Elaborating Domain Descriptions

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Elaborating Domain Descriptions

  1. 1. Elaborating domain descriptions Andreas Herzig and Laurent Perrussel and Ivan Varzinczak 1 Abstract. In this work we address the problem of elaborating do- Such cases of theory change are very important when one deals main descriptions (alias action theories), in particular those that are with logical descriptions of dynamic domains: it may always hap- expressed in dynamic logic. We define a general method based on pen that one discovers that an action actually has a behavior that is contraction of formulas in a version of propositional dynamic logic different from that one has always believed it had. with a solution to the frame problem. We present the semantics of Up to now, theory change has been studied mainly for knowledge our theory change and define syntactical operators for contracting a bases in classical logics, both in terms of revision and update. Only domain description. We establish soundness and completeness of the in a few recent works it has been considered in the realm of modal operators w.r.t. the semantics for descriptions that satisfy a principle logics, viz. in epistemic logic [12] and in action languages [7]. Re- of modularity that we have defined in previous work. cently, several works [31, 21] have investigated revision of beliefs about facts of the world. In our examples, this would concern e.g. 1 INTRODUCTION the current status of the switch: the agent believes it is up, but is wrong about this and might subsequently be forced to revise her be- Suppose a situation where an agent has always believed that if the liefs about the current state of affairs. Such revision operations do not light switch is up, then the room is light. Suppose now that someday, modify the agent’s beliefs about the action laws. In opposition to that, she observes that even if the switch is up, the light is off. In such a here we are interested exactly in such modifications. The aim of this case, the agent must change her beliefs about the relation between the paper is to make a step toward that issue and propose a framework propositions “the switch is up” and “the light is on”. This is an exam- that deals with the contraction of action theories. ple of changing propositional belief bases and is largely addressed in Propositional dynamic logic (PDL [13]), has been extensively the literature about belief change [10] and update [24]. used in reasoning about actions in the last years [2, 36, 8]. It has Next, let our agent believe that whenever the switch is down, after shown to be a viable alternative to situation calculus approaches be- toggling it, the room is light. This means that if the light is off, in cause of its simplicity and existence of proof procedures for it. In every state of the world that follows the execution of toggling the this work we investigate the elaboration of domain descriptions en- switch, the room is lit up. Then, during a blackout, the agent toggles coded in a simplified version of such a logical formalism, viz. the the switch and surprisingly the room is still dark. multimodal logic Kn . We show how a theory expressed in terms of Imagine now that the agent never worried about the relation be- static laws, effect laws and executability laws is elaborated: usually, tween toggling the switch and the material it is made of, in the sense a law has to be changed due to its generality, i.e., the law is too strong that she ever believed that just toggling the switch does not break and has to be weakened. It follows that elaborating an action theory it. Nevertheless, in a stressful day, she toggles the switch and then means contracting it by static, effect or executability laws, before ex- observes that she had broken it. panding the theory with more specific laws. Completing the wayside cross our agent experiments in discov- ering the world’s behavior, suppose she believed that it is always possible to toggle the switch, given some conditions e.g. being close 2 BACKGROUND enough to it, having a free hand, the switch is not broken, etc. How- Following the tradition in the reasoning about actions community, ever, in an April fool’s day, she discovers that someone has glued the action theories are collections of statements of the form: “if context, switch and, consequently, it is no longer possible to toggle it. then effect after every execution of action” (effect laws); and “if pre- The last three examples illustrate situations where changing the condition, then action executable” (executability laws). Statements beliefs about the behavior of the action of toggling the switch is mentioning no action at all represent laws about the world (static mandatory. In the first one, toggling the switch, once believed to be laws). Besides that, statements of the form “if context, then effect af- deterministic, has now to be seen as nondeterministic, or alternatively ter some execution of action” will be used as a causal notion to solve to have a different outcome in a specific context (e.g. if the power the frame and the ramification problems. station is overloaded). In the second example, toggling the switch is known to have side-effects (ramifications) one was not aware of. In the last example, the executability of the action under concern is 2.1 Logical preliminaries questioned in the light of new information showing a context that was Let Act a fa1 Y a2 Y X X Xg be the set of all atomic actions of a given not known to preclude its execution. Carrying out such modifications domain, an example of which is toggle. To each atomic action a there is what we here call elaborating a domain description, which has to is associated a modal operator ‘a“. Prop a fp1 Y p2 Y X X Xg denotes do with the principle of elaboration tolerance [28]. all the propositional constants (alias fluents or atoms). Examples of 1 The authors are with the Institut de Recherche en Informatique de Toulouse those are light (“the light is on”) and up (“the switch is up”). The set f g (IRIT), Toulouse, France. e-mail: herzig, perrusse, ivan @irit.fr of all literals is Lit a Prop ‘ fXp X p P Propg.
