action theory revision in dynamic logic

8/14/2019 Action Theory Revision in Dynamic Logic

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Action Theory Revision in Dynamic Logic

Ivan Jose Varzinczak

IRIT Universite de ToulouseToulouse, France

[email protected]

Meraka InstituteCSIR, Pretoria, South Africa

[email protected]

Abstract

Like any other logical theory, action theories in reason-ing about actions may evolve, and thus need revisionmethods to adequately accommodate new informationabout the behavior of actions. Here we give a semanticsthat complies with minimal change for revising actiontheories stated in a version ofPDL. We give algorithmsthat are proven correct w.r.t. the semantics for those the-ories that are modular.

Introduction

In logic-based approaches to reasoning about actions, theo-ries are collections of statements of the form: if context,then effect after every execution of action (effect laws);and if precondition, then action executable (executabil-ity laws). For example, in Propositional Dynamic Logic(PDL) (Harel, Tiuryn, and Kozen 2000), one could have thelaw (p1p2) [a]p1, saying that in every context wherep1p2 is the case, after every execution of action a we get

the effect p1; and (p1 p2) a, stating that p1 p2is a sufficient condition for as executability.These are examples of what we call action laws, as they

specify the behavior of the actions of a given domain. Be-sides that we can also have laws mentioning no action atall (static laws). They characterize the underlying structureof the world, i.e., its possible states. For instance, having

p1 p2 as a static law would mean p1 p2 is a forbiddenstate. Action theories will then be collections of laws, eachof them seen as a global axiom in PDL.

Well, it may happen that such descriptions have to be re-vised due e.g. to new incoming information about the be-havior of the world. In our example, we may learn that the

only valid states are those satisfying p1 p2, or that action ahas always p2 as outcome in p2-contexts, or even that p1is enough to guarantee as executability. Here we are inter-ested 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 alaw ? How to get minimal change, i.e., how to keep asmuch knowledge about other laws as possible?

Copyright c 2008, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

How to syntactically revise an action theory so that itsresult corresponds to the intended semantics?

Here we answer these questions.

Logical Preliminaries

Action Theories in Dynamic Logic

Our base formalism is PDL without the operator. LetAct = {a1, a2, . . .} be the set of atomic actions of a do-main. To each a there is associated a modal operator [a]. Wesuppose our multimodal logic is independently axiomatized,i.e., the logic is a fusion and there is no interaction betweenthe modal operators (Kracht and Wolter 1991).Prop = {p1,p2, . . .} denotes the set of all propositional

constants or atoms. The set of literals is Lit= {1, 2, . . .},where each i is either p or p, for some p Prop. In case = p, we identify with p. By || we will denote theatom in literal .

By , , . . . we denote Boolean formulas, examples ofwhich are p

1

p2

and p1

p2

. Fml is the set of allBoolean formulas. A propositional valuation v is a maxi-mally consistent set of literals. We denote v the factthat v satisfies . val() is the set of all valuations satisfy-ing . |=

CPLdenotes the classical consequence relation.

With IP() we denote the set of prime implicants (Quine1952) of. By we denote a prime implicant, and atm()is the set of atoms occurring in . For given and , abbreviates is a literal of.

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

A PDL-model is a tuple M = W,R where W is a setof valuations, and R maps action constants a to accessibility

relations Ra W W. Given a model M, |=M

wp (p is true at

world w of modelM) ifw p; |=M

w[a] if|=

M

w for every w

s.t. (w, w) Ra; truth conditions for the other connectivesare as 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


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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 ofactions. Following the tradition in the reasoning about ac-tions community, we here distinguish three types of them.

Static Laws A static law is a formula Fml. It charac-terizes the possible states of the world. The set of all staticlaws 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. The consequent is the effectwhich always obtains when a is executed in a state wherethe antecedent holds. If a is a nondeterministic action,then is typically a disjunction. If is inconsistent we havea special kind of effect law that we call an inexecutabilitylaw. For example, (p1 p2) [a] says that a cannot beexecuted (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 a has the form

a, with Fml. It stipulates the context in whicha is guaranteed to be executable. (InPDL, the operator a isused to express executability, a thus reads as executionis possible.) The set of all executability laws of a domainis denoted by X.

