Meta-theory of Actions: Beyond Consistency

Metatheory of actions: beyond consistency

Andreas Herzig a Ivan Varzinczak a,∗
a IRIT – Universit´ Paul Sabatier
e
118 route de Narbonne, 31062
Toulouse Cedex 9, France

Abstract

Traditionally, consistency is the only criterion for the quality of a theory in logic-
based approaches to reasoning about actions. This work goes beyond that and con-
tributes to the metatheory of actions by investigating what other properties a good
domain description should have. We state some metatheoretical postulates concern-
ing this sore spot. When all postulates are satisﬁed we call the action theory modu-
lar. Besides being easier to understand and more elaboration tolerant in McCarthy’s
sense, modular theories have interesting properties. We point out the problems that
arise when the postulates about modularity are violated, and propose algorithmic
checks that can help the designer of an action theory to overcome them.

Key words: Reasoning about actions, action theory, modularity, ramiﬁcations

1 Introduction

In logic-based approaches to knowledge representation, a given domain is de-
scribed by a set of logical formulas T , which we call a (non-logical) theory.
That is also the case for reasoning about actions, where we are interested in
theories describing particular actions (or, more precisely, action types). We
call such theories action theories.

A priori consistency is the only criterion that formal logic provides to check
the quality of such descriptions. In the present work we go beyond that, and
argue that we should require more than the mere existence of a model for a
given theory.
∗ Corresponding author.
Email addresses: herzig@irit.fr (Andreas Herzig), ivan@irit.fr (Ivan
Varzinczak).

Preprint submitted to Elsevier Science 12 March 2007

Our starting point is the fact that in reasoning about actions one usually
distinguishes several kinds of logical formulas. Among these are effect ax-
ioms, precondition axioms, and boolean axioms. In order to distinguish such
non-logical axioms from logical axioms, we prefer to speak of effect laws, ex-
ecutability laws, and static laws, respectively. Moreover we single out those
effect laws whose effect is ⊥, and call them inexecutability laws.

Given these types of laws, suppose the language is powerful enough to state
conditional effects of actions. For example, suppose that action a is inexe-
cutable in contexts where ϕ1 holds, and executable in contexts where ϕ2 holds.
It follows that there can be no context where ϕ1 ∧ ϕ2 holds. Now ¬(ϕ1 ∧ ϕ2 )
is a static law that does not mention a. It is natural to expect that ¬(ϕ1 ∧ ϕ2 )
follows from the static laws alone. By means of examples we show that when
this is not the case, then unexpected conclusions might follow from the theory
T , even in the case T is logically consistent.

This motivates postulates requiring that the different laws of an action theory
should be arranged modularly, i.e., in separated components, and in such a way
that interactions between them are limited and controlled. In essence, we argue
that static laws may entail new effects of actions (that cannot be inferred from
the effect laws alone), while effect laws and executability laws should never
entail new static laws that do not follow from the set of static laws alone. We
here formulate postulates that make these requirements precise. It will turn
out that in all existing accounts that allow for these four kinds of laws [1–6],
consistent action theories can be written that violate these postulates. In this
work we give algorithms that allow one to check whether an action theory
satisfies the postulates or not. With such algorithms, the task of correcting
flawed action theories can be made easier.

Although we here use the syntax of propositional dynamic logic (PDL) [7],
all we shall say applies as well to first-order formalisms, in particular to the
Situation Calculus [8]. All postulates we are going to present can be stated as
well for other frameworks, in particular for action languages such as A, AR [9–
11] and others, and for Situation Calculus based approaches. In [12] we have
given a Situation Calculus version of our analysis, while in [13] we presented
a similar notion for ontologies in Description Logics [14]. The present work is
the complete version of the one first appeared in [15].

This text is organized as follows: after some background definitions (Section 2)
we state some postulates concerning action theories (Section 3). In Sections 4
and 5, we study the two most important of these postulates, giving algorithmic
methods to check whether an action theory satisfies them or not. We then
generalize our postulates (Section 6) and discuss possible strengthening of
them (Section 7). In Section 8 we show interesting features of modular action
theories. Before concluding, we assess related work found in the literature on

2

metatheory of actions (Section 9).

2 Preliminaries

2.1 Dynamic logic

Here we establish an ontology of dynamic domains. As our base formalism we
use ∗-free PDL, i.e., PDL without the iteration operator ∗. For more details on
PDL, see [7,16].

Let Act = {a1 , a2 , . . .} be the set of all atomic action constants of a given
domain. Our running example is in terms of the Walking Turkey Scenario [4].
There, the atomic actions are load, shoot and tease. We use a as a variable for
atomic actions. To each atomic action a there is an associated modal operator
[a]. Here we suppose that the underlying multimodal logic is independently
axiomatized (i.e., the logic is a fusion and there is no interaction between the
modal operators [17,18]).

Prop = {p1 , p2 , . . .} denotes the set of all propositional constants, also called
fluents or atoms. Examples of those are loaded, alive and walking. We use p as
a variable for propositional constants.

We here suppose that both Act and Prop are nonempty and finite.

We use small Greek letters ϕ, ψ, . . . to denote classical formulas, also called
boolean formulas. They are recursively defined in the following way:

ϕ ::= p | ⊤ | ⊥ | ¬ϕ | ϕ ∧ ϕ | ϕ ∨ ϕ | ϕ → ϕ | ϕ ↔ ϕ

Fml is the set of all classical formulas.

Examples of classical formulas are walking → alive and ¬(bachelor ∧ married).

A classical formula is classically consistent if there is at least one valuation in
classical propositional logic that makes it true. Given ϕ ∈ Fml, valuations(ϕ)
denotes the set of all valuations of ϕ. We note |= the logical consequence in
CPL
classical propositional logic.

The set of all literals is Lit = Prop ∪ {¬p : p ∈ Prop}. Examples of literals
are alive and ¬walking. ℓ will be used as a variable for literals. If ℓ = ¬p, then
we identify ¬ℓ with p.

A clause χ is a disjunction of literals. We say that a literal ℓ appears in a

3

clause χ, written ℓ ∈ χ, if ℓ is a disjunct of χ.

We denote complex formulas (possibly with modal operators) by capital Greek
letters Φ1 , Φ2 , . . . They are recursively defined in the following way:

Φ ::= ϕ | [a]Φ | a Φ | ¬Φ | Φ ∧ Φ | Φ ∨ Φ | Φ → Φ | Φ ↔ Φ

where Φ denotes a complex formula. a is the dual operator of [a], defined
by: a Φ =def ¬[a]¬Φ. Sequential composition of actions is defined by the ab-
breviation [a1 ; a2 ]Φ =def [a1 ][a2 ]Φ. Examples of complex formulas are loaded →
[shoot]¬alive and hasGun → load; shoot (¬alive ∧ ¬loaded).

If T is a set of formulas (modal or classical), atm(T ) returns the set of all
atoms occurring in T . For instance, atm({¬¬¬p1 , [a]p2 }) = {p1 , p2 }.

For parsimony’s sake, whenever there is no confusion we identify a set of
formulas with the conjunction of its elements. The semantics is that for mul-
timodal K [19,20].

Definition 1 A PDL-model is a tuple M = W, R where W is a set of
valuations (alias possible worlds), and R : Act −→ 2W×W a function mapping
action constants a to accessibility relations Ra ⊆ W × W.

As an example, for Act = {a1 , a2 } and Prop = {p1 , p2 }, we have the PDL-
model M = W, R , where

W = {{p1 , p2 }, {p1 , ¬p2 }, {¬p1 , p2 }},
 

 ({p1 , p2 }, {p1 , ¬p2 }), ({p1 , p2 }, {¬p1 , p2 }), 

R(a1 ) = ,
1 , p2 }, {¬p1 , p2 }), ({¬p1 , p2 }, {p1 , ¬p2 })
 ({¬p
 


R(a2 ) = {({p1 , p2 }, {p1 , ¬p2 }), ({p1 , ¬p2 }, {p1 , ¬p2 })}

Figure 1 gives a graphical representation of M .

a1
a1
p1 , p2 ¬p1 , p2
a2
M : a1 a1

p1 , ¬p2

a2

Fig. 1. Example of a PDL-model for Act = {a1 , a2 }, and Prop = {p1 , p2 }.

4

Given M = W, R , a ∈ Act, and w, w ′ ∈ W, we write Ra instead of R(a),
and wRa w ′ instead of w ′ ∈ Ra (w).

Definition 2 Given a PDL-model M = W, R , the satisfaction relation is
defined as the smallest relation satisfying:
M
• |= p (p is true at world w of model M ) if p ∈ w;
w
M M
• |= [a]Φ if for every w ′ such that wRa w ′ , |= ′ Φ; and
w w
• the usual truth conditions for the other connectives.
M
Definition 3 A PDL-model M is a model of Φ (noted |= Φ) if and only if
M M
for all w ∈ W, |= Φ. M is a model of a set of formulas T (noted |= T ) if
w
M
and only if |= Φ for every Φ ∈ T .
M M
In the model depicted in Figure 1, we have |= p1 → [a2 ]¬p2 and |= p1 ∨ p2 .

Definition 4 A formula Φ is a consequence of the set of global axioms T in
the class of all PDL-models (noted T |= Φ) if and only if for every PDL-
PDL
M M
model M , if |= T , then |= Φ. 1

We here suppose that the logic under consideration is compact [21].

Having established the formal substratum our presentation will rely on, we
present in the next section the different types of formulas we use to describe
dynamic domains.

2.2 Describing action theories in PDL

Before elaborating a theory, we need to specify what we are about to describe,
i.e., what the formulas talk about. Following the tradition in the literature,
we identify a domain (alias scenario) with the actions we take into account
and the fluents they can change. More formally, we have:

Definition 5 A domain signature is a tuple Act, Prop .

An example of a domain is the well-known Yale Shooting Scenario (YSS) [22],
whose signature comprises the actions load, wait and shoot, and fluents loaded
and alive.
1 Instead of global consequence, in [5] local consequence is considered. For that
reason, a further modal operator 2 had to be introduced, giving a logic that is
multimodal K plus monomodal S4 for 2, and where axiom schema 2Φ → [a]Φ
holds.

5

Given a domain Act, Prop , we are interested in theories whose statements
describe the behavior of actions of Act on the fluents of Prop. PDL allows
for the representation of such statements, that we here call action laws. We
distinguish several types of them. We call effect laws formulas relating an
action to its effects. Statements of conditions under which an action cannot
be executed are called inexecutability laws. Executability laws in turn stipulate
the context where an action is guaranteed to be executable. Finally, static laws
are formulas that do not mention actions. They express constraints that must
hold in every possible state. These four types of laws are our fundamental
entities and we introduce them more formally in the sequel.

2.2.1 Static laws

Frameworks which allow for indirect effects of actions make use of logical
formulas that state invariant propositions about the world. Such formulas
delimit the set of possible states. They do not refer to actions, and we suppose
here that they are expressed as formulas of classical propositional logic.

Definition 6 A static law is a formula ϕ ∈ Fml.

In our running example, the static law walking → alive says that if a turkey
is walking, then it must be alive. Another one is saved ↔ (mbox1 ∨ mbox2),
which states that an e-mail message is saved if and only if it is in mailbox 1
or in mailbox 2 or both [23].

In some action languages, such as AR for example, we would write the state-
ment always alive → walking, and in the Situation Calculus it would be the
first-order formula

∀s.(Holds(walking, s) → Holds(alive, s)).

The set of all static laws of a given domain is denoted by S . At first glance,
no requirement concerning consistency of S is made. Of course, we want S to
be consistent, otherwise the whole theory is inconsistent. As we are going to
see in the sequel, however, consistency of S alone is not enough to guarantee
the consistency of a theory.

2.2.2 Effect laws

Logical frameworks for reasoning about actions contain expressions linking
actions and their effects. We suppose that such effects might be conditional,
and thus get a third component of such laws.

6

In PDL, the formula [a]Φ states that formula Φ is true after every possible
execution of action a.

Definition 7 An effect law 2 for action a is of the form ϕ → [a]ψ, where
ϕ, ψ ∈ Fml, with ψ classically consistent.

The consequent ψ is the effect which obtains when action a is executed in a
state where the antecedent ϕ holds. An example of an effect law is loaded →
[shoot]¬alive, saying that whenever the gun is loaded, after shooting the turkey
is dead. Another one is ⊤ → [tease]walking: in every circumstance, the result
of teasing is that the turkey starts walking. For parsimony’s sake, the latter
effect law will be written [tease]walking.

Note that the consistency requirement for ψ makes sense: if ψ is inconsistent
then we have an inexecutability law, that we consider as a separate entity and
which we are about to introduce formally in the sequel. On the other hand, if
ϕ is inconsistent then the effect law is obviously superfluous.

For the first example above, in action languages one would write the statement

shoot causes ¬alive if loaded,

and in the Situation Calculus formalism one would write the first-order formula

∀s.(Holds(loaded, s) → ¬Holds(alive, do(shoot, s))).

2.2.3 Inexecutability laws

We consider effect laws with inconsistent consequents as a particular kind
of law which we call inexecutability laws. (Such laws are sometimes called
qualifications [24].) This allows us to avoid mixing things that are conceptually
different: for an action a, an effect law mainly associates it with a consequent
ψ, while an inexecutability law only associates it with an antecedent ϕ, viz.
the context which precludes the execution of a.

Definition 8 An inexecutability law for action a is of the form ϕ → [a]⊥,
with ϕ ∈ Fml.

For example ¬hasGun → [shoot]⊥ expresses that shoot cannot be executed
if the agent has no gun. Another example is dead → [tease]⊥: a dead turkey
cannot be teased.

2 Effect laws are often called action laws, but we prefer not to use that term here
because it would also apply to executability laws that are to be introduced in the
sequel.

