu- ments. An interesting question, then, is whether it is possible to be more inclusive (i.e. being more credulous) in order to produce argumentation outcomes that are more socially desirable. For example, consider the simple argument graph in Figure 16.5 and suppose we have two agents with types A1 = {α1} and A2 = {α2} who both reveal their arguments. The grounded extension (Figure 16.5(a)) is empty here. α2 α1 (a) α2 α1 (b) α2 α1 (c) Fig. 16.5 Preferred extensions ‘dominate’ the grounded extension Suppose the judge chooses one of the preferred extensions instead (Figure 16.5(b) or (c)). Clearly, when compared to outcome (a), each preferred extension makes one agent better-off without making the other worse-off. Formally, we say that outcomes (b) and (c) each Pareto dominates outcome (a). An outcome that is not Pareto dominated is called Pareto optimal. Recently, Rahwan and Larson [11] presented an extensive analysis of Pareto op- timality in abstract argumentation, and established correspondence results between different semantics on one hand and the Pareto optimal outcomes on the other. 6.2 Glazer and Rubinstein’s Model Another game-theoretic analysis of argumentation was presented by Glazer and Ru- binstein [2]. The authors explore the mechanism design problem of constructing rules of debate that maximise the probability that a listener reaches the right con- clusion given arguments presented by two debaters. They study a very restricted setting, in which the world state is described by a vector ω = (w1,...,w5), where each ‘aspect’ wi has two possible values: 1 and 2. If wi = j for j ∈ {1,2}, we say that aspect wi supports outcome Oj. Presenting an argument amounts to revealing the value of some wi. The setting is modelled as an extensive-form game and anal- ysed. In particular, the authors investigate various combinations of procedural rules (stating in which order and what sorts of arguments each debater is allowed to state) and persuasion rules (stating how the outcome is chosen by the listener). In terms of procedural rules, the authors explore: (1) one-speaker debate in whichone debater
- 351. 16 Argumentation and Game Theory 337 chooses two arguments to reveal; (2) simultaneous debate in which the two debaters simultaneously reveal one argument each; and (3) sequential debate in which one debater reveals one argument followed by one argument by the other. Our mecha- nism is closer to the simultaneous debate, but is much more general as it enables the simultaneous revelation of an arbitrary number of arguments. Glazer and Rubin- stein investigate a variety of persuasion rules. For example, in one-speaker debate, one rule analysed by the authors states that ‘a speaker wins if and only if he presents two arguments from {a1,a2,a3} or {a4,a5}.’ In a sequential debate, one persuasion rule states that ‘if debater D1 argues for aspect a3, then debater D2 wins if and only if he counter-argues with aspect a4.’ These kinds of rules are arbitrary and do not fol- low an intuitive notion of persuasion (e.g. like scepticism). The sceptical mechanism presented in this chapter provides a more natural criterion for argument evaluation, supplemented by a strong solution concept that ensures all agents have incentive to reveal their arguments, and thus for the listener to reach the correct outcome. More- over, our framework for argumentation mechanism design is more general in that it can be used to model a variety of more complex argumentation settings. 6.3 Game Semantics It is worth contrasting our work with work on so-called game semantics for logic, which was pioneered by logicians such as Paul Lorenzen [4] and Jaakko Hintikka [3]. Although many specific instantiations of this notion have been presented in the literature, the general idea is as follows. Given some specific logic, the truth value of a formula is determined through a special-purpose, multi-stage dialogue game between two players, the verifier and falsifier. The formula is considered true precisely when the verifier has a winning strategy, while it will be false whenever the falsifier has the winning strategy. Similar ideas have been used to implement dialectical proof-theories for defeasible reasoning (e.g. by Prakken and Sartor [9]). In a related development, Matt and Toni recently proposed a game-theoretic ap- proach to characterise argument strength [6]. Thus, the acceptability of each argu- ment is rated between 0 and 1 by using a two-person zero-sum game with imperfect information between a proponent and an opponent. There is a fundamental difference between the aims of game semantics and our ArgMD approach. In game semantics, the goal is to interpret (i.e., characterise the truth value of) a specific formula by appealing to a notion of a winning strategy. As such, each player is carefully endowed with a specific set of formulae to en- able the game to characterise semantics correctly (e.g. the verifier may own all the disjunctions in the formula, while the falsifier is given all the conjunctions). In contrast, ArgMD is about designing rules for argumentation among self- interested players who may have incentives to manipulate the outcome given a va- riety of possible individual preferences (specified in arbitrary instantiations of a utility function). Our interest is in conditions that guarantee truth-revelation given different classes of preferences. Game semantics have no similar notion of strategic
- 352. 338 Iyad Rahwan and Kate Larson manipulation by hiding information. Moreover, our framework allows an arbitrary number of players (as opposed to two agents). 6.4 Argumentation and Cooperative Games Another notable early link between argumentation and cooperative game theory has been proposed by Dung in his seminal paper [1]. Let A = {a1,...,a|A|} be a set of agents. A cooperative game is defined by specifying a value to V(C) to each coalition C ⊆ A of agents. An outcome of the game is a vector u = (ua1 ,...,ua|A| ) ∈ R|A| specifying a vector of utilities, one per agent. Outcome u dominates outcome u′ if there is a (nonempty) coalition K ⊆ A in which agents get more utility (as a whole) in u than in u′. An outcome is said to be stable if no outcome dominates it (i.e. if no subset of agents has incentive to leave their own coalition and be all individually better off). A solution of the cooperative game is a set of outcomes S satisfying the following conditions: 1. No s ∈ S is dominated by an s′ ∈ S. 2. Every s / ∈ S is dominated by some s′ ∈ S. Dung argued that an n-person game can be seen as an argumentation framework A,R in which the set of arguments A is the set of all possible outcomes of the co- operative game, and the defeat relation is defined as R = {(u,u′) | u dominates u′}. Dung shows that with this characterisation, the set of solutions to the cooperative game corresponds to the set of stable extensions of an abstract argumentation frame- work. This enabled Dung to describe the well-known stable marriage problem as a problem of finding a stable extension. Another important notion in cooperative game theory is that of the core: the set of (feasible) outcomes which are not dominated by any other outcome. Dung showed that the core corresponds to F(∅) where F is the characteristic function of the corresponding argumentation framework. 7 Conclusion In this chapter, our aim was to demonstrate the importance of game theory as a tool for analysing strategic argumentation. We showed how normal form games and extensive-form games can be used to analyse equilibrium strategies in strategic ar- gumentation. We then introduced Argumentation Mechanism Design (ArgMD) as a new framework for designing and analysing argument evaluation criteria. With ArgMD, designing new argument acceptance criteria becomes akin to designing auction protocols in strategic multi-agent settings. The goal is to design rules that ensure, under precise conditions, that agents have no incentive to manipulate the
- 353. 16 Argumentation and Game Theory 339 outcome. We believe this approach will become increasingly important before argu- mentation can be applied in open agent systems. References 1. P. M. Dung. On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence, 77(2):321–358, 1995. 2. J. Glazer and A. Rubinstein. Debates and decisions: On a rationale of argumentation rules. Games and Economic Behavior, 36:158–173, 2001. 3. J. Hintikka and G. Sandu. Game-theoretical semantics. In J. van Benthem and A. ter Meulen, editors, Handbook of Logic and Language, pages 361–410. Elsevier, Amsterdam, The Nether- lands, 1997. 4. P. Lorenzen. Ein dialogisches konstruktivitätskriterium. In Infinitistic Methods, pages 193– 200. Pergamon Press, Oxford, UK, 1961. 5. A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, New York NY, USA, 1995. 6. P.-A. Matt and F. Toni. A game-theoretic measure of argument strength for abstract argumen- tation. In S. Hölldobler, C. Lutz, and H. Wansing, editors, Logics in Artificial Intelligence, 11th European Conference, JELIA 2008, volume 5293 of Lecture Notes in Computer Science, pages 285–297. 2008. 7. S. Modgil. Hierarchical argumentation. In Proceedings of the 10th European Conference on Logics in Artificial Intelligence. Liverpool, UK, 2006. 8. H. Prakken. Coherence and flexibility in dialogue games for argumentation. Journal of Logic and Computation, 15(6):1009–1040, 2005. 9. H. Prakken and G. Sartor. Argument-based logic programming with defeasible priorities. Journal of Applied Non-classical Logics, 7:25–75, 1997. 10. A. D. Procaccia and J. S. Rosenschein. Extensive-form argumentation games. In Proceedings of the Third European Workshop on Multi-Agent Systems (EUMAS-05), Brussels, Belgium, pages 312–322, 2005. 11. I. Rahwan and K. Larson. Pareto optimality in abstract argumentation. In D. Fox and C. Gomes, editors, Proceedings of the 23rd AAAI Conference on Artificial Intelligence (AAAI- 2008), Menlo Park CA, USA, 2008. 12. R. Riveret, H. Prakken, A. Rotolo, and G. Sartor. Heuristics in argumentation: A game- theoretical investigation. In P. Besnard, S. Doutre, and A. Hunter, editors, Proceedings of the 2nd International Conference on Computational Models of Argument (COMMA), pages 324–335. IOS Press, Amsterdam, The Netherlands, 2008. 13. J. von Neuman and O. Morgenstern. The Theory of Games and Economic Behaviour. Prince- ton University Press, Princeton NJ, USA, 1944.
