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 so-called knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P ---it is the well-founded 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...

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 t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to many-valued 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, introduced by M. Ginsberg, constitute an elegant family of multiple-valued 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 self-contained. 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 two-valued logic, and were introduced by Matt Ginsberg in [12], and more fully in [13]. Recently I have found t...

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...

Abstract. Kleene’s strong three-valued logic extends naturally to a four-valued 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 four-valued analogs of Kleene’s weak three-valued 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

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...

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Classical fixpoint semantics for logic programs is based on the TP immediate consequence operator. The Kripke/Kleene, three-valued, semantics uses #P , which extends TP to Kleene's strong three-valued 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 well-founded semantics has become influential for classical logic programs. We show how the well-founded 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 well-founded semantics, by stripping away inessential details. 1 Introduction There hav...

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 four-valued 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 non-monotonic reasoning, by means of negative litterals. Simply infering negative information from a positive program is already a form of non-monotonic inference that shows essential differences between the two main approaches to the model-theoretic semanti...

In this study we suggest a general formalism for nonmonotonic reasoning. The formalism, called biconsequence relations, provides a general framework of reasoning with respect to a pair of contexts. As is shown in [Bochman 1996], it allows to give a uniform representation for various semantics for logic programs involving negation as failure. Here we will consider a generalization of the formalism obtained by incorporating classical inference. This generalization turns out to be suciently expressive to provide a representation for such systems of nonmonotonic reasoning as various default and modal nonmonotonic logics. It allows also to account for some recent attempts to extend semantics suggested for logic programs to general nonmonotonic formalisms. Keywords : Representation formalisms, nonmonotonic logics.