7

In AR we would write the statement impossible shoot if ¬hasGun, and in the
Situation Calculus our example would be

∀s.(¬Holds(hasGun, s) → ¬Poss(shoot, s)).

2.2.4 Executability laws

With only static and effect laws one cannot guarantee that the action shoot
can be executed whenever the agent has a gun. We need thus a way to state
such conditions.

In dynamic logic the dual a ϕ, defined as ¬[a]¬ϕ, can be used to express
executability. a ⊤ thus reads “execution of action a is possible”.

Definition 9 An executability law for action a is of the form ϕ → a ⊤,
where ϕ ∈ Fml.

For instance hasGun → shoot ⊤ says that shooting can be executed whenever
the agent has a gun, and ⊤ → tease ⊤, also written tease ⊤, establishes that
the turkey can always be teased.

Some approaches (most prominently Reiter’s) use biconditionals ϕ ↔ a ⊤,
called precondition axioms. This is equivalent to ¬ϕ ↔ [a]⊥, which highlights
that they merge information about inexecutability with information about
executability. Here we consider these entities to be different and keep them
separate.

In action languages such laws are not represented, they are rather implicitly
inferred from inexecutability statements (cf. Section 7). In Situation Calculus
our example would be stated as

∀s.(Holds(hasGun, s) → Poss(shoot, s)).

Whereas all the extant approaches in the literature that allow for indirect
effects of actions contain static and effect laws, and provide a way for rep-
resenting inexecutabilities (in the form of implicit qualifications [25,2,4]), the
status of executability laws is less consensual. Some authors [26,27,3,4] more
or less tacitly consider that executability laws should not be made explicit but
rather inferred by the reasoning mechanism. Others [2,1,5,6] have executability
laws as first class objects one can reason about.

It seems a matter of debate whether one can always do without executabilities.
In principle it seems to be strange to just state information about necessary
conditions for action execution (inexecutabilities) without saying anything
about its sufficient conditions. The justification is that given an action we

8

have three possible situations: it is known to be executable, known to be
inexecutable, and unknown whether executable. This is the reason why we
think that we need executability laws. Indeed, in several domains one wants
to explicitly state under which conditions a given action is guaranteed to be
executable, e.g. that a robot never gets stuck and is always able to execute
a move action. And if we have a plan such as load; shoot (load followed by
shoot) of which we know that it achieves the goal ¬alive, then we would like
to be sure that it is executable in the first place! 3 In any case, allowing for
executability laws gives us more flexibility and expressive power.

2.2.5 Action theories

Given a domain Act, Prop , for an action a ∈ Act, we define E a as the set of
its effect laws, X a the set of its executability laws, and I a that of its inexe-
cutability laws.

Definition 10 An action theory for a is a tuple S , E a, X a , I a .

In our running scenario example, a theory for the action shoot would be

S = {walking → alive}, E shoot = {loaded → [shoot]¬alive},

X shoot = {hasGun → shoot ⊤}, I shoot = {¬hasGun → [shoot]⊥}

Given a dynamic domain we define E = a∈Act E a , X = a∈Act X a , and I =
a a a
a∈Act I . All these sets are finite, because Act is finite and each of the E , X ,
I a is finite.

Definition 11 An action theory is a tuple of the form S , E , X , I .

Given an action theory S , E , X , I and a formula Φ, we write S , E , X , I |=PDL
Φ instead of S ∪ E ∪ X ∪ I |= Φ.
PDL

When formalizing dynamic domains, we face the frame problem [8] and the
ramification problem [28]. In what follows we formally present the logical
framework in which action theories will henceforth be described.

2.3 Dynamic logic and the frame problem

As the reader might have already expected, the logical formalism of PDL alone
does not solve the frame problem. For instance, if S , E , X , I describes our

3 Of course this would require a solution to the qualification problem [24].

9

shooting domain, then

S , E , X , I |= hasGun → [load]hasGun.
PDL

The same can be said about the ramification problem in what concerns the
derivation of indirect effects not properly caused by the action under consid-
eration. For example,

S , E , X , I |= ¬alive → [tease]alive.
PDL

Thus, given an action theory S , E , X , I , we need a consequence relation pow-
erful enough to deal with the frame and ramification problems. This means
that the deductive power of PDL has to be augmented in order to ensure that
the only non-effects of actions that follow from the theory are those that are re-
ally relevant. The presence of static laws makes that this is a delicate task, and
starting with [2,3], several authors have argued that some notion of causality
is needed. In this work we opt for the dependence-based approach presented
in [5], which has been shown [29] to subsume Reiter’s solution to the frame
problem [30], and moreover at least partially accounts for the ramification
problem [31].

In the logical framework developed in [5], metalogical information, given in
the form of a dependence relation, is added to PDL.

Definition 12 (Dependence relation [5]) A dependence relation is a bi-
nary relation ; ⊆ Act × Lit.

The expression a ; ℓ denotes that the execution of action a may make the
literal ℓ true. In our example we have
 

 shoot, ¬loaded , shoot, ¬alive , 

;= ,
shoot, ¬walking , tease, walking 

 

which means that action shoot may make the literals ¬loaded, ¬alive and
¬walking true, and action tease may make walking true.

Semantically, the dependence-based approach relies on the explanation closure
assumption [26], and its solution to the frame problem consists in a kind of
negation as failure: because load, ¬hasGun ∈ ;, we have load ; ¬hasGun,
/
i.e., ¬hasGun is never caused by load. Thus, in a context where hasGun is
true, after every execution of load, hasGun still remains true. We also have
tease ; alive and tease ; ¬alive. The meaning of all these independences is
that the frame axioms hasGun → [load]hasGun, ¬alive → [tease]¬alive and
alive → [tease]alive hold.

10

We assume that ; is finite.

A dependence relation ; defines a class of possible worlds models:

Definition 13 A PDL-model M = W, R is a ;-model if and only if when-
ever wRa w ′ then:
M M
• if a ; p, then |= p implies |= ′ p; and
w w
M M
• if a ; ¬p, then |= p implies |= ′ p.
w w

Figure 2 depicts the dependence-based condition on models.

. . . , ¬p, . . .

a
a

. . . , ¬p, . . . a . . . , ¬p, . . . . . . , p, . . .
M :

a a

. . . , ¬p, . . .

Fig. 2. Dependence-based condition: preservation of literal ¬p under hypothesis
a ; p.
M M
Given a ;-model M , Φ and T , |= Φ and |= T are defined as in Definition 3.

Definition 14 A formula Φ is a ;-based consequence of the set of global
axioms T in the class of all ;-models (noted T |= Φ) if and only if for every
;
M M
;-model M , if |= T , then |= Φ.

In our example it thus holds

S , E , X , I |= hasGun → [load]hasGun
;

and
S , E , X , I |= ¬alive → [tease]¬alive.
;

In this way, the dependence-based approach solves the frame problem. How-
ever, it does not entirely solve the ramification problem: while indirect effects
such as loaded → [shoot]¬walking can be deduced with |= without explicitly
;
stating that in the set of effect laws for shoot, we nevertheless still have to
state indirect dependences such as shoot ; ¬walking. However, according to
Reiter’s view:

“what counts as a solution to the frame problem [. . .] is a systematic proce-

11

dure for generating, from the effect laws, [. . .] a parsimonious representation
for [all] the frame axioms” [32].

We comply with that as we can define a semi-automatic procedure for gen-
erating the dependence relation from the set of effect laws [33]. Moreover, as
it has been argued in [23,31], our approach is in line with the state of the
art because none of the existing solutions to the frame and the ramification
problems can handle domains with both indeterminate and indirect effects.

In the next section we turn to a metatheoretical analysis of action theories
and make a step toward formal criteria for theory evaluation. Before that, we
need a definition.

Definition 15 Let S , E a, X a , I a be an action theory for a, and ; a depen-
dence relation. Then M = W, R is the big (alias maximal/standard) model
for S , E a, X a , I a and ; if and only if:

• M is a ;-model;
• W = valuations(S ) (all valuations of S ); and
M M
• Ra = {(w, w ′) : for all ϕ → [a]ψ ∈ E a ∪ I a , if |= ϕ, then |= ′ ψ}.
w w

For an example, consider an action theory whose components are given by

S = ∅, E a = {p1 → [a]¬p2 }, X a = { a ⊤},

I a = {p2 → [a]⊥}, and ;= { a, ¬p1 , a, ¬p2 }

Figure 3 depicts one of its models and its associated big model.

a

p1 , ¬p2 p1 , ¬p2

M : a Mbig : a
p1 , p2 ¬p1 , p2

¬p1 , ¬p2 ¬p1 , ¬p2

a a
Fig. 3. A model of an action theory and its big model Mbig .

Big models contain all valuations consistent with S . Clearly, for a big model
M
M we have |= S ∧ E a ∧ I a . Because M maximizes executability, it is only
X a which might not be true in M .

12

In the rest of the paper we characterize when an action theory with a depen-
dence relation has a big model.

3 The postulates

“When does a given action theory have a model?”, and, more importantly, “is
that model intended?” are questions that naturally arise when we talk about
action theories. Here we claim that all the approaches that are put forward in
the literature are too liberal in the sense that we can have satisfiable action
theories that are intuitively incorrect. We argue that something beyond the
consistency notion is required in order to help us in answering these questions.

Our central thesis is that the different types of laws defined in Section 2.2
should be neatly separated in modules. Besides that, we want such laws to
interfere only in one sense: static laws together with action laws for a may
have consequences that do not follow from the action laws for a alone. The
other way round, action laws should not allow to infer new static laws; effect
laws should not allow to infer inexecutability laws; action laws for a should
not allow to infer action laws for a′ , etc. This means that our logical modules
should be designed in such a way that they are as specialized and as little
dependent on others as possible.

A first step in this direction has been the proposed division of our entities into
the sets S , E , X and I . In order to accomplish our goal, we have to diminish
interaction among such modules, rendering them the least interwoven we can.
The rest of the section contains postulates expressing this. We here only state
the postulates, and defer explanations and discussions to Sections 4–6.

PC (Logical consistency): S , E a , X a , I a |= ⊥
;

The theory of a given action should be logically consistent.

PS (No implicit static laws):

if S , E a, X a , I a |= ϕ, then S |= ϕ
; CPL

If a classical formula can be inferred from the action theory, then it should
be inferable from the set of static laws alone. (Note that on the left we use
the ;-based consequence, while on the right we use consequence in classical
logic: as both S and ϕ are classical, ϕ should be inferable from S in classical
logic.)

13

PI (No implicit inexecutability laws):

if S , E a , X a , I a |= ϕ → [a]⊥, then S , I a |= ϕ → [a]⊥
; PDL

If an inexecutability law for an action a can be inferred from its action theory,
then it should be inferable in PDL from the static laws and the inexecutability
laws for a alone. Note that we used |= instead of |= because we also suppose
PDL ;
that neither frame axioms nor indirect effects should be relevant to derive
inexecutability laws. The same remark holds for the postulate that follows:

PX (No implicit executability laws):

if S , E a , X a , I a |= ϕ → a ⊤, then S , X a |= ϕ → a ⊤
; PDL

If an executability law for a can be inferred from its action theory, then it
should already “be” in X a , in the sense that it should also be inferable in PDL
from the set of static and executability laws for a alone.

Postulate PC is obvious, for we are interested in consistent theories. It can
be shown that PX is a consequence of PS (see Corollary 41).

Thus, while PC is obvious and PX can be ensured by PS, things are less
obvious for Postulates PS and PI: it turns out that for all approaches in
the literature they are easily violated by action theories that allow to express
the four kinds of laws. We therefore study each of these postulates in the
subsequent sections by means of examples, give algorithms to decide whether
they are satisfied, and discuss about what to do in the case the answer is ‘no’.

4 No implicit static laws

While executability laws increase expressive power, they might conflict with
inexecutability laws. Consider, for example, the following action theory:
 

 [tease]walking, 

S1 = {walking → alive}, E1 = ,
 loaded → [shoot]¬alive 
 

X1 = { tease ⊤}, I1 = {¬alive → [tease]⊥}
and the dependence relation:
 

 shoot, ¬loaded , shoot, ¬alive , 

;=
shoot, ¬walking , tease, walking 

 

14

tease tease
From this description we have the unintuitive inference X1 , I1 |= alive:
PDL
the turkey is immortal! This is an implicit static law because alive does not
tease tease tease
follow from S1 alone: S1 , E1 , X1 , I1 violates Postulate PS.

Implicit static laws are not a drawback of our underlying logical formalism.
They also appear in Situation Calculus-based approaches and in causal laws
theories. To witness 4 , suppose in Lin’s framework we have

Holds(p1 , s) → Caused(p2 , true, s) (1)

and
Caused(p2 , false, s) (2)
Then from (2) we get
¬Holds(p2 , s) (3)
and then conclude
¬Caused(p2 , true, s) (4)
Finally, from (1) and (4) we get

¬Holds(p1 , s)

which is an implicit static law.

To see how implicit static laws show up in McCain and Turner’s causal laws
approach [3], let the causal law ϕ ⇒ ψ and T = {¬ψ}. Then ¬ϕ is an implicit
static law in such a description.

How can we ﬁnd out whether an action theory for a with a dependence relation
; satisﬁes Postulate PS?

Theorem 16 S , E a, X a , I a and ; satisfy Postulate PS if and only if the
big model for S , E a , X a , I a and ; is a model of S , E a, X a , I a and ;.

PROOF.