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- 355. Chapter 17 Belief Revision and Argumentation Theory Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari 1 Introduction Belief revision is the process of changing beliefs to adapt the epistemic state of an agent to a new piece of information. The logical formalization of belief revision is a topic of research in philosophy, logic, and in computer science, in areas such as databases or artificial intelligence. On the other hand, argumentation is concerned primarily with the evaluation of claims based on premises in order to reach conclu- sions. Both provide basic and substantial techniques for the art of reasoning, as it is performed by human beings in everyday life situations and which goes far beyond logical deduction. Reasoning, in this sense, makes possible to deal successfully with problems in uncertain, dynamic environments and has been promoting the develop- ment of human societies. The interest of computer scientists in both domains has increased considerably over the past years, as agent systems are to be endowed with similar capabilities. In an agent environment, belief revision describes the way in which an agent is supposed to change her beliefs when new information arrives, or changes in the world are observed; argumentation deals with strategies agents em- ploy for their own reasoning, or to change the beliefs of other agents, by providing reasons for such change. In this chapter, we will elaborate on the relationships between argumentation and belief revision, first recalling important work done by others and ourselves concern- Marcelo A. Falappa CONICET (National Council of Technical and Scientific Research) – Department of Computer Science and Engineering – Universidad Nacional del Sur – Bahı́a Blanca, Argentina, e-mail: mfalappa@cs.uns.edu.ar Gabriele Kern-Isberner Department of Computer Science – University of Dortmund – Dortmund, Germany – e-mail: gabriele.kern-isberner@cs.uni-dortmund.de Guillermo R. Simari Department of Computer Science and Engineering – Universidad Nacional del Sur – Bahı́a Blanca, Argentina, e-mail: grs@cs.uns.edu.ar I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 341 DOI 10.1007/978-0-387-98197-0 17, c Springer Science+Business Media, LLC 2009
- 356. 342 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari ing the links between both areas. Based on such insights, we will develop a con- ceptual view on this topic, which is based on the understanding of argumentation and belief revision being complementary disciplines for the broad picture sketched above. Each needs the other’s support if we want to model successful decision mak- ing in a real world application. We will also discuss how one area may contribute to the other, enriching the respective framework. 2 Basic facts on argumentation and belief revision An argument A for α is a set of interrelated pieces of knowledge supporting α from evidence. Abstract approaches to argumentation such as [13] make use of ar- gumentation frameworks A,R with is a finite set A of arguments and an attack relation R among arguments. The main issue of such argumentation frameworks is the selection of acceptable sets of arguments called extensions on the basis of which argumentation semantics can be defined. We assume that the reader is familiar with the basic concepts from argumentation theory such as conflict, attack, non-attack, defeat, evaluation of arguments, etc. In classical belief revision frameworks, a fixed finite language L with a complete set of boolean connectives is usually used. Formulae in L are be denoted by lower case Greek characters α,β,δ,..., while sets of formulae from L will be denoted by upper case letters A,B,C,.... The underlying logic contains a consequence operator Cn (Cn : 2L ⇒ 2L ) and it is assumed to be supraclassical (i.e., include classical propositional calculus), compact (Cn(A) = Cn(B) for some finite subset B of A) and to satisfy the deduction theorem (α → β ∈ Cn(A) if and only if β ∈ Cn(A∪{α})). Sometimes, the relation ⊢ is used as an alternative notation of the consequence op- erator: A ⊢ α if and only if α ∈ Cn(A). There are many different frameworks for belief revision with their respective epistemic models. The epistemic model is the formalism in which beliefs are rep- resented and in which different kinds of operators can be defined. The basic repre- sentation of epistemic states is through belief sets (sets of sentences closed under logical consequence) or belief bases (sets of sentences not necessarily closed). Op- erators may be presented in two ways: by giving an explicit construction (algorithm) for the operator, or by giving a set of rationality postulates to be satisfied. Rational- ity postulates determine constraints that the operators should satisfy. They treat the operators as black boxes by describing their response behavior to inputs in basic cases, but not the internal mechanisms used. The distinction between belief sets and belief bases is similar to the distinction between the coherence approach and the foundational approach to belief revision. The coherence approach focuses on logical relations among beliefs rather than on inferential relations, that is, no belief is more fundamental than others [11]. Beliefs provide each other with mutual support; therefore, a belief set represents the limit case of this approach. On the other hand, the foundational approach divides beliefs into two classes: explicit beliefs and those beliefs justified by the explicit beliefs.
- 357. 17 Belief Revision and Argumentation Theory 343 The explicit beliefs can be seen as “self-justified beliefs” whereas the other beliefs are considered as derived, justified or supported beliefs. The foundational approach provides explanation of beliefs by requiring that each belief is supportable by means of non-circular arguments from explicit or basic beliefs [11]. Since a belief α may be justified by several independent beliefs, if some of the justifications for α are removed, α may be retained because it is supported by other beliefs. Belief base revision provides a good example of the foundational approach. The AGM paradigm [1] has been extensively studied during the last three decades and it is the most influential model of belief revision so far, serving as a frame of reference for improvements, extensions or criticisms of the original proposal. The AGM model is conceived as an idealistic theory of rational change in which epis- temic states are represented by belief sets and the epistemic input is represented by a sentence. AGM theory studies the changes at the knowledge level whereas some others approaches studies the changes at the symbol level. The distinction about knowledge and symbol level was proposed by Allen Newell [43]. According to Newell, the knowledge level lies above the symbol level where all of the knowledge in the system is represented. Some belief bases with different symbol representa- tions may represent the same knowledge. Suppose that p and q are logically inde- pendent propositions. Then, K1 = {p,q} and K2 = {p, p → q} are different belief bases. Since their closure is the same Cn(K1) = Cn(K2) = Cn({p,q}) they repre- sent the same beliefs at the knowledge level. Although they are statically equivalent (they represent the same beliefs), they could be dynamically different: changes of K1 can be different from changes of K2 because p and q are totally independent in K1 but interrelated in K2 by the sentence p → q. Suppose that the set K represents the beliefs of an agent. The AGM theory defines three basic types of change operations for K: • Expansions: the result of expanding K by a sentence α is a larger set which infers α; • Contractions: the result of contracting K by α is a smaller set which does not infer α; • Revisions: the result of revising K by α is a set that neither extends nor is part of the set K. In general, if K infers ¬α then α is consistently inferred from the revision of K by α. In the classical AGM framework and all coherence approaches, the internal structure of beliefs is ignored and the focus is on belief sets. Given a belief set K1 and a sentence α, the expansion of K by α is denoted by K+α and is defined as K+α = Cn(K∪{α}). Suppose that ÷ is a contraction operator. The contraction of K by α is denoted as K÷α. The basic postulates for contraction are [1, 20] the following: • Closure: K÷α = Cn(K÷α). • Inclusion: K÷α ⊆ K. 1 Belief sets (i.e., logically closed sets of sentences) are typically denoted by Latin letters in bold- face.
- 358. 344 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari • Success: If α then α ∈K÷α. • Vacuity: If α ∈K then K÷α = K. • Recovery: K ⊆ (K÷α)+α. • Extensionality: If ⊢ α ↔ β then K÷α = K÷β. Recovery is one of the most controversial postulates of the AGM theory: it says that a sequence of first contracting by a sentence α, and then expanding by the same sentence α leaves the belief state unchanged. In other words, so much is retained in a belief set that everything can be recovered by reinstatement of the contracted sentence. Revision operators can be defined through Levi identity [38, 20] from contraction operators; in order to revise a belief set K with respect to a sentence α, we contract with respect to ¬α and then expand the new epistemic state with respect to α. That is, the revision of K by α, noted by K∗α, is defined as K∗α = (K÷¬α)+α. There is a set of basic postulates for revision that correspond to the above postulates for contraction. One of the most important (though not indisputable) properties is success (α ∈ K∗α) which specifies that the new information has primacy over the beliefs of an agent. 2.1 Changes on belief sets Partial meet contractions cover exactly the basic AGM-contractions. They are based on the concept of remainder sets. Definition 17.1. [2] Let K be a set of sentences and α a sentence. Then K⊥α is the set of all H such that H ⊆ K, H α and if H ⊂ K′ ⊆ K,H = K′, then K′ ⊢ α. The set K⊥α is called the remainder set of K with respect to α, and its elements are called the α-remainders of K. In order to define a partial meet contraction operator, a selection function is needed. This function makes a selection among the α-remainders, choosing those candidate sets for contraction which are preferred. Definition 17.2. Let K be a set of sentences. A selection function for K is a function γ defined on {K⊥α|α ∈ L} such that for any sentence α ∈ L, it holds that: 1) If K⊥α = ∅, then γ(K⊥α) is a non-empty subset of K⊥α. 2) If K⊥α = ∅, then γ(K⊥α) = K. Definition 17.3. Given a set of sentences K, a sentence α and a selection function γ for K, the partial meet contraction of K by α, denoted by K÷γα, is defined as K÷γα = ∩γ(K⊥α). That is, K÷γα is equal to the intersection of the α-remainders of K selected by γ. If γ selects a single element, then the induced contraction is called maxichoice contraction. If γ selects all elements of the remainder set, then the induced contraction is called full meet contraction.
