Domain Descriptions Should be Modular

Domain descriptions should be modular
Andreas Herzig and Ivan Varzinczak 1

Abstract. This work is about the metatheory of actions, and here we Static laws Frameworks which allow for indirect effects make use
address the problem of what a good domain description for reasoning of logical formulas that link invariant propositions about the world.
about actions should look like. We state some postulates concerning Such formulas characterize the set of possible states. They do not
this sore spot, which establishes the notion of a modular domain de- refer to actions, and we suppose they are expressed as formulas of
scription. We point out the problems that arise when modularity is classical propositional logic. PFOR = {A, B, . . .} is the set of all
violated and propose algorithms to overcome them. classical formulas.
A static law2 is a formula A ∈ PFOR that is consistent. An ex-
ample is Walking → Alive, saying that if a turkey is walking, then it
1 INTRODUCTION must be alive [13].

In logic-based approaches to reasoning about actions a domain is de- Effect laws Let ACT = {α, β, . . .} be the set of all actions of
scribed by a set of logical formulas Σ. At first glance satisfiability is a given domain. To speak about action effects we use the syntax of
the only criterion logic provides to check the quality of such a de- propositional dynamic logic (PDL) [5]. The formula [α]A expresses
scription. In this paper we go beyond that, and argue that we should that A is true after every possible execution of α.
require more than the mere existence of a model for Σ. Our start- An effect law3 for α is of the form A → [α]C , where A, C ∈
ing point is that in reasoning about actions one usually distinguishes PFOR, with A and C both classically consistent. (‘Classically
several kinds of logical formulas. Among these are effect axioms, consistent’ is a shorthand for ‘consistent in classical propositional
precondition axioms, and domain constraints. logic’.) The consequent C is the effect which obtains when α is
We prefer here to speak of effect, executability, and static laws, executed in a state where the antecedent A holds. An example is
respectively. Moreover we separate inexecutability laws from ef- Loaded → [shoot]¬Alive, saying that whenever the gun is loaded,
fect laws. Given these ingredients, suppose the language is power- after shooting the turkey is dead. Another one is [tease]Walking: in
ful enough to state that action α is inexecutable in contexts where A every situation, the result of teasing is that the turkey starts walking.
holds, and executable in contexts where B holds. It follows that there Note that the consistency requirements for A and C make sense:
can be no context where A ∧ B holds. Now ¬(A ∧ B) is a static law if A is inconsistent then the effect law is superfluous; if C is incon-
that is independent of α. It is therefore natural to expect that it fol- sistent then we have an inexecutability law, that we consider to be a
lows from these laws alone! By means of examples we show that if separate entity.
this is not the case then unexpected conclusions might follow from
Σ. A similar case can be made against implicit inexecutability laws. Inexecutability laws We suppose that effect laws with inconsis-
This motivates postulates requiring that the different ingredients of tent consequents are a particular kind of law. This allows us to avoid
domain descriptions should be arranged in a modular way, such that mixing things that are conceptually different: for an action α, an ef-
interactions between them are limited and controlled. It turns out that fect law mainly associates it with a consequent C , while an inexe-
in all existing accounts which allow for these four kinds of laws [8, 9, cutability law only associates it with an antecedent A.
13, 1, 15], consistent domain descriptions can be written that violate An inexecutability law for α is of the form A → [α]⊥, where A ∈
some of these postulates. We here give algorithms that allow one PFOR is classically consistent. For example ¬HasGun → [shoot]⊥
to check whether a domain description satisfies the postulates. With expresses that shoot cannot be executed if the agent has no gun.
such algorithms, the task of correcting badly written descriptions can
be made easier. Executability laws With only static and effect laws one cannot
This paper is organized as follows: after the preliminary defini- guarantee that shoot is executable if the agent has a gun. Whereas all
tions (Sections 2 and 3) we state (Section 4) and study (5–7) three the extant approaches in the literature that allow for indirect effects
postulates. Finally we discuss strengthenings (Section 8) and assess of actions contain static and effect laws, the status of executability
related work (Section 9). laws is less consensual. Some authors [12, 3, 9, 13] more or less
tacitly consider that executability laws should not be made explicit
but rather inferred by the reasoning mechanism. Others [8, 15] have
2 DOMAIN DESCRIPTIONS executability laws as first class objects one can reason about.