Let M = W, R be the big model of S , E a , X a, I a and ;.
M
(⇒): As M is a big model of S , E a, X a , I a and ;, we have |= S ∧E a ∧I a . It
M
remains to show that |= X a . Let ϕi → a ⊤ ∈ X a , and let w ∈ W be such that
M
|= ϕi . Therefore, for all ϕj ∈ Fml such that S , E a, X a , I a |= ϕj → [a]⊥, we
w ;
M
must have |= ϕj , because S , E a , X a, I a |= ¬(ϕi ∧ ϕj ), and as S , E a, X a , I a
w ;
M
and ; satisfy Postulate PS, S |= ¬(ϕi ∧ ϕj ), and hence |= ¬(ϕi ∧ ϕj ).
CPL

4 The examples are from [34].

15

M
Then, by the construction of M , there is some w ′ ∈ W such that |= ′ ψ, for
w
M
all ϕ → [a]ψ such that S , E a , I a |= ϕ → [a]ψ and |= ϕ, and wRa w ′. Hence,
; w
M
|= ϕi → a ⊤, and thus M is a model of S , E a , X a , I a and ;.
w

(⇐): Suppose S , E a, X a , I a and ; do not satisfy Postulate PS. Then there
must be ϕ ∈ Fml such that S , E a , X a , I a |= ϕ and S |= ϕ. This means that
; CPL
there is a valuation val of S that falsifies ϕ. As val ∈ W (because M contains
all possible valuations of S ), M is not a model of S , E a, X a , I a and ;.

We shall give an algorithm to find a finite characterization of all 5 implicit
static laws of a given action theory S , E a, X a , I a . The idea is as follows: for
each executability law ϕ → a ⊤ in the theory, construct from E a , I a and ;
a set of inexecutabilities {ϕ1 → [a]⊥, . . . , ϕn → [a]⊥} that potentially conflict
with ϕ → a ⊤. For each i, 1 ≤ i ≤ n, if ϕ ∧ ϕi is satisfiable w.r.t. S , mark
¬(ϕ∧ϕi ) as an implicit static law. Incrementally repeat this procedure (adding
all the implicit ¬(ϕ ∧ ϕi ) to S ) until no more implicit static law is obtained.

For an example of the execution of the algorithm, consider the action theory
tease tease tease
S1 , E1 , X1 , I1 with ; as above. For the action tease, we have the
tease tease
executability tease ⊤. Now, from E1 , I1 and ; we try to build an
inexecutability for tease. We take [tease]walking and compute then all indirect
effects of tease w.r.t. S1 . From walking → alive, we get that alive is an indirect
effect of tease, giving us [tease]alive. But tease, alive ∈ ;, which means the
/
frame axiom ¬alive → [tease]¬alive holds. Together with [tease]alive, this gives
us the inexecutability ¬alive → [tease]⊥. As S1 ∪ {⊤, ¬alive} is satisfiable (⊤
is the antecedent of the executability tease ⊤), we get ¬alive → ⊥, i.e., the
implicit static law alive. For this example no other inexecutability for tease
can be derived, so the computation stops.

Before presenting the pseudo-code of the algorithm we need some definitions.

Definition 17 Let ϕ ∈ Fml and χ a clause. χ is an implicate of ϕ if and
only if ϕ |= χ.
CPL

In our running example, walking ∨ alive and ¬walking ∨ alive are implicates of
the set of formulas {walking → alive, walking}.

Definition 18 Let ϕ ∈ Fml and χ a clause. χ is a prime implicate of ϕ if
and only if

• χ is an implicate of ϕ, and
5 Actually what the algorithm does is to find an interpolant of all implicit static
laws of the theory.

16

• for every implicate χ′ of ϕ, χ′ |= χ implies χ |= χ′ .
CPL CPL

The set of all prime implicates of a formula ϕ is denoted PI(ϕ).

For example, the set of prime implicates of p1 is just {p1 }, and that of p1 ∧
(¬p1 ∨ p2 ) ∧ (¬p1 ∨ p3 ∨ p4 ) is {p1 , p2 , p3 ∨ p4 }. In our shooting domain, alive is a
prime implicate of {walking → alive, walking}. For more on prime implicates,
their properties and how to compute them see [35].

Definition 19 Let ϕ, ψ ∈ Fml. Then NewCons(ψ, ϕ) = PI(ϕ ∧ ψ) PI(ϕ).

The function NewCons(ψ, ϕ) computes the new consequences of ϕ w.r.t. ψ:
the set of strongest clauses that follow from ϕ ∧ ψ, but do not follow from ϕ
alone (cf. e.g. [36]). It is computed by subtracting the prime implicates of ϕ
from those of ϕ ∧ ψ. For example, NewCons((¬p1 ∨ p2 ) ∧ (¬p1 ∨ p3 ∨ p4 ), p1 ) =
{p2 , p3 ∨ p4 }. And for our scenario, NewCons(walking, walking → alive) =
{alive, walking}.

The algorithm below improves the one in [37] by integrating a solution to the
frame problem (via the dependence relation ;). For convenience, we define
C a = E a ∪ I a as the set of all formulas expressing the direct consequences of
an action a, whether they are consistent or not.

Algorithm 1 (Finding all implicit static laws induced by a)
input: S , E a, X a , I a and ;
output: Simp* , the set of all implicit static laws of S , E a, X a , I a
Simp* := ∅
C a := E a ∪ I a
repeat
Simp := ∅
for all ϕ → a ⊤ ∈ X a do
ˆ ˆ
for all C a ⊆ C a such that C a = ∅ do
ˆ
ϕCâ := {ϕi : ϕi → [a]ψi ∈ C a }
ˆ
ψCâ := {ψi : ϕi → [a]ψi ∈ C a }
for all χ ∈ NewCons(ψCâ , S ) do
if S ∪ Simp* ∪ {ϕ, ϕCâ , ¬χ} |= ⊥ and ∀ℓi ∈ χ, a ; ℓi then
CPL
Simp := Simp ∪ {¬(ϕ ∧ ϕCâ ∧ ¬χ)}
Simp* := Simp* ∪ Simp
until Simp = ∅

This is the key algorithm of the paper. In each step of the algorithm, S ∪ Simp*
is the updated set of static laws (the original ones fed with the implicit laws
caught up to that point). At the end, Simp* collects all the implicit static laws.

Theorem 20 Algorithm 1 terminates.

17

PROOF. Let C a = E a ∪ I a . First, the set of candidates to be an implicit
static law that might be due to a and that are examined in the repeat-loop
is
ˆ
{¬(ϕ ∧ ϕCâ ∧ ¬χ) : C a ⊆ C a , ϕ → a ⊤ ∈ X a and χ ∈ NewCons(ψCâ , S )}

As E a , I a and X a are finite, this set is finite.

In each step, either the algorithm stops because Simp = ∅, or at least one of the
candidates is put into Simp in the outermost for-loop. (This one terminates,
because X a , C a and NewCons(.) are finite.) Such a candidate is not going
to be put into Simp in future steps of the algorithm, because once added to
S ∪ Simp* , it will be in the set of laws S ∪ Simp* of all subsequent executions
of the outermost for-loop, falsifying its respective if-test for such a candidate.
Hence the repeat-loop is bounded by the number of candidates, and therefore
Algorithm 1 terminates.

While terminating, our algorithm comes with considerable computational costs:
first, the number of formulas ϕCâ and ψCâ is exponential in the size of C a ,
and second, the computation of NewCons(ψCâ , S ) might result in exponential
growth. While we might expect C a to be reasonably small in practice (because
E a and I a are in general small), the size of NewCons(ψCâ , S ) is more difficult
to control.
tease tease tease
Example 21 For S1 , E1 , X1 , I1 , Algorithm 1 returns Simp* = {alive}.

Theorem 22 Let Simp* be the output of Algorithm 1 on input S , E a, X a , I a
and ;. Then S , E a , X a , I a with ; satisfies Postulate PS if and only if
Simp* = ∅.

PROOF. See Appendix A.

Theorem 23 Let Simp* be the output of Algorithm 1 on input S , E a, X a , I a
and ;. Then

(1) S ∪ Simp* , E a , X a , I a with ; satisfies PS (has no implicit static law).
(2) S , E a , X a , I a |=;
Simp* .

PROOF. Item (1) is straightforward from the termination of Algorithm 1 and
Theorem 22. Item (2) follows from the fact that by the if-test in Algorithm 1,
the only formulas that are put in Simp* at each execution of the repeat-
loop are exactly those that are implicit static laws of the current theory, and
therefore of the original theory, too.

18

Corollary 24 For all ϕ ∈ Fml, S , E a , X a , I a |= ϕ if and only if S ∪
;
Simp* |= ϕ.
CPL

PROOF. For the left-to-right direction, let S , E a , X a , I a |= ϕ, for given
;
ϕ ∈ Fml. Then S ∪ Simp* , E a , X a, I a |= ϕ, by monotonicity. By Theorem 23-
;
(1), S ∪ Simp* , E a , X a, I a has no implicit static law, hence S ∪ Simp* |= ϕ.
CPL

The right-to-left direction is straightforward by Theorem 23-(2).

What shall we do once we have discovered an implicit static law?

The existence of implicit static laws may indicate too strong executability laws:
in Example 21, we wrongly assumed that tease is always executable. Thus one
way of ‘repairing’ our theory would be to consider the weaker executability
alive → tease ⊤ instead of tease ⊤ in X tease .

On the other hand, implicit static laws may also indicate that the inexecutabil-
ity laws are too strong:

Example 25 Consider S = ∅, E shoot = {loaded → [shoot]¬alive}, X shoot =
{hasGun → shoot ⊤} and I shoot = {[shoot]⊥}, with the dependence relation
;= { shoot, ¬alive , shoot, ¬walking . For this theory Algorithm 1 returns
Simp* = {¬hasGun}.

In Example 25 we discovered that the agent never has a gun. The problem here
can be overcome by weakening [shoot]⊥ in I shoot with ¬hasGun → [shoot]⊥. 6

We can go further on this reasoning and also argue that the problem may
be due to a too strong set of effect laws, or even to too strong frame ax-
ioms (i.e., a too weak dependence relation). To witness, for Example 21, if
we replace the law [tease]walking by the weaker alive → [tease]walking, the
resulting action theory would satisfy Postulate PS. In the same way, stat-
ing the (unintuitive) dependence tease ; alive (which means the frame axiom
¬alive → [tease]¬alive is no longer valid) guarantees satisfaction of PS. (Note,
however, that this solution becomes intuitive when alive is replaced by awake.)

To finish, implicit static laws of course may also indicate that the static laws
are too weak:
6 Regarding Examples 21 and 25, one might argue that in practice such silly errors
will never be made. Nevertheless, the examples here given are quite simplistic, and
for applications of real interest, whose complexity will be much higher, we simply
cannot rely on the designer’s knowledge about all side effects the stated formulas
can have.

19

Example 26 Suppose a computer representation of the line of integers, in
which we can be at a strictly positive number, pos, or at a negative one or
zero, ¬pos. Let maxInt and minInt, respectively, be the largest and the smallest
representable integer number. Action goLeft is the action of moving to the
biggest integer strictly smaller than the one at which we are. Consider the
following action theory for this scenario (ati means we are at number i):

S = {ati → pos : 0 < i ≤ maxInt} ∪ {ati → ¬pos : minInt ≤ i ≤ 0}

E = {atminInt → [goLeft]underflow} ∪ {ati → [goLeft]ati−1 : i > minInt},
X = { goLeft ⊤}, I = ∅
with the dependence relation (minInt ≤ i ≤ maxInt):
 

 goLeft, ati , goLeft, pos , 

;=
goLeft, ¬pos , goLeft, underflow 

 

Applying Algorithm 1 to this action theory gives us the implicit static law
¬(at1 ∧ at2 ), i.e., we cannot be at numbers 1 and 2 at the same time.

To summarize, in order to satisfy Postulate PS, an action theory should con-
tain a complete set of static laws or, alternatively, should not contain too
strong action laws.

Remark 27 S ∪ Simp* in general is not intuitive.

Whereas in the latter example the implicit static laws should be added to S , in
the others the implicit static laws are unintuitive and due to an (in)executability
law that is too strong and should be weakened. Of course, how intuitive the
modified action theory will be depends mainly on the knowledge engineer’s
choice.

To sum it up, eliminating implicit static laws may require revision of S , E a or
;, or completion of X a and I a . Completing I a is the topic we address in the
next section.

5 No implicit inexecutability laws

Let S2 = S1 , E2 = E1 and X2 = I2 = ∅, and let ; be that for S1 , E1, X1 , I1 .
Note that S2 , E2 , X2 , I2 and ; satisfy Postulate PS. From [tease]walking it
follows with S2 that [tease]alive, i.e., in every situation, after teasing the turkey,
tease
it is alive: S2 , E2 |= [tease]alive. Now as tease ; alive, the status of alive
PDL
tease
is not modified by tease, and we have S2 , E2 |= ¬alive → [tease]¬alive.
;

20

From the above, it follows
tease tease tease
S2 , E2 , X2 , I2 |= ¬alive → [tease]⊥,
;

i.e., an inexecutability law stating that a dead turkey cannot be teased. But
tease
S2 , I2 |= ¬alive → [tease]⊥,
PDL

hence Postulate PI is violated. Here the formula ¬alive → [tease]⊥ is an
example of what we call an implicit inexecutability law.

In the literature, such laws are also known as implicit qualifications [25], and it
has been often supposed, in a more or less tacit way, that it is a positive feature
of frameworks to leave them implicit and provide mechanisms for inferring
them [2,38,39]. The other way round, one might argue as well that implicit
qualifications indicate that the domain has not been described in an adequate
manner: the form of inexecutability laws is simpler than that of effect laws,
and it might be reasonably expected that it is easier to exhaustively describe
them. 7 Thus, all inexecutabilities of a given action should be explicitly stated,
and this is what Postulate PI says.

