Action Theory Revision

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Action Theory Revision

Ivan Jose Varzinczak

Abstract

Like any other logical theory, action theories in reasoning about

actions may evolve, and thus need revision methods to adequatelyaccommodate new information about the behavior of actions. Herewe give a semantics that complies with minimal change for revisingaction theories stated in a version ofPDL. We give algorithms that areproven correct w.r.t. the semantics for those theories that are modular.

Keywords: Reasoning about actions, PDL, revision, minimal change.

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Contents

1 Introduction 3

2 Logical Preliminaries 32.1 Action Theories in Dynamic Logic . . . . . . . . . . . . . . . . 32.2 Elementary Atoms . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Prime Valuations . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 Closeness Between Models . . . . . . . . . . . . . . . . . . . . 6

3 Semantics of Revision 73.1 Revising a Model by a Static Law . . . . . . . . . . . . . . . . 7

3.2 Revising a Model by an Effect Law . . . . . . . . . . . . . . . 93.3 Revising a Model by an Executability Law . . . . . . . . . . . 103.4 Revising Sets of Models . . . . . . . . . . . . . . . . . . . . . 13

4 Syntactic Operators for Revision 144.1 Revision by a Static Law . . . . . . . . . . . . . . . . . . . . . 144.2 Revision by an Effect Law . . . . . . . . . . . . . . . . . . . . 154.3 Revision by an Executability Law . . . . . . . . . . . . . . . . 16

5 Correctness of the Algorithms 17

6 Conclusion and Perspectives 20

References 20

A Proof of Theorem 5.2 23

B Proof of Theorem 5.3 25

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1 Introduction

In logic-based approaches to reasoning about actions, theories are collec-tions of statements of the form: if context, then effect after every executionof action (effect laws); and if precondition, then action executable (exe-cutability laws). For example, in Propositional Dynamic Logic (PDL) [5], onecould have the law (p1 p2) [a]p1, saying that in every context wherep1 p2 is the case, after every execution of action a we get the effect p1;and (p1 p2) a, stating that p1 p2 is a sufficient condition for asexecutability.

These are examples of what we call action laws, as they specify the be-havior of the actions of a given domain. Besides that we can also have lawsmentioning no action at all (static laws). They characterize the underlyingstructure of the world, i.e., its possible states. For instance, having p1 p2as a static law would mean p1 p2 is a forbidden state. Action theories willthen be collections of laws, each of them seen as a global axiom in PDL.

Well, it may happen that such descriptions have to be revised due e.g. tonew incoming information about the behavior of the world. In our example,we may learn that the only valid states are those satisfying p1 p2, or thataction ahas always p2 as outcome in p2-contexts, or even that p1 is enoughto guarantee as executability. Here we are interested exactly in this kind of

theory change.The contributions of the present work are as follows:

What is the semantics of revising an action theory Tby a law ? Howto get minimal change, i.e., how to keep as much knowledge about otherlaws as possible? We answer these questions in Section 3.

How to syntactically revise an action theory so that its result corre-sponds to the intended semantics? We answer this question in Sec-tions 4 and 5.

2 Logical Preliminaries

2.1 Action Theories in Dynamic Logic

Our base formalism is PDL without the operator, which essentially amountsto the multimodal logic Km [17]. Let Act= {a1, a2, . . .} be the set of atomic

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actions of a domain. To each a there is associated a modal operator [a]. We

suppose our multimodal logic is independently axiomatized, i.e., the logic isa fusion and there is no interaction between the modal operators [11, 12].

Prop = {p1, p2, . . .} denotes the set ofpropositional constants, also knownas atoms or fluents. The set of literals is Lit= {1, 2, . . .}, where each i iseither p or p, for some p Prop. If = p, we identify with p. By ||we denote the atom in .

By , , . . . we denote Boolean formulas, examples of which are p1 p2and p1 p2. Fml is the set of all Boolean formulas. A valuation v is amaximally consistentset of literals. We denote v the fact that vsatisfies. val() is the set of all valuations satisfying . |=

CPLdenotes the classical

consequence relation, and Cn() is the set of all logical consequences of inclassical propositional logic.With IP() we denote the set of prime implicants [18, 14] of . By we

denote a prime implicant, and atm() is the set of atoms occurring in . Forgiven and , abbreviates is a literal occurring in .

We denote complex formulas (with modal operators) by , , . . . a is thedual operator of [a], (a =def [a]). An example of a complex formulais (p1 (p2 p3)) [a](p1 p3).

A PDL-model is a tuple M = W, R where W is a set of valuations, andR maps action constants a to accessibility relations Ra W W. Given a

model M,|=M

wp (p is true at world w of model M) if w p;

|=M

w[a] if

|=M

w

for every w s.t. (w, w) Ra; truth conditions for the other connectives areas usual. By M we will denote a set ofPDL-models.

M is a model of (noted |=M

) if and only if |=M

w for all w W. M

is a model of a set of formulas (noted |=M

) if and only if |=M

for every . is a consequence of the global axioms in all PDL-models (noted

|=PDL

) if and only if for every M, if |=M

, then |=M

.

With PDL we can state laws describing the behavior of actions. Followingthe tradition in the RAA community, we here distinguish three types of them.

Static Laws A static lawis a formula Fml. It characterizes the possiblestates of the world. The set of all static laws of a domain is denoted by S.

Effect Laws An effect law for a is of the form [a], where , Fml.Effect laws relate an action to its effects, which can be conditional. Theconsequent is the effect which always obtains when a is executed in a statewhere the antecedent holds. If a is a nondeterministic action, then is

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typically a disjunction. If is inconsistent we have a special kind of effect

law that we call an inexecutability law. For example, (p1 p2) [a] saysthat acannot be executed (there is no a-transition) in p1 p2-contexts. Theset of effect laws of a domain is denoted by E.