2 Static laws are often called domain constraints, but the different laws for
In this section we establish the ontology of domain descriptions. actions that we shall introduce in the sequel could in principle also be called
like that.
3 Effect laws are often called action laws, but we prefer not to use that term
1 The authors are with the Institut de Recherche en Informatique de Toulouse here because it would also apply to executability laws that are to be intro-
(IRIT), Toulouse, France. e-mail: {herzig,ivan}@irit.fr duced in the sequel.

It seems strange to us just stating information about necessary con- We here suppose that ; is finite. A given dependence relation ;
ditions for execution of an action (inexecutabilities) and saying noth- defines a class of possible worlds models M; : every M ∈ M;
ing about the sufficient ones. This is the reason why we think we need must satisfy that whenever wRα w then
executability laws. Indeed, in several domains one wants to explicitly
state under which conditions a given action is guaranteed to be exe- • α ; P and w ∈ I(P ) implies w ∈ I(P );
cutable, e.g. that a robot should never get stuck and should always be • α ; ¬P and w ∈ I(P ) implies w ∈ I(P ).
able to execute a move action. In any case, allowing for executability
laws gives us more flexibility and expressive power. The associated consequence relation is noted |=; . In our example
In dynamic logic the dual α A, defined as ¬[α]¬A, can be used we obtain S, E, X , I |=; HasGun → [load]HasGun.
to express executability. α thus reads “the execution of action α
is possible”. An executability law4 for α is of the form A → α ,
4 POSTULATES
where A ∈ PFOR is classically consistent. For instance HasGun →
shoot says that shooting can be executed whenever the agent has Our central hypothesis is that the different types of laws should be
a gun, and tease establishes that the turkey can always be teased. neatly separated, and should only interfere in one sense: static laws
Domain descriptions S ⊆ PFOR denotes the set of all static laws allow one to infer action laws that do not follow from the action laws
of a given domain. For a given action α ∈ ACT, Eα is the set of alone. The other way round, action laws should not allow to infer
its effect laws, Xα is the set of its executability laws, and Iα is the new static laws, effect laws should not allow to infer inexecutability
set of its inexecutability laws. We define E = α∈ACT Eα , X = laws, etc. Here are the postulates for that:
α∈ACT Xα , and I = α∈ACT Iα . A domain description is a tuple
of the form S, E, X , I . P0. Logical consistency: S, E, X , I |=; ⊥