- 359. 17 Belief Revision and Argumentation Theory 345 Representation theorems (also called axiomatic characterizations) characterize an operation in terms of axioms or postulates. They are widely used in belief re- vision in order to show an interrelation between constructions and postulates. The following is a representation theorem for partial meet contractions. Theorem 17.1. [1, 20] Let K be a belief set and ÷ be a contraction operator for K. Then ÷ is a partial meet contraction for K if and only if ÷ satisfies closure, inclusion, success, vacuity, recovery and extensionality. Partial meet revisions are defined from partial meet contractions using Levi iden- tity. On the other hand, partial meet contractions on belief sets can be defined from partial meet revisions using the Harper identity [20]: K÷α = K∩(K∗¬α). A rep- resentation theorem for partial meet revisions on belief sets can be found in [1]. 2.2 Changes on belief bases Following the seminal work of the AGM trio, almost all works on belief revision employed models in which epistemic states are represented by theories or belief sets, that is, set of sentences closed under logical consequence. However, it was observed by Alchourrón and Makinson [2] that “the intuitive processes of contraction and revision, contrary to casual impressions, are never really applied to theories as a whole, but rather to more or less clearly identified bases for them”. Perhaps inspired by this observation, operations on belief bases have been investigated, among other, by Hansson [26, 29, 30], Fuhrmann [17], and Nayak [42]. Fuhrmann [17] defined a contraction operator − on a belief base K and then studied the properties of its related contraction operator ÷ on belief sets K = Cn(K) defined as K÷α = Cn(K−α). That is, the definition of K÷α actually depends on K and α. Fuhrmann proposed a set of postulates for the operator − on belief bases based on the basic postulates on belief sets proposed by Gärdenfors, showing that recovery does not hold on belief bases. Hansson proposed an alternative approach, studying the changes on belief bases and not on their associated belief sets. He proposed an alternative contraction, called kernel contraction. Kernel contractions [31] are based on a selection among the sentences that imply the information to be retracted and they are a natural non- relational generalization of safe contractions [3]. Definition 17.4. [31] Let K be a set of sentences and α a sentence. Then K⊥ ⊥α is the set of all X such that X ⊆ K, X ⊢ α, and if K′ ⊂ X, then K′ α. The set K⊥ ⊥α is called the kernel set, and its elements are called the α-kernels of K. In order to retract some information α from K, some sentences in each α-kernel must be erased. This is done by incision functions. Definition 17.5. [31] Let K be a set of sentences. An incision function for K is a function σ defined on {K⊥ ⊥α|α ∈ L} such that for any sentence α ∈ L, the following statements hold:
- 360. 346 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari 1) σ(K⊥ ⊥α) ⊆ ∪(K⊥ ⊥α). 2) If X ∈ K⊥ ⊥α and X = ∅ then (X ∩σ(K⊥ ⊥α)) = ∅. In the limit case when K⊥ ⊥α = ∅ then σ(K⊥ ⊥α) = ∅. So, incision functions cut into each α-kernel, removing at least one sentence. Since all α-kernels are minimal subsets implying α, from the resulting sets it is no longer possible to derive α. Hence, incision functions may be used to derive (kernel) contraction operations. Definition 17.6. [31] Given a set of sentences K, a sentence α and an incision function σ for K, the kernel contraction of K by α, denoted by K÷σ α, is defined as: K÷σ α = K σ(K⊥ ⊥α). That is, K÷σ α can be obtained by erasing from K the sentences cut out by σ. In the following, we recall a set of postulates for contraction that can be consid- ered on every arbitrary set of sentences K: • Inclusion: K÷α ⊆ K. • Success: If α ∈Cn(∅) then α ∈Cn(K÷α). • Vacuity: If α ∈Cn(K) then K÷α = K. • Core-Retainment [27]: If β ∈ K and β ∈K÷α then there is a set K′ such that K′ ⊆ K such that α ∈Cn(K′) but α ∈ Cn(K′ ∪{β}). • Uniformity [28]: If it holds for all subsets K′ of K that α ∈ Cn(K′) if and only if β ∈ Cn(K′) then K÷α = K÷β. • Relative Closure [27]: K ∩Cn(K÷α) ⊆ K÷α. The following is a representation theorem for kernel contractions on belief bases. Theorem 17.2. [31] Let K a belief base and ÷ be a contraction operator for K. Then ÷ is a kernel contraction for K if and only if ÷ satisfies inclusion, success, core-retainment and uniformity. Smooth kernel contractions are kernel contractions that satisfy relative closure. Hansson showed that smooth kernel contractions and partial meet contractions are equivalent on belief sets. That is, smooth kernel contractions and partial meet con- tractions are just two different ways to construct the same class of operations on belief sets. However, kernel contractions are more general than partial meet con- tractions on belief bases. More relations about partial meet and kernel contractions can be found in [14]. The textbook [32] gives a good overview on axiomatic work in general (belief) base revision. 2.3 Alternative approaches and iterated revision There are other approaches to belief change realizing AGM change operations, for instance, Grove’s sphere system [23], epistemic entrenchment [22], and safe con- traction [3]. Grove’s sphere system is very similar to the sphere semantics for coun- terfactuals proposed by Lewis [39]. Concentric spheres allow the ordering of pos- sible worlds according to some notion of similarity or the like. Then, the result of
- 361. 17 Belief Revision and Argumentation Theory 347 revising by α can be semantically described by the set of α-models that are clos- est to the K-models. Epistemic entrenchment allows that a revision can be based on a relation ≤ among sentences that when forced to choose between two beliefs, an agent will give up the less entrenched one. Gärdenfors and Makinson [22] pro- posed an epistemic entrenchment relation from which it is possible to define AGM contractions. Belief updating [33] is a change operation of a different kind. Whereas in belief revision, both the old beliefs and the new information refer to the same situation, in updating the new information is about a current, possibly changed situation. Recent work relates update to actions [37]. Much current work in belief revision focuses on iterated revision that deals with changing epistemic relations such as epistemic entrenchment [7]. For a recent paper that considers iterated belief (base) revision in a broad epistemic framework, cf. [36]. 3 Related work Before entering into the discussion on the backgrounds of belief revision and argu- mentation theories, and possible connections in between, we will give an overview over work addressing this broad topic. 3.1 Truth Maintenance Systems [1979–1986] Doyle [10] presents initially the truth maintenance system, or TMS, which is a knowledge representation method for representing both beliefs and their justifica- tions. The aim of such systems is to restore consistency when a new justification has been added. A TMS associates a special data structure, called a node, with each problem solver datum which includes database entries, inference rules, and proce- dures. It records justifications, i.e., arguments, for potential inferred beliefs, so as to compute the current set of beliefs by manipulating the status of nodes repre- senting beliefs and evaluating justifications that represent reasons to believe. The truth maintenance process starts when a new justification for a node is added, and runs through basic steps of argumentation, such as evaluating the justifications to determine the status of nodes and checking justifications and contradictions, while avoiding circular relations among justifications and nodes. A different type of truth maintenance systems was proposed by de Kleer’s assumption-based truth maintenance systems (ATMS)[8]. Instead of evaluating jus- tifications in a global process, ATMS label each datum with the corresponding sets of assumptions, representing the contexts under which it holds. These assumption sets are computed by the ATMS from the justifications that are supplied by the prob- lem solver. The idea is that the assumptions provide contexts from which beliefs can be derived and hence may serve as arguments. The assumptions are primitive data,
- 362. 348 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari and all other data can be derived from them. There is no necessity that the overall database be consistent; it is easy to refer to contexts, and moving to different points in the search space requires very little effort. 3.2 Inconsistencies from multiple sources [1995] Benferhat et al. [4] present an article primarily oriented towards the treatment of in- consistency caused by the use of multiple sources of information. Knowledge bases are stratified, namely each formula in the knowledge base is associated with its level of certainty corresponding to the layer to which it belongs. The authors investigate two classes of approaches to deal with inconsistency in knowledge bases: coherence theories and foundation theories. The first insists on revising the knowledge base and restoring consistency, while the latter accepts in- consistency and copes with it. Coherence theories propose to give up some formulas of the knowledge base in order to get one or several consistent subbases, and to apply classical entailment on these consistent subbases to deduce plausible conclusions of the knowledge base. Foundation theories proceed differently since they retain all available information but each plausible conclusion inferred from the knowledge base is justified by some strong argumentative reason for believing in it. The claim of the paper is that it does not always make sense to revise an inconsis- tent knowledge base, in particular, if the information comes from multiple sources. It is not even necessary to restore consistency in order to make sensible inferences from an inconsistent knowledge base, since inference based on argumentation can derive conclusions and reasons to believe them, independently of the consistency of the knowledge base. 3.3 Belief revision and epistemology [2000] Pollock and Gillies [46] studied the dynamics of a belief revision system considering relations among beliefs in a “derivational approach” trying to derive a theory of belief revision from a more concrete epistemological theory. According to them, one of the goals of belief revision is to generate a knowledge base in which each piece of information is justified (by perception) or warranted by arguments containing previously held beliefs. The difficulty is that the set of justified beliefs can exhibit all kinds of logical incoherencies because it represents an intermediate stage in reasoning. Therefore, the authors propose a theory of belief revision concerned with warrant rather than justification. The set of warranted propositions already takes account of all possible inferences, so there is only one way to acquire new warranted propositions: through perception. However, perception adds a percept to the inference-graph, not a belief,
- 363. 17 Belief Revision and Argumentation Theory 349 so the effect of perception on warrant cannot be represented as the addition of a proposition to a belief state. 3.4 Deductive explanations and belief revision [2002] Falappa et al. [16] present a kind of non-prioritized revision operator based on the use of explanations. The idea is that an agent, before incorporating informa- tion which is inconsistent with its knowledge, requests an explanation supporting it. The authors build on the classical distinction between the explanandum, which is the final conclusion, and the explanans, represented by a set of sentences supporting the conclusion. The structure of an explanation is very similar to the structure of a deductive argument; the main difference is that every belief of an explanation is undefeasible (in a moment of time) whereas some beliefs of an argument may be defeasible or tentatively inferred. In this framework, every explanation contains rules and factual knowledge. If the sentences in the explanans are better or more plausible than the sentences in the original belief base, then the explanation is incorporated. Therefore, not beliefs, but explanations (and hence arguments) supporting a belief are used for the change process. The authors considered both kernel and partial meet revision by a set of sentences and gave representation theorems for them. These operators may partially accept the new information, so they are non-prioritized. 3.5 Data-oriented Belief Revision [2006] Paglieri and Castelfranchi [45] join argumentation and belief revision in the same conceptual framework by following Toulmin’s layout of argumentation, which is intended to be used to analyze the rationality of arguments typically found in court- rooms. The connection to belief revision is made by considering argumentation as “persuasion to believe”, hence argumentation is supposed to initiate successful re- vision processes. The authors propose Data-oriented Belief Revision (DBR) as an alternative to the AGM approach. Two basic informational categories, data and be- liefs, are put forward in their approach, to account for the distinction between pieces of information that are simply gathered and stored by the agent (data), and pieces of information that the agent considers (possibly up to a certain degree) truthful rep- resentations of states of the world (beliefs). The beliefs are a subset of the data: an agent might well be aware of a datum that he does not believe (i.e., he does not con- sider reliable enough). Data structures are conceived as networks of nodes (data), linked together by the relations of support, contrast and union. Data are selected (or rejected) as beliefs on the basis of their properties, described by relevance, credibil- ity, importance, and likeability.