Executability Laws An executability law for ahas the form a, with Fml. It stipulates the context in which a is guaranteed to be executable.(In PDL, the operator a is used to express executability, a thus readsas execution is possible.) The set of all executability laws of a domain isdenoted by X.

Action Theories T= S E X is an action theory.

Given an action a, Ea (resp. Xa) will denote the set of only those effect(resp. executability) laws about a.

For the sake of clarity, we here abstract from the frame [15] and rami-fication [3] problems, and assume Tcontains all frame axioms (cf. [6] for acontraction approach within a solution to the frame problem).

2.2 Elementary Atoms

Given Fml, E() denotes the elementary atoms actually occurring in .For example, E(p1 (p1 p2)) = {p1, p2}. An atom p is essential to if

and only if p E() for every such that |=CPL . For instance, p1 isessential to p1 (p1 p2). E!() will denote the essential atoms of . (If is not contingent, i.e., it is a tautology or a contradiction, then E!() = .)

For Fml, is the set of all Fml such that |=CPL

andE() E!(). For instance, p1 p2 / p1, as p1 |=CPLp1 p2 but E(p1 p2) E!(p1). We also have E() = E!(). Moreover whenever |=CPL

,E!() = E!() and also = .

Theorem 2.1 (Least atom-set theorem [16]) |=CPL

, and forevery s.t. |=

CPL , E() E().

The proof of this theorem is given in [16, 13] and we do not state ithere. Thus for each Fml there is a unique least set of elementary atomssuch that may equivalently be expressed using only atoms from that set.1

Hence, Cn() = Cn().

1The dual notion (redundant atoms) is addressed in [7], with similar purposes.

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2.3 Prime Valuations

Given a valuation v, v v is a subvaluation. For W a set of valuations, asubvaluation v satisfies Fml modulo W (noted v

W) if and only if

v for all v W such that v v. A subvaluation v essentially satisfies

(modulo W), noted v!

W, if and only ifv

W and {|| : v} E!(). If

v!

W, we call v an essential subvaluation of (modulo W).

Definition 2.1 Let Fml and W be a set of valuations. v is a prime

subvaluation of (modulo W) if and only if v !

W and there is no v

v s.t. v !

W.

Prime subvaluations of a formula are the weakest states of truth inwhich is true. They are just another way of seeing prime implicants of .By base(, W) we denote the set of all prime subvaluations of modulo W.

Theorem 2.2 Let Fml and W be a set of valuations. Then for allw W, w if and only if w

vbase(,W)

v.

Proof: Right to left direction is straightforward. For the left to right di-rection, if w , then w . Let w w be the least subset of w stillsatisfying . Clearly, w is a prime subvaluation of modulo W, and thenbecause w

w , the result follows.

2.4 Closeness Between Models

The distance between two PDL-models will depend upon the distance be-tween their sets of worlds and accessibility relations. These will be based onsymmetric difference between sets, defined as XY = (X\ Y) (Y \ X).

Definition 2.2 Let M = W, R. M = W, R is as close to M asM = W, R, notedM M M, if and only if

either WW WW

or WW = WW and RR RR

In what follows we use this closeness to compare models resulting fromthe semantical revision.

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3 Semantics of Revision

Contrary to action theory contraction [20], where we want the negation ofsome law to be satisfiable, in revision we want to make a new law valid. Thismeans one has to eliminate all cases satisfying its negation. This depicts theduality between revision and contraction: whereas in the latter one invali-dates a formula by making its negation satisfiable, in the former one makesa formula valid by forcing its negation to be unsatisfiable prior to adding thenew law to the theory.

The idea behind our semantics is as follows: we initially have a set ofmodels M in which a given formula is (potentially) not valid, i.e., is(possibly) not true in every model in M. In the result we want to have onlymodels of. Adding -models to M is of no help. Moreover, adding modelsmakes us to lose laws: the corresponding theory would be more liberal.

One solution amounts to delete from M those models that are not -models. Of course removing only some of them does not solve the problem,we must delete every such a model. By doing that, all resulting models willbe models of . (This corresponds to theory expansion, when the resultingtheory is satisfiable.) However, if M contains no model of, we will end upwith . Consequence: the resulting theory is inconsistent. (This is the mainrevision problem.) In this case the solution is to substitute each model M inM by its nearest modificationM that makes true. This lets us to keep as

close as possible to the original models we had. But, what if for one modelin M there are several minimal (incomparable) modifications of it validating? In that case we shall consider all of them. The result will also be a listof models M, all being models of.

Before defining revision of sets of models, we present what modificationsof (individual) models are.

3.1 Revising a Model by a Static Law

Let the model depicted in Figure 1, and suppose we want to revise it by the

Boolean formula p1 p2, i.e., we want such a formula to be a static law.In such a model, we do not want the formula p1 p2 to be satisfiable,

so the first step is to remove all worlds in which it is true. The second stepis to guarantee that all the remaining worlds satisfy the new law. Such anissue has been largely addressed in the literature on propositional belief base

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M :

p1,p2 p1, p2

p1,p2

a

aa

Figure 1: A model where p1 p2 is satisfiable.

revision and update [4, 21, 10, 7]. Here we can achieve that with a semanticssimilar to that of classical operators: basically one shall change the set ofpossible valuations, by removing or adding worlds.

The delicate point in removing worlds is that we may lose some executabil-ity laws: in the example, removing {p1, p2} also removes p2 a. Froma semantical point of view, this is intuitive: if the state of the world to whichwe could move is no longer possible, then we do not have a transition tothat state anymore. Hence, if that transition was the only one we had, it isnatural not to have a transition anymore.