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

Given an action a, Ea (resp. Xa) will denote the set ofonly those effect (resp. executability) laws about a. For thesake of clarity, we here abstract from the frame and rami-fication problems, and assume Tcontains all frame axioms(cf. (Herzig, Perrussel, and Varzinczak 2006) for a contrac-tion approach within a solution to the frame problem).

Elementary Atoms and Prime Valuations

Given Fml, E() denotes the elementary atoms actu-ally occurring in . For example, E(p1 (p1 p2)) ={p1,p2}. An atomp is essential to if and only ifp E(

)for every such that |=

CPL . For instance, p1 is es-

sential to p1 (p1 p2). E!() will denote the essentialatoms of. (If is not contingent, i.e., it is a tautology or acontradiction, then E!() = .)

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

CPL and E() E!(). For instance, p1 p2 / p1,

as p1 |=CPL p1 p2 but E(p1 p2) E!(p1). MoreoverE() = E!(), and whenever |=

CPL , E!() =

E!(

) and also =

.Theorem 1 (Least atom-set theorem (Parikh 1999))|=CPL

, and E() E() for every s.t.|=CPL

.

Thus for each Fml there is a unique least set of elemen-tary atoms such that may equivalently be expressed usingonly atoms from that set.1

1The dual notion (redundant atoms) is addressed in (Herzig andRifi 1999), with similar purposes.

Given a valuation v, v v is a subvaluation. For Wa setof valuations, a subvaluation 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 if v

W and {|| : v}

E!(). If v !

W, we call v an essential subvaluation of

(modulo W).

Definition 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 statesof truth in which is true. They are just another way ofseeing prime implicants of . By base(, W) we denote theset of all prime subvaluations of modulo W.

Theorem 2 Let Fml and W be a set of valuations. Thenfor all w W, w if and only ifw

vbase(,W)

v .

Closeness Between Models

When revising a model, we will perform a change in itsstructure. Because there can be several different ways ofmodifying a model (not all of them minimal), we need a no-tion of distance between models to identify those that areclosest to the original one.

As we are going to see in more depth in the sequel, chang-ing a model amounts to modifying its possible worlds orits accessibility relation. Hence, the distance between twoPDL-models will depend upon the distance between theirsets of worlds and accessibility relations. These here will bebased on the symmetric difference between sets, defined asXY = (X \ Y) (Y \ X).

Definition 2 LetM = W,R be a model. M = W,R

is as close to M as M = W,R, notedM M M,if and only if

either WW WW

or WW = WW and RR RR

(Notice that other distance notions are also possible, likee.g. considering the cardinality of symmetric differences.)

Semantics of Revision

Contrary to action theory contraction (Varzinczak 2008a),where we want the negation of some law to become satis-

fiable, in revision we want to make a new law valid. Thismeans that one has to eliminate all cases satisfying its nega-

tion. This depicts the duality between revision and contrac-tion: whereas in the latter one invalidates a formula by mak-ing its negation satisfiable, in the former one makes a for-mula valid by forcing its negation to be unsatisfiable prior toadding the new law to the theory.

The idea behind our semantics is as follows: we initiallyhave a set of models M in which a given formula is (po-tentially) not valid, i.e., is (possibly) not true in everymodel in M. In the result we want to have only models of. Adding -models to M is of no help. Moreover, adding


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models makes us to lose laws: the corresponding resultingtheory would be more liberal.