  2. 2. Fml is the set of all classical formulas. They are denoted by small 3 MODELS OF CONTRACTION Greek letters 9Y 2Y X X X An example is up 3 light. By val@9A we denote the set of valuations making 9 true. We view a valuation When an action theory has to be changed, the basic operation is that as a maximally-consistent set of literals. For Prop a flightY upg, of contraction. (In belief-base update [33, 24] it has also been called there are four valuations: flightY upg, flightY Xupg, fXlightY upg erasure.) In this section we define its semantics. and fXlightY Xupg. Given a set of formulas ¦, by lit@¦A we denote In general we might contract by any formula ¨. Here we focus on the set of all literals appearing in formulas of ¦. contraction by one of the three kinds of laws. We therefore suppose We denote complex formulas (with modal operators) by ¨Y ©Y X X X that ¨ is either 9, where 9 is classical, or 9 3 ‘a“2 , or 9 3 haib. hai is the dual operator of ‘a“, defined as hai¨ adef X‘a“X¨. An For the case of contracting static laws we resort to existing ap- example of a complex formula is Xup 3 ‘toggle“up. The semantics proaches to change the set of static laws. In the following, we con- sider any belief change operator like Forbus’ update method [9], the M is that of multimodal logic K [29]. A Kn -model is a tuple a hW Y R i where W is a set of valuations, possible models approach [33, 34], WSS [14] or MPMA [6]. Contraction by 9 corresponds to adding new possible worlds to M and R a function mapping action constants a to accessibility relations W ¢ W. Given a Kn -model M a hW Y R i, ja p (p is true at W. Let © be a given contraction operator for classical logic. Ra M w world w of model M ) if p P w; ja ‘a“¨ if for every wH such that Definition 4 Let hW Y R i be a Kn -model and 9 a classical formula. M w wRa wH , ja H ¨. Truth conditions for other connectives are as usual. The model resulting from contracting by 9 is hW Y R i  a fhWH Y Rig 9 such that WH a W © val@9A. M is a model of ¨ (noted jaM ¨) if for all w P M jaM ¨. M w is a model of a set of formulas ¦ (noted ja M ¦) if jaW, ¨wfor every Observe that R should, a priori, change as well, otherwise con- tracting a classical formula may conflict with ˆ .2 For instance, if ¨ P ¦. ¨ is a consequence of the set of global axioms   in the class M of all Kn -models (noted   ja n ¨) if for every Kn -model M , ja   X9 3 haib P ˆ and we contract by 9, the result may make ˆ M implies ja ¨. K untrue. However, given the amount of information we have at hand, we think that whatever we do with R (adding or removing edges), we will always be able to find a counter-example to the intuitiveness of the operation, since it is domain dependent. For instance, adding 2.2 Describing the behavior of actions in Kn edges for a deterministic action may render it nondeterministic. De- ciding on what changes to carry out on R when contracting static Kn allows for the representation of statements describing the behav- laws depends on the user’s intuition, and unfortunately this informa- ior of actions. They are called action laws. Here we distinguish sev- tion cannot be generalized and established once for all. We here opt eral types of them. The first kind of statement represents the static for a priori doing nothing with R and postponing correction of exe- laws, formulas that must hold in every possible state of the world. cutability laws. Action theories being defined in terms of effect and executability Definition 1 A static law is a formula 9 P Fml. laws, elaborating an action theory will mainly involve changes in these two sets of laws. Let us consider now both these cases. Suppose the knowledge engineer acquires new information re- An example of a static law is up 3 light: if the switch is up, then the garding the effect of