Similarly, one could ask what to do with the accessibility relation ifnew worlds are added (when expansion is not possible): shall new arrowsleave/arrive at the new world? If no arrow leaves the new added world, wemay lose an executability. If some arrow leaves it, we may lose an effect law,

the same holding if we add an arrow pointing to the new world. If no arrowarrives at the new world, what about the intuition? Do we want to have anunreachable state?

All this discussion shows how drastic a change in the static laws maybe: it is a change in the underlying structure (possible states) of the world!Besides their modification, changing them may have as consequence the lossof an effect law or an executability law.

The tradition in the RAA community says that executability laws are, ingeneral, more difficult to state than effect laws, and hence are more likely tobe incorrect. By adding no arrow to the resulting model we here comply with

that and postpone correction of executability laws, if needed (cf. [6, 20]). Itis controversial whether this approach is in line with intuition or not, but wethink that with the information we have at hand, this is the safest way ofchanging static laws.

The semantics for revision of one model by a static law is as follows:

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Definition 3.1 LetM = W, R. M = W, R M if and only if:

W = (W\ val()) val()

R R

Clearly |=M

for each M M . The minimal models resulting fromrevising a model M by are those closest to M w.r.t. M:

Definition 3.2 revise(M, ) =

min{M, M}

3.2 Revising a Model by an Effect Law

Let our language have three atoms and consider the model M in Figure 2.

M :

p1,p2,p3 p1, p2, p3

p1, p2,p3

aa

a

Figure 2: A model where p1 ap2 is satisfiable.

(Notice |=M

p2 p1 p3.) Suppose we want to revise M by p1 [a]p2.This means that we should guarantee the formula p1 ap2 is satisfiable innone of its worlds. To do that, we have to look at the worlds satisfying it (ifany) and either

make p1 false; or

make ap2 false by removing a-arrows leading to p2-worlds.

In our example, the worlds {p1, p2, p3} and {p1, p2, p3} satisfy p1 ap2 and both have to change. Flipping p1 would do the job but also has asconsequence the loss of a static law: we would violate p2 p1 p3. Here wethink that changing action laws should not have as side effect a change in thestatic laws. Given their special status, these should change only if explicitlyrequired. In this case, each world satisfying p1 ap2 has to be changed

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so that ap2 is no longer true in it. In our example, we should remove the

arrows ({p1, p2, p3}, {p1, p2, p3}) and ({p1, p2, p3}, {p1, p2, p3}).The semantics of one model revision for the case of a new effect law is:

Definition 3.3 LetM = W, R. M = W, R M[a] if and only if:

W = W

R R

If (w, w) R \ R, then |=M

w and |=

M

w

|=M

[a]

The minimal models resulting from the revision of a model M by a neweffect law are those that are closest to M w.r.t. M:

Definition 3.4 revise(M, [a]) =

min{M[a], M}

3.3 Revising a Model by an Executability Law

Let the model depicted in Figure 3 and suppose we want to revise it by the

new executability law p1 a.

M :

p1, p2

p1,p2 p1, p2

p1,p2

a

a

a

Figure 3: A model where p1 [a] is satisfiable.

Observe that (p1 a) is satisfiable in M, hence we must throwp1 [a] away to ensure the new formula is true. To remove p1 [a] wehave to look at all worlds satisfying it and modify M so that they no longersatisfy the formula.

Given world {p1, p2}, we have two options:

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change the interpretation of p1; or

add a new arrow leaving {p1, p2}.

A question that raises is what choice is more drastic: change a world oran arrow ? Again, here we think that changing the worlds content (thevaluation) is more drastic, as the existence of such a world was foreseen bysome static law and is hence assumed to be as it is, unless we have informationsupporting the contrary (cf. Section 3.1). Thus we shall add a new a-arrowfrom {p1, p2}. Having agreed on that, the issue now is: to which worldshould the new arrow point? Four options show up: point the arrow to{p1, p2}, {p1, p2}, {p1, p2} or {p1, p2} itself. The resulting model is

such that the unwanted formula is unsatisfiable and the new executabilityp1 a holds in all its worlds.

Whereas all these options make the new law true in the resulting model,not all of them comply with minimal change. To witness, putting an a-arrow from {p1, p2} to {p1, p2} or {p1, p2} makes us lose the effect lawp2 [a]p2; and pointing it to {p1, p2} deletes p1 [a]p1. Note that theselaws are preserved if we point the arrow to {p1, p2}. What would supportthe choice for the latter?

When pointing a new arrow leaving a world w we want to preserve asmany effects as we had before doing so. To achieve this, it is enough to

preserve old effects only in w (because the remaining structure of the modelremains unchanged after adding this new arrow). The operation we mustcarry out is to observe what is true in w and in the candidate target world w:

What changes from w to w (w \ w) must be what is obliged to do so.

What does not change from w to w (w w) must be what is eitherobliged or allowed to do so.

This means that every change outside what is forced to change is not anintended one. In our example, when putting the a-arrow from {p1, p2} to

{p1, p2}, p1 becomes a possible effect of a. As far as p1 is never causedby a, there is no justification for having it in a target world of {p1, p2}.Similarly, we want the literals preserved in the target world to be at mostthose that either are consequences of some effect or are usually preserved inthat context. Every preservation outside those may make us lose some law.For instance, when putting the new a-arrow from {p1, p2} to {p1, p2},

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p2 is preserved. Because p2 is not a necessary effect of a and is moreover

never preserved across as execution (in M), there is no reason to preserveit in this a-transition.