- 364. 350 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari According to Paglieri and Castelfranchi [45], the Toulmin’s layout schema is li- able of immediate implementation in DBR, since it uses a specific data structure. The union of data and warrant supports the claim, and the warrant is in turn sup- ported by its backing and contrasted by the rebuttal, i.e., supports of the rebuttal make the warrant less reliable. 3.6 Merging Dung argumentation systems [2007] Coste-Marquis et al. [6] proposed a general framework for merging Dung’s style argumentation systems. They presented a framework for deriving reasonable infor- mation from a collection of Dung argumentation systems. Their approach consists in merging such systems assuming that all agents do not share the same sets of ar- guments. No assumption is made concerning the meaning of the attack relations, so that such relations may differ not only because agents have different points of view on the way arguments interact but more generally may disagree on what an interaction is. Every argumentation system is expanded to a partial argumentation system, and such partial systems are built over the same set of arguments. Merging is used on the expanded systems as a way to solve the possible conflicts between them, and a set of argumentation systems which are as close as possible to the whole profile is generated. Finally, the last step consists in selecting the acceptable arguments at the group levels from the set of argumentation systems. 3.7 Prioritized revision by arguments [2008] Rotstein et al. [47] introduce an abstract theory that captures the dynamics of an argumentation framework through the application of belief revision concepts. They define a dynamic abstract argumentation theory including dialectical constraints, and then present argument revision techniques to describe the fluctuation of the set of active arguments (the ones considered by the inference process of the theory). Expansion, contraction, and revision operators are realized in this framework, where the revision can be expressed in terms of the other two, leading to an identity similar to the one defined by Isaac Levi [38]. Their abstract theory allows the introduction of an argument ensuring it can be believed afterwards. The expansion operator is quite straightforward, but the definition of the contraction operator allows a wide range of possibilities from affecting unrestrictedly any number of arguments in the system to keeping this perturbation to a minimum, following the minimal change principle of the AGM theory. Contraction may also have an indirect impact on the attack relation among arguments. In a second paper, Moguillansky et al. [41] proposed an instantiation of these change operators to Defeasible Logic Programming, DeLP [19]. In particular, a
- 365. 17 Belief Revision and Argumentation Theory 351 warrant-prioritized argument revision operator (WPA) is defined that implements successful change. To be more precise, when a program is revised by an argument A,α (where A is an argument for α), the revised program will be such that A is an undefeated argument, and α will therefore be warranted. The main issue under- lying WPA revision lies in the selection of arguments and the incisions that have to be made over them. An argument selection criterion determines which arguments should not be present and once this selection is made, incisions (in the form of deletion of rules) will make those arguments “disappear” following some minimal change principle. 3.8 Adding arguments to Dung systems [2008] Cayrol et al. [5] proposed a Dung-style abstract argumentation system that allows the addition of a new argument which may interact with previous arguments. An argumentation framework A,R is identified with an associated attack graph G. The revision process produces a new framework represented by a graph G′ and a new set of extensions. By considering how the set of extensions is modified under the revision process, the authors propose a typology of different revisions including decisive revision, when there is only one acceptable set of arguments in the revised framework, and expansive revision, when the revision simply adds the new argument to the existing extensions. 3.9 Relating reinstatement and recovery [2008] Boella et al. [24] try to show a direct relation between argumentation and belief revision on the level of abstract properties. Like Paglieri and Castelfranchi [45], the authors also consider argumentation as persuasion to believe, and persuasion should be related to belief revision. They establish a link between reinstatement and recovery. Reinstatement plays an important role in argumentation; it refers to the situation that an argument that is not acceptable because of the existence of an attacking argument become acceptable again when an attacker to the attacking argument exists. Recovery reflects the central idea of minimal change in AGM belief revision; according to it, expansion by α should recover what was lost when α was contracted. More recently, Boella et al. [25] present the application of a belief revision ma- chinery to argumentation in a multi-agent system. The authors define an argument base revision that gives priority to the last introduced argument. As they are inter- ested particularly in the persuasive power of argumentation, they express appealing- ness of arguments in terms of belief revision.
- 366. 352 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari 4 A conceptual view on Argumentation and Belief Revision An investigation of the multifaceted relationships between argumentation and belief revision makes it necessary to consider cross-links between different aspects on ei- ther side, while at the same time taking the whole context of reasoning into account. The works sketched in section 3 have contributed to clarify this broad picture, in one way or another. In this chapter, we will add some pieces and links to the complex scenario. 4.1 The big picture In order to obtain a clear view on possible connections between argumentation and belief revision, we start with making the basic steps of reasoning, as it was sketched in the introduction, more precise, pointing out the way from receiving (new) infor- mation to coming up with adequate plausible beliefs on which a decision can be based. We hereby assume that the current epistemic state is given and represented within some chosen framework. • Receiving new information: The new information I may come in very different shapes and forms. In the simplest scenario, I is a propositional fact. This is the scope of the basic AGM theory, assuming the epistemic state of the agent to be given by a belief set, i.e., a deductively closed set of (propositional) formulas. But I might be much more complex. It can be equipped with a degree of plausibility, or have the form of a rule or even a complete argument, or consist of a set of such entities. • Evaluating new information: For the further processing of I, it is crucial for the agent to know its origin, as this knowledge will influence decisively her willing- ness to adopt I. For instance, if I is based on an observation made by the agent herself, she will usually be convinced of it being true. However, if I is conveyed to her by another agent, be it as part of the official news, in personal communi- cation or found as written material, the agent will require some justification for I. In any case, as a mandatory step for rational, critical thinking, she will evalu- ate both I and a possibly given justification on the basis of her own beliefs and decide if I is to be incorporated into her stock of beliefs or not. • Changing beliefs: If the agent has decided to adopt I in the previous step, she employs strategies to incorporate I consistently into her beliefs. For this, she has to use belief revision techniques that allows her to change her epistemic state accordingly. • Inference: From her new epistemic state, the agent derives (most) plausible be- liefs that guide her behavior. This scenario can also be applied in parts if I is not a new information but a query to which the agent is expected to reply. In most of such cases, the change step would be obsolete, and evaluation and inference would collapse. However, if I is a conditional
- 367. 17 Belief Revision and Argumentation Theory 353 query “Suppose α holds, would you believe β?”, whatever α and β are, reasoning would include a hypothetical change process that the current epistemic state of the agent must undergo. From this embedding into a complex reasoning process, the complementary char- acters of argumentation and belief revision become obvious: while argumentation can make substantial contributions to the evaluation step, belief revision theory should be employed in the belief change part. But this is not the end of the story. Evaluation might include hypothetical change processes, considering what would happen if the new information were to be believed, and belief change implicitly relies on logical links between pieces of information which can be represented by arguments. In the end, both from argumentation processes and from belief revision processes, plausible beliefs can be obtained, but both areas focus only on parts of the dynamic reasoning process while at the same time providing general and ver- satile frameworks. Revision operators can not only be applied to beliefs, but also to intentions, preferences, theories, ontologies, law codes, etc. Argumentation can be used for negotiation (when the agents have conflicting interests and they try to make the best out of a deal for themselves), inquiry (when the agents have general ignorance in some subject and they try to find a proof or destroy one), deliberation (when the agents collaborate to decide what course of action to take) or information seeking (when the agents have personal ignorance on some subject and every agent seeks the answer to some questions from other agents). This is a much more general view than considering argumentation as persuasion to believe, as has been proposed in [45, 25]. These contemplations result in a most complex, highly interrelated view on ar- gumentation and belief revision. Consequently, discussing links between both fields must reflect this complexity, taking into regard that proposed frameworks on either side might only implement some but not all aspects of the corresponding field. In particular, the famous AGM theory [1] in belief revision is more concerned with judging the results of change in a very abstract way than with the change process itself; moreover, argumentative evaluation of justifications to believe the given in- formation in the sense described above is not at all a topic of AGM. That is why suc- cess is one of the basic postulates of the AGM theory according to which the agent is forced to believe the new information. Here, it is implicitly assumed that evaluation has been done beforehand. Hence, a comparison between AGM and some approach that makes use of a sophisticated argumentative evaluation of information to select beliefs, as has been developed e.g. in [45], is very likely to shed a bad light on AGM. AGM is no more on argumentative reasons for belief than is Dung’s framework [13] on change processes. Both are highly abstract, declarative approaches to the respec- tive field, hence (necessarily) over-simplifying in some respects but nevertheless extremely valuable as reference points. Therefore, an investigation of the connections between belief revision and ar- gumentation theory that is to do justice to both areas must go beneath the surface of abstract frameworks. It must study methods and the rationale underlying these methods, as well as purposes and intentions guiding the application of techniques.
