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Incomplete Causal Laws in the Situation Calculus Using Free Fluents

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Incomplete Causal Laws in the Situation Calculus Using Free Fluents Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI-16) Incomplete Causal Laws in the Situation Calculus Using Free Fluents Marcelo Arenas, Jorge A. Baier, Juan S. Navarro Sebastian Sardina Universidad Catolica´ de Chile RMIT University, Australia marenas, jabaier, juansnn @ing.puc.cl [email protected] { } Abstract theories (GATs) [De Giacomo and Levesque, 1999], allow both the representation of the occlusion principle [Sandewall, We propose a simple relaxation of Reiter’s basic 1994] and non-deterministic actions. An important limitation action theories, based on fluents without succes- of GATs is that they do not come with a complete regres- sor state axioms, that accommodates incomplete- sion procedure. The Probabilistic Situation Calculus [Pinto ness beyond the initial database. We prove that fun- et al., 2000] is major variant of Reiter’s BATs which can rep- damental results about basic action theories can be resent non-deterministic and probabilistic actions, requiring fully recovered and that the generalized framework different foundational axioms, a different meta-theory, and allows for natural specifications of various forms of a re-work of important theorems. Finally, Delgrande and incomplete causal laws. We illustrate this by show- Levesque [2013] propose treatment a for non-deterministic ing how the evolution of incomplete databases, actions that builds on an epistemic view of the Situation Cal- guarded action theories, and non-deterministic ac- culus, which also requires different foundational axioms. tions can be conveniently specified. In this work, we show how non-determinism in causal laws can be accommodated in the Situation Calculus by simply al- 1 Introduction lowing some fluents to have no successor state axiom. We The situation calculus [McCarthy and Hayes, 1969; Reiter, call such fluents free fluents, which are analogous to “deter- [ ] 2001] is a popular and well-established second-order logical mining fluents” in the Event Calculus Shanahan, 1999 . Our formalism for representing and reasoning about dynamic sys- approach, which is a small modification to the simplest form tems. Basic action theories (BATs) [Reiter, 1991] are axiom- of the Situation Calculus, does not need any changes to foun- atizations in the situation calculus of a certain shape describ- dational axioms. Moreover, it allows the modeling of a range ing how the world evolves under the effects of actions. Im- of applications, such as incomplete databases, guarded action portantly, BATs provide a parsimonious and effective way to theories, and non-deterministic actions. solve the frame problem within classical logic, a problem that Below we prove that fundamental theorems on BATs can resisted solution for decades [McCarthy and Hayes, 1969; be fully recovered in our relaxation. Specifically, relative sat- Shanahan, 1997]. This type of theories have been studied isfiability and the regression theorem still apply in our gener- at depth, and have been shown elaboration tolerant via mul- alized theories. Then we show how free fluents can be applied tiple extensions, such as time [Pinto, 1994], high-level pro- to model non-deterministic actions and databases with incom- grams [Levesque et al., 1997], ramifications [Pinto, 1999; plete knowledge. We continue by showing that GATs can McIlraith, 2000], and concurrency and natural actions [Re- be mapped to our BATs. This has significance since known iter, 1996]. forms of regression for GATs are limited to certain types of BATs enjoy a number of good properties. Among them regressable formulas, while our regression is not. We final- are the relative satisfiability and regression theorems. Both ize with a discussion of related work in which we analyze reduce important tasks—including the projection task [Pirri a few advantages, besides simplicity, that our approach has and Reiter, 1999; Reiter, 2001]—to first-order theorem prov- over Pinto et al.’s and Delgrande and Levesque’s approaches. ing on the axiomatization for the initial situation. This is im- Moreover, we explain that, surprisingly, free fluents can be portant because the situation calculus is a second-order logic. “compiled away” provided we allow infinite distinguished el- In BATs there ought to be exactly one successor state ax- ements in the domain, albeit yielding substantially more con- iom per fluent. Each of these axioms characterizes precisely voluted theories. how each fluent evolves with actions. Possibly because of such a requirement, there is an informal understanding that 2 Action Theories in the Situation Calculus BATs are not able to accommodate incomplete knowledge on The Situation Calculus [Reiter, 2001] sitcalc, is a many- the causal