How can we check whether PI is violated? We can conceive an algorithm
to find implicit inexecutability laws of a given action a. The basic idea is as
follows: for every combination of effect laws of the form (ϕ1 ∧ . . . ∧ ϕn ) →
[a](ψ1 ∧ . . . ∧ ψn ), with each ϕi → [a]ψi ∈ E a , if ϕ1 ∧ . . . ∧ ϕn is consistent w.r.t.
to S , ψ1 ∧ . . . ∧ ψn inconsistent w.r.t. S , and S , I a |= (ϕ1 ∧ . . . ∧ ϕn ) → [a]⊥,
PDL
then output (ϕ1 ∧ . . . ∧ ϕn ) → [a]⊥ as an implicit inexecutability law. Our
algorithm basically does this, and moreover takes into account dependence
information.
tease tease tease
For an example of the execution of the algorithm, take S2 , E2 , X2 , I2
tease
with ; as given above. From E2 we get the law ⊤ → [tease]walking, whose
antecedent is consistent with S . As long as |= ¬alive → [tease]¬alive and
;
tease
S ∪ {walking} |= alive, and because S , I2
CPL
|= (⊤ ∧ ¬alive) → [tease]⊥,
PDL
we caught an implicit inexecutability. As there is no other combination of
effect laws for tease, we end the simulation here.

Below is the pseudo-code of the algorithm for that (the reason X a is not used
in the computation will be made clear in the sequel):

Algorithm 2 (Finding implicit inexecutability laws for a)
input: S , E a, X a , I a and ;

7 Note that this concerns the necessary conditions for executability, and thus it is
not related to the qualification problem, which basically says that it is difficult to
state all the sufficient conditions for executability.

21

a
output: Iimp , the set of implicit inexecutability laws for a
a
Iimp := ∅
ˆ
for all E a ⊆ E a do
ˆ
ϕEâ := {ϕi : ϕi → [a]ψi ∈ E a }
ˆ
ψEâ := {ψi : ϕi → [a]ψi ∈ E a }
for all χ ∈ NewCons(ψEâ , S ) do
if ∀ℓi ∈ χ, a ; ℓi and S , I a |= (ϕEâ ∧ ¬χ) → [a]⊥ then
PDL
a a
Iimp := Iimp ∪ {(ϕEâ ∧ ¬χ) → [a]⊥}

Theorem 28 Algorithm 2 terminates.

PROOF. Straightforward, as we have assumed S , E , I and ; finite, and
NewCons(.) is finite (because S and ψEâ are finite).

tease tease tease
Example 29 Consider S2 , E2 , X2 , I2 and ; as given above. Then
tease
Algorithm 2 returns Iimp = {¬alive → [tease]⊥}.

Nevertheless, applying Algorithm 2 is not enough to guarantee Postulate PI,
as illustrated by the following example:

Example 30 (Incompleteness of Algorithm 2 without PS) Let S = ∅,
E a = {p1 → [a]p2 }, X a = { a ⊤}, I a = {p2 → [a]⊥}, and ;= ∅. Then we
have S , E a , X a , I a |= p1 → [a]⊥, but running Algorithm 2 on S , E a, X a , I a
;
a
we get S , Iimp |= p1 → [a]⊥.
PDL

Example 30 shows that the presence of implicit static laws (induced by ex-
ecutabilities) implies the existence of implicit inexecutabilities that are not
caught by Algorithm 2. One way of getting rid of this is requiring S , E a, X a , I a
and ; to satisfy Postulate PS prior to running Algorithm 2. This gives us
the following result:

Theorem 31 Let Iimp be the output of Algorithm 2 on input S , E a, X a , I a
a

and ;. If S , E , X , I a with ; satisfies Postulate PS, then it satisfies Pos-
a a
a
tulate PI if and only if Iimp = ∅.

PROOF. See Appendix B.

With Algorithm 2, not only do we decide whether Postulate PI is satisfied,
but we also get information on how to “repair” the action theory. The set of
implicit inexecutabilities so obtained provides logical and metalogical infor-
mation concerning the correction that must be carried out: in the first case,
elements of Iimp can be added to I a ; in the second one, Iimp helps in properly
a a

22

changing E a or ;. For instance, to correct the action theory of our example,
the knowledge engineer would have the following options:
tease
(1) Add the qualification ¬alive → [tease]⊥ to I2 ; or
(2) Add the (unintuitive) dependence tease, alive to ;; or
tease
(3) Weaken the effect law [tease]walking to alive → [tease]walking in E2 .

It is easy to see that whatever she opts for, the resulting action theory for
tease will satisfy Postulate PI (while still satisfying PS).

Example 32 (Drinking coffee [12]) Suppose a situation in which we rea-
son about the effects of drinking a cup of coffee:
 
 sugar

→ [drink]happy, 

S = ∅, E drink = , X drink = I drink = ∅
 salt → [drink]¬happy
 


and the dependence relation

;= { drink, happy , drink, ¬happy }

Observe that S , E drink , X drink , I drink with ; satisfies PS. Then, running Al-
drink
gorithm 2 on this action theory will give us Iimp = {(sugar∧salt) → [drink]⊥}.

Remark 33 I a ∪ Iimp is not always intuitive.
a

Whereas in Example 29 we have got an inexecutability that could be safely
tease
added to I2 , in Example 32 we got an inexecutability that is unintuitive
(just the presence of sugar and salt in the coffee precludes drinking it). In
that case, revision of other parts of the theory should be considered in order
to make it intuitive. Anyway, the problem pointed out in the depicted scenario
just illustrates that intuition is beyond syntax. The scope of this work relies
on the syntactical level. Only the knowledge engineer can judge about how
intuitive a formula is.

In what follows we revisit our postulates in order to strengthen them to the
case where more than one action is under concern and thus get results that
can be applied to whole action theories.

6 Generalizing the postulates

We have seen the importance that satisfaction of Postulates PC, PS and PI
may have in describing the action theory of a particular action a. However,
in applications of real interest more than one action is involved, and thus a

23

natural question that could be raised is “can we have similar metatheoretical
results for multiple action theories”?

In this section we generalize our set of postulates to action theories as a
whole, i.e., considering all actions of a domain, and prove some interesting
results that follow from that. As we are going to see, some of these results
are straightforward, while others must rely on some additional assumptions in
order to hold.

A generalization of Postulate PC is quite easy and has no need for justiﬁcation:

PC* (Logical consistency): S , E , X , I |= ⊥
;

The whole action theory should be logically consistent.

Generalizing Postulate PS will give us the following:

PS* (No implicit static laws):

if S , E , X , I |= ϕ, then S |= ϕ
; CPL

If a classical formula can be inferred from the whole action theory, then it
should be inferable from the set of static laws alone. We have the following
results:

Theorem 34 S , E , X , I and ; satisfy PS* if and only if S , E a, X a , I a
and ; satisﬁes PS for all a ∈ Act.

PROOF.

(⇒): Straightforward: Suppose that for some a ∈ Act S , E a, X a , I a does
not satisfy PS. Then there is ϕ ∈ Fml such that S , E a , X a , I a |= ϕ and
;
S |= ϕ. Of course S , E , X , I |= ϕ, by monotonicity, but still S |= ϕ.
CPL ; CPL
Hence S , E , X , I does not satisfy PS*.

(⇐): Suppose S , E , X , I with ; does not satisfy PS*. Then there is ϕ ∈ Fml
such that S , E , X , I |= ϕ and S |= ϕ. ϕ is equivalent to ϕ1 ∧ . . . ∧ ϕn , with
; CPL
ϕ1 , . . . , ϕn ∈ Fml and such that there is at least one ϕi such that S |= ϕi
CPL
(otherwise S |= ϕ). Because the logic is independently axiomatized, there
CPL
must be some a ∈ Act such that S , E a , X a , I a |= ϕi . From this and S |= ϕi
; CPL
it follows that S , E a , X a , I a and with ; do not satisfy PS.

Corollary 35 S , E , X , I and ; satisfy Postulate PS* if and only if the big
model for S , E , X , I and ; is a model of S , E , X , I .

24

PROOF. The proof follows from Theorems 16 and 34.

Theorem 36 If S , E , X , I with ; satisfies PS*, then S , E , X , I with ;
satisfies PC* if and only if S , E a , X a , I a and ; satisfy PC for all a ∈ Act.

PROOF. Let S , E , X , I and ; satisfy PS*.

(⇒): Suppose that S , E a, X a , I a with ; does not satisfy PC, for some a ∈
Act. Because S , E , X , I and ; satisfy PS*, S , E a, X a , I a and ; satisfy
Postulate PS (Theorem 34), and then S |= ⊥. From this it follows that
CPL
S , E , X , I does not satisfy Postulate PC*.

(⇐): Suppose S , E , X , I and ; do not satisfy PC*. Then S , E , X , I |= ⊥.
;
Because S , E , X , I with ; satisfies Postulate PS*, S |= ⊥. Since Act = ∅,
CPL
there is some a ∈ Act such that S , E a , X a, I a |= ⊥.
;

A more general form of Postulate PI can also be stated:

PI* (No implicit inexecutability laws):

if S , E , X , I |= ϕ → [a]⊥, then S , I |= ϕ → [a]⊥
; PDL

If an inexecutability law can be inferred from the whole action theory, then it
should be inferable in PDL from the static and inexecutability laws alone.

Note that having that S , E a , X a , I a with ; satisfies PI for all a ∈ Act is
not enough to S , E , X , I with ; satisfy PI* if there are implicit static
laws. To witness, let S = E a1 = ∅, and X a1 = { a1 ⊤}, I a1 = {ϕ → [a1 ]⊥}.
Let also E a2 = X a2 = I a2 = ∅. Observe that both S , E a1 , X a1 , I a1 and
S , E a2 , X a2 , I a2 with ; satisfy PI, but S , E , X , I |= ϕ → [a2 ]⊥ and
;
S , I |= ϕ → [a2 ]⊥.
PDL

Nevertheless, under PS* the result follows:

Theorem 37 Let S , E , X , I with ; satisfy PS*. S , E , X , I with ; sat-
isfies PI* if and only if S , E a, X a , I a and ; satisfy PI for all a ∈ Act.

PROOF. See Appendix C.

In the next section we make a step toward an attempt of amending our mod-
ularity criteria by investigating possible extensions of our set of postulates.

25

7 Can we ask for more?

Can we augment our set of postulates to take into account other modules
of action theories, or even other metatheoretical issues in reasoning about
actions? That is the topic we discuss in what follows.

7.1 Postulates about action effects

It seems to be in line with our postulates to require action theories not to allow
for the deduction of new effect laws: if an effect law can be inferred from an
action theory (and no inexecutability for the same action in the same context
can be derived), then it should be inferable from the set of static and effect
laws alone. This means we should have:

PE (No implicit effect laws):
if S , E , X , I |= ϕ → [a]ψ and S , E , X , I |= ϕ → [a]⊥,
; ;
then S , E |= ϕ → [a]ψ
;

But consider the following intuitively correct action theory:
 

 loaded → [shoot]¬alive, 

S3 = ∅, E3 =
 (¬loaded ∧ alive)

→ [shoot]alive 


X3 = {hasGun → shoot ⊤}, I3 = {¬hasGun → [shoot]⊥}
together with the dependence shoot ; ¬alive. It satisfies Postulates PS* and
PI*, but does not satisfy PE. Indeed:

S3 , E3 , X3 , I3 |= ¬hasGun ∨ loaded → [shoot]¬alive
;

and
S3 , E3 , X3 , I3 |= ¬hasGun ∨ loaded → [shoot]⊥,
;
but
S3 , E3 |= ¬hasGun ∨ loaded → [shoot]¬alive
;
So, Postulate PE would not help us to deliver the goods.

Another possibility of improving our modularity criteria could be:

P⊥ (No unattainable effects):

if ϕ → [a]ψ ∈ E , then S , E , X , I |= ϕ → [a]⊥
;

26

This expresses that if we have explicitly stated an effect law for a in some
context, then there should be no inexecutability law for the same action in
the same context. It is straightforward to design an algorithm which checks
whether this postulate is satisfied. We do not investigate this further here,
but just observe that the slightly stronger version below leads to unintuitive
consequences:

P⊥’ (No unattainable effects – strong version):

if S , E |= ϕ → [a]ψ, then S , E , X , I |= ϕ → [a]⊥
; ;

Indeed, for the above action theory we have

E3 |= (¬hasGun ∧ loaded) → [shoot]¬alive,
;

but
S3 , E3 , X3 , I3 |= (¬hasGun ∧ loaded) → [shoot]⊥.
;
This is certainly too strong. Our example also illustrates that it is sometimes
natural to have ‘redundancies’ or ‘overlaps’ between E and I . Indeed, as we
have pointed out, inexecutability laws are a particular kind of effect laws,
and the distinction here made is conventional. The decision of considering
them as strictly different entities or not depends mainly on the context. At a
representational level we prefer to keep them separated, while in Algorithm 1
we have mixed them together in order to compute the consequences of an
action.

In what follows we address the problem of completing the set of executability
laws of an action theory.

7.2 Maximizing executability

As we have seen, implicit static laws only show up when there are executability
laws. So, a question that naturally raises is “which executability laws can be
consistently added to a given action theory?”.

A hypothesis usually made in the literature is that of maximization of ex-
ecutabilities: in the absence of a proof that an action is inexecutable in a
given context, assume its executability for that context. Such a hypothesis is
captured by the following postulate that we investigate in this section:

PX+ (Maximal executability laws):

if S , E a, X a , I a |= ϕ → [a]⊥, then S , X a |= ϕ → a ⊤
; PDL

27

Such a postulate expresses that if in context ϕ no inexecutability for a can
be inferred, then the respective executability should follow in PDL from the
executability and static laws.

Postulate PX+ generally holds in nonmonotonic frameworks, and can be en-
forced in monotonic approaches such as ours by maximizing X a . We never-
theless would like to point out that maximizing executability is not always
intuitive. To witness, suppose we know that if we have the ignition key, the
tank is full, . . ., and the battery tension is beyond 10V, then the car (neces-
sarily) will start. Suppose we also know that if the tension is below 8V, then
the car will not start. What should we conclude in situations where we know
that the tension is 9V? Maximizing executabilities makes us infer that it will
start, but such reasoning is not what we want if we would like to be sure that
all possible executions lead to the goal.