One solution amounts to deleting from M those modelsthat are not -models. Of course removing only some ofthem does not solve the problem, we must delete every sucha model. By doing that, all resulting models will be mod-els of . (This corresponds to theory expansion, when theresulting theory is satisfiable.) However, if M contains no

model of, we will end up with . Consequence: the result-ing theory is inconsistent. (This is the main revision prob-lem.) In this case the solution is to substitute each modelM in M by its nearest modification M that makes true.This lets us to keep as close as possible to the original mod-els we had. But, what if for onemodel in M there are severalminimal (incomparable) modifications of it validating ? Inthat case we shall consider all of them. The result will alsobe a list of models M, all being models of.

Before defining revision of sets of models, we presentwhat modifications of (individual) models are.

Revising a Model by a Static Law

Consider the model depicted in Figure 1, and suppose wewant to revise it by the Boolean formula p1 p2, i.e., wewant such a formula to be a static law.

M :

p1,p2 p1,p2

p1,p2

a

aa

Figure 1: A model where p1 p2 is satisfiable.

In such a model, we do not want the formula p1 p2to be satisfiable, so the first step is to remove all worlds inwhich it is true. The second step is to guarantee that allthe remaining worlds satisfy the new law. Such an issuehas been largely addressed in the literature on propositionalbelief base revision and update (Gardenfors 1988; Winslett1988; Katsuno and Mendelzon 1992; Herzig and Rifi 1999).Here we can achieve that with a semantics similar to thatof classical operators: basically one shall change the set ofpossible valuations, by removing or adding worlds.

The delicate point in removing worlds is that we maylose some executability laws: in the example, removing

{p1, p2} also removes p2 a. From a semanticpoint of view, this is intuitive: if the state of the world towhich we could move is no longer possible, then we do nothave a transition to that state anymore. Hence, if that transi-tion was the only one we had, it is natural to lose it.

Similarly, one could ask what to do with the accessibil-ity relation if new worlds are added (when expansion is notpossible): shall new arrows leave/arrive at the new world? Ifno arrow leaves the new added world, we may lose an exe-cutability law. If some arrow leaves it, we may lose an effectlaw, the same holding if we add an arrow pointing to the new

world. If no arrow arrives at this new world, what about theintuition? Do we want to have an unreachable state?

All this discussion shows how drastic a change in thestatic laws may be: it is a change in the underlying struc-ture (possible states) of the world! Changing it may have asconsequence the loss of an effect law or an executability law.

The tradition in the reasoning about actions communitysays that executability laws are, in general, more difficult

to state than effect laws, and hence are more likely to beincorrect. By adding no arrow to the resultingmodel we herecomply with that and postpone correction of executabilitylaws, if needed (cf. (Herzig, Perrussel, and Varzinczak 2006;Varzinczak 2008a)).

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

Definition 3 LetM = W,R. M = W,R M ifand only if:

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

R R

Clearly |=M

for each M

M. The minimal models

resulting from revising a model M by are those closest toM w.r.t. M:

Definition 4 revise(M, ) =

min{M, M}

Revising a Model by an Effect Law

Let our language now have three atoms and consider themodel 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 that |=M

p2 p1 p3.) Suppose we want to reviseM by p1 [a]p2. This means that we should guaranteethe formula p1 ap2 is satisfiable in none of its worlds. Todo that, we have to look at the worlds satisfying it (if any)and either make p1 false, or make ap2 false by removinga-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.Flippingp1 would do the job but also has as consequence theloss of a static law: we would violate p2 p1p3. Here wethink that changing action laws should not have as side effecta change in the static laws. Given their special status, theseshould change only if explicitly required (see above). In thiscase, each world satisfying p1 ap2 has to be changedso that ap2 is no longer true in it. In our example, weshould remove the arrows ({p1, p2, p3}, {p1,p2,p3})and ({p1,p2, p3}, {p1,p2, p3}).


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The semantics of one model revision for the case of a neweffect law is:

Definition 5 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 amodel M by a new effect law are those that are closest toM w.r.t. M:

Definition 6 LetM be a model and [a] an effect law.Then revise(M, [a]) =

min{M[a], M}.