This looks like prime implicants [18, 14], and that is where prime subval-uations (see Section 2.3) play their role: the worlds to which the new arrowshall point will be those whose difference w.r.t. the departing world are liter-als that are relevant, and whose similarity w.r.t. it are literals that we knowdo not change.

Before giving a formal definition for that, we need to consider two impor-tant issues: First, when checking satisfaction of these two conditions, lookingjust at what is true in the model M we want to modify is not enough. It

can be a model in which a contingent, i.e., not true in all models formulais true. Hence we shall consider all the models in M. Second, ifa is neverexecutable in w, i.e., Ra(w) = for every M = W, R M, then lots ofeffects for a trivially hold in w, and then not all of them should be takeninto account in deciding what has to be changed or preserved. In this case,one should instead look at the effects that hold for those worlds w such thatRa(w) = (because everything that holds in these worlds also holds triviallyin those worlds with no transition by a).

Definition 3.5 LetM = W, R be a model, w, w W, M a set of modelssuch thatM M, and a an executability law. Thenw is arelevanttarget world of w w.r.t. a forM in M if and only if:

|=M

w

If there is M = W, R M such that Ra(w) = :

for all w \ w, there is Fml s.t. there is v base(, W)

s.t. v w, v, and for everyMi M, |=Mi

w[a]

for all w w, either there is Fml s.t. there is v

base(, W) s.t. v w, v, and for allMi M, |=Mi

w[a]; or

there is Mi M s.t. |=Mi

w[a]

If Ra(w) = for everyM = W, R M:

for all w \ w, there is Mi = Wi, Ri M s.t. there isu, v Wi s.t. (u, v) Ria and v \ u

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for all w w, there is Mi = Wi, Ri M s.t. there is u, v

Wi s.t. (u, v) Ria and u v, or for allMi = Wi, Ri M,if (u, v) Ria, then / v \ u

By RelTarget(w, a,M, M) we denote the set of all relevant targetworlds of w w.r.t. a forM in M.

The semantics of model revision by a new executability law is given by:

Definition 3.6 LetM = W, R. M = W, R Ma if and only if:

W = W

R R

If (w, w) R \ R, then w RelTarget(w, a,M, M)

|=M

a

The minimal models resulting from revising a model M by a new exe-cutability law are those closest to M w.r.t. M:

Definition 3.7 revise(M, a) = min{Ma, M}

3.4 Revising Sets of Models

Now we are ready to define revision of a set of models M by a new law :

Definition 3.8 LetM be a set of models and a law. Then

M =

M \ {M :|=

M}, if there is M M s.t. |=

M

MM revise(M, ), otherwise

Observe that Definition 3.8 comprises both expansion and revision: in thefirst one, simple addition of the new law gives a satisfiable theory; in thelatter a deeper change is needed to get rid of inconsistency (see the motivatingdiscussion in the beginning of Section 3).

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4 Syntactic Operators for Revision

We now turn our attention to the syntactical counterpart of revision. Supposethat we have an action theory Tand a formula we want to revise Twith.If T {} is satisfiable, then adding to T(expansion) will do the job.Otherwise, if T {} |=

PDL, then we have to modify the laws in T to

accommodate with the new incoming law (proper revision). Our endeavorhere is to perform minimal change at the syntactical level. By T we denotethe result of revising Twith .

4.1 Revision by a Static Law

Looking at the semantics of revision by Boolean formulas, we see that revisingan action theory by a new static law may conflict with the executability laws:some of them may be lost and thus have to be changed as well. The approachhere is to preserve as many executabilities as we can in the old possible states.Algorithm 1 deals with that (S denotes the classical revision of S by using any method from the literature [21, 10, 7]).

Algorithm 1 Revision by a static lawinput: T, output: T

1: if T {} |=PDL then2: T:= T {}3: else4: S:= S , E:= E, X:= 5: for all IP(S) do6: for all A atm() do7: A:=

piatm()

piA

pi

piatm()pi /A

pi

8: if S |=CPL

( A) then9: if S |=

CPL( A) then

10: if T |=PDL

( A) a and S, E, X |=

PDL

( A) then11: Xa

:= {(i A) a : i a Xa}12: else13: E:= E {( A) [a]}14: T:= S

E X

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The basic idea is as follows: we look at each possible valuation that is

common to the new S and the old one. Everytime an executability used tohold in that state and no inexecutability holds there in the new theory, wemake the action executable in such a context. For those contexts not allowedby the old S, we make a inexecutable (see Section 3.1).

4.2 Revision by an Effect Law

When revising a theory by a new effect law [a], we want to eliminateall possible executions of a leading to -states. To achieve that, we look atall -contexts and every time a transition to some -context is not always

the case, i.e., T |=PDL a, we can safely force [a] for that context.On the other hand, if in such a context there is always an execution of a to, then we should strengthen the executability laws to make room for thenew effect in that context we want to add. Algorithm 2 below does the job.

Algorithm 2 Revision by an effect law

input: T, [a]output: T[a]

1: if T { [a]} |=PDL

then2: T[a]:= T { [a]}3: else4: T:= T5: for all IP(S ) do6: for all A atm() do7: A:=

piatm()

piA

pi

piatm()pi /A

pi

8: if S |=CPL

( A) then9: for all IP(S ) do

10: if T |=PDL

( A) a then

11: T:=(T \ Xa)

{(i ( A)) a : i a X

a}12: T:= T {( A) [a]}13: if T |=

PDL( A) [a] then

14: T:= T {(i A) a : i a T}15: T[a]:= T

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4.3 Revision by an Executability Law

Revising a theory by a new executability law will have as immediate con-sequence a change in the set of effect laws: all those laws preventing theexecution of a shall be weakened. Besides that, in order to comply withminimal change, we shall ensure that in all models of the resulting theorythere will be at most one transition by action a from those worlds in whichTprecluded as execution.