- 368. 354 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari We will first identify characteristic and prototypic concepts on either side and find basic correspondences as well as crucial differences. Afterwards, we will investigate how either domain may enrich the other domain’s framework. 4.2 Comparing belief revision and argumentation At first glance, the differences between argumentation and belief revision prevail. To begin with representational issues, the syntactic and semantic foundations of both areas have not much in common. In standard belief revision, logical formulas are used for knowledge representation, and the result of change processes are logi- cal formulas. In more advanced frameworks, epistemic states are changed in which some qualitative relation allows for ranking worlds or sentences with respect to en- trenchment, plausibility, and the like. In order to verify the results of the change processes, classical logical semantics is used. On the contrary, argumentation the- ory focuses on the interactions of arguments as pieces of information that may attack one another, and a relation between arguments may give priority to one argument or another. The arguments themselves, however, are very heterogeneous. In most approaches, they are quite complex argument structures built up by some kind of rules, using standard or non-standard derivation to implement reasoning. However, they might as well be abstract objects as in Dung’s framework [13] without any in- ternal structure. A special semantics makes precise what good arguments are, and there are several such semantics possible, implemented by preferred, stable, and grounded extensions. As to the common grounds, both disciplines aim at resolving conflicts which are usually based on logical grounds, i.e., on contradictions, and make use of prefer- ence relations to achieve this aim. However, belief revision theory provides a highly declarative framework for that, based on postulates, whereas argumentation theory is more concerned with practical, justification-based techniques. For a direct comparison of argumentation and belief revision, it is tempting to take the standard AGM approach [1], as no other work has influenced the develop- ment of belief revision theory in a similar way. And the AGM postulates offer a clear framework that makes fundamental views explicit. However, one of the most basic assumptions of the standard AGM theory, namely its focus on deductively closed sets of formulas as representing belief sets, does not fit at all to argumentation the- ory, as it abstracts from the basic steps of reasoning, blurring the distinction between what is explicitly given or serves as assumptions, and what is justified belief. Hence, works on belief base revision (e.g. [32, 14]), or on iterated epistemic change (e.g. [7, 34, 35, 9]) offer a much richer base for comparison, since they allow one to take deeper insights into change processes. Belief revision methods and most argumentation frameworks can be used directly for reasoning, although this is not their principal concern. The connection between belief revision and nonmonotonic inference has been made clear by [40] and further developed for iterative change operations in [36]; this is the BRDI view on belief
- 369. 17 Belief Revision and Argumentation Theory 355 revision, as Dubois [12] calls it. As to argumentation, those approaches that are based on rules or use some sort of derivation provide logical chains that establish links between what is presupposed and what is concluded. Also with respect to that aspect, belief revision theories remain on an abstract level, describing by axioms what good inferences are, while argumentation is more concerned with how and why conclusions are drawn, making reasons for belief apparent. These considerations make obvious that belief revision and argumentation theory are basically complementary areas of research. In the following, we will discuss how they may complement each other. 4.3 Argumentation in Belief Revision In this subsection we study how argumentation concepts and techniques can be used in belief revision theory. Among the works relating argumentation and belief revi- sion, many of them add belief revision capabilities to an argumentation system. However, there are some papers that propose to use an “argumentative machin- ery” for a belief revision process, like e.g. [46, 45] (see also section 3). The first approach to use argumentation in belief revision is justification-based truth mainte- nance systems (TMS’s) proposed by Doyle [10] (cf. section 3.1). Doyle studies the interactions between justifications when a new justification has been added, to find out which conclusions can be justified. He does not consider retraction of justifica- tions. Assumption-based truth maintenance systems (ATMS) [8] are more concerned with managing assumptions instead of implementing change processes. As in DBR [45], we have data and beliefs: a data is believed in some contexts represented by assumptions sets. As in [4], there is no necessity that the overall database be consis- tent; it is easy to refer to contexts and move to different points in the search space representing the proper context. Both types of truth maintenance systems work on belief bases and are very early approaches to belief revision, dating before AGM theory. In [16], we combine the ATMS idea with base revision and propose a system that uses argumentative structures in the form of explanations for nonprioritized revisions of belief bases K. Different from standard approaches to belief revision, the new information not only consists of a proposition α but of reasons to believe a proposition, i.e., of rules and prerequisites A from which α can be deductively derived. So, A can be considered an explanans for the explanandum α. In order to integrate an argumentative evaluation of the new information into the revision process, we defined partial acceptance revision operators in the following way: i) the epistemic input is the set of sentences A as explanans for the explanandum α, ii) A is initially accepted, that is, A is joined to K (possibly producing an inconsistent intermediate state), iii) all possible inconsistencies of K ∪A are removed, returning a consistent revised belief base K ◦A. This operator is an operator of external revision. The name “external” indicates that the revision process takes place outside of the original set.
- 370. 356 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari Whether α is accepted or not depends on the evaluation of its explanans A with respect to the current belief base. The acceptance of the explanans forces the ac- ceptance of the explanandum in the revised set, since explanation here is based on classical deduction. However, there is an important remark to be made regarding the degree of acceptance of the explanans and the explanandum. While the explanans can be explicitly included in the revised set, the explanandum may be inferred from it without actually being included. So, the distinction between explicitly given in- formation (as data in [45]) and inferred beliefs is respected and can be implemented only when working with belief bases instead of belief sets. The main difference between the process of revision by a set of sentences and the process of argumentation is significant: in revision, external beliefs are compared with internal beliefs and, after a selection process, some sentences are discarded, other ones are accepted. In argumentation, the process is more procedural: we have an argument, attack this by counterarguments, defend it by counterarguments to counterarguments, and so on. Nevertheless, the rationale behind partial acceptance revision operators matches the intentions leading argumentation in that not the new information itself, but reasons to believe it, are evaluated. 4.4 Belief Revision in Argumentation Belief revision theory offers nice methods to implement dynamical features in an argumentation framework. The papers by Rotstein, Moguillansky et al. [47, 41] and Boella et al. [25] are among the most comprehensive approaches to address such a revision theory for argument systems (cf. section 3). Several different ways of applying belief revision in argumentation may be distinguished: • Changing by adding or deleting an argument. • Changing by adding or deleting a set of arguments. • Changing the attack (and/or defeat) relation among arguments. • Changing the status of beliefs (as conclusions of arguments). • Changing the type of an argument (from strict to defeasible, or vice versa). The distinction between the two first ways is similar to the distinction among sin- gle change (as in AGM model) and multiple change (as in multiple contraction [18, 44]). Adding or removing an argument or a set of arguments may trigger a change in the justified conclusions. The consideration of such changes may lead to a base revision theory which deals with changes of argumentative bases, and in which deduction is replaced by an argumentation process. The method of kernel contraction making use of incision functions to eliminate base elements (cf. section 2) is of particular interest here; first steps towards this direction can be found in [41]. Changing the attack and/or defeat relation among arguments may lead to a com- pletely different behavior of the system, the understanding and the control of which constitutes substantially new challenges for belief revision theory. Boella et al. [25] have proposed an approach that changes the attack relation in a dictatorial way in
- 371. 17 Belief Revision and Argumentation Theory 357 favor of the new information. More sophisticated change processes are conceivable, taking ideas from the methods for epistemic belief change (or iterated belief change) that deal with modifying relations on possible worlds (cf. e.g. [7, 35]). Although ar- guments are very different from possible worlds, those works might be considered as approaches to realize minimal change processes for general relations. However, before applying those techniques from epistemic belief revision to argumentative at- tack or defeat relations, a careful investigation has to be done concerning conceptual differences between both frameworks. Changing the status of beliefs may be a consequence of previous changes to the argument system. In classical, Dung-style argument systems, the criterion to be a justified belief is based on extensions. Cayrol et al. [5] have studied how extensions change when new arguments are added. In argumentation frameworks where ar- gumentation is based on conclusions (e.g. DeLP [19]), the addition of an argument may change the status of a claim from unwarranted to warranted and vice versa. This touches basic concerns of the reasoning aspect of belief revision, and investigations on the level of belief revision postulates might be very useful. The link between reinstatement (in argumentation) and recovery (in belief revision) proposed in [24] is in this line of research. Finally, changing the type of an argument from strict to defeasible, or vice versa, addresses novel issues in belief revision. Such modifications are not dealt with prop- erly by chaining retraction and addition of arguments, as the old and the new ar- gument are linked by a syntactic or semantic relation which gets lost when these operations are carried out independently. The basic idea is that inconsistencies that arise when new information has to be incorporated into the stock of beliefs can be eliminated not only by removing arguments (resp. beliefs), but by weakening strict beliefs to defeasible rules (resp. conditionals). The procedure of dynamic classifica- tion of generic rules has been frequently used in the evolution of humanity’s knowl- edge. The possibility of changing the status of beliefs from undefeasible to defea- sible induces revision operations of a new quality, with important consequences for argumentation, as arguments are formed very often by defeasible beliefs. In [16], we have introduced a new type of base revision that implements such a dynamic classification of beliefs and is likewise interesting for argumentation and belief re- vision theory. We proposed a framework in which defeasible conditionals can be generated by revising belief structures composed of defeasible rules and undefeasi- ble knowledge. The approach preserves consistency in the undefeasible knowledge and it provides a mechanism to dynamically qualify the beliefs as undefeasible or defeasible, providing a more complete set of epistemic attitudes and extending the inference power of knowledge based systems. As a particular case, Falappa et al. [15] propose to extend the application of this non-prioritized revision operator to Defeasible Logic Programming (DeLP) [19].