action a. Then it means that the law under con- light is on. ƒ Fml denotes all the static laws of a domain. sideration is probably too strong, i.e., the expected effect may not The second kind of law we consider are the effect laws. They are occur and thus the law has to be weakened. Consider e.g. Xup 3 formulas relating an action to its effects, which can be conditional. ‘toggle“light, and suppose it has to be weakened to the more specific @Xup ” XblackoutA 3 ‘toggle“light.3 In order to carry out such a Definition 2 An effect law for action a has the form 9 3 ‘a“2, weakening, first the designer has to contract the set of effect laws where 9Y 2 P Fml. and second to expand the resulting set with the weakened law. Contraction by 9 3 ‘a“2 amounts to adding some ‘counterex- ample’ arrows from 9-worlds to X2 -worlds. To ease such a task, The consequent 2 is the effect always obtained when a is executed we need a definition. Let PI@9A denote the set of prime implicates in a state where the antecedent 9 holds. i denotes the set of all ef- of 9. If 91 Y 92 P Fml, NewCons91 @92 A a PI@91 ” 92 A n PI@91 A fect laws of a domain, an example of which is Xup 3 ‘toggle“light: computes the new consequences of 92 w.r.t. 91 : the set of strongest clauses that follow from 91 ” 92 , but do not follow from 91 alone whenever the switch is down, after toggling it, the room is lit up. If 2 is inconsistent, we have a special kind of effect law that we call an inexecutability law. For example, broken 3 ‘toggle“c says that (cf. e.g. [20]). For example, the set of prime implicates of p1 is just fp1 g, that of p1 ” @Xp1 • p2 A ” @Xp1 • p3 • p4 A is fp1 Y p2 Y p3 • p4 g, hence NewConsp1 @@Xp1 • p2 A ” @Xp1 • p3 • p4 AA a fp2 Y p3 • p4 g. toggle cannot be executed if the switch is broken. Finally, we also define executability laws, which stipulate the con- text for an action to be executable. In Kn , we use hai to express Definition 5 Let hW Y R i be a Kn -model and 9 3 ‘a“2 an effect executability. haib thus reads “the execution of a is possible”. law. The set of models that result from contracting by 9 3 ‘a“2 is hW Y R i 3[a]2 a fhWY R ‘ RHa i X RHa f@wY wH A X jaW YR i 9Y jaWH YR i 9 h w h w Definition 3 An executability law for action a is of the form 93 X2 and w H n w lit@NewConsƒ @X2AAgg. haib, where 9 P Fml. In our context, wH n w lit@NewConsƒ @X2 AA means that for all For instance, Xbroken 3 htoggleib says that toggling can be exe- the added arrows, the new/extra effects of action a are limited to cuted whenever the switch is not broken. The set of all executability 2 We are indebted to the anonymous referees for pointing this out to us. laws of a given domain is denoted by ˆ . : ! 3 Replacing the law by up [toggle](light _: light) looks silly.
  3. 3. the consequences of the static laws combined with X2 , i.e., all the Definition 9 An action theory is a tuple of the form hƒ Y i Y ˆ Y ;i. ramifications that action a can produce. Suppose now the knowledge engineer learns new information In our running example, the corresponding action theory is about the executability of a. This usually occurs when some exe- cutabilities are too strong, i.e., the condition in the theory guarantee- ƒ a fup 3 lightgY i a fXup 3 ‘toggle“upY up 3 ‘toggle“Xupg ˆ a fhtoggleibgY ; a htoggleY lightiiYYhhtoggleYYXlightiY ing the executability of a is too weak and should be made more re- strictive. Let e.g. htoggleib be the law to be contracted, and suppose htoggleY up toggle Xupi it has to be weakened to the more specific Xbroken 3 htoggleib. To do that, the designer first contracts the executability laws and then And we have ƒ Y i Y ˆ ja Xup 3 ‘toggle“light. (For parsimony’s ; expands the resulting set with the weakened law. sake, we write ƒ Y i Y ˆ ja ¨ instead of ƒ ‘ i ‘ ˆ ja ¨.) ; ; Contraction by 9 