8 Exploiting modularity

In this section we present other properties related to consistency and modu-
larity of action theories, emphasizing the main results that we obtain when
Postulate PS* is satisfied.

Theorem 38 If S , E , X , I with ; satisfies PS*, then S , E , X , I |= ⊥ if
;
and only if S |= ⊥.
CPL

This theorem says that if there are no implicit static laws, then consistency
of an action theory can be checked by just checking consistency of S .

Theorem 39 If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |=;
ϕ → [a]ψ if and only if S , E a , I a |= ϕ → [a]ψ.
;

PROOF. See Appendix D.

This means that under PS* we have modularity inside E , too: when deducing
the effects of a we need not consider the action laws for other actions. Versions
for executability and inexecutability can be stated as well:

Theorem 40 If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |=;
ϕ → a ⊤ if and only if S , X a |= ϕ → a ⊤.
;

PROOF. See Appendix E.

28

Corollary 41 PX is a consequence of PS.

PROOF. Straightforward.

Theorem 42 If S , E , X , I and ; satisfy Postulates PS* and PI*, then
S , E , X , I |= ϕ → [a]⊥ if and only if S , I a |= ϕ → [a]⊥.
; PDL

PROOF.

(⇒): If S , E , X , I |= ϕ → [a]⊥, then from PS* and Theorem 39 we have
;
S , E a , I a |= ϕ → [a]⊥. From this and PI* we get S , I a |= ϕ → [a]⊥.
; PDL

(⇐): Suppose S , E , X , I |= ϕ → [a]⊥. Then there is a ;-model M such that
;
M M M
|= S ∧E ∧X ∧I and |= ϕ → [a]⊥. Then, given a, we have |= S ∧E a ∧X a ∧I a ,
M
and then |= S ∧ I a . Moreover, by definition, M is a PDL-model. Hence
S , I a |= ϕ → [a]⊥.
PDL

In Theorems 40 and 42, modularity guarantees moreover that no dependence
is needed to derive, respectively, executabilities and inexecutabilities.

Let E a1 ,...,an = 1≤i≤n E ai , X a1 ,...,an = 1≤i≤n X ai , and I a1 ,...,an = 1≤i≤n I ai .
Under Postulate PS*, deduction of an effect of a sequence of actions a1 ; . . . ; an
(prediction) needs neither the effect and inexecutability laws for actions other
than a1 , . . . , an , nor the executability laws of the domain:

Theorem 43 If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |= ;
ϕ → [a1 ; . . . ; an ]ψ if and only if S , E a1 ,...,an , I a1 ,...,an |= ϕ → [a1 ; . . . ; an ]ψ.
;

PROOF. See Appendix F.

The same result holds for testing inexecutability of a sequence of actions:

Corollary 44 If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |= ;
ϕ → [a1 ; . . . ; an ]⊥ if and only if S , E a1 ,...,an , I a1 ,...,an |= ϕ → [a1 ; . . . ; an ]⊥.
;

PROOF. Straightforward, as a special case of Theorem 43.

The next theorem shows that our notion of modularity is also fruitful in plan
validation:

29

Theorem 45 Let S , E , X , I and ; satisfy Postulate PS*. Then we have
S , E , X , I |= ϕ → a1 ; . . . ; an ψ if and only if S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an |=
; ;
ϕ → a1 ; . . . ; an ψ.

PROOF. See Appendix G.

And as a consequence, we also optimize testing executability of a plan:

Corollary 46 Let S , E , X , I and ; satisfy Postulate PS*. Then we have
S , E , X , I |= ϕ → a1 ; . . . ; an ⊤ if and only if S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an |=
; ;
ϕ → a1 ; . . . ; an ⊤.

PROOF. Straightforward, as a special case of Theorem 45.

Theorems 43 and 45 together with Corollaries 44 and 46 suggest that we
can simulate modularization by sub-domains [40]: If {a1 , . . . , an }, Prop′ is a
sub-domain for some Prop′ ⊆ Prop, then S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an with
; corresponds to the module for {a1 , . . . , an }, Prop′ in Lifschitz and Ren’s
sense (see the next section).

9 Related work

Pirri and Reiter have investigated the metatheory of the Situation Calcu-
lus [41]. In a spirit similar to ours, they use executability laws and effect laws.
Contrarily to us, their executability laws are equivalences and are thus at the
same time inexecutability laws. As they restrict themselves to domains with-
out ramifications, there are no static laws, i.e., S = ∅. For this setting they give
a syntactical condition on effect laws guaranteeing that they do not interact
with the executability laws in the sense that they do not entail implicit static
laws. Basically, the condition says that when there are effect laws ϕ1 → [a]ψ
and ϕ2 → [a]¬ψ, then ϕ1 and ϕ2 are inconsistent (which essentially amounts
to having in their theories a kind of “implicit static law schema” of the form
¬(ϕ1 ∧ ϕ2 )).

This then allows them to show that such theories are always consistent. More-
over they thus simplify the entailment problem for this calculus, and show
for several problems such as consistency or regression that only some of the
modules of an action theory are necessary.

30

Amir [42] focuses on design and maintainability of action descriptions apply-
ing many of the concepts of the object-oriented paradigm in the Situation
Calculus. In that work, guidelines for a partitioned representation of a given
theory are presented, with which the inference task can also be optimized, as
it is restricted to the part of the theory that is really relevant to a given query.
This is observed specially when different agents are involved: the design of an
agent’s theory can be done with no regard to others’, and after the integration
of multiple agents, queries about an agent’s beliefs do not take into account
the belief state of other agents.

In the referred work, executabilities are as in [41] and the same condition on
effect laws is assumed, which syntactically precludes the existence of implicit
static laws. The frame problem is solved using Reiter’s solution [32] and then
is also restricted to domains without static laws. Ramifications are dealt with
by compiling them away ` la Reiter and Lin [43] based on the method given
a
in [44], which takes into account only some restricted state constraints.

In spite of using many of the object-oriented paradigm tools and techniques,
no mention is made to the concepts of cohesion and coupling [45,46], which
are closely related to modularity [12]. In the approach presented in [42], even
if modules are highly cohesive, they are not necessarily lowly coupled, due to
the dependence between objects in the reasoning phase. We do not investigate
this further here, but conjecture that this could be done there by, during the
reasoning process defined for that approach, avoiding passing to a module a
formula of a type different from those it contains.

The present work generalizes and extends Pirri and Reiter’s result to the
case where S = ∅ and both these works where the syntactical restriction
on effect laws is not made. This gives us more expressive power, as we can
reason about inexecutabilities, and a better modularity in the sense that we
do not combine formulas that are conceptually different (viz. executabilities
and inexecutabilities). It also constitutes a better approach for domains with
ramifications as we do not impose any restriction on the domain laws we can
deal with.

Zhang et al. [47] have also proposed an assessment of what a good action
theory should look like. They develop the ideas in the framework of EPDL [6],
an extended version of PDL which allows for propositions as modalities to
represent a causal connection between literals. We do not present the details
of that, but concentrate on the main meta-theoretical results.

Zhang et al. propose a normal form for describing action theories, 8 and in-
8 But not as expressive as one might think: For instance, in modeling the non-
deterministic action of dropping a coin on a chessboard, we are not able to
state [drop](black ∨ white). Instead, we should write something like [dropblack ]black,

31

vestigate three levels of consistency. Roughly speaking, a set of laws T is
uniformly consistent if it is globally consistent (i.e., T |=EPDL
⊥); a formula Φ
is T -consistent if T |=EPDL
¬Φ, for T a uniformly consistent theory; T is uni-
versally consistent if (in our terms) every logically possible world is accessible.

Furthermore, two assumptions are made to preclude the existence of implicit
qualifications. Satisfaction of such assumptions means the theory under con-
sideration is safe, i.e., it is uniformly consistent. Such a normal form justifies
the two assumptions made and on which their notion of good theories relies.

Given this, they propose algorithms to test the different versions of consistency
for a theory T that is in normal form. This test essentially amounts to checking
whether T is safe, i.e., whether T |= EPDL
a ⊤, for every action a. Success of
this check should mean that the theory under analysis satisfies the consistency
requirements.

Although they are concerned with the same kind of problems that have been
discussed in this work, they take an overall view of the subject, in the sense
that all problems are dealt with together. This means that in their approach
no special attention (in our sense) is given to the different components of the
theory, and then every time something is wrong with it this is taken as a
global problem inherent to the theory as a whole. Whereas such a “systemic”
view of action theories is not necessarily a drawback (we have just seen the
strong interaction that exists between the different sets of laws composing
an action theory), being modular in our sense allows us to better identify the
“problematic” laws and take care of them. Moreover, the advantage of allowing
to find the set of laws which must be modified in order to achieve the desired
consistency is made evident by the algorithms we have proposed (while their
results only allow to decide whether a given theory satisfies some consistency
requirement).

Lang et al. [48] address consistency of action theories in a version of the causal
laws approach [3], focusing on the computational aspects.

To solve the frame problem, they suppose an abstract notion of completion.
Given a theory T a containing logical information about a’s direct effects as well
as the indirect effects that may follow (expressed in the form of causal laws),
the completion of T a , roughly speaking, is the original theory T a amended of
some axioms stating the persistence of all non-affected (directly nor indirectly)
literals. (Note that such a notion of completion is close to the underlying
semantics of the dependence relation used throughout the present work, which

[dropwhite ]white, [dropblack,white ]black and [dropblack,white ]white, where dropblack is the
action of dropping the coin on a black square (analogously for the others) and
drop = dropblack ∪ dropwhite ∪ dropblack,white , with “∪” the nondeterministic compo-
sition of actions.

32

essentially amounts to the explanation closure assumption [26].)

Their executability problem is to check whether action a is executable in
all possible initial states (Zhang et al.’s safety property). This amounts to
testing whether every possible state w has a successor w ′ reachable by a such
that w and w ′ both satisfy the completion of T a . For the Walking Turkey
Scenario, the formalization of action tease with causal laws is given by:

 
tease

 ⊤ ⇒ walking, 

T tease =
 ¬alive

⇒ ¬walking 


where the first formula is a conditional effect law for tease, and the latter a
causal law in McCain and Turner’s sense. We will not dive in the technical
details, and just note that the executability check will return “no” for this
example as tease cannot be executed in a state satisfying ¬alive.

In the mentioned work, the authors are more concerned with the complexity
analysis of the problem of doing such a consistency test and no algorithm for
performing it is given, however. Despite the fact their motivation is the same
as ours, again what is presented is a kind of “yes-no tool” which can help in
doing a meta-theoretical analysis of a given action theory, and many of the
comments concerning Zhang et al.’s approach could be repeated here.

Another criticism that could be made about both these approaches concerns
the assumption of full executability they rely on. We find it too strong to
require all actions to be always executable (cf. Section 7), and to reject as bad
an action theory admitting situations where some action cannot be executed
at all. As an example, consider a very simple action theory S , E , X , I , where
S = {walking → alive}, E = {[tease]walking}, X = { tease ⊤}, and I = ∅,
with a dependence relation given by ;= { tease, walking }. Observe that, with
our approach, it suffices to derive the implicit inexecutability law ¬alive →
[tease]⊥, change I , and the system will properly run in situations where ¬alive
is the case.

On the other hand, if we consider the equivalent representation of such an
action theory in the approach of Lang et al., after computing the completion
of T tease , if we test its executability, we will get the answer “no”, the reason
being that tease is not executable in the possible state where ¬alive holds. Such
an answer is correct, but note that with only this as guideline we have no idea
about where a possible modification in the action theory should be carried out
in order to achieve full executability for tease. The same observation holds for
Zhang et al.’s proposal.

Just to see how things can be even worse, let the same action theory as above,

33

but with X = {alive → tease ⊤}, obtained by its correction with the algo-
rithms we proposed. Observe that the resulting theory satisfies all our postu-
lates. It is not hard to see, however, that the representation of such an action
theory in the above frameworks, when checked by their respective consistency
tests, is still considered to have a problem.

This problem arises because Lang et al.’s proposal do not allow for executabil-
ity laws, thus one cannot make the distinction between X = { tease ⊤},
X = {alive → tease ⊤} and X = ∅. By their turn, Zhang et al.’s allows for
specifying executabilities, however their consistency definitions do not distin-
guish the cases alive → tease ⊤ and tease ⊤.

A concept similar to that of implicit static laws was firstly addressed, as far as
we are concerned, in the realm of regulation consistency with deontic logic [49].
Indeed, the notions of regulation consistency given in the mentioned work
and that of modularity presented in [37] and refined here can be proved to
be equivalent. The main difference between the mentioned work and the ap-
proach in [37] relies on the fact that in [49] some syntactical restrictions on
the formulas have to be made in order to make the algorithm to work.

Lifschitz and Ren [40] propose an action description language derived from
C+ [50] in which domain descriptions can also be decomposed in modules.
Contrarily to our setting, in theirs a module is not a set of formulas for given
action a, but rather a description of a subsystem of the theory, i.e., each
module describes a set of interrelated fluents and actions. As an example, a
module describing Lin’s suitcase [2] should contain all causal laws in the sense
of C+ that are relevant to the scenario. Actions or fluents having nothing
to do, neither directly nor indirectly, with the suitcase should be described in
different modules. This feature makes such a decomposition somewhat domain-
dependent, while here we have proposed a type-oriented modularization of the
formulas, which does not depend on the domain.

In the referred work, modules can be defined in order to specialize other mod-
ules. This is done by making the new module to inherit and then specialize
other modules’ components. This is an important feature when elaborations
are involved. In the suitcase example, adding a new action relevant to the
suitcase description can be achieved by defining a new module inheriting all
properties of the old one and containing the causal laws needed for the new
action. Such ideas are interesting from the standpoint of software and knowl-
edge engineer: reusability is an intrinsic property of the framework, and easy
scalability promotes elaboration tolerance.