Revising a Model by an Executability Law

Let the model depicted in Figure 3 and suppose we want torevise 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 wemust throwp1[a] away to ensure the new formula is true.

To removep1 [a] we have to look at all worlds satisfyingit and modify M so that they no longer satisfy the formula.Given world {p1, p2}, we have two options: change the in-terpretation of p1 or add a new arrow leaving this world. Aquestion that raises is what choice is more drastic: change aworld or an arrow? Again, here we think that changing theworlds content (the valuation) is more drastic, as the exis-tence of such a world was foreseen by some static law and ishence assumed to be as it is, unless we have information sup-porting the contrary (see above). Thus we shall add a newa-arrow from {p1, p2}. Having agreed on that, the issuenow is: to which world should the new arrow point? Fouroptions show up: point the arrow to {p1,p2}, {p1,p2},

{p1, p2} or {p1, p2} itself. The resulting model is suchthat the unwanted formula is unsatisfiable and p1 aholds in all its worlds.

Whereas all these options make the new law true inthe resulting model, not all of them comply with minimalchange. 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} also deletes fromthe model p1 [a]p1. Note that these laws are preservedif we point the arrow to {p1,p2}. What would support thechoice for the latter?

When pointing a new arrow leaving a world w we wantto preserve as many effects as we had before doing so. Toachieve this, it is enough to preserve old effects only in w(because the remaining structure of the model remains un-changed after adding this new arrow). The operation wemust carry out is to observe what is true in w and in thecandidate target world w:

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

What does not change from w to w (ww) must be whatis either obliged or allowed to do so.

This means that every change outside what is forced tochange is not an intended one. In our example, when puttingthe a-arrow from {p1, p2} to {p1,p2}, p1 becomes apossible effect of a. As far as p1 is never caused by a,there is no justification for having it in a target world of{p1, p2}. Similarly, we want the literals preserved in thetarget world to be at mostthose that either are consequencesof some effect or are usually preserved in that context. Ev-ery preservation outside those may make us lose some law.For instance, when putting the new a-arrow from {p1, p2}to {p1, p2}, p2 is preserved. Because p2 is not a nec-essary effect ofa and is moreover never preserved across asexecution (in M), there is no reason to preserve it in thisnew a-transition.

This looks like prime implicants, and that is where primesubvaluations play their role: the worlds to which the newarrow shall point are those whose difference w.r.t. the depart-ing world are literals that are relevant, and whose similarityw.r.t. it are literals that we know do not change.

Before giving a formal definition for that, we need to con-sider two important issues: First, when checking satisfac-tion of these two conditions, looking just at what is true inthe model M we want to modify is not enough. It can be

a model in which a contingent, i.e., not true in all modelsformula is true. Hence we shall consider all the models inM. Second, if a is never executable in w, i.e., Ra(w) = for every M = W,R M, then lots of effects for a triv-ially hold in w, and then not all of them should be taken intoaccount in deciding what has to be changed or preserved. Inthis case, one should instead look at the effects that hold forthose worlds w such that Ra(w) = (because everythingthat holds in these worlds also holds trivially in those worldswith no transition by a).

Definition 7 LetM = W,R be a model, w, w W, Ma set of models such that M M , and a anexecutability law. Then w is a relevant target world of ww.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 every Mi

M, |=Mi

w[a]

for all w w, either there is Fml s.t. thereis v base(, W) s.t. v w, v , and for all

Mi M, |=Mi

w[a]; or there is Mi M s.t. |=

Mi

w[a]


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

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

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 all Mi = Wi,Ri M, if (u, v) Ria , then / v \ u

By RelTgt(w, a,M, M) we denote the set of allrelevant target worlds ofw w.r.t. a forM in M.

The semantics of one model revision by a new executabil-ity law is given by:

Definition 8 Let M = W,R. M = W,R M

a if and only if:

W = W

R R

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

|=M

a

The minimal models resulting from revising a model Mby a new executability law are those closest toMw.r.t. M:

Definition 9 Let M be a model and a bean executability law. Then revise(M, a) =

min{Ma, M}.

