Towards an Expressive Practical Logical Action Theory

Abstract

In the area of reasoning about actions, one of the key computational problems is the projection problem: to find whether a given logical formula is true after performing a sequence of actions. This problem is undecidable in the general situation calculus; however, it is decidable in some fragments. We consider a fragment P of the situation calculus and Reiter's basic action theories (BAT) such that the projection problem can be reduced to the satisfiability problem in an expressive description logic ALCO(U) that includes nominals (O), the universal role (U), and constructs from the well-known logic ALC. It turns out that our fragment P is more expressive than previously explored description logic based fragments of the situation calculus. We explore some of the logical properties of our theories. In particular, we show that the projection problem can be solved using regression in the case where BATs include a general "static" TBox, i.e., an ontology that has no occurrences of fluents. Thus, we propose seamless integration of traditional ontologies with reasoning about actions. We also show that the projection problem can be solved using progression if all actions have only local effects on the fluents, i.e., in P, if one starts with an incomplete initial theory that can be transformed into an ALCO(U) concept, then its progression resulting from execution of a ground action can still be expressed in the same language. Moreover, we show that for a broad class of incomplete initial theories progression can be computed efficiently.

Deterministic planning in incompletely known domains with local effects

“…Hereafter, we assume that the successor state axioms have the following more specific syntax, which incorporates the Reiter's solution [38] to the frame problem -the problem of specifying the behaviour of the fluents which are unaffected by the execution of an action. This syntax is a de facto standard in the existing approaches [26,41,10] to planning in situation calculus. Here and below,x is an abbreviation for x 1 , .…”

“…The restricted syntax of local-effect SSA can be beneficially exploited with the help of the unique name axioms for actions and a logically equivalent transformation. To that end, we introduce the notion of a transformed SSA, similarly to [41], and related concepts used in the definition of local-effect progression.…”

“…In either case, there is no obstacle for expressing the respective theories as proper + for the purposes of progression. According to Theorem 1, computing progression involves reasoning about situations only in a very limited sense, so the situation terms can be safely suppressed from D S0 for the forgetting stage of the computation and reinstated afterwards; the notion of situation-suppressed terms and formulas is formally defined in [37] and notably used in [41] for a similar purpose.…”

“…D ss [Ω] can be expressed as a finite set of ground clauses, which becomes a subset of proper + when the situation arguments are suppressed as described above. However, following a remark in [41], note that a naive suppression of all situation terms from D ss [Ω] leads to a collision of atoms F (c, S α ) and F (c, S 0 ), for each fluent symbol F . The solution to this is discussed in relation to the next result.…”

Deterministic planning in incompletely known domains with local effects

“…Hereafter, we assume that the successor state axioms have the following more specific syntax, which incorporates the Reiter's solution [38] to the frame problem -the problem of specifying the behaviour of the fluents which are unaffected by the execution of an action. This syntax is a de facto standard in the existing approaches [26,41,10] to planning in situation calculus. Here and below,x is an abbreviation for x 1 , .…”