- 372. 358 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari 4.5 Possible future work To alleviate orientation within the complex picture and promote further work, we will take up some of our ideas and suggestions developed above and propose a list of possible important topics for future work on the connection between argumentation and belief revision. • Development of new change operators for argumentative systems: this point may include non-prioritized revision as well as symmetrical revision operators. Non- prioritized revision means revision in which the new information (argument) could be accepted, partially accepted or rejected. Symmetric revision means merging of argumentative systems. • Elaborating a set of postulates or properties for changes of argumentative sys- tems: this may include postulates for expansions, contractions, revisions as well as postulates for consolidations (for instance, changes for cleaning blocking de- featers) or merging the beliefs of several argumentative systems. • Finding representation theorems for changes on argumentative systems: repre- sentation theorems establish links between constructive approaches to revision that propose mechanisms and algorithms for change, and black box approaches that specify the properties that a change operator should have to show equiva- lence. • Epistemic entrenchment and defeat: epistemic entrenchment relations among beliefs may be derived from attack/defeat relations among arguments and vice versa. Basically, warranted beliefs should be more entrenched than non-warranted beliefs. • Argumentation and epistemic revision: interesting new views that alleviate the combination of belief revision and argumentation might arise from studying change operators within the broader framework of iterated revision. In partic- ular, it would be interesting to find parallels in changing relational structures on either side. 5 Conclusion In this chapter, we discuss the relationships between argumentation and belief revi- sion, first recalling important work done, then analyzing the current state of the art and proposing some tentative future research lines. We developed a conceptual view on argumentation and belief revision based on the understanding of both areas as being complementary disciplines in the art of reasoning. While argumentation seems to be more appropriate for the evaluation of (rule-based) information, belief revision proves to be useful for the handling of dynamic beliefs. Hence they should not be regarded as competitive alternatives that may replace each other. Just to the contrary – combining argumentation and belief revision allows the modelling of reasoning processes in greater variety and
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- 374. 360 Marcelo A. Falappa, Gabriele Kern-Isberner and Guillermo R. Simari 18. A. Fuhrmann and S. O. Hansson. A Survey of Multiple Contractions. The Journal of Logic, Language and Information, 3:39–76, 1994. 19. A. J. Garcı́a and G. R. Simari. Defeasible Logic Programming: an Argumentative Approach. Theory and Practice of Logic Programming, 4(1):95–138, 2004. 20. P. Gärdenfors. Knowledge in Flux: Modelling the Dynamics of Epistemic States. The MIT Press, Bradford Books, Cambridge, Massachusetts, 1988. 21. P. Gärdenfors. Belief Revision. Gärdenfors, Cambridge University Press, 1992. 22. P. Gärdenfors and D. Makinson. Revisions of Knowledge Systems using Epistemic Entrench- ment. Second Conference on Theoretical Aspects of Reasoning About Knowledge, pages 83– 95, 1988. 23. A. Grove. Two Modellings for Theory Change. The Journal of Philosophical Logic, 17:157– 170, 1988. 24. G. Guido Boella, C. d. Costa Perera, A. Tettamanzi, and L. van der Torre. Dung Argumentation and AGM Belief Revision. In Fifth International Workshop on Argumentation in Multi-Agent Systems, ArgMAS 2008, 2008. 25. G. Guido Boella, C. d. Costa Perera, A. Tettamanzi, and L. van der Torre. Making others believe what they want. Artificial Intelligence in Theory and Practice II, pages 215–224, 2008. 26. S. O. Hansson. New Operators for Theory Change. Theoria, 55:114–132, 1989. 27. S. O. Hansson. Belief Contraction without Recovery. Studia Logica, 50:251–260, 1991. 28. S. O. Hansson. A Dyadic Representation of Belief. In Belief Revision [21], pages 89–121. 29. S. O. Hansson. In Defense of Base Contraction. Synthese, 91:239–245, 1992. 30. S. O. Hansson. Theory Contraction and Base Contraction Unified. The Journal of Symbolic Logic, 58(2), 1993. 31. S. O. Hansson. Kernel Contraction. The Journal of Symbolic Logic, 59:845–859, 1994. 32. S. O. Hansson. A Textbook of Belief Dymanics: Theory Change and Database Updating. Kluwer Academic Publishers, 1999. 33. H. Katsuno and A. Mendelzon. On the Difference between Updating a Knowledge Database and Revising it. In Belief Revision [21], pages 183–203. 34. G. Kern-Isberner. Postulates for Conditional Belief Revision. In Proceedings of IJCAI’99, pages 186–191. Morgan Kaufmann, 1999. 35. G. Kern-Isberner. A Thorough Axiomatization of a Principle of Conditional Preservation in Belief Revision. Annals of Mathematics and Artificial Intelligence, 40(1-2):127–164, 2004. 36. G. Kern-Isberner. Linking Iterated Belief Change Operations to Nonmonotonic Reasoning. In G. Brewka and J. Lang, editors, Proceedings of KR 2008, pages 166–176, Menlo Park, CA, 2008. AAAI Press. 37. J. Lang. Belief Update Revisited. In Proceedings of IJCAI 2007, pages 2517–2522, 2007. 38. I. Levi. Subjunctives, Dispositions, and Chances. Synth̀ese, 34:423–455, 1977. 39. D. Lewis. Counterfactuals. Harvard University Press, Cambridge, Massachusetts, 1973. 40. D. Makinson and P. Gärdenfors. Relations between the Logic of Theory Change and Non- monotonic Logic. Lecture Notes in Computer Science, 465:183–205, 1991. 41. M. O. Moguillansky, N. D. Rotstein, M. A. Falappa, A. J. Garcı́a, and G. R. Simari. Argument Theory Change: Revision Upon Warrant. In Proceedings of The Twenty-Third Conference on Artificial Intelligence, AAAI 2008, pages 132–137, 2008. 42. A. Nayak. Foundational Belief Change. Journal of Philosophical Logic, 23:495–533, 1994. 43. A. Newell. The Knowledge Level. Readings from the AI Magazine, pages 357–377, 1988. 44. R. Niederée. Multiple Contraction: a further case against Gärdenfors’ Principle of Recovery. Lecture Notes in Computer Science, 465:322–334, 1991. 45. F. Paglieri and C. Castelfranchi. The Toulmin Test: Framing Argumentation within Belief Revision Theories, pages 359–377. Berlin, Springer, 2006. 46. J. L. Pollock and A. S. Gillies. Belief Revision and Epistemology. Synthese, 122(1–2):69–92, 2000. 47. N. D. Rotstein, M. O. Moguillansky, M. A. Falappa, A. J. Garcı́a, and G. R. Simari. Argument Theory Change: Revision upon Warrant. In Proceedings of The Computational Models of Argument, COMMA 2008, pages 336–347, 2008.
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- 377. Chapter 18 Argumentation in Legal Reasoning Trevor Bench-Capon, Henry Prakken and Giovanni Sartor 1 Introduction A popular view of what Artificial Intelligence can do for lawyers is that it can do no more than deduce the consequences from a precisely stated set of facts and legal rules. This immediately makes many lawyers sceptical about the usefulness of such systems: this mechanical approach seems to leave out most of what is important in legal reasoning. A case does not appear as a set of facts, but rather as a story told by a client. For example, a man may come to his lawyer saying that he had developed an innovative product while working for Company A. Now Company B has made him an offer of a job, to develop a similar product for them. Can he do this? The lawyer firstly must interpret this story, in the context, so that it can be made to fit the framework of applicable law. Several interpretations may be possible. In our exam- ple it could be seen as being governed by his contract of employment, or as an issue in Trade Secrets law. Next the legal issues must be identified and the pros and cons of the various interpretations considered with respect to them. Does his contract in- clude a non-disclosure agreement? If so, what are its terms? Was he the sole devel- oper of the product? Did Company A support its development? Does the product use commonly known techniques? Did Company A take measures to protect the secret? Some of these will favour the client, some the Company. Each interpretation will require further facts to be obtained. For example, do the facts support a claim that the employee was the sole developer of the product? Was development work carried Trevor Bench-Capon Department of Computer Science, University of Liverpool, UK e-mail: tbc@liverpool.ac. uk Henry Prakken Department of Information and Computing Sciences, Utrecht University, and Faculty of Law, Uni- versity of Groningen, The Netherlands e-mail: henry@cs.uu.nl Giovanni Sartor European University Institute, Law Department, Florence, and CIRSFID, University of Bologna, Italy e-mail: giovanni.sartor@eui.eu I. Rahwan, G. R. Simari (eds.), Argumentation in Artificial Intelligence, 363 DOI 10.1007/978-0-387-98197-0 18, c Springer Science+Business Media, LLC 2009
- 378. 364 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor out in his spare time? What is the precise nature of the agreements entered into? Once an interpretation has been selected, the argument must be organised into the form considered most likely to persuade, both to advocate the client’s position and to rebut anticipated objections. Some precedents may point to one result and others to another. In that case, further arguments may be produced to suggest following the favourable precedent and ignoring the unfavourable one. Or the rhetorical presenta- tion of the facts may prompt one interpretation rather than the other. Surely all this requires the skill, experience and judgement of a human being? Granted that this is true, much effort has been made to design computer programs that will help people in these tasks, and it is the purpose of this chapter to describe the progress that has been made in modelling and supporting this kind of sophisticated legal reasoning. We will review1 systems that can store conflicting interpretations and that can propose alternative solutions to a case based on these interpretations. We will also describe systems that can use legal precedents to generate arguments by drawing analogies to or distinguishing precedents. We will discuss systems that can argue why a rule should not be applied to a case even though all its conditions are met. Then there are systems that can act as a mediator between disputing parties by struc- turing and recording their arguments and responses. Finally we look at systems that suggest mechanisms and tactics for forming arguments. Much of the work described here is still research: the implemented systems are prototypes rather than finished systems, and much work has not yet reached the stage of a computer program but is stated as a formal theory. Our aim is therefore to give a flavour (certainly not a complete survey) of the variety of research that is going on and the applications that might result in the not too distant future. Also for this reason we will informally paraphrase example inputs and outputs of systems rather than displaying them in their actual, machine-readable format; moreover, because of space limitations the examples have to be kept simple. 2 Early systems for legal argumentation In this section we briefly discuss some early landmark systems for legal argumenta- tion. All of them concern the construction of arguments and counterarguments. 2.1 Conflicting interpretations Systems to address conflicting interpretations of legal concepts go back to the very beginnings of AI and Law. Thorne McCarty (e.g. [25, 27]) took as his key problem a landmark Supreme Court Case in US tax law which turned on differing interpreta- tions of the concept of ownership, and set himself the ambitious goal of reproducing, 1 This chapter is a revised and updated version of [6].