Revising Sets of Models

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

Definition 10 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 10 comprises both expansion andrevision: in the first one, simple addition of the new lawgives a satisfiable theory; in the latter a deeper change isneeded to get rid of inconsistency.

Syntactic Operators for Revision

We now turn our attention to the syntactical counterpart ofrevision. Suppose we have an action theory Tand a law wewant to revise Twith. IfT {} is satisfiable, adding toT(expansion) will do the job. Otherwise, ifT {} |=

PDL,

then we have to modify the laws in Tto accommodate withthe new incoming law (proper revision). Our endeavor hereis to perform minimal change at the syntactical level. By Twe denote the result of revising Twith .

Revision by a Static Law

Looking at the semantics of revision by Boolean formulas,we see that revising an action theory by a new static lawmay conflict with the executability laws: some of them maybe lost and thus have to be changed as well. The approachhere is to preserve as many executabilities as we can in theold possible states. To do that, we look at each possible

valuation that is common to the new S and the old one. Ev-ery time an executability used to hold in that state and noinexecutability holds there in the new theory, we make theaction executable in such a context. For those contexts notallowed by the old S, we make a inexecutable (cf. the se-mantics). Algorithm 1 deals with that (S denotes theclassical revision ofS by using any standard method fromthe literature (Winslett 1988; Katsuno and Mendelzon 1992;

Herzig and Rifi 1999)).

Algorithm 1 Revision by a static law

input: T, output: T

ifT {} |=PDL

thenT:= T {}

elseS:= S , E:= E, X:= for all IP(S) do

for all A atm() doA:=

Vpiatm()

piA

pi V

piatm()pi /A

pi

ifS |=CPL

( A) then

ifS |=CPL( A) thenif T |=PDL

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

PDL

( A) thenXa

:= {(i A) a : i a Xa}

elseE:= E {( A) [a]}

T:= S E X

Revision by an Effect Law

When revising a theory by a new effect law [a], wewant to eliminate all possible executions ofa leading to -states. To achieve that, we look at all -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 al-ways an execution ofa to , then we should strengthen theexecutability laws to make room for the new effect in thatcontext we want to add. Algorithm 2 below does the job.

Revision by an Executability Law

Revising a theory by a new executability law will have asimmediate consequence a change in the set of effect laws:all those laws preventing the execution of a shall be weak-ened. Besides that, in order to comply with minimal change,we shall ensure that in all models of the resulting theory

there will be at most one transition by a from those worldsin which Tprecluded as execution.

Let E,a denote a minimum subset of Ea such thatS, E,a |=PDL [a]. In the case the theory is modu-lar (Herzig and Varzinczak 2005) (see further), interpolationguarantees that this set always exists. Moreover, note thatthere can be more than one such a set, in which case wedenote them (E,a )1, . . . , (E

,a )n. Let

Ea =

1in

(E,a )i


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Algorithm 2 Revision by an effect law

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

ifT { [a]} |=PDL

thenT[a]:= T { [a]}

elseT:= Tfor all IP(S ) do


Vpiatm()

piA

pi V

piatm()pi /A

pi

ifS |=CPL

( A) thenfor all IP(S ) do

ifT |=PDL

( A) a then

T:=(T \ Xa) {(i ( A)) a :i a X

a}

T:= T {( A) [a]}ifT |=

PDL( A) [a] then

T:= T{(i A) a : i a T}

T

[a]:= T

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

The idea behind our algorithm is as follows: to force a to be true in all models of the resulting theory,we visit every possible -context allowed by it and make thefollowing operations to ensure a is the case for that con-text: Given a -context, if Tnot always precludes a frombeing executed in it, we can safely force a without mod-ifying other laws. On the other hand, if a is always inexe-cutable in that context, then we should weaken the laws inEa . The first thing we must do is to preserve all old ef-fects in all other -worlds. To achieve that we specialize theabove laws to each possible valuation (maximal conjunctionof literals) satisfying but the actual one. Then, in the cur-rent -valuation, we must ensure that action a may have anyeffect, i.e., from this -world we can reach any other pos-sible world. We achieve that by weakening the consequentof the laws in Ea to the exclusive disjunction of all possi-ble contexts in T. Finally, to get minimal change, we mustensure that all literals in this -valuation that are not forcedto change are preserved. We do this by stating a conditionalframe axiom of the form (k ) [a], where k is the

