Dualities between alternative semantics for logic programming and nonmonotonic reasoning

Abstract

The Gelfond-Lifschitz operator associated with a logic program (and likewise the operator associated with default theories by Reiter) exhibits oscillating behavior. In the case of logic programs, there is always at least one finite, nonempty collection of Herbrand interpretations around which the Gelfond-Lifschitz operator ‘bounces around’. The same phenomenon occurs with default logic when Reiter's operator Г^Δ is considered. Based on this, a ‘stable class’ semantics and ‘extension class’ semantics has been proposed. The main advantage of this semantics was that it was defined for all logic programs (and default theories), and that this definition was modelled using the standard operators existing in the literature such as Reiter's Г^Δ operator. In this paper our primary aim is to prove that there is a very interestingduality between stable class theory and the well-founded semantics for logic programming. In the stable class semantics, classes that were minimal with respect to Smyth's power-domain ordering were selected. We show that the well-founded semantics precisely corresponds to a class that is minimal w.r.t. Hoare's power domain ordering: the well-known dual of Smyth's ordering. Besides this elegant duality, this immediately suggests how to define a well-founded semantics for default logic in such a way that the dualities that hold for logic programming continue to hold for default theories. We show how the same technique may be applied to ‘strong’ autoepistemic logic: the logic of strong expansions proposed by Marek and Truszczynski.