Abduction in Logic Programming started in the late 80s, early 90s, in an attempt to extend logic programming into a framework suitable for a variety of problems in Artificial Intelligence and other areas of Computer Science. This paper aims to chart out the main developments of the field over the last ten years and to take a critical view of these developments from several perspectives: logical, epistemological, computational and suitability to application. The paper attempts to expose some of the challenges and prospects for the further development of the field.

We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the proof-theoretic and model-theoretic issues and the relationships between them.

The most general notion of canonical model for a logic program with negation is the one of stable model [9]. In [7] the stable models of a logic program are characterized by the well-supported Herbrand models of the program, and a new fixed point semantics that formalizes the bottom-up truth maintenance procedure of [4] is based on that characterization. Here we focus our attention on the abstract notion of well-supportedness in order to derive sufficient conditions for the existence of stable models. We show that if a logic program \Pi is positive-order-consistent (i.e. there is no infinite decreasing chain w.r.t. the positive dependencies in the atom dependency graph of \Pi) then the Herbrand models of comp(\Pi) coincide with the stable models of \Pi. From this result and the ones of [10] [17] [2] on the consistency of Clark's completion, we obtain sufficient conditions for the existence of stable models for positive-order-consistent programs. Then we show that a negative cycle free ...

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The definition of a stable model has provided a declarative semantics for Prolog programs with negation as failure and has led to the development of answer set programming. In this paper we propose a new definition of that concept, which covers many constructs used in answer set programming (including disjunctive rules, choice rules and conditional literals) and, unlike the original definition, refers neither to grounding nor to fixpoints. Rather, it is based on a syntactic transformation, which turns a logic program into a formula of secondorder logic that is similar to the formula familiar from the definition of circumscription. 1

this paper. Similarly, we have shown that integrity constraints can apply to temporal conditions, but that there is no one uniform approach to handling temporal databases. Many areas of the use of integrity constraints still need investigating. Below, we discuss some aspects associated with this topic that require additional work. ffl Implement semantic query optimization and cooperative answering systems. Current relational and deductive database systems do not provide these capabilities, but the current 30 April 1997 ICs: Semantics and Applications---Godfrey, Grant, Gryz, & Minker p. 36 of 46 standards for SQL provide for the incorporation of some aspects of integrity constraints.

The introduction of negation into logic programming brings the benefit of enhanced syntax and expressibility, but creates some semantical problems. Specifically, certain operators which are monotonic in the absence of negation become non-monotonic when it is introduced, with the result that standard approaches to denotational semantics then become inapplicable. In this paper, we show how generalized metric spaces can be used to obtain fixed-point semantics for several classes of programs relative to the supported model semantics, and investigate relationships between the underlying spaces we employ. Our methods allow the analysis of classes of programs which include the acyclic, locally hierarchical, and acceptable programs, amongst others, and draw on fixed-point theorems which apply to generalized ultrametric spaces and to partial metric spaces.

This paper presents declarative semantics of possibly inconsistent disjunctive logic programs. We introduce the paraconsistent minimal and stable model semantics for extended disjunctive programs, which can distinguish inconsistent information from others in a program. These semantics are based on lattice-structured multi-valued logics, and are characterized by a new fixpoint semantics of extended disjunctive programs. Applications of the paraconsistent semantics for reasoning in inconsistent programs are also presented. Keywords: Extended disjunctive programs, inconsistency, multi-valued logic, paraconsistent stable model semantics. 3 Journal of Logic and Computation 5: 265-285, Oxford University Press, 1995. 1 1

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In [Prz91], Przymusinski introduced the partial (or 3-valued) stable model semantics which extends the (2-valued) stable model semantics defined originally by Gelfond and Lifschitz [GL88]. In this paper we describe a procedure to compute the collection of all partial stable models of an extended disjunctive logic program. This procedure consists in transforming an extended disjunctive logic program into a constrained disjunctive program free of negation-by-default whose set of 2-valued minimal models corresponds to the set of partial stable models of the original program. 1 Introduction The partial (or 3-valued) stable model semantics defined by Przymusinski in [Prz91] is a three-valued semantics for the class of extended disjunctive logic programs (edlps). This class of programs consists of disjunctive logic programs that may contain two kinds of negations: negation-by-default and explicit negation. The definition of this semantics extends the (2-valued) stable model semantics given...