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Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence (IJCAI 2015) Characterizing Causal Action Theories and Their Implementations in Answer Set Programming: Action Languages B, C and Beyond∗ Haodi Zhang and Fangzhen Lin Department of Computer Science and Engineering Hong Kong University of Science and Technology Clearwater Bay, Kowloon, Hong Kong Abstract have been proposed. These different approaches basically all agree when the set of causal rules is stratified, and in this case We consider a simple language for writing causal yields a complete action theory that can be represented, for action theories, and postulate several properties for example, by a set of successor state axioms. However, when the state transition models of these theories. We there are cycles in the rules, it is not always clear how these then consider some possible embeddings of these rules are going to be represented according to these different causal action theories in some other action for- approaches, and what the correct results are supposed to be. malisms, and their implementations in logic pro- For instance, do we allow cyclic rules to produce indetermi- grams with answer set semantics. In particular, we nate effects on actions? propose to consider what we call permissible trans- This motivated us to do a computer experiment that would lations from these causal action theories to logic systematically enumerate all possible causal theories in a programs. We identify two sets of properties, and small language, and look at their desirable models. As it prove that for each set, there is only one permissi- turned out, what counts as a desirable model depends on what ble translation, under strong equivalence, that can properties we want to have about causal theories. This led us satisfy all properties in the set. As it turns out, for to consider various properties of transition models of causal one set, the unique permissible translation is essen- theories. We then consider how these various properties fit tially the same as Balduccini and Gelfond’s transla- into some existing action languages that allow static causal tion from Gelfond and Lifschitz’s action language rules. To make the comparisons systematic, and also for B to logic programs. For the other, it is essen- computational reasons, we consider what we call permissible tially the same as Lifschitz and Turner’s translation translations of our causal action theories to logic programs C from the action language to logic programs. This under the answer set semantics [Gelfond and Lifschitz, 1988; work provides a new perspective on understanding, 1991]. Specifically, given a set of properties on the transi- evaluating and comparing action languages by us- tion models, we consider all possible permissible translations ing sets of properties instead of examples. It will from our causal theories to logic programs so that the answer be interesting to see if other action languages can sets of these programs, when mapped back to the transition be similarly characterized, and whether new action models of the causal theories, will satisfy all properties in the formalisms can be defined using different sets of set. In this paper, we identify two such sets of properties. properties. For both of them, our computer experiment shows that when there are at most three fluents in the language, the permissible 1 Introduction translations are unique up to a notion of strong equivalence. The general case is proved by induction with the three fluent Formal reasoning about action has been a central topic in language as the base case. logic-based AI for a long time, and motivated much of the Furthermore, for one set of properties, the translation is early work on nonmonotonic logics. The main difficulties essentially the same as Balduccini and Gelfond’s translation have been the frame and the ramification problems. Current from Gelfond and Lifschitz’s action language B to logic pro- consensus in the community is that to solve the ramification grams. For the other, the translation is essentially the same as problem, a notion of causality is needed. As a result, there has Lifschitz and Turner’s translation from the action language C [ been much work on causal action theories (e.g. Lifschitz, to logic programs. These results are significant in that they 1987; Lin, 1995; McCain and Turner, 1995; Baral, 1995; provide a new perspective on understanding, evaluating and Thielscher, 1997; Lifschitz, 1997; Turner, 1999; Gelfond and comparing action languages by using sets of properties in- Lifschitz, 1998; Lin, 2003; Herzig and Varzinczak, 2007; stead of examples. It is possible that other action languages et al. ] Lee , 2013 ), and a variety of languages and semantics can be similarly characterized, and new action languages de- ∗This work was supported in part by HK RGC under GRF fined using different sets of properties. 616013. The rest of the paper is organized as follows. In Section 3285 2 we formally introduce the syntax of the causal action the- s to yield s0. ories that we use in this work. Next we list some reasonable Definition 1 A state is a set of fluent atoms, and a transition properties that we expect the semantics of the causal action is a pair of states. A semantic function δ is a mapping from theories to satisfy in Section 3. Then in Section 4 we consider causal action theories to sets of transitions. some straightforward mappings of our theories to some exist- ing formalisms such as the action languages B [Gelfond and Thus a semantic function δ gives a semantics to each causal Lifschitz, 1998] and C [Giunchiglia and Lifschitz, 1998], and action theory. We say that two causal action theories T1 and the situation calculus. In Section 5 we consider the problem T2 are equivalent under δ if they have the same transactions: of mapping causal action theories to logic programs under the δ(T1) = δ(T2). In the next section, we are going to discuss answer set semantics, define the notion of permissible trans- some properties about semantic functions. lations, and give our main results. In Section 6 we discuss some related work, and finally in Section 7 we conclude the 3 Properties paper. We assume a fixed semantic function δ below. Thus when we say that (s; s0) is a transition of T , we mean that (s; s0) 2 2 Simple causal action theories δ(T ). We list below some interesting properties about δ. When reasoning about, say whether a switch is open or We assume a finite set F of propositional atoms called flu- closed, we can use fluent closed to mean that the switch ents. We also assume two distinguished symbols “>” for tau- is closed and represent the fact that the switch is open by tology, and “?” for contradiction. A fluent literal is either f :closed. Or we can do it the other way around, use open to or :f where f 2 F. So far in work on causal action theory, mean that the switch is open and represent closed by :open. the focus is on the formalization of the effects of primitive ac- Our following property says that choosing which one to use tions, and how the causal rules are used in this formalization. as primitive is immaterial as long as one does this systemati- The actions are assumed to be independent, in the sense that cally. the effects of one action are independent of the effects of any For any other actions. So to make our formalism as simple and to the Property 1 (Fluent literals are interchangeable) causal action theory T , any fluent f, and any pair of states s1 point as possible, we assume that there is just one unnamed f f action in our language, and when we talk about the effect of and s2, (s1; s2) is a transition of T , iff (s1 ; s2 ) is a transition an action, we refer to the effect of this implicitly assumed, of T f , where for any state s, unnamed action. s n ffg if f 2 s sf = Syntactically, a causal action theory is a pair (S; D), where s [ ffg otherwise S is a set of static causal rules, and D a set of dynamic causal rules. Both static and dynamic causal rules are pairs of the and for any causal action theory T , T f is the causal action form (l; G), where l is a fluent literal, and G a set of fluent theory obtained from T by replacing every occurrence of f literals. As a static causal rule, (l; G) means that in every by :f. situation, whenever all fluent literals in G hold, l is caused Our next property is that the states in a transition must sat- to be true. As a dynamic causal rule, it means that in every isfy all static causal rules. situation where all fluent literals in G hold, if the action is successfully executed, then l will be true in the new situation. Property 2 (Static causal rules are state constraints) If 0 Thus our dynamic causal rules are essentially direct action (s; s ) is a transition of a causal action theory (S; D), then effect axioms. Notice that in the dynamic causal rules, action for every static causal rule (l; G) in S, we have that argument is omitted here as we have assumed that there is just ^ one action. In the following, we call l the head, and G the s j= (l _: li) premise of the static or dynamic causal rule. We assume that li2G G is consistent, i.e. it does not contain both f and :f, and and 0 ^ does not contain ?.

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