above -valuation.Algorithm 3 gives the pseudo-code for that.

Correctness of the Algorithms

Suppose we have two atoms p1 and p2, and only one actiona. Let the action theory T1 = {p2,p1 [a]p2, a}.The only model of T1 is M in Figure 4. Revising such amodel by p1 p2 gives us the models M

i , 1 i 3, in

Figure 4. Now, revising T1 by p1p2 will give us T1

p1p2=

{p1 p2,p1 [a]p2}. The only model ofT1

p1p2is M1

Algorithm 3 Revision by an executability law

input: T, aoutput: Ta

ifT { a} |=PDL

thenTa:= T { a}

elseT:= Tfor all IP(S ) do


Vpiatm()

piA

pi V

piatm()pi /A

pi

ifS |=CPL

( A) thenifT |=

PDL( A) [a] then

T:=

(T \ Ea )

{(i ( A)) [a]i :i [a]i E

a }

{(i A) [a]L

IP(S )

Aatm()

( A) :

i [a]i E

a }

for all L LitdoifS |=

CPL( A)

VL then

for all L doif T |=PDL

[a] or (T PDL

[a]and T |=

PDL [a]) then

T:= T {( A ) [a]}T:= T {( A) a}

Ta:= T

in Figure 4. This means that the semantic revision producesmodels (viz. M2 and M

3 in Figure 4) that are not models of

the revised theories.

M : p1,p2

a

M1 : p1,p2

M2 : p1,p2 M3 : p1,p2

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

The other way round, the algorithms may produce theo-ries whose models do not result from the semantic revisionof some model of the original theory. As an example, con-sider T2 = {(p1 p2) [a], a}, whose only model isM in Figure 4. The revision ofM by p1 p2 is as above.However T2

p1p2

= {p1 p2, (p1 p2) [a]} has amodel M = {{p1,p2}, {p1, p2}, {p1,p2}}, that isnot in Mp1p2 .

This happens because the possible states are not com-pletely characterized by the static laws in S. Fortunately


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we get the right result by requiring S to be big enough.This is connected with the principle of modularity (Herzigand Varzinczak 2005):

Definition 11 (Modularity (Herzig and Varzinczak 2005))Tis modular if and only if for every Fml, ifT |=

PDL,

then S |=CPL

.

Under modularity, revision of models of Tby a law

in the semantics produces models of the output of the algo-rithms T:

Theorem 3 LetTbe modular and be a law. For all mod-

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

T} , then

|=M

T.

Also under modularity, models ofT result from revisionof models ofTby :

Theorem 4 Let Tbe modular and a law. For every M,

if|=M

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

MT}.

In (Herzig and Varzinczak 2005) algorithms are given tocheck whether Tsatisfies the principle of modularity andalso to make Tsatisfy it, if that is not the case.

Modular theories have other interesting properties (Herzigand Varzinczak 2007): for example, consistency amounts tothat of S; deduction of effect laws does not need the exe-cutability ones and vice versa; prediction of an effect of asequence of actions a1; . . . ; an does not need the effect lawsfor actions other than a1, . . . , an. This also applies to planvalidation when deciding whether a1; . . . ; an is the case.