3 haib corresponds to removing some arrows Let hƒ Y i Y ˆ Y ; i be an action theory and ¨ a Kn -formula. leaving worlds where 9 holds. Removing such arrows has as conse- hƒ Y i Y ˆ Y ; i  denotes the action theory resulting from the contrac- ¨ quence that a is no longer always executable in context 9. tion of hƒ Y i Y ˆ Y ; i by ¨. Contracting a theory by a static law 9 amounts to using any exist- Definition 6 Let hW Y R i be a Kn -model and 9 3 haib an exe- ing contraction operator for classical logic. Let © be such an opera- cutability law. The set of models that result from the contraction by tor. Moreover, based on [19], we also need to guarantee that 9 does 9 3 haib is hW Y R i 3haib a fhWY RH i X RH a R n RHH Y RHH 9 a a ; not follow from i , ˆ and . We define contraction of a domain f@wY wH A X wRa wH and jaW YR i 9gg. h w description by a static law as follows: ; ; Definition 10 hƒ Y i Y ˆ Y i  a hƒ   Y i Y ˆ   Y i, where ƒ   9 a 4 CONTRACTING AN ACTION THEORY ƒ © 9 and ˆ   a f@9i ” 9A 3 haib X 9i 3 haib P ˆ g. Having established the semantics of action theory contraction, we can turn to its syntactical counterpart. Nevertheless, before doing that We now consider contraction by an executability law 9 3 haib. we have to consider an important issue. As the reader might have For every executability in ˆ , we ensure that a is executable only in expected, Kn alone does not solve the frame problem. For instance, contexts where X9 is true. The following operator does the job. up 3 lightY Xup 3 ‘toggle“upY Tja n broken 3 ‘toggle“brokenX Definition 11 hƒ Y i Y ˆ Y ; i 3haib a hƒ Y i Y ˆ   Y ;i, where 9 up 3 ‘toggle“XupY htoggleib K ˆ   a f@9i ” X9A 3 haib X 9i 3 haib P ˆ g. For instance, contracting glued 3 htoggleib in our example would We need thus a consequence relation powerful enough to deal with give us ˆ   a fXglued 3 htoggleibg. the frame and ramification problems. Hence the deductive power of Kn has to be augmented to ensure that all the relevant frame axioms apply. Following the framework developed in [2], we consider meta- Finally, to contract a theory by 9 3 ‘a“2 , for every effect law logical information given in the form of dependence: in i , we first ensure that a still has effect 2 whenever 9 does not hold, second we enforce that a has no effect in context X9 except on those literals that are consequences of X2 . Combining this with the ; Definition 7 (Dependence relation [2]) A dependence relation is a binary relation Act ¢ Lit. new dependence relation also linking a to literals involved by X2 , we have that a may now produce X2 as outcome. In other words, the ; ; The expression a l denotes that the execution of action a may effect law has been contracted. The operator below formalizes this: change the truth value of the literal l. On the other hand, haY li P a ; (written a T l) means that l can never be caused by a. In our example hƒ Y i Y ˆ Y ; i 3[a]2 a hƒ Y i   Y ˆ Y ;  i, where ; ; 9 Definition 12 we have toggle light and toggle Xlight, which means that action toggle may cause a change in literals light and Xlight. We do ; a; ‘fhaY li X l P lit@NewConsƒ @X2AAg and i   a f@9i ” X9A 3 ‘a“2 X 9i 3 ‘a“2 P ig ‘ f@X9 ” XlA 3 ‘a“Xl X haY li P not have toggle ; Xbroken, for toggling the switch never repairs it. @;  n ;Ag. M fXl 3 ‘a“Xl X a T; lg. Definition 8 A model of a dependence relation ; is a Kn-model M For instance, contracting the law blackout 3 ‘toggle“light from our such that ja theory would give us i   a f@Xup ”XblackoutA 3 ‘toggle“upY @up ” ; ; XblackoutA 3 ‘toggle“Xupg. ; We assume is finite. Given a dependence relation , the asso- ciated consequence relation in the set of models for is noted ja . ; For our example we obtain 5 RESULTS up 3 lightY Xup 3 ‘toggle“upY We here present the main results that follow from our framework. up 