laws of the domain. In the literature, this reflected sorted second-order language with equality,L designed for rep- by at least three extensions to Reiter’s formalism for handling resenting dynamically changing worlds whose changes are incomplete knowledge at the causal level. Guarded action the result of actions. Terms of sort situation are finite se- 907 quences of actions: the empty sequence is denoted by the dis- 2.1 Two Key Results: Regression & Satisfiability tinguished constant S0, and do(a, s) denotes the situation that A fundamental task in the situation calculus is the projec- results from performing action a in situation s. For legibility, tion task: given a BAT and a query formula ' uniform in we write do([A ,...,A ],s) to denote the situation resulting D 1 k do([A1,...,An],s), check whether = ' holds. from executing actions A1,...,Ak in situation s. Sort object Reiter [1991] proves that the projectionD| task, for a gen- is a catch-all sort for terms that denote objects of the world. eral class of sentences '—the regressable sentences—can In sitcalc, fluents describe the dynamic aspects of the first-order L be carried out only using theorem proving on the world. A relational (functional) fluent is a predicate (func- initial situation. Roughly speaking, a sentence W in sitcalc tion) whose last argument is a situation, and thus whose truth is regressable iff each term of sort situation has theL form value (value) can change from situation to situation. For ex- do([A1,...,An],S0) and each atom of the form Poss(↵, s) is ample, relational fluent Broken(x, s) can be used to denote such that ↵ has the form A(t1,...,tn). For such sentences, a that object x is broken in situation s, and functional fluent regression operator ['] can be defined to transform ' into R salary(x, s) to denote the salary of employee x in situation s. a formula—the regression of '—that is uniform in S0 (i.e., it Actions need not be executable in all situations. Predicate only talks about what is true in the initial situation). Poss(a, s) is used to state that action a is executable in sit- Theorem 1 (Regression Theorem; Reiter, 1991). If ' is an uation s. Finally, the distinguished binary predicate s @ s0 regressable query sentence and is a BAT, then: states that situation s represents a sub-history of situation s, Lsitcalc D that is, s0 = do([A1,...,Ak],s), for some non-empty se- = ' iff = [']. D| DS0 [Duna | R quence of actions A1,...,Ak. This is an important result from a practical standpoint in that A formula ' of sitcalc is called uniform in situation term σ [Reiter, 2001] ifL the only situation term mentioned in ' second-order axioms in ⌃ are not needed: a query against the initial situation and unique name axioms is sufficient. is σ, neither Poss, @, nor equalities between situations are mentioned, and there is no quantification over situations. The other fundamental result for BATs states that the con- In this paper we embrace Reiter’s basic action theories sistency of a BAT depends, basically, on building a consistent (BATs), which is a well-established form to build initial situation: Lsitcalc theories. Let = poss ssa una S0 ⌃ such A D D [D [D [D [ Theorem 2 (Relative Satisfiability; Reiter, 1991, 2001). that: (1) poss is the set of precondition axiom of the form BAT is satisfiable iff is satisfiable. D una S0 Poss(A(~x ),s) ⇧A(~x , s ), where ⇧A(~x , s ) is a formula uni- D D [D form in s; (2) ⌘ is the set of successor state axiom charac- Dssa 3 Basic Actions Theories with Free Fluents terizing how fluents evolve w.r.t. actions (see below); (3) una D In this section, we lift the requirement of having one succes- is the set of unique names axioms for actions; (4) S is a set D 0 sor state axiom per fluent in a basic action theory. It turns out of formulae uniform in S0, defining the initial situation; and (5) ⌃ is a set of domain-independent foundational axioms. that the fundamental results for BATs can be be recovered. In particular, the set contains one successor state ax- A basic action theory with free fluents (BAT with free flu- Dssa ents) is defined exactly as a basic action theory but where iom (SSA) per relational fluent F and functional fluent f of D the set ssa contains at most one successor state axiom per the form F (~x , do(a, s)) ΦF (~x , a , s ) and f(~x , do(a, s)) = ⌘ fluent.D A free fluent is one that has no successor state ax- y Φf (~x , y , a , s ), resp., where ΦF and Φf are formulas uniform⌘ in s characterizing how F and f evolve, resp., in sit- iom, and hence, as “determining fluents” in the Event Cal- [ ] uation s when action a is performed. Reiter [1991] defined a culus Shanahan, 1999 , they are not subject to the common way to generate successor SSAs from a set of so-called effect law of inertia. Intuitively, a free fluent can be used to denote a axioms, stating how an action changes the value of a fluent.
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