Consistency of a given theory and how to prevent conflicts between modules
(independent or inherited) however is not addressed.

In this work we have illustrated by some examples what we can do in order

34

to make a theory intuitive. This involves theory modification. Action theory
change has been addressed in the recent literature on revision and update [51–
53]. In [54] we have investigated this issue and shown the importance that
modularity has in such a task.

10 Conclusion

Our contribution is twofold: general, as we presented postulates that apply to
all reasoning about actions formalisms; and specific, as we proposed algorithms
for a dependence-based solution to the frame problem.

We have defined here the concept of modularity of an action theory and
pointed out some of the problems that arise if it is not satisfied. In particular
we have argued that the non-dynamic part of action theories could influence
but should not be influenced by the dynamic one. 9

We have put forward some postulates, and in particular tried to demonstrate
that when there are implicit static and inexecutability laws then one has
slipped up in designing the action theory in question. As shown, a possible
solution comes into its own with Algorithms 1 and 2, which can give us some
guidelines in correcting an action theory if needed. By means of examples we
have seen that there are several alternatives of correction, and choosing the
right module to be modified as well as providing the intuitive information that
must be supplied is up to the knowledge engineer.

Given the difficulty of exhaustively enumerating all the preconditions under
which a given action is executable (and also those under which such an action
cannot be executed), it is reasonable to expect that there is always going to be
some executability precondition ϕ1 and some inexecutability precondition ϕ2
that together lead to a contradiction, forcing, thus, an implicit static law
¬(ϕ1 ∧ ϕ2 ). This is the reason we propose to state some information about
both executabilities and inexecutabilities, and then run the algorithms in order
to improve the description.

It could be argued that unintuitive consequences in action theories are mainly
due to badly written axioms and not to the lack of modularity. True enough,
but what we have presented here is the case that making a domain description
modular gives us a tool to detect at least some of such problems and correct it.
(But note that we do not claim to correct badly written axioms automatically
9 It might be objected that it is only by doing experiments that one learns the static
laws that govern the universe. But note that this involves learning, whereas here –
as always done in the reasoning about actions field – the static laws are known once
forever, and do not evolve.

35

and once for all.) Besides this, having separate entities in the ontology and
controlling their interaction help us to localize where the problems are, which
can be crucial for real world applications.

In this work we have illustrated by some examples what we can do in order
to make a theory intuitive. This involves theory modification. In [31] we have
presented a general method for changing a domain description given a formula
we want to contract. Their we have defined a semantics for theory contraction
and also presented its syntactical counterpart through contraction operators.
Soundness and completeness of such operators with respect to the semantics
have been established. In that work we have also shown that modularity is a
sufficient condition for contraction to be successful. This gives further evidence
that the notion of modularity is fruitful.

Our postulates can be formulated in any reasoning about actions framework,
but the algorithms require a decidable logic (in particular Algorithm 2). PDL is
one candidate for that, as we have seen along the paper. For first-order-based
frameworks the consistency checks of Algorithms 1 and 2 are undecidable. We
can get rid of this by assuming that there is no function symbol in the language.
In this way, the result of NewCons(.) is finite and the algorithm terminates.
Hence another candidate for our ideas would have been the Situation Calculus
fragment with only propositional fluents.

The present paper is also a step toward a solution to the problem of indirect
dependences: indeed, if the implicit dependence shoot ; ¬walking is not in ;,
then after running Algorithm 2 we get an indirect inexecutability (loaded ∧
walking) → [shoot]⊥, i.e., shoot cannot be executed if loaded ∧ walking holds.
Such an unintuitive inexecutability is not in I and thus indicates the missing
indirect dependence.

The general case is nevertheless more complex, and it seems that such indirect
dependences cannot be computed automatically in the case of indeterminate
effects (cf. the example in [23]). We are currently investigating this issue.

The first work on formalizing modularity in logical systems in general is due to
Garson [55]. Modularity of theories in reasoning about actions was originally
defined in [15]. Modularization of ontologies in description logics is addressed
in [56]. A different viewpoint of the work we presented here can be found
in [12], where modularity of action theories is assessed from a software en-
gineering perspective. A modularity-based approach for narrative reasoning
about actions is given in [57]. In [13] we show that our modularity notion can
also be extended to ontologies in the description logic ALC.

Our postulates do not take into account causality statements linking propo-
sitions such as those defined in [2,3,58,39]. This could be a topic for further
investigation.

36

Acknowledgments

The authors are grateful to the anonymous referees for useful comments on
an earlier version of this paper.

Ivan Varzinczak has been supported by a fellowship from the government of
the Federative Republic of Brazil. Grant: CAPES BEX 1389/01-7.

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A Proof of Theorem 22

Let Simp* be the output of Algorithm 1 on input S , E a, X a , I a and ;. Then
S , E a , X a, I a with ; satisfies Postulate PS if and only if Simp* = ∅.

We recall that |= is logical consequence in classical propositional logic, and
CPL
PI(A) is the set of prime implicates of a set A of classical formulas.

Before giving the proof of the theorems, we recall some properties of prime
implicates [59,35] and of the function NewCons(.) [36] (see Section 4). Let
ϕ ∈ Fml, A ⊆ Fml finite (identified with the conjunction of its formulas), and
χ be a clause. Then

(1) |= ϕ ↔ PI(ϕ) [35, Corollary 3.2].
CPL
(2) PI(A) ∪ NewCons(ϕ, A) = PI(A ∧ ϕ) (by definition of NewCons(.)).
(3) |= (A ∧ ϕ) ↔ (A ∧ NewCons(ϕ, A)) (from 1 and 2)
CPL
(4) If PI(ϕ) |= χ, then there is χ′ ∈ PI(ϕ) such that χ′ |= χ [35, Propo-
CPL CPL
sition 3.4].

41

Let S , E a , X a , I a with ; be an action theory for a, and let ϕ → a ⊤ ∈ X a ,
ˆ
C a = E a ∪ I a , and C a ⊆ C a . We define:

ϕCâ = ˆ
{ϕi : ϕi → [a]ψi ∈ C a }

ψCâ = ˆ
{ψi : ϕi → [a]ψi ∈ C a }

Moreover, let indepa = {¬ℓ : a ; ℓ}.

Lemma 47 Let indep′a ⊆ indepa . S ∪ {ψCâ } ∪ indep′a |= ⊥ if and only if
CPL
S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥.
CPL

PROOF.
S ∪ {ψCâ } ∪ indep′a |= ⊥
CPL
if and only if

PI(S ∪ {ψCâ }) ∪ indep′a |= ⊥ (by Property 1)
CPL

if and only if

PI(S ) ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥ (by Property 2)
CPL

if and only if

S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥ (by Property 1).
CPL

Lemma 48 Let indep′a ⊆ indepa . If S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥,CPL
then there exists χ ∈ NewCons(ψCâ , S ) such that S ∪ {χ} ∪ indep′a |= ⊥.
CPL

PROOF.
S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥
CPL
if and only if

PI(S ) ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥ (by Property 1)
CPL

if and only if

PI(S ∪ {ψCâ }) ∪ indep′a |= ⊥ (by Property 2)
CPL

if and only if

PI(S ∪ {ψCâ }) |= ¬
CPL
{¬ℓi : ¬ℓi ∈ indep′a }

42

if and only if
PI(S ∪ {ψCâ }) |=CPL
{ℓi : ¬ℓi ∈ indep′a }
if and only if there exists χ ∈ PI(S ∪ {ψCâ }) such that

χ |=CPL
{ℓi : ¬ℓi ∈ indep′a } (by Property 4)

if and only if
{χ} ∪ indep′a |= ⊥
CPL
if and only if
S ∪ {χ} ∪ indep′a |= ⊥.
CPL

Lemma 49 Let indep′a ⊆ indepa . If we have S ∪ {ϕ, ϕCâ } ∪ indep′a |= ⊥ and
CPL
S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥, then there exists χ ∈ NewCons(ψCâ , S )
CPL
such that S ∪ {χ} ∪ indep′a |= ⊥.
CPL

PROOF. By Lemma 48 and classical logic.

Lemma 50 Let indep′a ⊆ indepa . If we have S ∪ {ϕ, ϕCâ } ∪ indep′a |= ⊥ and
CPL
S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥, then there exists χ ∈ NewCons(ψCâ , S )
CPL
such that both S ∪ {ϕ, ϕCâ } ∪ indep′a |= ⊥ and S ∪ {χ} ∪ indep′a |= ⊥.
CPL CPL

PROOF. Trivially, by Lemma 49.

Lemma 51 Let indep′a ⊆ indepa . If χ ∈ NewCons(ψCâ , S ) is such that S ∪
{ϕ, ϕCâ }∪indep′a |= ⊥ and S ∪{χ}∪indep′a |= ⊥, then both S ∪{ϕ, ϕCâ }∪
CPL CPL
{¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥ and S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ;
CPL
ℓi } |= ⊥.
CPL

PROOF. Let S ∪ {ϕ, ϕCâ } ∪ indep′a |= ⊥ and χ ∈ NewCons(ψCâ , S ) be
CPL
such that S ∪ {χ} ∪ indep′a |= ⊥.
CPL

If χ = ⊥, the result is trivial. Otherwise, we have the following cases:

• If atm(χ) ⊂ atm(indep′a ), then the premise is false (and the lemma trivially
holds).
• If atm(χ) = atm(indep′a ), the lemma holds.
• Let atm(χ) ⊂ atm(indep′a ). Then, from

S ∪ {ϕ, ϕCâ } ∪ indep′a |= ⊥ (the hypothesis)
CPL

43

it follows

S ∪ {ϕ, ϕCâ } ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥.
CPL

From
S ∪ {χ} ∪ indep′a |= ⊥ (hypothesis)
CPL
and because
S ∪ indep′a |= ⊥,
CPL
it follows
S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥.
CPL

Lemma 52 If χ ∈ NewCons(ψCâ , S ) is such that both S ∪ {ϕ, ϕCâ } ∪ {¬ℓi :
ℓi ∈ χ and a ; ℓi } |= ⊥ and S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥,
CPL CPL
then S ∪ {ϕ, ϕCâ , ¬χ} |= ⊥ and for all ℓi ∈ χ, a ; ℓi .
CPL

PROOF. From

S ∪ {ϕ, ϕCâ } ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥
CPL

we conclude
S ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥.
CPL

From this and the hypothesis

S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥,
CPL

it follows
S ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ¬χ.
CPL

If S |= ¬χ, then S ∪ {ψCâ } |= ¬χ, and because χ ∈ NewCons(ψCâ , S ), we
CPL CPL
have χ |= ¬χ, a contradiction. Hence S ∪ {χ} |= ⊥.
CPL CPL

Suppose now there is a literal ℓ ∈ χ such that ¬ℓ ∈ {¬ℓi : ℓi ∈ χ and a ; ℓi }.
/
Then, the propositional valuation in which χℓ←true satisfies

S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi },

and then
S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥.
CPL

Hence there cannot be such a literal, and then for all ℓi ∈ χ, a ; ℓi .

Now, from a ; ℓi for all ℓi ∈ χ, we have |=CPL
{¬ℓi : ℓi ∈ χ and a ; ℓi } ↔

44

¬χ. From this and the hypothesis

S ∪ {ϕ, ϕCâ } ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥
CPL

it follows S ∪ {ϕ, ϕCâ , ¬χ} |= ⊥.
CPL

Proof of Theorem 22

We are about to prove that S , E a , X a , I a with ; satisfies Postulate PS if
and only if Simp* = ∅.

(⇒): Suppose Simp* = ∅. Then at the first step of the algorithm there has
ˆ
been some ϕ → a ⊤ ∈ X a and some C a ⊆ C a such that for some χ ∈
a a a
NewCons(ψCâ , S ), S , E , X , I |= ¬(ϕ∧ϕCâ ∧¬χ) and S |= ¬(ϕ∧ϕCâ ∧¬χ).
; CPL
Hence S , E a, X a , I a does not satisfy Postulate PS.

(⇐): Suppose that Simp* = ∅. Therefore for all ϕ′ → a ⊤ ∈ X a and for all
ˆ
subsets C a ⊆ C a , we have that

for all χ ∈ NewCons(ψCâ , S ), if
S ∪ {ϕ′ , ϕCâ , ¬χ} |= ⊥,
CPL
(A.1)
then there exists ℓi ∈ χ such that a ; ℓi

From (A.1) and Lemma 52, we get

for all χ ∈ NewCons(ψCâ , S ), if
S ∪ {ϕ, ϕCâ } ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥,
CPL
then S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥ CPL

From this and Lemma 51, it follows that

for all χ ∈ NewCons(ψCâ , S ), if S ∪ {ϕ′ , ϕCâ } ∪ indep′a |= ⊥,
CPL
then S ∪ {χ} ∪ indep′a |= ⊥ CPL

This and Lemma 50 give us

if S ∪ {ϕ′ , ϕCâ } ∪ indep′a |= ⊥, then
CPL
S ∪ NewCons(ψCâ , S ) ∪ indep′a |= ⊥
CPL

From this and Lemma 47, it follows that for all indep′a ⊆ indepa , for every
ˆ
ϕ′ → a ⊤ ∈ X a and all C a ⊆ C a ,

if S ∪ {ϕ′ , ϕCâ } ∪ indep′a |= ⊥, then S ∪ {ψCâ } ∪ indep′a |= ⊥.
CPL CPL
(A.2)

45

Now, suppose S |= ϕ for some propositional ϕ. We will build a model M
CPL
such that M is a ;-model for S , E a, X a , I a that does not satisfy ϕ.