3 ‘toggle“XupY htoggleib ja broken 3 ‘toggle“brokenX ; These require the action theory under analysis to be modular [19]. ; We have toggle T Xbroken, i.e., Xbroken is never caused by toggle. An action theory is modular if a formula of a given type entailed by the whole theory can also be derived solely from its respective Hence in all contexts where broken is true, after every execution of module (the set of formulas of the same type) with the static laws ƒ . As shown in [19], to make a domain description satisfy such a ; toggle, broken still remains true. This independence means that the frame axiom broken 3 ‘toggle“broken is valid in the models of . property it is enough to guarantee that there is no classical formula Such a dependence-based approach has been shown [5] to sub- entailed by the theory that is not entailed by the static laws alone. sume Reiter’s solution to the frame problem [30] and moreover treats Definition 13 (Implicit static law [17]) 9 P Fml is an implicit ; the ramification problem, even when actions with both indeterminate and indirect effects are involved [3, 16]. static law of hƒ Y i Y ˆ Y i if and only if ƒ Y i Y ˆ ja 9 and ƒ Tja 9. ;
  4. 4. A theory is modular if it has no implicit static laws. Modularity of ; Lemma 2 Let hƒ Y i Y ˆ Y i be modular, and let ¨ be a formula of theories was originally defined in [19], but similar notions have also ; the form of one of the three laws. Then hƒ Y i Y ˆ Y i  is modular. ¨ been addressed in the literature [4, 1, 35, 25]. A modularity-based approach for narrative reasoning about actions is given in [22]. The third one establishes the required link between the contraction To witness how implicit static laws can show up, consider the sim- operators and contraction of ‘big’ models. ple theory below, depicting the walking turkey scenario [32]: ; Lemma 3 Let hƒ Y i Y ˆ Y i be modular, and let ¨ be a formula of ‘tease“walkingY M the form of one of the three laws. If H a hval@ƒ AY RH i is a model of ƒ a fwalking 3 alivegY i a loaded 3 ‘shoot“Xalive ; M ; hƒ YH i Y ˆ Y   i  , then there is a model of hƒ Y i Y ˆ Y i such that M M P ¨. ¨ hteaseibY Y ; a hshootY XloadediY hshootY XaliveiY Putting the three above lemmas together we get: ˆ a hshootib hshootY XwalkingiY hteaseY walkingi With this description we have ƒ Y i Y ˆ ja alive.4 As ƒ Tja alive, the Theorem 2 Let hƒ Y i Y ˆ Y ; i be modular, ¨ be a formula of ; the form of one of the three laws, and hƒ   Y i   Y ˆ   Y ;  i be formula alive is an implicit static law of hƒ Y i Y ˆ Y ; i. hƒ Y i Y ˆ Y ; i  . If ƒ   Y i   Y ˆ   ja   © , then for every model M ¨ ; Modular theories have several advantages [17, 15]. For example, of hƒ Y i Y ˆ Y ; i and every M H P M¨ it holds that ja © .   MH ing consistency of ƒ : if hƒ Y i Y ˆ Y ; i is modular, then ƒ Y i Y ˆ ja consistency of a modular action theory can be checked by just check- ; c if and only if ƒ ja c. Deduction of an effect of a sequence of ac- Our two theorems together establish correctness of the operators: tions a1 Y X X X Y an (prediction) needs not to take into account the effect laws for actions other than a1 Y X X X Y an . This applies in particular to Corollary 1 Let hƒ Y i Y ˆ Y ; i be modular, ¨ be a formula of plan validation when deciding whether ha1 Y X X X Y an i9 is the case. the form of one of the three laws, and hƒ   Y i   Y ˆ   Y ;  i be Throughout this work we use multimodal logic Kn . For an assess- hƒ Y i Y ˆ Y ; i  . Then ƒ   Y i   Y ˆ   ja   © if and only if for every ¨ ; ment of the modularity principle in the Situation Calculus, see [18].   