Let E,a denote a minimum subset of Ea such that S, E,a |=PDL [a].

In the case the theory is modular [8] (see further), interpolation guaranteesthis set always exists. Moreover, note that there can be more than one such

a set, in which case we denote them (E,

a )1, . . . , (E,

a )n. Let

Ea =

1in

(E,a )i

The effect laws in Ea will serve as guidelines to get rid of [a] in each -worldallowed by the theory: they are the laws to be weakened to allow for a.

The idea behind our algorithm is as follows. To force the new executabil-ity law a to be true in all models of the resulting theory, we visitevery possible -context allowed by it and make the following operations toensure that a is the case for that context:

if Tnot always precludes a from being executed in this -context, wecan safely force a without modifying the other laws

on the other hand, ifa is always inexecutable in that context, then weshould weaken the laws in Ea .

When weakening the laws in Ea , the first thing we must do is to preserveall old effects in all other -contexts. To achieve that we specialize the abovelaws to each possible valuation (maximal conjunction of literals) satisfying but the actual one. Then, in the current -valuation, we must ensure that

action a may have any effect, i.e., from this -world we can reach any otherpossible world. We achieve that by weakening the consequent of the laws inEa to the exclusive disjunction of all possible contexts in T.

Finally, to get minimal change, we must ensure that all literals in this-valuation that are not forced to change are preserved (see Section 3.3). We

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do this by stating a conditional frame axiom [1] of the form (k ) [a],

where k is the above -valuation.

Algorithm 3 below gives the pseudo-code for that:

Algorithm 3 Revision by an executability law

input: T, aoutput: Ta

1: if T { a} |=PDL

then2: Ta:= T { a}3: else4: T:= T

5: for all IP(S ) do6: for all A atm() do7: A:=

piatm()

piA

pi

piatm()pi /A

pi

8: if S |=CPL

( A) then9: if T |=

PDL( A) [a] then

10: T:=

(T \ Ea) {(i ( A)) [a]i : i [a]i E

a}

{(i A) [a]

IP(S )

Aatm()

( A) : i [a]i Ea}

11: for all L Litdo12: if S |=

CPL( A)

L then

13: for all L do14: if T |=

PDL [a] or (T

PDL [a] and T |=

PDL

[a]) then15: T:= T {( A ) [a]}16: T:= T {( A) a}17: Ta:= T

5 Correctness of the Algorithms

Suppose we have two atoms p1 and p2, and only one action a. Let theaction theory T1 = {p2, p1 [a]p2, a}. The only model of T1 is M

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in Figure 4. Revising such a model by p1 p2 gives us the models Mi ,

1 i 3, in Figure 4. Now, revising T1 by p1 p2 will give us T1p1p2 ={p1 p2, p1 [a]p2}. The only model of T1

p1p2

is M1 in Figure 4. Thismeans the semantical revision produces models (viz. M2 and M

3) that are

not models of the revised theories.

M : p1,p2

a

M1 : p1,p2 M2 : p1, p2 M

3 : p1, p2

Figure 4: The model M of Tand the semantical revision ofM by p1 p2.

The other way round, the algorithms may produce theories whose modelsdo not result from the semantical revision of some model of the originaltheory. Consider T2 = {(p1 p2) [a], a}, whose only model is M inFigure 4. The revision ofM by p1 p2 is as above. However T2

p1p2

= {p1 p2, (p1 p2) [a]} has a model M

= {{p1, p2}, {p1, p2}, {p1, p2}}, that is not in Mp1p2.

This happens because the possible states are not completely characterizedby the static laws in S. Fortunately we get the right result by requiring S to

be big enough. This means considering models whose possible worlds areall allowed by S:

Definition 5.1 LetT= S E X be an action theory. M = W, R is thebig model of Tif and only if:

W= val(S); and

Ra = {(w, w) : for all [a] Ea, if |=M

w then |=

M

w}.

Big models thus contain all allowed valuations and maximize executability.They are connected with the principle of modularity [8]:

Definition 5.2 ([8]) Tis modular if and only if for every Fml, ifT |=PDL

, then S |=CPL

.

Theorem 5.1 ([19]) Tis modular if and only if its big model is a model of T.

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Proof: Let Mbig = Wbig, Rbig be the big model of T.

(): By definition, Mbig is such that |=Mbig S E. It remains to show that

|=MbigX. Let i a Xa, and let w Wbig be such that |=

Mbig

wi. There-

fore for all j Fml such that T |=PDL

j [a], we must have |=Mbig

wj ,

because T |=PDL

(i j), and as Tis modular, S |=CPL

(i j), and hence

|=Mbig(i j). Then by the construction ofMbig, there is some w Wbig

such that |=Mbig

w for all [a] Ea such that |=

Mbig

w. Thus Ra(w) = and

|=Mbigi a. Hence |=

MbigT.

(): Suppose Tis not modular. Then there must be some Fml suchthat T |=

PDL and S |=

CPL. This means that there is v val(S) such that

v . As v Wbig (because Wbig contains all possible valuations ofS), Mbigis not a model ofT.

Under modularity, revision of models ofTby in the semantics producesmodels of T (the output of the algorithms):

Theorem 5.2 Let T be modular and be a law. For all models M, if

M M, for some M = {M :|=M

T}, then |=M

T.

Proof: See Appendix A.

Also under modularity, models ofT result from revision of models ofTby :

Theorem 5.3 Let Tbe modular and a law. For all models M, if |=M

T,

thenM M, for some M = {M :|=M

T}.

