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Bilattices and the Semantics of Logic Programming
, 1989
"... Bilattices, due to M. Ginsberg, are a family of truth value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnap's fourvalued logic, based on classical twovalued logic. Among other examples are those based on finite manyvalued logics, and on prob ..."
Abstract

Cited by 381 (13 self)
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Bilattices, due to M. Ginsberg, are a family of truth value spaces that allow elegantly for missing or conflicting information. The simplest example is Belnap's fourvalued logic, based on classical twovalued logic. Among other examples are those based on finite manyvalued logics, and on probabilistic valued logic. A fixed point semantics is developed for logic programming, allowing any bilattice as the space of truth values. The mathematics is little more complex than in the classical twovalued setting, but the result provides a natural semantics for distributed logic programs, including those involving confidence factors. The classical twovalued and the Kripke/Kleene threevalued semantics become special cases, since the logics involved are natural sublogics of Belnap's logic, the logic given by the simplest bilattice. 1 Introduction Often useful information is spread over a number of sites ("Does anybody know, did Willie wear a hat when he left this morning?") that can be speci...
Logic Programming and Knowledge Representation
 Journal of Logic Programming
, 1994
"... In this paper, we review recent work aimed at the application of declarative logic programming to knowledge representation in artificial intelligence. We consider exten sions of the language of definite logic programs by classical (strong) negation, disjunc tion, and some modal operators and sh ..."
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Cited by 227 (21 self)
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In this paper, we review recent work aimed at the application of declarative logic programming to knowledge representation in artificial intelligence. We consider exten sions of the language of definite logic programs by classical (strong) negation, disjunc tion, and some modal operators and show how each of the added features extends the representational power of the language.
Negation as Refutation
, 1989
"... A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A fourvalued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The f ..."
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Cited by 28 (5 self)
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A refutation mechanism is introduced into logic programming, dual to the usual proof mechanism; then negation is treated via refutation. A fourvalued logic is appropriate for the semantics: true, false, neither, both. Inconsistent programs are allowed, but inconsistencies remain localized. The fourvalued logic is a wellknown one, due to Belnap, and is the simplest example of Ginsberg's bilattice notion. An e#cient implementation based on semantic tableaux is sketched; it reduces to SLD resolution when negations are not involved. The resulting system can give reasonable answers to queries that involve both negation and free variables. Also it gives the same results as Prolog when there are no negations. Finally, an implementation in Prolog is given. 1 Introduction The most common treatment of negation in logic programming is negationasfailure. This leads to problems that are now familiar: meanings of programs become di#cult to specify; program operators need not reach fix...
DomainTheoretic Semantics For Disjunctive Logic Programs
"... . We propose three equivalent domaintheoretic semantics for disjunctive logic programs in the style of Gelfond and Lifschitz. These are (i) a resolutionstyle inference semantics; (ii) a modeltheoretic semantics; and (iii) a (nondeterministic) statetransition semantics. We show how these three ..."
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Cited by 2 (1 self)
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. We propose three equivalent domaintheoretic semantics for disjunctive logic programs in the style of Gelfond and Lifschitz. These are (i) a resolutionstyle inference semantics; (ii) a modeltheoretic semantics; and (iii) a (nondeterministic) statetransition semantics. We show how these three semantics generalize "for free" to disjunctive logic programming over any Scott domain. We also relate these semantics to default logic, showing how ZhangRounds powerdefault reasoning is isomorphic to clausal default logic, and finally by briefly mentioning stablemodel semantics in Scott domains. The main technical tool used is the Smyth powerdomain, a way to specify denotational semantics of nondeterminism. To make this tool work, we give a representation theorem for the Smyth powerdomain as a domain of clausal theories over a Scott domain. 1. Introduction Gelfond and Lifschitz, in [GL91], introduced a class of disjunctive logic programs incorporating what they called "classica...
THE OPTIMAL MODEL OF A PROGRAM WITH NEGATION
"... Abstact: The aim of this paper is to provide another denotational semantics than the least firpoint semantics for a logic program with negation. It takes place in a more general work about threevalued (partial) logic and logic programming. We have developped a logical and algebrical theory which se ..."
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Abstact: The aim of this paper is to provide another denotational semantics than the least firpoint semantics for a logic program with negation. It takes place in a more general work about threevalued (partial) logic and logic programming. We have developped a logical and algebrical theory which seems wellsuited to logic programs with negation. In this theory, we have extended the "consequence " operator of Van Emden and Kowalski associated with a program without negation to a "consequence " operator for programs with negation taking inconsistent programs in account. As the models of a program are exactly the postfixpoints of this operator, the first provided denotational semantics of a program is the least fixpoint semantics. As an intersection of threevalued Herbrand models of a program is still a model of this program, this least ftxpoint is also the set of all the ground literals of a program which are threevalued logical consequences of this program. Even if the least model is thus unique and defined for every consistent program, it is a partial model and may contain only few literals. A second denotational semantics is given by considering the threevalued Herbrand interpretations as partial functions from the set of ground atoms to the set of the truth values {T, F}; the theory of Manna and Shamir can be adapted in two ways: The first one is the notion of optimalftrpoint of the operator. The second one seems more interesting for our purpose, since the models of a logic program with negation are exactly the postf'vcpoints of the consequence operator. It is the notion of optimal postftxpoint of our "consequence " operator, which is the optimal model of our program. This notion is about to strengthen the threevalued logic. This optimal model is still unique and defined for every consistent program. It is still a partial model but generally strictly contains the least model, and thus really provides another denotational semantics than the least fixpoint semantics for a program with181 negation. It may even be more interesting since it contains more informations on the program itself. Situation of the work.
www.elsevier.com/locate/tcs Fixpoint semantics for logic programming a survey
"... The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close para ..."
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The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and manyvalued logic, lattice theory, game theory, and topology. One source of this richness is the inherent nonmonotonicity of its negation, something that does not have close parallels with the machinery of other programming paradigms. Nonetheless, much of the work on logic programming semantics seems to exist side by side with similar work done for imperative and functional programming, with relatively minimal contact between communities. In this paper we summarize one variety of approaches to the semantics of logic programs: that based on 0xpoint theory. We do not attempt to cover much beyond this single area, which is already remarkably fruitful. We hope readers will see parallels with, and the divergences from the better known 0xpoint treatments developed for other programming methodologies. c © 2002 Elsevier Science B.V. All rights reserved. 1.