Let M = W, Ra be such that W = valuations(S ), and Ra be such that for
all w, w ′ ∈ W, wRa w ′ if and only if
M M
• |= ′ ψi for every ϕi → [a]ψi ∈ C a such that |= ϕi ; and
w w
M M
• |= ′ ¬ℓ for all ℓ such that a ; ℓ and |= ¬ℓ.
w w

We have that M is a ;-model, by the definition of Ra . By the definition of
W, M is a model of S . We have that M is a model of E a and I a , too: for every
M
ϕi → [a]ψi ∈ C a and every world w ∈ W, if |= ϕi , then, by the definition of
w
M
Ra , |= ′ ψi for all w ′ ∈ W such that wRa w ′. Moreover, M is also a model of
w
M
X a : for every ϕi → a ⊤ ∈ X a and every world w ∈ W, if |= ϕi , then
w

M
E a (w) = {ϕi → [a]ψi ∈ E a : |= ϕi },
w

and
M
indepa (w) = {¬ℓ : a ; ℓ and |= ¬ℓ}
w
are such that S ∪ {ϕi , ϕE a(w) } ∪ indepa (w) |= ⊥, where
CPL

ϕE a (w) = {ϕi : ϕi → [a]ψi ∈ E a (w)}

From this and (A.2), we have S ∪ {ψE a (w) } ∪ indepa (w) |= ⊥, where
CPL

ψE a (w) = {ψi : ϕi → [a]ψi ∈ E a (w)}
M
As W is maximal, there exists w ′ such that |= ′ ψE a (w) ∧ indepa (w). As Ra is
w
maximal by definition, we have wRa w ′ . Hence there exists at least one w ′ such
M
that wRa w ′ , and |= a ⊤.
w

M
Hence, M is a model of S , E a, X a , I a and ;. Clearly |= ϕ, by the definition
of W. Hence S , E a , X a , I a |= ϕ. Therefore S , E a, X a , I a with ; satisfies
;
Postulate PS.

B Proof of Theorem 31

Let Iimp be the output of Algorithm 2 on input S , E a, X a , I a and ;. If
a

S , E , X a, I a with ; satisfies Postulate PS, then it satisfies Postulate PI if
a
a
and only if Iimp = ∅.

46

Let S , E a, X a , I a be an action theory for a with a dependence relation ;.
ˆ
For every E a ⊆ E a we define:

ϕEâ = ˆ
{ϕi : ϕi → [a]ψi ∈ E a }

ψEâ = ˆ
{ψi : ϕi → [a]ψi ∈ E a }

Moreover, let indepa = {¬ℓ : a ; ℓ}.

Lemma 53 If S , I a |= (ϕEâ ∧ indep′a ) → [a]⊥ and S ∪ {ψEâ } ∪ indep′a |=
PDL CPL
⊥, then there is χ ∈ NewCons(ψEâ , S ) such that S , I a |= (ϕEâ ∧ ¬χ) → [a]⊥
PDL
and a ; ℓi for all ℓi ∈ χ.

PROOF. Let S , I a |= (ϕEâ ∧ indep′a ) → [a]⊥. Then there is a PDL-model
PDL
M M
M = W, Ra such that |= S ∧I a and |= (ϕEâ ∧indep′a ) → [a]⊥. This means
M M
that there is a possible world v ∈ W such that |= ϕEâ ∧ indep′a and |= [a]⊥.
v v
M
From |= ϕEâ ∧ indep′a , it follows
v

S ∪ {ϕEâ } ∪ indep′a |= ⊥
CPL
(B.1)

From hypothesis S ∪ {ψEâ } ∪ indep′a |= ⊥ and Lemma 47, we get
CPL

S ∪ NewCons(ψEâ , S ) ∪ indep′a |= ⊥
CPL

and from this and Lemma 48 we have that there is χ ∈ NewCons(ψEâ , S ) such
that
S ∪ {χ} ∪ indep′a |= ⊥
CPL
(B.2)

From (B.1), (B.2) and classical logic, there is χ ∈ NewCons(ψEâ , S ) such that

S ∪ {ϕEâ } ∪ indep′a |= ⊥ and S ∪ {χ} ∪ indep′a |= ⊥
CPL CPL

From this and Lemma 51 it follows that there is χ ∈ NewCons(ψEâ , S ) such
that
S ∪ {ϕ, ϕEâ } ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥
CPL
and
S ∪ {χ} ∪ {¬ℓi : ℓi ∈ χ and a ; ℓi } |= ⊥CPL

This and Lemma 52 gives us that for all ℓi ∈ χ, a ; ℓi .
M
Now, because M above is such that |= ϕEâ ∧ indep′a , from this and S ∪ {χ} ∪
v
M M
indep′a |= ⊥, we have that |= ϕEâ ∧ ¬χ. Because |= [a]⊥, we therefore
CPL v v
have S , I a |= (ϕEâ ∧ ¬χ) → [a]⊥.
PDL

47

Proof of Theorem 31

We are about to prove that if S , E a , X a , I a with ; satisfies Postulate PS,
a
then it satisfies Postulate PI if and only if Iimp = ∅.

(⇒): Straightforward, as every time S , E a , X a , I a |= ϕ → [a]⊥, we have
;
S , I a |= ϕ → [a]⊥, and then Iimp never changes.
PDL
a

ˆ
(⇐): Suppose that Iimp = ∅. Therefore for all subsets E a ⊆ E a , we have that
a

for all χ ∈ NewCons(ψEâ , S ), if S , I a |= (ϕEâ ∧ ¬χ) → [a]⊥,
PDL (B.3)
then there exists ℓi ∈ χ such that a ; ℓi

ˆ
From (B.3) and Lemma 53, it follows that for all E a ⊆ E a ,

if S , I a |= (ϕEâ ∧ indep′a ) → [a]⊥,
PDL (B.4)
then S ∪ {ψEâ } ∪ indep′a |= ⊥.
CPL

Suppose S , I a |= ϕ → [a]⊥ for some ϕ ∈ Fml. Then there exists a PDL-
PDL
M M
model M = W, Ra such that |= S ∧ I a and |= ϕ → [a]⊥. This means that
M M
there is a possible world v ∈ W such that |= ϕ and |= [a]⊥.
v v

(We are going to build a ;-model of S , E a , X a , I a , and hence conclude that
S , E a , X a , I a |= ϕ → [a]⊥.)
;

For given w ∈ W, we define:
M
I a (w) = {ϕi → [a]⊥ ∈ I a : |= ϕi }
w

Because S , E a , X a , I a with ; satisfies Postulate PS, we can extend M to
a big model M ′ = W′ , R′a such that W = valuations(S ), and R′a is defined
such that for all w, w ′ ∈ W′ , wR′a w ′ if and only if

M′ M′
• |= ′ ¬ℓ for all ℓ such that a ; ℓ and |= ¬ℓ;
w w
M′ M′
• |= ′ ψi for every ϕi → [a]ψi ∈ E a such that |= ϕi ; and
w w
• I a (w) = ∅.

M′
By definition, M ′ is a ;-model. We also have |= S , by the definition of W′ .
M ′ is a model of E a , too: for every ϕi → [a]ψi ∈ E a and every w ∈ W′ , if
M′ M′
|= ϕi , then |= ′ ψi for all w ′ ∈ W′ such that wR′a w ′ . Clearly M ′ is also a
w w
M′
model of I a : for every ϕi → [a]⊥ ∈ I a and every w ∈ W′ , if |= ϕi , then
w
I a (w) = ∅ and R′a (w) = ∅. M ′ is a model of X a , too: for every ϕi → a ⊤ ∈ X a

48

M′
and every w ∈ W′ , if |= ϕi , then
w

M
E a (w) = {ϕi → [a]ψi ∈ E a : |= ϕi }
w

and
M
indepa (w) = {¬ℓ : a ; ℓ and |= ¬ℓ}
w
are such that S , I a |= (ϕE a (w) ∧ indepa (w)) → [a]⊥, where
PDL

ϕE a (w) = {ϕi : ϕi → [a]ψi ∈ E a (w)}

The justification is that S , I a |= (ϕE a (w) ∧ indepa (w)) → [a]⊥ would imply
PDL
S , E a , X a , I a |= (ϕE a (w) ∧indepa (w)) → [a]⊥, and as long as ϕi → a ⊤ ∈ X a ,
;
S , E a , X a , I a |= ¬(ϕi ∧ ϕE a (w) ∧ indepa (w)). As by hypothesis S , E a, X a , I a
;
satisfies PS, S |= ¬(ϕi ∧ ϕE a (w) ∧ indepa (w)), and then w ∈ W′ .
CPL
/

Hence, from S , I a |= (ϕE a (w) ∧ indepa (w)) → [a]⊥ and (B.4), it follows that
PDL
S ∪ {ψE a (w) } ∪ indepa (w) |= ⊥, where
CPL

ψE a (w) = {ψi : ϕi → [a]ψi ∈ E a (w)}

M′
As W′ is maximal, there exists w ′ such that |= ′ ψE a (w) ∧ indepa (w). As R′a
w
is maximal by definition, we have wR′a w ′ . Hence there exists at least one w ′
M′
such that wR′a w ′, and then |= ′ a ⊤.
w

Therefore, M ′ is a model of S , E a, X a , I a and ;.

Looking at v ∈ W′ , we must have S , I a |=PDL
(ϕE a (v) ∧ indepa (v)) → [a]⊥,
M
because otherwise Ra (v) = ∅, against the hypothesis that |= [a]⊥. Hence,
v
from (B.4) it follows that S ∪ {ψE a (v) } ∪ indepa (v) |= ⊥, and then there
CPL
M′
exists at least one v ′ such that vR′a v ′ , and then |= ′ a ⊤. From this it fol-
v
lows that S , E a , X a , I a |= ϕ → [a]⊥. Therefore S , E a , X a, I a and ; satisfy
;
Postulate PI.

C Proof of Theorem 37

Let S , E , X , I with ; satisfy PS*. S , E , X , I with ; satisfies PI* if and
only if S , E a , X a , I a and ; satisfy PI for all a ∈ Act.

(⇒): Suppose that S , E a, X a , I a |= ϕ → [a]⊥. By monotonicity, S , E , X , I |=
; ;
ϕ → [a]⊥, too. Now suppose that S , I a |= ϕ → [a]⊥. Then there exists a
PDL
M
possible worlds model M = W, Ra such that |= S ∧ I a and there is a pos-

49

M M
sible world v ∈ W such that |= ϕ and |= [a]⊥. Let M ′ = W′ , R′ be such
v v
M′
that W′ = W, and R′a′ = ∅, for all a′ = a, and R′a = Ra . Then |= S ∧ I , and
then S , I |= ϕ → [a]⊥. Hence S , E , X , I with ; does not satisfy PI*.
PDL

(⇐): Suppose S , E , X , I and ; do not satisfy Postulate PI*. Then there
exists ϕ ∈ Fml such that S , E , X , I |= ϕ → [a]⊥ and S , I |= ϕ → [a]⊥.
; PDL

Claim: S , E a , X a, I a |= ϕ → [a]⊥.
;

(Proof of the claim): Suppose S , E a , X a , I a |= ϕ → [a]⊥. Then there exists a
;
M M
;-model M = W, Ra such that |= S ∧ E a ∧ X a ∧ I a and |= ϕ → [a]⊥.
M M
This means that there is a possible world v ∈ W such that |= ϕ and |= [a]⊥,
v v
i.e., there is v ′ ∈ W such that Ra (v) = v ′ .

(We extend M to all other actions S , E , X , I speaks of and obtain a ;-
model of S , E , X , I .)

Given w ∈ W, for each ai ∈ Act we define:
M
I ai (w) = {ϕj → [ai ]⊥ ∈ I ai : |= ϕj }
w
M
X ai (w) = {ϕj → ai ⊤ ∈ X ai : |= ϕj }
w

Let M ′ = W′ , R′ be such that W′ = W, and R′ = Ra ∪ a′ =a Ra′ , where for
each a′ = a and every w, w ′ ∈ W′ , wRa′ w ′ if and only if
M′ M′
• |= ′ ¬ℓ for all ℓ such that a′ ; ℓ and |= ¬ℓ;
w w
M′ ′ M′
• |= ′ ψi for every ϕi → [a′ ]ψi ∈ E a such that |= ϕi ; and
w w
′
• I a (w) = ∅.

By definition, M ′ is a model of the dependence relation ;. Because, by hy-
pothesis, S , E , X , I with ; satisfies PS*, there is no implicit static law, i.e.,
for every ai ∈ Act and every w ∈ W′ , if I ai (w) = ∅, then X ai (w) = ∅. Then,
as W′ = valuations(S ), M ′ is a model of S . We have that M ′ is a model
of E , too: it is a model of E a , and given a′ = a, for every ϕi → [a′ ]ψi ∈ E
M′ M′
and every w ∈ W′ , if |= ϕi , then |= ′ ψi for all w ′ ∈ W′ such that wRa′ w ′ .
w w
Clearly M ′ is also a model of I : it is a model of I a , and given a′ = a, for
M′
every ϕi → [a′ ]⊥ ∈ I and every w ∈ W′ , if |= ϕi , then I a (w) = ∅ and
′

w
Ra′ (w) = ∅. M ′ is a model of X , too: besides being a model of X a , for every
a′ = a and all worlds w ∈ W′ such that X a (w) = ∅ there is a world acces-
′

′
sible by Ra′ , because Ra′ (w) = ∅ in this case would preclude X a (w) = ∅,
and otherwise w ∈ W′ , which is impossible as long as PS* is satisfied. Thus
/
M′
|= S ∧ E ∧ X ∧ I , but if this is the case, S , E , X , I |= ϕ → [a]⊥, hence we
;
must have S , E a, X a , I a |= ϕ → [a]⊥. (End of the proof of the claim.)
;

50

From S , I |=PDL
ϕ → [a]⊥ it follows S , I a |=PDL
ϕ → [a]⊥. Putting all the
a a a
results together, we have that S , E , X , I with ; does not satisfy Postu-
late PI.