A domain description should be logically consistent.
3 DYNAMIC LOGIC AND THE FRAME
PROBLEM
P1. No implicit executability laws:
Given a domain description S, E, X , I , we need a consequence
relation solving the frame problem. To this end we now give the se- if S, E, X , I |=; A → α , then S, X |=PDL A → α
mantics of PDL, and extend it with dependence relations.
P1 , P2 , . . . denote propositional constants, L1 , L2 , . . . literals, and If an executability law can be inferred from the domain description,
Φ, Ψ, . . . formulas. (We recall that A, B, . . . denote classical formu- then it should already “be” in X , in the sense that it should also be
las.) If L = ¬P then we identify ¬L with P . inferable in PDL from the set of executability and static laws alone.
A PDL-model is a triple M = W, R, I where W is a set of
possible worlds, R maps action constants α to accessibility rela- P2. No implicit inexecutability laws:
tions Rα ⊆ W × W , and I maps propositional constants to sub-
sets of W . Given a PDL-model M = W, R, I , |=M Φ if for if S, E, X , I |=; A → [α]⊥, then S, I |=PDL A → [α]⊥
all w ∈ W , w |=M Φ; w |=M [α]Φ if w |=M Φ for every
w such that wRα w . A formula Φ is a consequence of the set of If an inexecutability law can be inferred from the domain description,
global axioms {Φ1 , . . . , Φn } in the class of all PDL-models (noted then it should be inferable in PDL from the static and inexecutability
Φ1 , . . . , Φn |=PDL Φ) if and only if for every PDL-model M , if laws alone.
|=M Φi for every Φi , then |=M Φ.
PDL alone does not solve the frame problem. For instance, if P3. No implicit static laws: if S, E, X , I |=; A, then S |=PDL A.
S, E, X , I describes our shooting domain then S, E, X , I |=PDL
HasGun → [load]HasGun. The deductive power of PDL has to be If a static law can be inferred from the domain description, then it
augmented in order to ensure that the relevant frame axioms follow. should be inferable in PDL (and even classically) from the set of
The presence of static constraints makes that this is a delicate task, static laws alone.
and starting with [8, 9], several authors have argued that some no- Postulate P0 is obvious. P1 can be ensured by maximizing X . This
tion of causality is needed. We here opt for the dependence based suggests a stronger version of P1:
approach presented in [1], where dependence information has been
added to PDL. In [2] it has been shown how Reiter’s solution to the P4. Maximal executability laws:
frame problem can be recast in PDL. α ; L denotes that the exe-
cution of action α may change the truth value of the literal L. In our if S, E, X , I |=; A → [α]⊥, then S, X |=PDL A → α
example we have
It expresses that if in context A no inexecutability for α can be in-
shoot, ¬Loaded , shoot, ¬Alive ,
;= ferred, then the respective executability follows in PDL from the exe-
shoot, ¬Walking , tease, Walking
cutability and static laws. P4 generally holds in nonmonotonic frame-
Because load, ¬HasGun ∈ ;, we have load ; ¬HasGun, i.e.,
/ works, and can be enforced in monotonic approaches such as ours by
¬HasGun is never caused by load. We also have tease ; Alive and maximizing X .
tease ; ¬Alive. Things are less obvious for Postulates P2 and P3. They are violated
4
by domain descriptions designed in all approaches in the literature
Some approaches (most prominently Reiter’s) use biconditionals A ↔
α , called precondition axioms. This is equivalent to ¬A ↔ [α]⊥,
that allow to express the four kinds of laws. We therefore discuss
which illustrates that they thus merge information about inexecutability each of them in the subsequent sections by means of examples, and
with information about executability. give algorithms to decide whether they are satisfied.