- 379. 18 Argumentation in Legal Reasoning 365 in his TAXMAN system, both the majority and the dissenting opinions express- ing these interpretations. This required highly sophisticated reasoning, constructing competing theories and reasoning about the deep structure of legal concepts to map the specific situation onto paradigmatic cases. Although some aspects of the system were prototyped, the aim was perhaps too ambitious to result in a working sys- tem, certainly given the then current state of the art. This was not McCarty’s goal, however: his motivation was to gain insight into legal reasoning through a computa- tional model. McCarty’s main contribution was the recognition that legal argument involves theory construction as well as reasoning with established knowledge. Mc- Carty [26] summarises his position as follows: “The task for a lawyer or a judge in a “hard case” is to construct a theory of the disputed rules that produces the desired legal result, and then to persuade the relevant audience that this theory is preferable to any theories offered by an opponent” (p. 285). Note also the emphasis on persua- sion, indicating that we should expect to see argumentation rather than proof. Both the importance of theory construction and the centrality of persuasive argument are still very much part of current thinking in AI and Law. Another early system was developed by Anne Gardner [15] in the field of of- fer and acceptance in American contract law. The task of the system was “to spot issues”: given an input case, it had to determine which legal questions arising in the case were easy and which were hard, and to solve the easy ones. The system was essentially rule based, and this simpler approach offered more possibilities for practical exploitation than did McCarty’s system. One set of rules was derived from the Restatement of Contract Law, a set of 385 principles abstracting from thousands of contract cases. These rules were intended to be coherent, and to yield a single answer if applicable. This set of rules was supplemented by a set of interpretation rules derived from case law, common sense and expert opinion, intended to link these other rules to the facts of the case. Gardner’s main idea was that easy ques- tions were those where a single answer resulted from applying these two rule sets, and hard questions, or issues, were either those where no answer could be produced, because no interpretation rule linked the facts to the substantive rules, or where con- flicting answers were produced by the facts matching with several rules. Some of the issues were resolved by the program with a heuristic that gives priority to rules derived from case law over restatement and commonsense rules. The rationale of this heuristic is that if a precedent conflicts with a rule from another source, this is usually because that rule was set aside for some reason by the court. The remaining issues were left to the user for resolution. Consider the following example, which is a very much simplified and adapted version of Gardner’s own main example2. The main restatement rule is R1: An offer and an acceptance constitute a contract Suppose further that there are the following commonsense (C) and expert (E) rules on the interpretation of the concepts of offer and acceptance: 2 We in particular abstract from Gardner’s refined method for representing knowledge about (speech act) events.
- 380. 366 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor C1: A statement “Will supply ...” in reply to a request for offer is an offer. C2: A statement “Will you supply ...” is a request for offer. C3: A statement “I accept ...” is an acceptance. E1: A statement “I accept” followed by terms that do not match the terms of the offer is not an acceptance. Suppose that Buyer sent a telegram to Seller with “Will you supply carload salt at $2.40 per cwt?” to which Seller replied with “Will supply carload at $2.40, terms cash on delivery”, after which Buyer replied with her standard “Purchase Order” indicating “I accept your offer of 12 July” but which also contained a standard pro- vision “payment not due until 30 days following delivery”. Applying the rules to these events, the “offer” antecedent of R1 can be estab- lished by C1 combined with C2, since there are no conflicting rules on this issue. However, with respect to the “acceptance” antecedent of R1 two conflicting rules apply, viz. C3 and E1. Since we have no way of giving precedence to C3 or E1, the case will be a hard one, as there are two conflicting notions of “acceptance”. If the case is tried and E1 is held to have precedence, E1 will now be a precedent rule, and any subsequent case in which this conflict arises will be easy, since, as a precedent rule, E1 will have priority over C3. 2.2 Reasoning with precedents The systems described in the last section do recognise the importance of precedent cases as a source of legal knowledge, but they make use of them by extracting the rationale of the case and encoding it as a rule. To be applicable to a new case, however, the rule extracted may need to be analogised or transformed to match the new facts. Nor is extracting the rationale straightforward: judges often leave their reasoning implicit and in reconstructing the rationale a judge could have had in mind there may be several candidate rationales, and they can be expressed at a variety of levels of abstraction. These problems occur especially in so-called ‘factor- based’ domains, i.e., domains where problems are solved by considering a variety of factors that plead for or against a solution. In such domains a rationale of a case often just expresses the resolution of a particular set of factors in a specific case. A main source of conflict in such domains is that a new case often will not exactly match a precedent but will share some features with it, lack some of its other features, and/or have some additional features. Moreover, cases are more than simple rationales: matters such as the context and the procedural setting can influence the way the case should be used. In consequence, some researchers have attempted to avoid using rules and rationales altogether, instead representing the input, often interpreted as a set of factors, and the decisions of cases, and defining separate argument moves for interpreting the relation between the input and decision (e.g. [23, 1], both to be discussed below). This approach is particularly associated with researchers in US, where the common law tradition places a greater stress on precedent cases and their particular features than is the case with the civil law jurisdictions of Europe. None
- 381. 18 Argumentation in Legal Reasoning 367 the less cases are also used in civil law jurisdictions and the reasoning techniques are similar. The most influential system of this sort is HYPO [2], developed by Edwina Riss- land and Kevin Ashley in the domain of US Trade Secrets Law, which can be seen as a factor-based domain. In HYPO cases are represented according to a number of dimensions. A dimension is some aspect of the case relevant to the decision, for example, the security measures taken by the plaintiff. One end of the dimension represents the most favourable position for the plaintiff (e.g. specific non-disclosure agreements), while the other end represents the position most favourable to the de- fendant (e.g. no security measures at all). Typically a case will lie somewhere be- tween the two extremes and will be more or less favourable accordingly. HYPO then uses these dimensions to construct three-ply arguments. First one party (say the plaintiff) cites a precedent case decided for that side and offers the dimensions it shares with the current case as a reason to decide the current case for that side. In the second ply the other party responds either by citing a counter example, a case decided for the other side which shares a different set of dimensions with the cur- rent case, or distinguishing the precedent by pointing to features which make the precedent more, or the current case less, favourable to the original side. In the third ply the original party attempts to rebut the arguments of the second ply, by distin- guishing the counter examples, or by citing additional precedents to emphasise the strengths or discount the weaknesses in the original argument. Subsequently Ashley went on, with Vincent Aleven, to develop CATO (most fully reported in [1]), a system designed to help law students to learn to reason with precedents. CATO simplifies HYPO in some respects but extends it in others. In CATO the notion of dimensions is simplified to a notion of factors. A factor can be seen as a specific point of the dimension: it is simply present or absent from a case, rather than present to some degree, and it always favours either the plaintiff or defen- dant. A new feature of CATO is that these factors are organised into a hierarchy of increasingly abstract factors, so that several different factors can be seen as meaning that the same abstract factor is present. One such abstract factor is that the defendant used questionable means to obtain the information, and two more specific factors in- dicating the presence of this factor are that the defendant deceived the plaintiff and that he bribed an employee of the plaintiff: both these factors favour the plaintiff. The hierarchy allows for argument moves that interpret the relation between a case’s input and its decision, such as emphasising or downplaying distinctions. To give an example of downplaying, if in the precedent defendant used deception while in the new case instead defendant bribed an employee, then a distinction made by the de- fendant at this point can be downplayed by saying that in both cases the defendant used questionable means to obtain the information. To give an example of empha- sising a distinction, if in the new case defendant bribed an employee of plaintiff while in the precedent no factor indicating questionable means was present, then the plaintiff can emphasise the distinction “unlike the precedent, defendant bribed an employee of plaintiff” by adding “and therefore, unlike the precedent defendant used questionable means to obtain the information”.
- 382. 368 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor Perhaps the most elaborate representation of cases was produced in Karl Brant- ing’s Grebe system in the domain of industrial injury, where cases were represented as semantic networks [13]. The program matched portions of the network for the new case with parts of the networks of precedents, to identify appropriate analogies. Of all this work, HYPO in particular was highly influential, both in the explicit stress it put on reasoning with cases as constructing arguments, and in providing a dialectical structure in which these arguments could be expressed, anticipating much other work on dialectical procedures. 3 Logical accounts of reasoning under disagreement The systems discussed in the previous section were (proposals for) implemented systems, based on informal accounts of some underlying theory of reasoning. Other AI Law research aims at specifying theories of reasoning in a formal way, in order to make general reasoning techniques from nonmonotonic logic and formal argumentation available for implementations. The first AI Law proposals in this vein, for example, [16, 30], can be regarded as formal counterparts of Gardner’s ideas on issue spotting. Recall that Gardner allows for the presence in the knowledge base of conflicting rules governing the interpretation of legal concepts and that she defines an issue as a problem to which either no rules apply at all, or conflicting rules apply. Now in logical terms an issue can be defined as a proposition such that either there is no argument about this proposition or there are both arguments for the proposition and for its negation. Some more recent work in this research strand has utilised a very abstract AI framework for representing systems of arguments and their relations developed by Dung [14]. For Dung, the notion of argument is entirely abstract: all that can be said of an argument is which other arguments it attacks, and which it is attacked by. Given a set of arguments and the attack relations between them, it is possible to determine which arguments are acceptable: an argument which is not attacked will be acceptable, but if an argument has attackers it is acceptable only if it can be defended, against these attackers, by acceptable arguments which in turn attack those attackers. This framework has proved a fruitful tool for understanding non- monotonic logics and their computational properties. Dung’s framework has also been made use of in AI Law. It was first applied to the legal domain by Prakken Sartor [35], who defined a logic for reasoning with conflicting rules as an instan- tiation of Dung’s framework. (See below and Chapter 8 of this book) Bench-Capon [3] has explored the potential of the fully abstract version of the framework to rep- resent a body of case law. he uses preferred semantics, where arguments can defend themselves: in case of mutual attack this gives rise to multiple sets of acceptable arguments, which can explain differences in the application of law in different juris- dictions, or at different times in terms of social choices. Dung’s framework has also been extended to include a more formal consideration of social values (see Chapter 3 of this book).