Proof: See Appendix B.

In [8] algorithms are given to check whether Tsatisfies the principle ofmodularity and also to make Tsatisfy it, if that is not the case.

Modular theories also have other interesting properties [9]: for example,consistency of the whole theory amounts to that ofS; deduction of effect lawsdoes not need the executability ones and vice versa; prediction of an effect ofa sequence of actions a1; . . . ; an does not need the effect laws for actions otherthan a1, . . . , an. This also applies to plan validation when deciding whethera1; . . . ; an is the case.

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6 Conclusion and Perspectives

The problem of action theory change has only recently received attention inthe litterature, both in action languages [2] and in dynamic logic [6, 20].

Here we have studied what revising action theories by a law means, bothin the semantics and at the syntactical level. We have defined a semanticsbased on distances between models that also captures minimal change w.r.t.the preservation of effects of actions. With our algorithms and the correctnessresults under modularity we have established the link between the semanticsand the syntax, and have also shown that the modularity notion is fruitful.Since modularity is preserved across revision (Lemma B.1), it has to beensured only once during the evolution of the action theory.

Here we have presented the case for revision. In [20] we also define thecontraction counterpart of action theory change. There we show that more-over our constructions satisfy all Katsuno and Mendelzons postulates forcontraction [10].

Our next step on the subject is to define a general framework in whichto revise a theory by any formula of the language and not only laws. Webelieve such a definition will use as basic operations semantical modificationslike those we studied here (addition/removal of arrows and worlds). Henceour constructions will help us in better understanding what revision by ageneral formula means.

Acknowledgements

This work has been partially supported by the government of the Federa-tive Republic of Brazil. Grant: CAPES BEX 1389/01-7.

The author is thankful to Andreas Herzig and Laurent Perrussel for in-teresting discussions on the subject of this work.

References

[1] M. Castilho, O. Gasquet, and A. Herzig. Formalizing action and changein modal logic I: the frame problem. J. of Logic and Computation,9(5):701735, 1999.

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[2] T. Eiter, E. Erdem, M. Fink, and J. Senko. Updating action domain de-

scriptions. In L. Kaelbling and A. Saffiotti, editors, Proc. 19th Intl. JointConf. on Artificial Intelligence (IJCAI05), pages 418423, Edinburgh,2005. Morgan Kaufmann Publishers.

[3] J. Finger. Exploiting constraints in design synthesis. PhD thesis, Stan-ford University, Stanford, 1987.

[4] P. Gardenfors. Knowledge in Flux: Modeling the Dynamics of EpistemicStates. MIT Press, Cambridge, MA, 1988.

[5] D. Harel, J. Tiuryn, and D. Kozen. Dynamic Logic. MIT Press, Cam-

bridge, MA, 2000.

[6] A. Herzig, L. Perrussel, and I. Varzinczak. Elaborating domain descrip-tions. In G. Brewka, S. Coradeschi, A. Perini, and P. Traverso, editors,Proc. 17th Eur. Conf. on Artificial Intelligence (ECAI06), pages 397401, Riva del Garda, 2006. IOS Press.

[7] A. Herzig and O. Rifi. Propositional belief base update and minimalchange. Artificial Intelligence, 115(1):107138, 1999.

[8] A. Herzig and I. Varzinczak. On the modularity of theories. In

R. Schmidt, I. Pratt-Hartmann, M. Reynolds, and H. Wansing, edi-tors, Advances in Modal Logic, volume 5, pages 93109. Kings CollegePublications, 2005. Selected papers of AiML 2004 (also available athttp://www.aiml.net/volumes/volume5).

[9] A. Herzig and I. Varzinczak. Metatheory of actions: beyond consistency.Artificial Intelligence, 171:951984, 2007.

[10] H. Katsuno and A. Mendelzon. On the difference between updating aknowledge base and revising it. In P. Gardenfors, editor, Belief revision,pages 183203. Cambridge University Press, 1992.

[11] M. Kracht and F. Wolter. Properties of independently axiomatizablebimodal logics. J. of Symbolic Logic, 56(4):14691485, 1991.

[12] M. Kracht and F. Wolter. Simulation and transfer results in modal logic:A survey. Studia Logica, 59:149177, 1997.

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[13] D. Makinson. Friendliness and sympathy in logic. In J.-Y. Beziau,

editor, Logica Universalis. Springer-Verlag, 2nd edition, 2007.

[14] P. Marquis. Consequence finding algorithms. In D. Gabbay and Ph.Smets, editors, Algorithms for Defeasible and Uncertain Reasoning, inS. Moral, J. Kohlas (Eds), Handbook of Defeasible Reasoning and Uncer-tainty Management Systems, volume 5, chapter 2, pages 41145. KluwerAcademic Publishers, 2000.

[15] J. McCarthy and P. Hayes. Some philosophical problems from the stand-point of artificial intelligence. In B. Meltzer and D. Mitchie, editors,Machine Intelligence, volume 4, pages 463502. Edinburgh University

Press, 1969.

[16] R. Parikh. Beliefs, belief revision, and splitting languages. In L. Moss,editor, Logic, Language and Computation, volume 2 of CSLI LectureNotes, pages 266278. CSLI Publications, 1999.

[17] S. Popkorn. First Steps in Modal Logic. Cambridge University Press,1994.

[18] W. V. O. Quine. The problem of simplifying truth functions. AmericanMathematical Monthly, 59:521531, 1952.

[19] I. Varzinczak. What is a good domain description? Evaluating andrevising action theories in dynamic logic. PhD thesis, Universite PaulSabatier, Toulouse, 2006.