5 NO IMPLICIT INEXECUTABILITY LAWS NewCons S (CJ ) is more difficult to control. Note that the algorithm
terminates because we have assumed ; finite.
Consider the following domain description (and ; as above): The algorithm not only decides whether the postulate is satisfied,
[tease]Walking, its output I I also can provide a way to “repair” the domain descrip-
S1 = {Walking → Alive}, E1 = , tion. Basically there are three options, that we illustrate with our ex-
Loaded → [shoot]¬Alive
ample: 1) add ¬Alive → [tease]⊥ to I1 ; 2) add the (unintuitive) de-
X1 = I 1 = ∅ pendence tease, Alive to ;; or 3) weaken the law [tease]Walking
to Alive → [tease]Walking. It is easy to see that whatever we opt for,
From [tease]Walking it follows with S1 that [tease]Alive, i.e., in every the new domain description will satisfy P2.
situation, after teasing the turkey is alive: S1 , E1 |=PDL [tease]Alive.
Now as tease ; Alive, the status of Alive is not modified by the
tease action, and we have S1 , E1 |=; ¬Alive → [tease]¬Alive. 6 NO IMPLICIT STATIC LAWS
From the above, it follows S1 , E1 , X1 , I1 |=; ¬Alive → [tease]⊥, Executability laws increase expressive power, but might conflict
i.e., the turkey cannot be teased if it is dead. But S1 , I1 |=PDL with inexecutability laws. For instance, let S2 = S1 , E2 = E1 ,
¬Alive → [tease]⊥, hence Postulate P2 is violated. The formula X2 = { tease }, and I2 = {¬Alive → [tease]⊥}. (Note that Pos-
¬Alive → [tease]⊥ is an example of what we call an implicit in- tulate P2 is satisfied.) We have the unintuitive X2 , I2 |=PDL Alive:
executability law. the turkey is immortal! This is an implicit static law because Alive
In the literature, such laws are also known as implicit qualifica- does not follow from S2 alone: P3 is violated.
tions [4], and it has been argued that it is a positive feature of rea- How can we find out whether there are implicit static laws? We as-
soning about actions frameworks to leave them implicit and provide sume that Postulate P2 is satisfied, i.e., all inexecutabilities are cap-
mechanisms for inferring them [8, 13]. The other way round, one tured by I.
might argue as well that implicit qualifications indicate that the do-
main has not been described in an adequate manner: inexecutability Algorithm 2 (Finding implicit static laws)
laws have a form simpler than that of effect laws, and it might be input: S, X , I
output: a set of implicit static laws S I
reasonably expected that it is easier to exhaustively describe them.5 S I := ∅
Thus, all the inexecutabilities should be explicitly stated, and this is for all α ∈ ACT do
what Postulate P2 says. for all A → [α]⊥ ∈ I and A → α ∈ X do
How can we check whether P2 is violated? First we need a defini- if S |=PDL ¬(A ∧ A ) then
S I := S I ∪ {¬(A ∧ A )}
tion. Given classical formulas A and B, the function NewCons A (B)
computes the set of strongest clauses that follow from A ∧ B,
Example 2 For S2 , E2 , X2 , I2 , Algorithm 2 returns S I =
but do not follow from A alone (cf. e.g. [6]). It is known that {Alive}.
NewCons A (B) can be computed by subtracting the prime impli-
cates of A from those of A ∧ B. For example, the set of prime im- The existence of implicit static laws may thus indicate too strong
plicates of P is just {P }, that of P ∧ (¬P ∨ Q) ∧ (¬P ∨ R ∨ T ) is executability laws: in our example, we wrongly assumed that tease is
{P, Q, R ∨ T }, hence NewCons P ((¬P ∨ Q) ∧ (¬P ∨ R ∨ T )) = always executable. It may also indicate that the inexecutability laws
{Q, R∨T }. And for our example, NewCons Walking→Alive (Walking) = are too strong, or that the static laws are too weak:
{Alive, Walking}.
Example 3 Suppose a computer representation of the line of inte-
Algorithm 1 (Finding implicit inexecutability laws) gers, in which we can be at a strictly positive number, Positive, or
input: S, E, I, ;
output: a set of implicit inexecutability laws I I at a negative one or zero, ¬Positive. Let MaxInt and MinInt, respec-
I I := ∅ tively, be the largest and the smallest representable integer number.
for all α ∈ ACT do goleft is the action of moving to the biggest integer smaller than the
for all J ⊆ Eα do one at which we are. Consider the following domain description for
AJ := {Ai : Ai → [α]Ci ∈ J}
CJ := {Ci : Ai → [α]Ci ∈ J} this scenario (Ati means we are at number i):
if S ∪ {AJ } is classically consistent then
for all Li ∈ NewCons S (CJ ) do S3 = {Ati → Positive : i > 0} ∪ {Ati → ¬Positive : i ≤ 0}
if ∀i, α ; Li and S, I |=PDL (AJ ∧ ¬Li ) → [α]⊥ then
I I := I I ∪ {(AJ ∧ ¬Li ) → [α]⊥} {AtMinInt → [goleft]Underflow}∪
E3 =
{Ati → [goleft]Ati−1 : i > MinInt}
Example 1 Consider S1 , E1 , I1 and ; as given above. Then Algo-
rithm 1 returns I I = {¬Alive → [tease]⊥}. X3 = { goleft }, I3 = ∅
with the dependence relation (MinInt ≤ i ≤ MaxInt):
Theorem 1 S, E, X , I satisfies Postulate P2 if and only if I I = ∅.
goleft, Ati , goleft, Positive ,
;=
This is the key algorithm of the paper. We are aware that it comes goleft, ¬Positive , goleft, Underflow
with considerable computational costs: first, the number of formulas In order to satisfy Postulate P2, we run Algorithm 1 and obtain I3 =
AJ and CJ is exponential in the size of Eα , and second, the compu- {(At1 ∧ At2 ) → [goleft]⊥}. Now applying Algorithm 2 to this action
tation of NewCons S (CJ ) might result in exponential growth. While theory gives us the implicit static law ¬(At1 ∧ At2 ), i.e., we cannot
we might expect Eα to be reasonably small in practice, the size of be at 1 and 2 at the same time.
5 Note that nevertheless this is not related to the qualification problem, which
basically says that it is difficult to state all the executability laws of a do- Theorem 2 Suppose S, E, X , I satisfies P2. Then Postulate P3 is
main. satisfied if and only if S I = ∅.

