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Logic Programming and Negation: A Survey
 JOURNAL OF LOGIC PROGRAMMING
, 1994
"... We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the prooftheoretic and modeltheoretic issues and the relationships between them. ..."
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Cited by 242 (8 self)
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We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the prooftheoretic and modeltheoretic issues and the relationships between them.
Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation
 Information and Computation
, 1997
"... Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunct ..."
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Cited by 75 (17 self)
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Disjunctive logic programs have become a powerful tool in knowledge representation and commonsense reasoning. This paper focuses on stable model semantics, currently the most widely acknowledged semantics for disjunctive logic programs. After presenting a new notion of unfounded sets for disjunctive logic programs, we provide two declarative characterizations of stable models in terms of unfounded sets. One shows that the set of stable models coincides with the family of unfoundedfree models (i.e., a model is stable iff it contains no unfounded atoms). The other proves that stable models can be defined equivalently by a property of their false literals, as a model is stable iff the set of its false literals coincides with its greatest unfounded set. We then generalize the wellfounded WP operator to disjunctive logic programs, give a fixpoint semantics for disjunctive stable models and present an algorithm for computing the stable models of functionfree programs. The algor...
The Value of the Four Values
 Artificial Intelligence
, 1998
"... In his wellknown paper "How computer should think" ([Be77b]) Belnap argues that four valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the fourvalued structure has among Ginsberg's wellknown bilatti ..."
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Cited by 62 (6 self)
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In his wellknown paper "How computer should think" ([Be77b]) Belnap argues that four valued semantics is a very suitable setting for computerized reasoning. In this paper we vindicate this thesis by showing that the logical role that the fourvalued structure has among Ginsberg's wellknown bilattices is similar to the role that the twovalued algebra has among Boolean algebras. Specifically, we provide several theorems that show that the most useful bilatticevalued logics can actually be characterized as fourvalued inference relations. In addition, we compare the use of threevalued logics with the use of fourvalued logics, and show that at least for the task of handling inconsistent or uncertain information, the comparison is in favor of the latter. Keyworkds: Bilattices, Paraconsistency, Multiplevalued systems, Preferential logics, Reasoning. 1 Introduction In [Be77a, Be77b] Belnap introduced a logic intended to deal in a useful way with inconsistent and incomplete information....
Reasoning with Logical Bilattices
, 1995
"... . The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We als ..."
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Cited by 55 (10 self)
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. The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. The outcome are paraconsistent logics with a lot of desirable properties 1 . 1. Introduction When using multiplevalued logics, it is usual to order the truth values in a lattice structure. In most cases such a partial order intuitively reflects differences in the "measure of truth" that the lattice elements are supposed to represent. There exist, however, other intuitive criteria of ordering that might be useful. Another reasonable ordering might reflect, for example, differences in the amoun...
Logic programming revisited: logic programs as inductive definitions
 ACM Transactions on Computational Logic
, 2001
"... Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to ..."
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Cited by 34 (21 self)
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Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a nonmonotonic reasoning formalism strongly related to Default Logic and Autoepistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent nonmonotone inductive definitions. It is argued that this thesis results in an alternative justification of the wellfounded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning
Antitonic Logic Programs
, 2001
"... In a previous work we have de ned Monotonic Logic Programs which extend definite logic programming to arbitrary complete lattices of truthvalues with an appropriate notion of implication. We have shown elsewhere that this framework is general enough to capture Generalized Annotated Logic Programs, ..."
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Cited by 32 (10 self)
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In a previous work we have de ned Monotonic Logic Programs which extend definite logic programming to arbitrary complete lattices of truthvalues with an appropriate notion of implication. We have shown elsewhere that this framework is general enough to capture Generalized Annotated Logic Programs, Probabilistic Deductive Databases, Possibilistic Logic Programming, Hybrid Probabilistic Logic Programs and Fuzzy Logic Programming [3, 4]. However, none of these semantics define a form of nonmonotonic negation, which is fundamental for several knowledge representation applications. In the spirit of our previous work, we generalise our framework of Monotonic Logic Programs to allow for rules with arbitrary antitonic bodies over general complete lattices, of which normal programs are a special case. We then show that all the standard logic programming theoretical results carry over to Antitonic Logic Programs, defining Stable Model and Wellfounded Model alike semantics.
Kleene’s threevalued logics and their children
 Fundamenta Informaticae
, 1994
"... Abstract. Kleene’s strong threevalued logic extends naturally to a fourvalued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it fourvalued analogs of Kleene’s weak threevalued logic, and the asymmetri ..."
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Cited by 25 (4 self)
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Abstract. Kleene’s strong threevalued logic extends naturally to a fourvalued logic proposed by Belnap. We introduce a guard connective into Belnap’s logic and consider a few of its properties. Then we show that by using it fourvalued analogs of Kleene’s weak threevalued logic, and the asymmetric logic of Lisp are also available. We propose an extension of these ideas to the family of distributive bilattices. Finally we show that for bilinear bilattices the extensions do not produce any new equivalences. 1
Paraconsistent Stable Semantics for Extended Disjunctive Programs
 Journal of Logic and Computation
, 1995
"... 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 ..."
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Cited by 24 (0 self)
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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 latticestructured multivalued 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, multivalued logic, paraconsistent stable model semantics. 3 Journal of Logic and Computation 5: 265285, Oxford University Press, 1995. 1 1
Logical Bilattices and Inconsistent Data
 Proc. 9th IEEE Annual Symp. on Logic in Computer Science, IEEE Press
, 1994
"... The notion of a bilattice was first proposed by Ginsberg as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting. In the present paper we develop proof systems, which correspond to bilattices in an essential way. We then show how to ..."
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Cited by 20 (4 self)
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The notion of a bilattice was first proposed by Ginsberg as a general framework for many applications. This notion was further investigated and applied for various goals by Fitting. In the present paper we develop proof systems, which correspond to bilattices in an essential way. We then show how to use those bilattices for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii concerning inconsistencies, which happen to suit well the framework of bilattices. The outcome is a paraconsistent logic with a lot of desirable properties. 1 Introduction When using multiplevalued logics, it is usual to order the truth values in a lattice structure, where its partial order, t , describes intuitively differences in the "measure of truth" that the lattice elements are supposed to represent. However, these elements (the "truth values") can be ordered differently. Another reasonable ordering, k , reflects (again, intuitively) differenc...
DisLog  A System for Reasoning in Disjunctive Deductive Databases
 Proc. Intl. Workshop on the Deductive Approach to Information Systems and Databases
, 1994
"... DisLog is a system for reasoning in disjunctive deductive databases. It seeks to combine features of disjunctive logic programming, such as the support for incomplete information, with those of deductive databases, such as allresult inference capabilities. Several basic operators are provided for ..."
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Cited by 19 (6 self)
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DisLog is a system for reasoning in disjunctive deductive databases. It seeks to combine features of disjunctive logic programming, such as the support for incomplete information, with those of deductive databases, such as allresult inference capabilities. Several basic operators are provided for logical and nonmonotonic reasoning: The logical consequence operator derives all logically implied disjunctive clauses from a disjunctive logic program. The non monotonic operators are semantically founded on generalizations of the well known closedworldassumption. Reasoning in disjunctive deductive databases is very complex, even for small examples. Many different optimization techniques are integrated in DisLog to speed up the application performance. The main techniques rely on a clause tree data structure allowing for an efficient and transparent evaluation. The operators of DisLog can be loaded from a library into a Prolog application. DisLog itself is implemented as a metai...