- 383. 18 Argumentation in Legal Reasoning 369 3.1 Reasoning about conflicting rules Generally speaking, the proposed systems discussed so far attempt to identify con- flicting interpretations and arguments, but do not attempt to resolve them, leaving it to the user to choose which argument will be accepted. As we saw above, Gardner’s system went somewhat further in that it gave priority to rules derived from case law over restatement and commonsense rules. Thus her system was able to solve some of the cases to which conflicting rules apply. This relates to much logical work in Artificial Intelligence devoted to the resolution of rule conflicts in so-called com- monsense reasoning. If we have a rule that birds can fly and another that ostriches cannot fly, we do not want to let the user decide whether Cyril the ostrich can fly or not: we want the system to say that he cannot, since an ostrich is a specific kind of bird. Naturally attempts have been made to apply these ideas to law. One approach was to identify general principles used in legal systems to establish which of two conflicting rules should be given priority. These principles included preferring the more specific rule (as in the case of the ostrich above, or where a law expresses an exception to a general provision), preferring the more recent rule, or preferring the rule deriving from the higher legislative authority (for instance, ‘federal law precedes state law’). To this end the logics discussed above were ex- tended with the means to express priority relations between rules in terms of these principles so that rule conflicts would be resolved. Researchers soon realised, how- ever, that general priority principles can only solve a minority of cases. Firstly, as for the specificity principle, whether one rule is more specific than another often depends on substantive legal issues such as the goals of the legislator, so that the specificity principle cannot be applied without an intelligent appreciation of the par- ticular issue. Secondly, general priority principles usually only apply to rules from regulations and not to, for instance, case rationales or interpretation rules derived from cases. Accordingly, in many cases the priority of one rule over another can be a matter of debate, especially when the rules that conflict are unwritten rules put forward in the context of a case. For these reasons models of legal argument should allow for arguments about which rule is to be preferred. As an example of arguments about conflicting case rationales, consider three cases discussed by, amongst others, [10, 7, 32] and [8] concerning the hunting of wild animals. In all three cases, the plaintiff (P) was chasing wild animals, and the defendant (D) interrupted the chase, preventing P from capturing those animals. The issue to be decided is whether or not P has a legal remedy (a right to be compensated for the loss of the game) against D. In the first case, Pierson v Post, P was hunting a fox on open land in the traditional manner using horse and hound, when D killed and carried off the fox. In this case P was held to have no right to the fox because he had gained no possession of it. In the second case, Keeble v Hickeringill, P owned a pond and made his living by luring wild ducks there with decoys, shooting them, and selling them for food. Out of malice, D used guns to scare the ducks away from the pond. Here P won. In the third case, Young v Hitchens, both parties were commercial fisherman. While P was closing his nets, D sped into the gap, spread his
- 384. 370 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor own net and caught the fish. In this case D won. The rules we are concerned with here are the rationales of these cases: R1 Pierson: If the animal has not been caught, the defendant wins R2 Keeble: If the plaintiff is pursuing his livelihood, the plaintiff wins R3 Young: If the defendant is in competition with the plaintiff and the animal is not caught, the defendant wins. Note that R1 applies in all cases and R2 in both Keeble and Young. In order to explain the outcomes of the cases we need to be able to argue that R3 R2 R1. To start with, note that if, as in HYPO, we only look at the factual similarities and differences, none of the three precedents can be used to explain the outcome of one of the other precedents. For instance, if we regard Young as the current case, then both Pierson and Keeble can be distinguished. A way of arguing for the desired priorities, first mooted in [10], is to refer to the purpose of the rules, in terms of the social values promoted by following the rules. The logic of [35] provides the means to formalise such arguments. Consider an- other case in which only plaintiff was pursuing his livelihood and in which the ani- mal was not caught. In the following (imaginary) dispute the parties reinterpret the precedents in terms of the values promoted by their outcomes, in order to find a con- trolling precedent (we leave several details implicit for reasons of brevity; a detailed formalisation method can be found in [32]; see also [8]. Plaintiff: I was pursuing my livelihood, so (by Keeble) I win Defendant: You had not yet caught the animal, so (by Pierson) I win Plaintiff: following Keeble promotes economic activity, which is why Keeble takes precedence over Pierson, so I win. Defendant: following Pierson protects legal certainty, which is why Keeble does not take precedence over Pierson, so you do not win. Plaintiff: but promoting economic activity is more important than protecting legal certainty since economic development, not legal certainty is the basis of this coun- try’s prosperity. Therefore, I am right that Keeble takes precedence over Pierson, so I still win. This dispute contains priority debates at two levels: first the parties argue about which case rationale should take precedence (by referring to values advanced by following the rationale), and then they argue about which of the conflicting pref- erence rules for the rationales takes precedence (by referring to the relative order of the values). In general, a priority debate could be taken to any level and will be highly dependent on the context and jurisdiction. Various logics proposed in the AI Law literature are able to formalise such priority debates, such as [17, 35, 19] and [21].
- 385. 18 Argumentation in Legal Reasoning 371 3.2 Other arguments about rules Besides priority debates in case of conflicting rules, these logics can also model debates about certain properties of rules, such as their legal validity or their appli- cability to a legal case. The most fully developed logical theory about what it takes to apply a rule is reason-based logic, developed jointly by Jaap Hage and Bart Ver- heij (e.g. [19, 45]). They claim that applying a legal rule involves much more than subsuming a case under the rule’s conditions. Their account of rule application can be briefly summarised as follows. First in three preliminary steps it must be deter- mined whether the rule’s conditions are satisfied, whether the rule is legally valid, and whether the rule’s applicability is not excluded in the given case by, for instance, a statutory exception. If these questions are answered positively (and all three are open to debate), it must finally be determined that the rule can be applied, i.e., that no conflicting rules or principles apply. On all four questions reason-based logic al- lows reasons for and against to be provided and then weighed against each other to obtain an answer. Consider by way of illustration a recent Dutch case (HR 7-12-1990, NJ 1991, 593) in which a male nurse aged 37 married a wealthy woman aged 97 whom he had been nursing for several years, and killed her five weeks after the marriage. When the woman’s matrimonial estate was divided, the issue arose whether the nurse could retain his share. According to the relevant statutes on Dutch matrimo- nial law the nurse was entitled to his share since he had been the woman’s husband. However, the court refused to apply these statutes, on the grounds that applying it would be manifestly unjust. Let us assume that this was in turn based on the legal principle that no one shall profit form his own wrongdoing (the court did not explic- itly state this). In reason-based logic this case could be formalised as follows (again the full details are suppressed for reasons of brevity). Claimant: Statutory rule R is a valid rule of Dutch law since it was enacted ac- cording to the Dutch constitution and never repealed. All its conditions are satisfied in my case, and so it should be applied to my case. The rule entitles me to my late wife’s share in the matrimonial estate. Therefore, I am entitled to my wife’s share in the matrimonial estate. Defendant: Applying rule R would allow you to profit from your own wrongdoing: therefore rule R should not be applied in this case. Court: The reason against applying this rule is stronger than that for applying the rule, and so the rule does not apply. Of course, in the great majority of cases the validity or applicability of a statute rule is not at issue but instead silently presumed by the parties (recall the differ- ence between arguments and proofs described in the introduction). The new logical techniques alluded to above can also deal with such presumptions, and they can be incorporated in reason-based logic. Reason-based logic also has a mechanism for ‘accruing’ different reasons for conclusions into sets and for weighing these sets against similar sets for conflict-
- 386. 372 Trevor Bench-Capon, Henry Prakken and Giovanni Sartor ing conclusions. Thus it captures that having more reasons for a conclusions may strengthen one’s position. Prakken [33] formalises a similar mechanism for accruing arguments in the context of Dung’s framework. Prakken also proposes three prin- ciples that any model of accrual should satisfy. In Chapter 12 of this book Gordon and Walton show that the Carneades logic of [18] has an accrual mechanism that satisfies these principles. One way to argue about the priority of arguments is to claim that the argument is preferred if it is grounded in the better or more coherent legal theory3. While there has been considerable progress in seeing how theories can be constructed on the basis of a body of past cases, evaluation of the resulting theories in terms of their coherence is more problematic, since coherence is a difficult notion to define pre- cisely4. Bench-Capon Sartor [8] describe some features of a theory which could be used in evaluation, such as simplicity of a theory or the number of precedent cases explained by the theory. As an (admittedly somewhat simplistic) example of the last criterion, consider again the three cases on hunting animals, and imagine two theories that explain the case decisions in terms of the values of promotion of economic activity and protection of legal certainty. A theory that gives precedence to promoting economic activity over protecting legal certainty explains all three precedents while a theory with the reverse value preference fails to explain Keeble. The first theory is therefore on this criterion the more coherent one. However, how several coherence criteria are to be combined is a matter for further resear