What shall we do with an implicit static law? Again, several op- Theorem 6 There exist domain descriptions S, E, X , I not satis-
tions show up: whereas in the latter example the implicit static law fying P3 such that S, E, X , I |=; A → [α]C and S, Eα , Iα |=;
should be added to S, in the former the implicit static law is due to A → [α]C .
an executability law that is too strong and should be weakened.
So, in order to satisfy Postulate P3, a domain description should For example, we have S2 , E2 , X2 , I2 |=; ¬Alive → [shoot]Alive,
contain a complete set of static laws or, alternatively, should not make but S2 , E2shoot , I2 shoot |=; ¬Alive → [shoot]Alive.
so strong assumptions about executability. This means that eliminat- Now we turn to postulates that are too strong. First, it seems to be
ing implicit static laws may require revision of S or completion of in line with the other postulates to require domain descriptions not to
X . In the next section we approach the latter option. allow for the deduction of new effect laws: if an effect law follows
from a domain description, and no inexecutability law for the same
action in the same context can be derived, then it should follow from
7 MAXIMAL EXECUTABILITY LAWS the set of static and effect laws alone. This means we should have:
Implicit static laws only show up when there are executability laws. P5. No implicit effect laws:
Which executability laws can be consistently added to a given do-
main description? if S, E, X , I |=; A → [α]C and S, E, X , I |=; A → [α]⊥,

Algorithm 3 (Finding implicit executability laws) then S, E |=; A → [α]C
input: S, X , I
output: a set of implicit executability laws X I But consider the following intuitively correct domain description:
X I := ∅
for all α ∈ ACT do Loaded → [shoot]¬Alive,
Aα := {Ai : Ai → [α]⊥ ∈ Iα } S5 = ∅, E5 =
if S |=PDL Aα and S, X |=PDL ¬Aα → α then
(¬Loaded ∧ Alive) → [shoot]Alive
X I := X I ∪ {¬Aα → α }
X5 = {HasGun → shoot }, I5 = {¬HasGun → [shoot]⊥}
Example 4 Suppose S4 = {Walking → Alive}, X4 = ∅ and I4 = together with the dependence relation ; of Example 1. It satisfies
{¬Alive → [tease]⊥}. Then Algorithm 3 yields X I = {Alive → Postulates P1, P2, P3, and P4, but does not satisfy P5. Indeed, we
tease }. have that S5 , E5 , X5 , I5 |=; ¬HasGun ∨ Loaded → [shoot]¬Alive
and S5 , E5 , X5 , I5 |=; ¬HasGun ∨ Loaded → [shoot]⊥, but
Theorem 3 Suppose S, E, X , I satisfies P2 and P3. Postulate P4 S5 , E5 |=; ¬HasGun ∨ Loaded → [shoot]¬Alive. So, Postulate P5
is satisfied if and only if X I = ∅. would not help us to deliver the goods.
Another though obvious possibility of amending our modularity
What Theorem 3 says is that it suffices to take the ‘complement’ criteria could be by stating the following postulate:
of I to obtain all the executability laws of the domain. Note that this
P6. No unattainable effects:
counts as a solution to the qualification problem given that all pre-
conditions for guaranteeing executability of actions are thus known. if A → [α]C ∈ E, then S, E, X , I |=; A → [α]⊥

