A Closed-Form Solution to the Ramification Problem

Our general concern is with how to integrate a representation of action into an existing set of state constraints. As has been observed in the literature, state constraints implicitly define indirect effects of actions as well as indirectly imposing further preconditions on the performance of actions. Thus, any representation scheme we propose must address the ramification and qualification problems, as well as the frame problem. In this paper we achieve such a representation for a syntactically restricted class of situation calculus theories. This paper presents two major technical contributions. The first contribution is provision of an axiomatic closed-form solution to the frame, ramification and qualification problems for a common class of ramification constraints. The solution is presented in the form of an automatable procedure that compiles a syntactically restricted set of situation calculus ramification constraints and effect axioms into a set of successor state axioms. The second major contributionof this paper is provision of an independent semantic justification for this closed-form solution. In particular, we present a semantic specification for a solution to the frame and ramification problems in terms of a prioritized minimization policy, and show that the successor state axioms of our closed-form solution adhere to this specification. Observing that our minimization policy is simply an instance of prioritized circumscription, we exploit results of Lifschitz on computing circumscription [6] to show that computing the prioritized circumscription yields our successor state axioms. In the special case where there are no ramification constraints, computing the circumscription yields Reiter’s earlier successor state axiom solution to the frame problem [17]. I would like to acknowledge Fangzhen Lin for providing helpful comments on the circumscription results, and particularly for mathematically articulating my intuitions regarding the existence of a lemma that was essential to the proof of Theorem 3. yA subset of the material presented in this paper also appears in Representing Actions and State Constraints in Model-Based Diagnosis, Proceedings of Fourteenth National Conference on Artificial Intelligence c AAAI Press, Menlo Park, CA, 1997. zCurrent Address: Xerox PARC, 3333 Coyote Hill Road, Palo Alto, CA 94304, E-mail: [email protected], Phone: 415-812-4332, Fax: 415-812-4334, and Knowledge Systems Laboratory, Stanford University, Gates Sciences Building 2A-248, Stanford, CA 94305-9020, E-mail: [email protected].