D Proof of Theorem 39

If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |= ϕ → [a]ψ if
;
and only if S , E a, I a |= ϕ → [a]ψ.
;

(⇒): Let S , E , X , I and ; satisfy Postulate PS*, and also suppose that
S , E a , I a |= ϕ → [a]ψ. Then there exists a ;-model M = W, Ra , such that
;
M M
|= S ∧ E a ∧ I a and |= ϕ → [a]ψ. This means that there is a possible world
M M
v ∈ W such that |= ϕ and |= [a]ψ, i.e., there is v ′ ∈ W such that Ra (v) = v ′
v v
M
and |= ′ ψ.
v

(We will extend M to obtain a ;-model of S , E , X , I and thus show that
S , E , X , I |= ϕ → [a]ψ.)
;

M
I ai (w) = {ϕj → [ai ]⊥ ∈ I ai : |= ϕj }
w
M
X ai (w) = {ϕj → ai ⊤ ∈ X ai : |= ϕj }
w

M′ M′
w w
M′ ′ M′
w w
′
• I a (w) = ∅.

pothesis, S , E , X , I satisﬁes PS*, there is no implicit static law, i.e., for
every ai ∈ Act and every w ∈ W′ , if I ai (w) = ∅, then X ai (w) = ∅. Then, as
W′ = valuations(S ), M ′ is a model of S . We have that M ′ is a model of E ,
too: it is a model of E a , and given a′ = a, for every ϕi → [a′ ]ψi ∈ E and every
M′ M′
w ∈ W′ , if |= ϕi , then |= ′ ψi for all w ′ ∈ W′ such that wRa′ w ′. Clearly
w w
M ′ is also a model of I : besides being a model of I a , given a′ = a, for every
M′
ϕi → [a′ ]⊥ ∈ I and every w ∈ W′ , if |= ϕi , then I a (w) = ∅ and Ra′ (w) = ∅.
′

w
M ′ is a model of X , too: it is a model of X a , and for every a′ = a and all
worlds w ∈ W′ such that X a (w) = ∅ there is a world accessible by Ra′ , because
′

Ra′ (w) = ∅ in this case would preclude X a (w) = ∅, and otherwise w ∈ W′ ,
′
/
M′
which is impossible as long as PS* is satisﬁed. Thus |= S ∧ E ∧ X ∧ I .

51

M′ M′
Because there are v, v ′ ∈ W′ such that |=v
ϕ, vRa v ′ and |= ′
v
ψ, we have
S , E , X , I |= ϕ → [a]ψ.
;

(⇐): Suppose S , E , X , I |= ϕ → [a]ψ. Then there is a ;-model M such that
;
M M M
|= S ∧E ∧X ∧I and |= ϕ → [a]ψ. Then, given a, we have |= S ∧E a ∧X a ∧I a ,
M
and then |= S ∧ E a ∧ I a . Hence S , E a , I a |= ϕ → [a]ψ.
;

E Proof of Theorem 40

If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |= ϕ → a ⊤ if
;
and only if S , X a |= ϕ → a ⊤.
;

(⇒): Let S , E , X , I and ; satisfy Postulate PS*, and suppose S , X a |=PDL
M
ϕ → a ⊤. Then there exists a PDL-model M = W, Ra , such that |= S ∧X a
M
and |= ϕ → a ⊤. This means that there is a possible world v ∈ W such that
M M
|= ϕ and |= a ⊤.
v v

(We extend M to build a ;-model of S , E , X , I and then conclude that
S , E , X , I |= ϕ → a ⊤.)
;

M
I ai (w) = {ϕj → [ai ]⊥ ∈ I ai : |= ϕj }
w
M
X ai (w) = {ϕj → ai ⊤ ∈ X ai : |= ϕj }
w

M′ M′
w w
M′ ′ M′
w w
′
• I a (w) = ∅.

every ai ∈ Act and every w ∈ W′ , if X ai (w) = ∅, then I ai (w) = ∅. Then,
M′ M′
w w
M′
′

w

52

′

′
and otherwise w ∈ W′ , which is impossible as long as PS* is satisfied. Hence
/
M′ M′ M′
|= S ∧ E ∧ X ∧ I . Because there is v ∈ W′ such that |= ϕ and |= a ⊤,
v v
we have S , E , X , I |= ϕ → a ⊤.
;

(⇐): Suppose S , E , X , I |= ϕ → a ⊤. Then there is a ;-model M such that
;
M M M
|= S ∧E ∧X ∧I and |= ϕ → a ⊤. Then, given a, we have |= S ∧E a ∧X a ∧I a ,
M
and then |= S ∧ X a . Moreover, by definition, M is a PDL-model. Hence
S , X a |= ϕ → a ⊤.
PDL

F Proof of Theorem 43

If S , E , X , I and ; satisfy Postulate PS*, then S , E , X , I |= ϕ → [a1 . . . ; an ]ψ
;
if and only if S , E a1 ,...,an , I a1 ,...,an |= ϕ → [a1 ; . . . ; an ]ψ.
;

Lemma 54 If S , E , X , I |= ϕ → [a1 ; . . . ; an ]ψ, then there is ϕ′ ∈ Fml such
;
that S , E , X , I |= ϕ → [a1 ; . . . ; an−1 ]ϕ′ and S , E , X , I |= ϕ′ → [an ]ψ.
; ;

PROOF. Let S , E , X , I |= ϕ → [a1 ; . . . ; an ]ψ. In the case we have S , E , X , I |=
; ;
ϕ → [a1 ; . . . ; an ]⊥, the result immediately follows. Then, given a model M =
M M
W, R of S , E , X , I such that |= ϕ for some w ∈ W, if |= a1 ; . . . ; an ⊤,
w w
′ M ′
there must be at least one wn−1 such that |= ′ [an ]ψ. Take all such wn−1 and
wn−1
let ϕ′ be
′
wn−1
M
|= ′ [a ]ψ
wn−1 n

Then we have both S , E , X , I |= ϕ → [a1 ; . . . ; an−1 ]ϕ′ , and S , E , X , I |=
; ;
ϕ′ → [an ]ψ.

Proof of Theorem 43

(⇒): The proof is by induction on the number of action operators.

Base: n = 1. As S , E , X , I satisfies Postulate PS*, the result follows from
Theorem 39.

Induction hypothesis: for any k < n, if S , E , X , I |= ϕ → [a1 ; . . . ; ak ]ψ, then
;
S , E a1 ,...,ak , I a1 ,...,ak |= ϕ → [a1 ; . . . ; ak ]ψ.
;

53

Step: let S , E , X , I |= ϕ → [a1 ; . . . ; an ]ψ. By Lemma 54, there is a classical
;
formula ϕ′ such that S , E , X , I |= ϕ → [a1 ; . . . ; an−1 ]ϕ′ and S , E , X , I |=
; ;
ϕ′ → [an ]ψ. From the induction hypothesis, we have S , E a1 ,...,an−1 , I a1 ,...,an−1 |= ;
ϕ → [a1 ; . . . ; an−1 ]ϕ′ and S , E an , I an |= ϕ′ → [an ]ψ. Thus S , E a1 ,...,an , I a1 ,...,an |=
; ;
ϕ → [a1 ; . . . ; an ]ψ.

(⇐): Suppose S , E , X , I |= ϕ → [a1 ; . . . ; an ]ψ. Then there is a ;-model M
;
M M
such that |= S ∧ E ∧ X ∧ I and |= ϕ → [a1 ; . . . ; an ]ψ. Then, given a1 , . . . , an ,
M M
we have |= S ∧E a1 ,...,an ∧X a1 ,...,an ∧I a1 ,...,an , and then |= S ∧E a1 ,...,an ∧I a1 ,...,an .
Hence S , E a1 ,...,an , I a1 ,...,an |= ϕ → [a1 ; . . . ; an ]ψ.
;

G Proof of Theorem 45

Let S , E , X , I and ; satisfy Postulate PS*. Then we have S , E , X , I |= ;
ϕ → a1 ; . . . ; an ψ if and only if S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an |= ϕ → a1 ; . . . ; an ψ.
;

Lemma 55 Let S , E , X , I and ; satisfy Postulate PS*. If S , E , X , I |=;
ϕ → a ψ, then S , E a, X a , I a |= ϕ → a ψ.
;

PROOF. Let S , E , X , I satisfy Postulate PS* and suppose S , E a , X a , I a |=;
ϕ → a ψ. Then there exists a ;-model M = W, Ra , such that both
M M
|= S ∧ E a ∧ X a ∧ I a and |= ϕ → a ψ. This means that there is a pos-
M M
sible world v ∈ W such that |= ϕ and |= a ψ.
v v

(We extend M to build a model of S , E , X , I and then conclude that
S , E , X , I |= ϕ → a ψ.)
;

M
I ai (w) = {ϕj → [ai ]⊥ ∈ I ai : |= ϕj }
w
M
X ai (w) = {ϕj → ai ⊤ ∈ X ai : |= ϕj }
w

Let M ′ = W′ , R′ be such that W′ = W, and R′ = Ra ∪ a′ =a Ra′ (we
extend M to all other actions S , E , X , I speaks of), where for each a′ = a
and every w, w ′ ∈ W′ , wRa′ w ′ if and only if
M′ M′
w w
M′ ′ M′
w w
′
• I a (w) = ∅.


54

every ai ∈ Act and every w ∈ W′ , if X ai (w) = ∅, then I ai (w) = ∅. Then,
M′ M′
w w
M′
′

w
′

′
and otherwise w ∈ W′ , which is impossible as long as PS* is satisﬁed. Hence
/
M′ M′ M′
|= S ∧ E ∧ X ∧ I . Because there is v ∈ W′ such that |= ϕ and |= a ψ,
v v
we have S , E , X , I |= ϕ → a ψ.
;

Lemma 56 If S , E , X , I |= ϕ → a1 ; . . . ; an ψ, then there is ϕ′ ∈ Fml such
;
that S , E , X , I |= ϕ → a1 ; . . . ; an−1 ϕ′ and S , E , X , I |= ϕ′ → an ψ.
; ;

PROOF. The proof is by induction on the number of action operators.

Base: n = 2. Suppose S , E , X , I |= ϕ → a1 ; a2 ψ. Then S , E , X , I |= ϕ →
; ;
a1 a2 ψ. For every model M = W, R of S , E , X , I and for every w ∈ W
M M
such that |= ϕ, there is w ′ ∈ W such that wRa1 w ′ and |= ′ a2 ψ. Let ϕ′ be
w w
{ℓ : ℓ ∈ w ′ } and the result follows.

Induction hypothesis: for any k < n, if S , E , X , I |= ϕ → a1 ; . . . ; ak ψ,
;
then there is ϕ′ ∈ Fml such that S , E , X , I |= ϕ → a1 ; . . . ; ak−1 ϕ′ and
;
S , E , X , I |= ϕ′ → ak ψ.
;

Step: let S , E , X , I |= ϕ → a1 ; . . . ; an ψ. Then we have that S , E , X , I |=
; ;
ϕ → a1 ; . . . ; an−1 ⊤. By the induction hypothesis, there is ϕ′ ∈ Fml such
that S , E , X , I |= ϕ → a1 ; . . . ; an−2 ϕ′ and S , E , X , I |= ϕ′ → an−1 ⊤.
; ;
Because S , E , X , I |= ϕ → a1 ; . . . ; an ψ, given a model M = W, R of
;
M ′
S , E , X , I such that |= ϕ for some w ∈ W, there must be wn−2 ∈ W such
w
M
that |= ′ an−1 an ψ. Then we can safely take ϕ′ as ′
{ℓ : ℓ ∈ wn−2 }.
wn−2
Now, S , E , X , I |= ϕ′ → an−1 an ψ. By the base step, there is ϕ′′ ∈ Fml
;
such that S , E , X , I |= ϕ′ → an−1 ϕ′′ and S , E , X , I |= ϕ′′ → an ψ.
; ;
Putting all the results together, we get S , E , X , I |= ϕ → a1 ; . . . ; an−1 ϕ′′
;
and S , E , X , I |= ϕ′′ → an ψ, for some ϕ′′ ∈ Fml.
;

Proof of Theorem 45

(⇒): The proof is by induction on the number of action operators.

55

Base: n = 1. As S , E , X , I satisﬁes Postulate PS*, the result follows from
Lemma 55.

Induction hypothesis: for any k < n, if S , E , X , I |= ϕ → a1 ; . . . ; ak ψ, then
;
S , E a1 ,...,ak , X a1 ,...,ak , I a1 ,...,ak |= ϕ → a1 ; . . . ; ak ψ.
;

Step: let S , E , X , I |= ϕ → a1 ; . . . ; an ψ. By Lemma 56, there is ϕ′ ∈ Fml
;
such that S , E , X , I |= ϕ → a1 ; . . . ; an−1 ϕ′ and S , E , X , I |= ϕ′ → an ψ.
; ;
By the induction hypothesis, we have S , E a1 ,...,an−1 , X a1 ,...,an−1 , I a1 ,...,an−1 |=;
ϕ → a1 ; . . . ; an−1 ϕ′ and also S , E an , X an , I an |= ϕ′ → an ψ. Then, this
;
gives us S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an |= ϕ → a1 ; . . . ; an ψ.
;

(⇐): Suppose S , E , X , I |= ϕ → a1 ; . . . ; an ψ. Then there is a ;-model M
;
M M
such that |= S ∧E ∧X ∧I and |= ϕ → a1 ; . . . ; an ψ. Then, given a1 , . . . , an ,
M
|= S ∧E a1 ,...,an ∧X a1 ,...,an ∧I a1 ,...,an , and hence S , E a1 ,...,an , X a1 ,...,an , I a1 ,...,an |=;
ϕ → a1 ; . . . ; an ψ.

56

Meta-theory of Actions: Beyond Consistency

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