This expresses that if we have explicitly stated an effect law for α
8 DISCUSSION in some context, then there should be no inexecutability law for the
In this section we discuss other properties related to consistency same action in the same context. We do not investigate this further
and modularity of domain descriptions. Some will follow from ours, here, but just observe that the slightly stronger version below leads
while some others look natural at first glance, but turn out to be too to unintuitive consequences:
strong.
P6 . No unattainable effects (strong version):
Theorem 4 If S, E, X , I satisfies P3, then S, E, X , I |=; ⊥ iff if S, E |=; A → [α]C , then S, E, X , I |=; A → [α]⊥
S |=PDL ⊥.
Indeed, for the above domain description we have that E5 |=;
This means that if there are no implicit static laws then consistency (¬HasGun ∧ Loaded) → [shoot]¬Alive, but S5 , E5 , X5 , I5 |=;
of a domain description (P0) can be checked by just checking con- (¬HasGun ∧ Loaded) → [shoot]⊥. This is certainly too strong. Our
sistency of S. example also illustrates that it is sometimes natural to have some ‘re-
dundancies’ or ‘overlaps’ between I and E.
Theorem 5 If S, E, X , I satisfies P3, then S, E, X , I |=; A →
[α]C iff S, Eα , Iα |=; A → [α]C .
9 RELATED WORK
This means that under P3 we have modularity inside E, too: when Pirri and Reiter have investigated the metatheory of the situation cal-
deducing the effects of α we need not consider the action laws for culus [11]. In a spirit similar to ours, they simplify the entailment
other actions. Versions for executability and inexecutability can be problem for this calculus, and show for several problems such as con-
stated as well. sistency or regression that only some of the modules of a domain de-
scription are necessary. Note that in their domain descriptions S = ∅.
Remark 8.1 Although in the present paper concurrency is not taken This allows them to show that such theories are always consistent.
into account, we conjecture that Theorem 5 holds when we have con- Zhang et al. [14] have also proposed an assessment of what a good
current action execution. domain description should look like. They develop the ideas in the

framework of EPDL [15], an extended version of PDL which allows own with Algorithms 1, 2 and 3, which can give us some guidelines
for propositions as modalities to represent causal connection between in correcting a domain description if needed.
literals. We do not present the details of that, but concentrate on the Given the difficulty of exhaustively enumerating all the precondi-
main metatheoretical results. tions under which a given action is executable and also those under
Zhang et al. propose a normal form for describing action theories, which such an action cannot be executed, there is always going to be
and investigate three levels of consistency. Roughly speaking, a do- some executability precondition A or some inexecutability precondi-
main description Σ is uniformly consistent if it is globally consistent tion B that together lead to a contradiction, forcing, thus, an implicit
(i.e., Σ |=EPDL ⊥); a formula Φ is Σ-consistent if Σ |=EPDL ¬Φ, static law ¬(A ∧ B). This is the reason we propose to state some in-
for Σ a uniformly consistent theory; Σ is universally consistent if formation about both executabilities and inexecutabilities, complete
Σ |=EPDL A implies |=EPDL A. the latter and then, after deriving all implicit static laws, complete the
Given these definitions, they propose algorithms to test the dif- former. As a final result we will have complete S, X and I.
ferent versions of consistency for a domain description Σ that is in Throughout this work we used a weak version of PDL, but our
normal form. This test essentially amounts to checking whether Σ notions and results can be applied to other frameworks as well. It
is safe, i.e., whether Σ |=EPDL α , for every α. Success of this is worth noting however that for first-order based frameworks the
check should mean the domain description under analysis satisfies consistency check of Algorithm 1 is undecidable. (We can get rid of
the consistency requirements. this by assuming that S, E, X , I is finite and there is no function
Nevertheless, this is only a necessary condition: it is not hard symbol in the language. In this way, the result of NewCons is finite
to imagine domain descriptions that are uniformly consistent but and the algorithm terminates.)
in which we can still have implicit inexecutabilities that are not Our postulates do not take into account causality statements link-
caught by the algorithm. Consider for instance a scenario with ing propositions. This could be a topic for further investigation.
a lamp that can be turned on and off by a toggle action, and
its EPDL representation given by {On → [toggle]¬On, Off → ACKNOWLEDGMENTS
[toggle]On, [On]¬Off, [¬On]Off}.
The causal statement [On]¬Off means On causes ¬Off. Such a do- Ivan Varzinczak has been supported by a fellowship from the gov-
main description satisfies each of the consistency requirements (in ernment of the Federative Republic of Brazil, grant CAPES.
particular it is uniformly consistent, as Σ |=EPDL ⊥). However, Σ is
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6 It might be objected that it is only by doing experiments that one learns the (2002).
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