Well-Founded and Partial Stable Semantics Logical Aspects

Abstract

This paper is devoted to logical aspects of two closely related semantics for logic programs: the partial stable model semantics of Przymusinski [20] and the well-founded semantics of Van Gelder, Ross and Schlipf [24]. For many years the following problem remained open: Which (non-modal) logic can be regarded as yielding an adequate foundation for these semantics in the sense that its minimal models (appropriately defined) coincide with the partial stable models of a logic program? Initial work on this problem was undertaken by Cabalar [5] who proposed a frame-based semantics for a suitable logic which he called HT 2. Preliminary axiomatics of HT 2 was presented in [6]. In this paper we analyse HT 2 frames and identify them as structures of a logic N * having intuitionistic positive connectives and Routley negation and give a natural axiomatics for HT 2. We define a notion of minimal, total HT 2 model which we call partial equilibrium model . We show that for logic programs these models coincide with partial stable models, and we propose the resulting partial equilibrium logic as a logical foundation for partial stable and well-founded semantics. Finally, we discuss the strong equivalence for theories and programs in partial equilibrium logic.