MH model M of hƒ Y i Y ˆ Y ; i and every M H P M¨ , ja © . Here we show that our operators are correct w.r.t. the semantics. ; ; The first theorem establishes that the semantical contraction of mod- We give a sufficient condition for contraction to be successful. els of hƒ Y i Y ˆ Y i by ¨ produces models of hƒ Y i Y ˆ Y i  . ¨ Theorem 3 Let ¨ be an effect or an executability law such that Theorem 1 Let hW Y R i be a model of hƒ Y i Y ˆ Y i, and let ¨ be a ; ; ; ƒ Tja n ¨, and let hƒ   Y i   Y ˆ   Y   i be hƒ Y i Y ˆ Y i  . If ¨ M K ; hƒ Y i Y ˆ Y i is modular, then ƒ   Y i   Y ˆ   Tja   ¨. ; M M ; formula that has the form of one of the three laws. For all models , if P hW Y R i  , then is a model of hƒ Y i Y ˆ Y i  . ¨ ¨ 6 DISCUSSION AND CONCLUSION hƒ Y i Y ˆ Y ; i  result from the semantical contraction of models of It remains to prove that the other way round, the models of ¨ hƒ Y i Y ˆ Y ; i by ¨. This does not hold in general, as shown by the In this work we have presented a general method for changing a do- main description (alias action theory) given any formula we want ; following example: suppose there is only one atom p and one ac- to contract. We have defined a semantics for theory contraction and tion a, and consider the theory hƒ Y i Y ˆ Y i such that ƒ a Y, i a fp 3 ‘a“cg, ˆ a fhaibg, and a Y. The only model of ; also presented its syntactical counterpart through contraction opera- M a hffXpggY f@fXpgY fXpgAgi. By defini- tors. Soundness and completeness of such operators with respect to M M ; that action theory is tion, p  aib a f g. On the other hand, hƒ Y i Y ˆ Y i  aib a the semantics have been established (Corollary 1). 3h p3h We have also shown that modularity is a sufficient condition for a hYY fp 3 ‘a“cgY fXHp 3 haibgY Yi. The contracted theory has two M M a hffpgY fXpggY @fXpgY fXpgAi. While Xp is contraction to be successful (Theorem 3). This gives further evidence ; models: and that the notion of modularity is fruitful. valid in the contraction of the models of hƒ Y i Y ˆ Y i, it is invalid in the models of hƒ Y i Y ˆ Y i  aib . p3h ; What is the status of the AGM-postulates for contraction in our framework? First, contraction of static laws satisfies all the postu- Fortunately, we can establish a result for those action theories that lates, as soon as the underlying classical contraction operation © sat- are modular. The proof requires three lemmas. The first one says that isfies all of them. for a modular theory we can restrict our attention to its ‘big’ models. ; ; In the general case, however, our constructions do not satisfy the central postulate of preservation hƒ Y i Y ˆ Y i  a hƒ Y i Y ˆ Y i if Lemma 1 Let hƒ Y i Y ˆ Y ;i be modular. Then ƒY i Y ˆ ja; ¨ if and ¨ ƒ Y i Y ˆ Tja ¨. Indeed, suppose we have only one atom p, and a ; only if ja h Yi ¨ for every model hW Y R i of hƒ Y i Y ˆ Y ; i such that W R model M with two worlds w a fpg and wH a fXpg such that M M W a val@ƒ A. wRa wH , wH Ra w, and wH Ra wH . Then ja p 3 ‘a“Xp and Tja ‘a“Xp, i.e., M is a model of the effect law p 3 ‘a“Xp, but not of ‘a“Xp. Now the contraction M  X yields the model M H such that R a ; Note that the lemma does not hold for non-modular theories (the set fhW Y R i X hW Y R i is a model of hƒ Y i Y ˆ Y i and W a val@ƒ Ag is M p 3] ‘a“Xp, i.e., the effect law p 3 ‘a“Xp is [ a H a p empty then). W ¢ W. Then Tja not preserved. Our contraction operation thus behaves rather like an The second lemma says that contraction preserves modularity. update operation. f 4 Because first walking! alive tease walkingg j ; tease alive, second ;[ ] = [ ] Now let us focus on the other postulates. Since our operator has j ; :alive ! tease :alive (from the independence tease 6; alive), and = [ ] a behavior which is close to the update postulate, we focus on the then S E j ; :alive ! tease ?. As long as S E X j ; hteasei, we = [ ] = following basic erasure postulates introduced in [23]. Let Cn@„ A be must have S E X j ; alive. ; ; ; ; ; = the set of all logical consequences of a theory „ .