Conclusion and PerspectivesContrary to classical belief change, the problem of actiontheory change has only recently received attention in theliterature, both in action languages (Baral and Lobo 1997;Eiter et al. 2005) and in dynamic logic (Herzig, Perrussel,

and Varzinczak 2006; Varzinczak 2008a).Here we have studied what revising action theories by

a law means, both in the semantics and at the syntacticallevel. We have defined a semantics based on distances be-tween models that also captures minimal change w.r.t. thepreservation of effects of actions. With our algorithms andthe correctness results under modularity we have establishedthe link between the semantics and the syntax, and have alsoshown that the modularity notion is fruitful. Since modular-ity is preserved across revision (see Lemma 1 in the appen-dices), it has to be ensured only once during the evolution ofthe action theory.

Here we presented the case for revision. In (Varzinczak

2008a) we also define the contraction counterpart of ac-tion theory change. There we show that moreover our con-structions satisfy all Katsuno and Mendelzons postulates forcontraction (Katsuno and Mendelzon 1992).

Our next step on the subject is to define a general frame-work in which to revise a theory by any formula of thelanguage and not only laws. We believe that such a def-inition will use as basic operations semantic modificationslike those we studied here (addition/removal of arrows andworlds). Hence our constructions will help us in better un-derstanding what revision by a general formula means.

Acknowledgements

The author is thankful to Andreas Herzig and Laurent Per-russel for interesting discussions on the subject of this work.

This work has been partially supported by the governmentof the FEDERATIVE REPUBLIC OF BRAZIL. Grant: CAPESBEX 1389/01-7.

References

Baral, C., and Lobo, J. 1997. Defeasible specifications inaction theories. In Proc. IJCAI, 14411446.

Eiter, T.; Erdem, E.; Fink, M.; and Senko, J. 2005. Updat-ing action domain descriptions. In Proc. IJCAI, 418423.

Gardenfors, P. 1988. Knowledge in Flux: Modeling theDynamics of Epistemic States. MIT Press.

Harel, D.; Tiuryn, J.; and Kozen, D. 2000. Dynamic Logic.MIT Press.

Herzig, A., and Rifi, O. 1999. Propositional beliefbase update and minimal change. Artificial Intelligence115(1):107138.

Herzig, A., and Varzinczak, I. 2005. On the modularity oftheories. In Advances in Modal Logic, volume 5. KingsCollege Publications. 93109.

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

Herzig, A.; Perrussel, L.; and Varzinczak, I. 2006. Elabo-rating domain descriptions. In Proc. ECAI, 397401.

Katsuno, H., and Mendelzon, A. 1992. On the differencebetween updating a knowledge base and revising it. In Be-lief revision. Cambridge. 183203.

Kracht, M., and Wolter, F. 1991. Properties of indepen-dently axiomatizable bimodal logics. J. of Symbolic Logic56(4):14691485.

Parikh, R. 1999. Beliefs, belief revision, and splitting lan-guages. In Logic, Language and Computation, 266278.

Quine, W. V. O. 1952. The problem of simplifying truthfunctions. American Mathematical Monthly 59:521531.

Varzinczak, I. 2008a. Action theory contraction and mini-mal change. To appear in Proc. KR 2008.

Varzinczak, I. 2008b. Action theory revision. TechnicalReport IRIT/RT2008-1FR, IRIT, Toulouse.

Winslett, M.-A. 1988. Reasoning about action using apossible models approach. In Proc. AAAI, 8993.

Proof of Theorem 3

Let be a law, M M, and let T be the output of ouralgorithms on input theory Tand law .

IfT {} |=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 whereW = (W\ val()) val() is minimal w.r.t. Wand R Ris maximal w.r.t. R, for some M = W,R M.


8/9

As we have assumed the syntactical classical revision op-erator is sound and 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

w

iA , 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) =, and then S E |=

PDL( A) [a]. Because

T |=PDL

( A) a, and S |=CPL

( A) andS |=

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

PDL(A),

and then (i A) a / T. Hence R

a(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. ThenM = W,R for some M = W,R M s.t. W = Wand R R, where R is maximal w.r.t. R.

