Soundness and Completeness of a Logic Programming Approach to Default Logic

We present a method of representing some classes of default theories as normal logic programs. The main point is that the standard semantics (i.e. SLDNF-resolution) computes answer substitutions that correspond exactly to the extensions of the represented default theory. We explain the steps of constructing a logic program LogProg(P,D) from a given default theory (P,D), and present the proof ideas of the soundness and completeness results for the approach. Introduction In last years much work is done to establish relationships between nonmonotonic reasoning and logic programming (Bidoit&Froidevaux 1991a, 199lb, Gelfond&Lifschitz 1988, 1991, Marek&Truszczynski 1989, Marek&Subrahmanian 1992, Pereira&Nerode 1993). This is usually done by defining a new semantics for logic programming (in particular of negation) and deriving a relationship to some nonmonotonic logic. In this paper we go the other way around: We maintain the classical standard semantics of logic programming (SLDNF-resolution) and try to translate a nonmonotonic logic into appropriate logic programs. In particular, 0 we translate portions of Reiter’s default logic (Reiter 1980) into normal logic programs, and 0 prove that the translation LogProg(T) of a default theory T computes answer substitutions that exactly correspond to the extensions of T. What are the benefits of such an approach? l Standard semantics of logic programming is well analyzed and understood. 0 Prolog is a powerful implementation of standard semantics. 0 Therefore, we provide an implementational para934 Nonmonotonic Reasoning digm for nonmonotonic reasoning that might prove valuable in practice. For example, we might use parallel logic programming systems to achieve efficient reasoning systems. Throughout the paper we assume familiarity with notation and basic notions of predicate logic and logic programming. In case of discomfort, please refer to (Lloyd 1987, Sperschneider&Antoniou 1991). Basics of default logic A default 6 is a string (p:wl,. . .,w,lx with closed ftrstorder formulas (p,wl,. . .,vn and x (n>O>. We call cp the prerequisite, wl,...,yn the justiJ-kations, and x the consequent of 6. A default schema is a string of the form <p:wl,. . .,w,lx with arbitrary formulas. Such a schema defines a set of defaults, namely the set of all ground instances ~po:~~o,. . .,~,cJ/xo of <p:wl,. . .,v&, where o is an arbitrary ground substitution. A default theory T is a pair (W,D) consisting of a set of closed formulas W (the set of truths) and a denumerable set of defaults D. The default set D may be defined using default schemata. Let 6=(p:v1,. . ., w,lx be a default, and E and F sets of formulas. We say that 6 is applicable to F with respect to belief set E iff <PEF, and lvlgE,. ..,++r,@ E. F is closed under D with respect to E iff, for every default (p:wl,. . ., w,lx in D that is applicable to F with respect to belief set E, its consequent x is also contained in F. Given a default theory T=(W,D) and a set of closed fOm&iS E, let AT(E) be the least Set of closed formulas that contains W, is closed under logical conclusion and closed under D with respect to E. A set of closed formulas E is called an extension of T iff h+)=E. From: AAAI-94 Proceedings. Copyright © 1994, AAAI (www.aaai.org). All rights reserved.