  5. 5. KM1 Cn@hƒ Y i Y ˆ Y ;i A Cn@hƒY i Y ˆ Y ;iA ¨ [4] L. Cholvy, ‘Checking regulation consistency by using SOL-resolution’, in Proc. Int. Conf. on AI and Law, pp. 73–79, Oslo, (1999). Postulate KM1 does not always hold because it is possible to make [5] R. Demolombe, A. Herzig, and I. Varzinczak, ‘Regression in modal the formula 9 3 ‘a“c valid in the resulting theory by removing logic’, J. of Applied Non-Classical Logics, 13(2), 165–185, (2003). [6] P. Doherty, W. Łukaszewicz, and E. Madalinska-Bugaj, ‘The PMA and elements of Ra (cf. Definition 6). relativizing change for action update’, in Proc. KR’98, pp. 258–269. KM2 ¨ P Cn@hƒ Y i Y ˆ Y a ; i  A ¨ Morgan, (1998). [7] T. Eiter, E. Erdem, M. Fink, and J. Senko, ‘Updating action domain Under the condition that hƒ Y i Y ˆ Y ;i is modular, Postulate KM2 is descriptions’, in Proc. IJCAI’05, pp. 418–423. Morgan, (2005). [8] N. Y. Foo and D. Zhang, ‘Dealing with the ramification problem in the satisfied (cf. Theorem 3). extended propositional dynamic logic’, in Advances in Modal Logic, ; ; volume 3, 173–191, World Scientific, (2002). KM3 If Cn@hƒ1 Y i1 Y ˆ1 Y 1 iA a Cn@hƒ2 Y i2 Y ˆ2 Y 2 iA [9] K. D. Forbus, ‘Introducing actions into qualitative simulation’, in Proc. and ja n ¨1 6 ¨2 , then Cn@hƒ1 Y i1 Y ˆ1 Y 1 i 2 A a K ¨ ; IJCAI’89, pp. 1273–1278. Morgan, (1989). ; Cn@hƒ2 Y i2 Y ˆ2 Y 2 i 1 A. ¨ [10] P. G¨ rdenfors, Knowledge in Flux: Modeling the Dynamics of Epis- a temic States, MIT Press, 1988. ; ; [11] M. Gelfond and V. Lifschitz, ‘Action languages’, ETAI, 2(3–4), 193– Theorem 4 If hƒ1 Y i1 Y ˆ1 Y 1 i and hƒ2 Y i2 Y ˆ2 Y 2 i are modu- 210, (1998). lar and the propositional contraction operator © satisfies Postu- [12] S. O. Hansson, A Textbook of Belief Dynamics: Theory Change and late KM3, then Postulate KM3 is satisfied for every ¨1 Y ¨2 P Fml. Database Updating, Kluwer, 1999. [13] D. Harel, ‘Dynamic logic’, in Handbook of Philosophical Logic, vol- ume II, 497–604, D. Reidel, Dordrecht, (1984). Following [26, 27], Eiter et al. [7] have investigated update of ac- [14] A. Herzig and O. Rifi, ‘Propositional belief base update and minimal tion theories in a fragment of the action description language g [11] change’, Artificial Intelligence, 115(1), 107–138, (1999). and given complexity results showing how hard such a task can be. [15] A. Herzig and I. Varzinczak. Metatheory of actions: beyond consis- tency. To appear. Update of action descriptions in their sense is always relative to [16] A. Herzig and I. Varzinczak, ‘An assessment of actions with indetermi- some conditions (interpreted as knowledge possibly obtained from nate and indirect effects in some causal approaches’, Technical Report earlier observations and that should be kept). This characterizes a 2004–08–R, IRIT – Universit´ Paul Sabatier, (May 2004). e constraint-based update, like ours. [17] A. Herzig and I. Varzinczak, ‘Domain descriptions should be modular’, Even though they do not explicitly state postulates for their kind in Proc. ECAI’04, pp. 348–352. IOS Press, (2004). [18] A. Herzig and I. Varzinczak, ‘Cohesion, coupling and the meta-theory of theory update, they establish conditions for the update operator to of actions’, in Proc. IJCAI’05, pp. 442–447. Morgan, (2005). be successful. Basically, they claim for consistency of the resulting [19] A. Herzig and I. Varzinczak, ‘On the modularity of theories’, in Ad- theory; maintenance of the new knowledge and the invariable part of vances in Modal Logic, volume 5, 93–109, King’s College, (2005). the description; satisfaction of the constraints; and minimal change. [20] K. Inoue, ‘Linear resolution for consequence finding’, Artificial Intelli- gence, 56(2–3), 301–353, (1992). With their method we can also contract by a static and an effect [21] Y. Jin and M. Thielscher, ‘Iterated belief revision, revised’, in Proc. law. Contraction