[20] I. Varzinczak. Action theory contraction and minimal change. To appearin Proc. KR, 2008.

[21] M.-A. Winslett. Reasoning about action using a possible models ap-proach. In R. Smith and T. Mitchell, editors, Proc. 7th Natl. Conf. onArtificial Intelligence (AAAI88), pages 8993, St. Paul, 1988. Morgan

Kaufmann Publishers.

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A Proof of Theorem 5.2

Let be a law, M M, and let T be the output of our algorithms on

input theory Tand law .

If T {} |=PDL

, then M M \ {M :|=M

} and M is a model ofT = T {}.

Let T {} |=PDL

. We analyze each case.

Let be some Fml. Then M = W, R where W = (W\val())val() is minimal w.r.t. W and R R is maximal w.r.t. R, for some M =W, R M.

As we have assumed the syntactical classical revision operator is soundand complete w.r.t. its semantics and is moreover minimal, we have |=

M

S.

Because R R, |=M

E. Thus it is enough to show that M is a model of theadded laws.

Given (i A) a T, for every w W

, if |=M

wi A, then

w W (because S |=CPL

( A) ). From w i and i a Xa,

we have Ra(w) = . Suppose Ra(w) = . As |=

M

S E and R is

maximal, every M = W, R s.t. |=M

S E is s.t. Ra(w) = , andthen S E |=

PDL( A) [a]. Because T |=

PDL( A) a, and

S |=CPL

( A) and S |=CPL

( A) , we get S , E, X |=PDL

( A), and then (i A) a / T. Hence Ra(w) = , and

|=M

(i A) a.If ( A) [a] T

, then S |=CPL ( A) . Thus, for every

w W, if |=M

w A, R

a(w) = and the result follows.

Let now have the form [a], for , Fml. Then M = W, Rfor some M = W, R M s.t. W = W and R R, where R is maximalw.r.t. R.

From W = W, |=M

S. As R R, |=M

E. Because S E T[a], itsuffices to show that M is a model of the added laws.

By definition, |=M

[a], and then |=M

( A) [a] for every IP(S ).

If (iA) a T[a], then for every w W

, ifw iA,we have w i. As w W, and i a Xa, Ra(w) = . If R

a(w) = ,

then w for every w Ra(w). Thus as far as we added ( A) [a]to T[a], we must have T

[a] |=PDL( A) [a]. Hence R

a(w) = .

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Let (i T|=PDL(A)a ( A)) a T [a]. For everyw W, if |=M

wi

T|=PDL

(A)a( A), then w i, and as w W

and i a Xa, we have Ra(w) = . If Ra(w) = , because |=

M

S E

and R is maximal, every M = W, R s.t. |=M

S E is s.t. Ra(w) = .Then S, E |=

PDL

w [a]. But then T |=PDL

w [a], and as

i a Xa, T |=PDL(

w i), and then w / W, a contradiction.Hence Ra(w) = .

Finally, let be of the form a, for some Fml. Then M =W, R for some M = W, R M s.t. W = W and R = R R,a , with

R,a = {(w, w) : w RelTarget(w, a,M, M)}

such that R is minimal w.r.t. R.From W = W, |=

M

S. As R R, |=M

X. As far as S X T a, itis enough to show that M satisfies the added laws.

By definition, |=M

a, and then |=M

( A) a for every IP(S ).

If (i A) [a](i

IP(S )

Aatm()

( A)) Ta, then for

every w W, if w i A, then w i. Because |=M

i [a]i,

we have |=M

w i for all w

W s.t. (w, w

) Ra, and then |=M

w i for everyw W s.t. (w, w) Ra \ R

,a . Now, given (w, w

) R,a , we have

|=M

w

IP(S )

Aatm()

( A), and the result follows.

Let (i T|=PDL

(A)[a]( A)) [a]i T

a. For every w

W, if|=M

wi

T|=PDL

(A)[a]( A), then w i, and as |=

Mi [a]i,

we have |=M

wi for all w

W s.t. (w, w) Ra. Thus |=M

wi for every

w W s.t. (w, w) Ra \ R,a . Now, if w , then R

,a = and the

result follows. Otherwise, if w , then T |=PDL

( A) [a], and then

(i T|=PDL(A)[a] ( A)) [a]i has not been put in T

a, acontradiction.

Let now ( A ) [a] Ta. For every w W

, if|=M

w A ,

then |=M

w, and then |=

M

w. From ( A ) [a] T

a, we have

T |=PDL

[a] or T |=PDL

[a] and T |=PDL

[a]. In both cases, |=M

w

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for every w Ra(w), and then |=M

w for every w s.t. (w, w) R \ R,a . It

remains to show that |=M

w for every w W s.t. (w, w) R,a .

Suppose |=M

w. Then w \ w. From the construction ofM, there

is M = W, R M s.t. there is (u, v) Ra and v \ u, i.e., |=M

u

and |=M

v. From (u, v) Ra, we do not have T |=PDL [a]. From

|=M

v, we do not have T |=

PDL [a]. Thus the algorithm has not put

( A ) [a] in Ta, a contradiction.

B Proof of Theorem 5.3

Lemma B.1 Let be a law. If Tis modular and T {} |=PDL

, then T ismodular.

Proof: Let be nonclassical. Suppose T is not modular. Then there is Fml s.t. T |=PDL

and S |=CPL

, where S is the set of static lawsin T. Suppose T |=PDL

. Then we must have T |=PDL [a] and

T |=PDL a.