From W = W, |=M

S. As R R, |=M

E. BecauseS E T [a], it suffices 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 (i A) a T [a], then for every

w W, if w i A, 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) = .

Let (i T|=PDL

(A)a( A)) a

T[a]. For every w 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) = . ThenS, 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 = Wand R = R R,a , with

R,a = {(w, w) : w RelTgt(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 asS X T a, it is 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

wi for all

w Ws.t. (w, w) Ra, and then |=M

wi for every w

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

Ta. 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

Ws.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. Other-wise, if w , then T |=

PDL( A) [a], and then

(i T|=PDL

(A)[a] ( A)) [a]i has not beenput in Ta, a contradiction.

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

w W, if|=M

w A , then |=

M

w, and then |=

M

w. From

( A ) [a] Ta, we have T |=PDL [a]

or T |=PDL

[a] and T |=PDL

[a]. In both cases,

|=M

w 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 T

a, a contradiction.

Proof of Theorem 4

Lemma 1 Let be a law. IfTis modular andT {} |=PDL

, then T is modular.

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

and S |=CPL

, where

S is static laws in 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. Then T |=PDL

if and only if

S |=CPL

, a contradiction.

Suppose is of the form a, for Fml. Thenfor all -contexts such that T |=PDL(

) a,

T |=PDL ( ) [a] is impossible as far as Ea has


9/9

been weakened. Then T |=PDL if and only ifS |=

CPL, a

contradiction.Hence we have T |=

PDL. Because is nonclassical,

S = S. Then T |=PDL

and S |=CPL

, and hence Tis notmodular.

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 , we must have T |=PDL

[a] andT |=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.

If v / val(S), then T |=PDL a, as no ex-

ecutability for context has been put into T. HenceT |=PDL

, a contradiction.

Lemma 2 If Mbig = Wbig,Rbig is a model of T , thenfor every M = W,R such that |=

MTthere is a mini-

mal (w.r.t. set inclusion) extension R Rbig \ R such thatM = val(S),R R is a model ofT.

Proof: See (Varzinczak 2008b).

Lemma 3 LetTbe modular, and be a law. Then T |=PDL

if and only if every M = 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 |=M

T, in particular for those that are extensions

of some model ofT.

(): 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 2 thereis 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 Fml or an effect

law, any extension 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 Wget a newleaving arrow. Thus (R R)a(w) = , and then |=

M

w.

Lemma 4 Let T be modular and a law. If M =

val(S),R is a model ofT, 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 eachcase.

Let be of the form [a], for , Fml. LetM = {M : M = val(S),R}. As Tis modular, byLemmas 2 and 3, M is non-empty and contains only modelsofT.

SupposeM isnot a minimal model ofT[a], i.e., thereis M such that M M M for some M M. ThenM andM differ only in the effect ofa 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 isminimal w.r.t. M.

When revising by an effect law, S = S. Hence tak-ing 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 2and 3, M is non-empty and contains only models ofT.Suppose thatM is not a minimal model ofTa, 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 exe-cutability ofa in a given -world, i.e., a A-context, forsome IP(S ) and A =

piatm()

piA

pi

piatm()pi /A

pi,

such that A atm(). This means M has no arrow leav-

ing this A-world. Then |=M

( A) [a], and

hence |=M

a. Hence M is a minimal model ofTa w.r.t. M.

When revising by executability laws, S

= S. Thustaking 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 RelTgt(w, a,M, M)},

we get M M and then M Ma.

Finally, let be some Fml. Then M is such thatfor every w W, if Ra(w) = , then w val(S) and

Ra(w) = for every M = W,R M. Choosing theright M M the result follows.

Proof of Theorem 4

Let T be the output of our algorithms on input theory T

and law . IfT

= T {}, then T {} |=PDL, and henceeveryM such that |=

M

T is such that M M\{M :|=

M

} and the result follows.

Suppose T {} |=PDL

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

action theory revision in dynamic logic

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