of executabilities are not explicitly addressed. M E IJCAI’05, pp. 478–483. Morgan, (2005). A main difference between the approach in [7] and ours is that [22] A. Kakas, L. Michael, and R. Miller, ‘ odular- : an elaboration tol- we do not need to add new fluents at every elaboration stage: we erant approach to the ramification and qualification problems’, in Proc. still work on the same set of fluents, refining their behavior w.r.t. LPNR’05, pp. 211–226. Springer, (2005). [23] H. Katsuno and A. O. Mendelzon, ‘Propositional knowledge base re- an action a. In Eiter et al.’s proposal an update forces changing all vision and minimal change’, Artificial Intelligence, 52(3), 263–294, the variable rules appearing in the action theory by adding to each (1991). one a new update fluent. This is a constraint when elaborating action [24] H. Katsuno and A. O. Mendelzon, ‘On the difference between updating theories. a knowledge base and revising it’, in Belief revision, ed., P. G¨ rdenfors, a 183–203, Cambridge, (1992). Here we have presented the case for contraction, but our defini- [25] J. Lang, F. Lin, and P. Marquis, ‘Causal theories of action – a computa- tions can be extended to revision, too. Our results can also be gener- tional core’, in Proc. IJCAI’03, pp. 1073–1078. Morgan, (2003). alized to the case where learning new actions or fluents is involved. [26] R. Li and L.M. Pereira, ‘What is believed is what is explained’, in Proc. AAAI’96, pp. 550–555. AAAI Press/MIT Press, (1996). This means in general that more than one simple formula should be [27] P. Liberatore, ‘A framework for belief update’, in Proc. JELIA’2000, added to the belief base and must fit together with the rest of the the- pp. 361–375, (2000). ory with as little side-effects as possible. We are currently defining [28] J. McCarthy, Mathematical logic in artificial intelligence, Daedalus, algorithms based on our operators to achieve that. 1988. [29] S. Popkorn, First Steps in Modal Logic, Cambridge, 1994. ACKNOWLEDGEMENTS [30] R. Reiter, ‘The frame problem in the situation calculus: A simple so- lution (sometimes) and a completeness result for goal regression’, in We are grateful to the anonymous referees for useful comments on Artificial Intelligence and Mathematical Theory of Computation: Pa- pers in Honor of John McCarthy, 359–380, Academic Press, (1991). an earlier version of this paper. Ivan Varzinczak has been supported [31] S. Shapiro, M. Pagnucco, Y. Lesp´ rance, and H. J. Levesque, ‘Iterated e by a fellowship from the government of the F EDERATIVE R EPUBLIC belief change in the situation calculus’, in Proc. KR’2000, pp. 527–538. OF B RAZIL . Grant: CAPES BEX 1389/01-7. Morgan, (2000). [32] M. Thielscher, ‘Computing ramifications by postprocessing’, in Proc. REFERENCES IJCAI’95, pp. 1994–2000. Morgan, (1995). [33] M.-A. Winslett, ‘Reasoning about action using a possible models ap- [1] E. Amir, ‘(De)composition of situation calculus theories’, in Proc. proach’, in Proc. AAAI’88, pp. 89–93. Morgan, (1988). AAAI’2000, pp. 456–463. AAAI Press/MIT Press, (2000). [34] M.-A. Winslett, ‘Updating logical databases’, in Handbook of Logic in [2] M. A. Castilho, O. Gasquet, and A. Herzig, ‘Formalizing action and AI and Logic Programming, volume 4, 133–174, Oxford, (1995). change in modal logic I: the frame problem’, J. of Logic and Computa- [35] D. Zhang, S. Chopra, and N. Y. Foo, ‘Consistency of action descrip- tion, 9(5), 701–735, (1999). tions’, in Proc. PRICAI’02. Springer, (2002). [3] M. A. Castilho, A. Herzig, and I. J. Varzinczak, ‘It depends on the con- [36] D. Zhang and N. Y. Foo, ‘EPDL: A logic for causal reasoning’, in Proc. text! a decidable logic of actions and plans based on a ternary depen- IJCAI’01, pp. 131–138. Morgan, (2001). dence relation’, in NMR’02, pp. 343–348, (2002).

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