Suppose has the form [a], for , Fml. Then for all -

contexts, as far as T |=PDL( ) [a], ( ) a / T. ThenT |=PDL

if and only if S |=CPL

, a contradiction.Suppose is of the form a, for Fml. Then for all -

contexts such that T |=PDL( ) a, T |=PDL(

) [a] isimpossible as far as Ea has been weakened. Then T

|=PDL

if and only ifS |=

CPL, a contradiction.

Hence we have T |=PDL

. Because is nonclassical, S = S. ThenT |=

PDL and S |=

CPL, and hence Tis not modular.

Let now be some Fml. Suppose T is not modular, i.e., there is Fml s.t. T |=PDL

and S = S |=CPL

.

From S |=CPL, there is v val(S) s.t. v .If v val(S), as Tis modular, T |=

PDL. From this and T |=PDL

, wemust have T |=PDL

[a] and T |=PDL a. From the latter,

we get T |=PDL

a, and from the first we have T |=PDL

[a].Putting both results together we get T |=

PDL. As S |=

CPL, we have a

contradiction.

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If v / val(S), then T |=PDL a, as no executability for context

has been put into T. Hence T |=PDL, a contradiction.

Lemma B.2 If Mbig = Wbig, Rbig is a model of T, then for every M =

W, R such that |=M

Tthere is a minimal (w.r.t. set inclusion) extensionR Rbig\ R such thatM

= val(S), R R is a model of T.

Proof: Let Mbig = Wbig, Rbig be a model ofT, and let M = W, R be such

that |=M

T. Consider M = val(S), R. If |=M

T, we have R = Rbig \ R

that is minimal. Suppose then |=M

T. We extend M to a model ofTthat is

a minimal extension ofM. As |=M

T, there is v val(S) \ Wsuch that |=M

v

T.

Then there is Tsuch that |=M

v. If is some Fml, as v Wbig,

Mbig is not a model of T. If is of the form [a], for , Fml,there is v val(S) such that (v, v) Ra and v

, a contradiction sinceRa(v) = . Let now have the form a for some Fml. Then

|=M

v. As v Wbig, if |=

Mbig

v a, then |=

Mbig T. Hence, Rbiga(v) = .

Thus taking any (v, v) Rbiga gives us a minimal R = {(v, v)} such that

M = val(S), R R is a model of T.

Lemma B.3 Let Tbe modular, and be a law. Then T |=PDL

if and only

if everyM = val(S), R such that |=W,R

Tand R R is a model of .

Proof:(): Straightforward, as T |=

PDL implies |=

M for every M such that |=

MT,

in particular for those that are extensions of some model of T.

(): Suppose T |=PDL

. Then there is M = W, R such that |=M

Tand |=M

.As Tis modular, the big model Mbig = Wbig, Rbig of Tis a model of T.Then by Lemma B.2 there is a minimal extension R of R w.r.t. Rbig such

that M = val(S), R R is a model of T. Because |=M

, there is w W

such that |=M

w . If is some propositional Fml or an effect law, anyextension M ofM is such that |=

M

w. If is of the form a, then

|=M

w and Ra(w) = . As any extension ofM is such that (u, v) R

if and

only if u val(S) \ W, only worlds other than those in W get a new leaving

arrow. Thus (R R)a(w) = , and then |=M

w.

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Lemma B.4 LetTbe modular and a law. IfM = val(S), R is a model

of T, then there is M = {M :|=M

T} s.t. M M.

Proof: Let M = val(S), R be such that |=M

T. If |=M

T, the result

follows. Suppose |=M

T. We analyze each case.Let be of the form [a], for , Fml. Let M = {M : M =

val(S), R}. As Tis modular, by Lemmas B.2 and B.3, M is non-emptyand contains only models of T.

Suppose M is not a minimal model of T[a], i.e., there is M such

that M M M for some M M. Then M and M differ only in the

effect of a in a given -world, viz. a A-context, for some IP(S )

and A =

piatm()piA

pi

piatm()pi /A

pi such that A atm().

Because |=M

( A) a, we must have |=M

( A) a, and

then |=M

[a]. Hence M is minimal w.r.t. M.When revising by an effect law, S = S. Hence taking the right R and

R,a such that M = val(S), R and R = R \ R,a , for some R

,a

{(w, w) :|=M

w, |=

M

w and (w, w) Ra}, we have M M and then M

M[a].

Let have the form a, for Fml. Let M = {M : M =val(S), R}. As Tis modular, by Lemmas B.2 and B.3, M is non-empty

and contains only models of T.Suppose that M is not a minimal model of Ta, i.e., there is M

such that |=M

Ta and M M M for some M M. Then M and

M differ only on the executability of a in a given -world, i.e., a A-context, for some IP(S ) and A =

piatm()

piA

pi

piatm()pi /A

pi, such

that A atm(). This means M has no arrow leaving this A-world.

Then |=M

( A) [a], and hence |=M

a. Hence M is a minimalmodel of Ta w.r.t. M.

When revising by executability laws, S = S. Thus taking the right R

and a minimal R,a such that M = val(S), R and R

= R R,a , for

some R,a {(w, w) :|=

M

w and w RelTarget(w, a,M, M)}, we

get M M and then M Ma.

Finally, let be some Fml. Then M is such that for every w W,if Ra(w) = , then w val(S) and Ra(w) = for every M = W, R M.Choosing the right M M the result follows.

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Proof of Theorem 5.3

Let T be the output of our algorithms on input theory Tand law . If

T = T {}, then T {} |=PDL, and hence every M such that |=

M

T is

such that M M \ {M :|=M

} and the result follows.

Suppose T {} |=PDL

. From the hypothesis that Tis modular andLemma B.1, T is modular. Then M = val(S), R is a model of T, byLemma B.2. From this and Lemma B.3 the result follows.

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Action Theory Revision