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25
The Family of Stable Models
, 1993
"... The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the well ..."
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Cited by 54 (4 self)
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The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the wellfounded model. There is also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain co...
A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 52 (3 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
Bilattices In Logic Programming
, 1990
"... Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiplevalued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to l ..."
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Cited by 42 (4 self)
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Bilattices, introduced by M. Ginsberg, constitute an elegant family of multiplevalued logics. Those meeting certain natural conditions have provided the basis for the semantics of a family of logic programming languages. Now we consider further restrictions on bilattices, to narrow things down to logic programming languages that can, at least in principle, be implemented. Appropriate bilattice background information is presented, so the paper is relatively selfcontained. 1 Introduction Logic programming is more than just Prolog. It is a distinctive way of thinking about computers and programming that has led to the creation of a whole family of programming languages, mostly experimental. Some time ago I found that bilattices provided a uniform semantics for a rich and interesting group of logic programming languages [9]. Bilattices are a natural generalization of classical twovalued logic, and were introduced by Matt Ginsberg in [12], and more fully in [13]. Recently I have found t...
Symbolic Trajectory Evaluation
 Formal Hardware Verification
, 1996
"... ion The main problem with model checking is the state explosion problem  the state space grows exponentially with system size. Two methods have some popularity in attacking this problem: compositional methods and abstraction. While they cannot solve the problem in general, they do offer significa ..."
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Cited by 26 (6 self)
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ion The main problem with model checking is the state explosion problem  the state space grows exponentially with system size. Two methods have some popularity in attacking this problem: compositional methods and abstraction. While they cannot solve the problem in general, they do offer significant improvements in performance. The direct method of verifying that a circuit has a property f is to show the model M satisfies f . The idea behind abstraction is that instead of verifying property f of model M , we verify property f A of model MA and the answer we get helps us answer the original problem. The system MA is an abstraction of the system M . One possibility is to build an abstraction MA that is equivalent (e.g. bisimilar [48]) to M . This sometimes leads to performance advantages if the state space of MA is smaller than M . This type of abstraction would more likely be used in model comparison (e.g. as in [38]). Typically, the behaviour of an abstraction is not equivalent...
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
Model Checking Partially Ordered State Spaces
, 1995
"... The state explosion problem is the fundamental limitation of verification through model checking. In many cases, representing the state space of a system as a lattice is an effective way of ameliorating this problem. The partial order of the state space lattice represents an information ordering. Th ..."
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Cited by 16 (3 self)
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The state explosion problem is the fundamental limitation of verification through model checking. In many cases, representing the state space of a system as a lattice is an effective way of ameliorating this problem. The partial order of the state space lattice represents an information ordering. The paper shows why using a lattice structure is desirable, and why a quaternary temporal logic rather than a traditional binary temporal logic is suitable for describing properties in systems represented this way. The quaternary logic not only has necessary technical properties, it also expresses degrees of truth. This is useful to do when dealing with a state space with an information ordering defined on it, where in some states there may be insufficient or contradictory information available. The paper presents the syntax and semantics of a quaternary valued temporal logic. Symbolic trajectory evaluation (STE) [32] has been used to model check partially ordered state spaces with some succes...
An inconsistency tolerant model for belief representation and belief revision
 In Proceedings of the 16th International Joint Conference on Arti Intelligence
, 1999
"... ..."
WellFounded Semantics, Generalized
 In Proceedings of International Symposium on Logic Programming
, 1991
"... Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, threevalued, semantics uses #P , which extends TP to Kleene's strong threevalued logic. Both these approaches generalize to cover logic programming systems based on a wide class of ..."
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Cited by 13 (2 self)
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Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, threevalued, semantics uses #P , which extends TP to Kleene's strong threevalued logic. Both these approaches generalize to cover logic programming systems based on a wide class of logics, provided only that the underlying structure be that of a bilattice. This was presented in earlier papers. Recently wellfounded semantics has become influential for classical logic programs. We show how the wellfounded approach also extends naturally to the same family of bilatticebased programming languages that the earlier fixpoint approaches extended to. Doing so provides a natural semantics for logic programming systems that have already been proposed, as well as for a large number that are of only theoretical interest. And finally, doing so simplifies the proofs of basic results about the wellfounded semantics, by stripping away inessential details. 1 Introduction There hav...
Combinators for paraconsistent attitudes
 Logical Aspects of Computational Linguistics
, 2001
"... Abstract. In order to analyse the semantics of natural language sentences a translation into a partial type logic using lexical and logical combinators is presented. The sentences cover a fragment of English with propositional attitudes like knowledge, belief and assertion. A combinator is a closed ..."
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Cited by 8 (8 self)
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Abstract. In order to analyse the semantics of natural language sentences a translation into a partial type logic using lexical and logical combinators is presented. The sentences cover a fragment of English with propositional attitudes like knowledge, belief and assertion. A combinator is a closed term of the lambda calculus possibly containing lexical and/or logical constants. Such combinators seem promising from both a cognitive and computational point of view. There is approximately one lexical combinator for each word, but just eleven logical combinators for the present fragment. The partiality is only used for embedded sentences expressing propositional attitudes, thereby allowing for inconsistency without explosion (also called paraconsistency), and is based on a few key equalities for the connectives giving four truth values (truth, falsehood, and undefinedness with negative and positive polarity; only the first truth value is designated, i.e. yields the logical truths). 1
Combining Explicit Negation and Negation by Failure via Belnap's Logic
, 1994
"... This paper deals with logic programs containing two kinds of negation: negation as failure and explicit negation, allowing two different forms of uncertainty reasoning in the presence of incomplete information. Such programs have been introduced by Gelfond and Lifschitz and called extended programs. ..."
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Cited by 6 (0 self)
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This paper deals with logic programs containing two kinds of negation: negation as failure and explicit negation, allowing two different forms of uncertainty reasoning in the presence of incomplete information. Such programs have been introduced by Gelfond and Lifschitz and called extended programs. We provide them with a logical semantics in the style of Kunen, based on Belnap's fourvalued logic, and an answer sets' semantics that is shown to be equivalent to that of Gelfond and Lifschitz. The proofs rely on a translation into normal programs, and on a variant of Fitting's extension of logic programming to bilattices. 1 INTRODUCTION One of the striking features of logic programming is that it naturally supports various forms of nonmonotonic reasoning, by means of negative litterals. Simply infering negative information from a positive program is already a form of nonmonotonic inference that shows essential differences between the two main approaches to the modeltheoretic semanti...