Recently several authors have stressed and showed the importance of having a second kind of negation in logic programs for use in deductive databases, knowledge representation, and nonmonotonic reasoning [6, 7, 8, 9, 13, 14, 15, 24]. Different semantics for logic programs extended with :-negation (extended logic programs) have appeared [1, 4, 6, 9, 11, 12, 17, 19, 24] but, contrary to what happens with semantics for normal logic programs, there is no general comparison among them, specially in what concerns the use and meaning of the newly introduced :-negation. The goal of this paper is to contrast a variety of these semantics in what concerns their use and meaning of :-negation, and its relation to classical negation and to the default negation of normal programs, here denoted by not : To this purpose we define a parametrizeable schema to encompass and characterize a diversity of proposed semantics for extended logic programs, where the parameters are two: one the axioms AX: defin...

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

In this chapter we motivate the use of paraconsistency, and survey the most salient paraconsistent semantics for (extended) logic programs, which are briefly defined and explained. Most of the semantics are accompanied with their multi-valued model theory, giving them a new perspective. The survey also presents new results regarding the embedding of part of these semantics into normal logic programs under Well-Founded Semantics [20], Partial Stable Model Semantics (or stationary semantics) [48], and Stable Model Semantics [21]. Furthermore, a concise recapitulation of other related paraconsistent formalisms is made. The reader is assumed to have a good knowledge of the semantics of normal logic programs. We believe a comprehensive coverage of the topic as it stands at present is attained here.

We provide a simple formulation of a framework where some extensions of logic programming with non-monotonic reasoning are treated uniformly, namely two kinds of negation and abduction. The resulting semantics is purely model-theoretic, and gives meaning to any noncontradictory abductive logic program. Moreover, it embeds and generalizes some existing semantics which deal with negation and abduction. The framework is equipped with a correct top-down proof procedure. Keywords: Programming languages, Logic programming, Non-monotonic reasoning, Negation, Abduction. Dipartimento di Informatica, Universit`a di Pisa, Corso Italia 40, Pisa, Italy. brogi@di.unipi.it y DEIS, Universit`a di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. elamma@deis.unibo.it z Dipartimento di Informatica, Universit`a di Pisa, Corso Italia 40, Pisa, Italy. paolo@di.unipi.it x DEIS, Universit`a di Ferrara, Via Saragat, 41100 Ferrara, Italy. pmello@ing.unife.it Contents 1 Introduction and Motiva...

We present a coherent, flexible, unifying, and intuitive framework for the study of explicit negation in logic programs, based on the notion of admissible scenaria and the "coherence principle". With this support we introduce, in a simple way, a proposed "ideal sceptical semantics", as well as its well-- founded counterpart. Another result is a less sceptical "complete scenaria semantics", and its proof of equivalence to the well--founded semantics with explicit negation (WFSX). This has the added benefict of bridging complete scenaria to default theory via WFSX, defined here based on Gelfond--Lifschitz \Gamma operator.. Finally, we characterize a variety of more and less sceptical or credulous semantics, including answer--sets, and give sufficient conditions for equivalence between those semantics. Introduction In general, approaches to semantics follow two major intuitions: scepticism and credulity [30]. In logic programming, the credulous approach includes such semantics as stabl...

this paper, provides a representational expressivity that captures a wide variety of logical reasoning forms [6, 16, 22]. The two forms of negation, default and explicit, are not unrelated: the "coherence principle" stipulates that the latter entails the former. Of course, introducing explicit negation requires, in addition, dealing with contradiction. Default negation and the revising of such default assumptions in the face of contradiction (or integrity constraint violation) are both non-monotonic reasoning mechanisms available in logic programming. One major contribution of this paper is that of tackling contradiction, with some generality, and in particular within our choice semantics of extended logic programs, WFSX. Although such issues can be appproached from a belief revision point of view, they can likewise, and equivalently, be thought of within

This paper is the continuation of [1] in this volume. There we present a sceptical semantics which avoids contradiction for extended logic programs plus integrity contraints in the form of denials, based on the notion of optative hypotheses --an abductive approach. In this part we define a program revision method for removing contradiction from contradictory programs under WFSX, based on the notion of revisable hypotheses --a belief revision approach-- and show the equivalence between the contradiction avoidance semantics and the WFSX of revised programs obtained by contradiction removal. The motivation, as well as some preliminary definitions can be found in [1]. Proofs of all theorems are omitted for brevity, but exist in an extended version of this work.

Abstract. We provide a nine-valued logic to characterize the models of logic programs under a paraconsistent well-founded semantics with explicit negation WFSX p. We define a truth-functional logic, N IN E, based on the bilattice construction of Ginsberg and Fitting. The models identified by WFSX p are models of logic N IN E. We conclude with a discussion on the conditions to obtain an isomorphism between the two definitions, and thereby characterizing WFSX p model-theoretically. 1

In this work, we present Paraconsistent Well-Founded Semantics with eXplicit negation (WFSXp ) by introducing a frame-based semantics. As far as we know, this is the first time that a declarative approach for WFSXp is e#ectively characterised in terms of a model theory.

We begin by motivating the use of paraconsistency and the detection of contradiction supported conclusions by recourse to examples. Next we overview WFSX p and present its embedding into WFS. We then address the problem of detecting contradiction support and relate it to WFSX p's intrinsic properties. Afterwards, we show how to implement two recent modal contradiction related constructs in the language of extended logic programs in order to gain explicit control of contradiction propagation. We finish by making comparisons and drawing some conclusions.