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The alternating fixpoint of a logic program with negation is defined constructively. The underlying idea is monotonically to build up a set of negative conclusions until the least fixpoint is reached, using a transformation related to the one that defines stable models. From a fixed set of negative conclusions, the positive conclusions follow (without deriving any further negative ones), by traditional Horn clause semantics. The union of positive and negative conclusions is called the alternating xpoint partial model. The name "alternating" was chosen because the transformation runs in two passes; the first pass transforms an underestimate of the set of negative conclusions into an (intermediate) overestimate; the second pass transforms the overestimate into a new underestimate; the composition of the two passes is monotonic. The principal contributions of this work are (1) that the alternating fixpoint partial model is identical to the well-founded partial model, and (2) that alternating xpoint logic is at least as expressive as xpoint logic on all structures. Also, on finite structures, fixpoint logic is as expressive as alternating fixpoint logic.

The variety of semantical approaches that have been invented for logic programs is quite broad, drawing on classical and many-valued logic, lattice theory, game theory, and topology. One source of this richness is the inherent non-monotonicity 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 fixpoint 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 fixpoint treatments developed for other programming methodologies.

We study the expressive powers of two semantics for deductive databases and logic programming: the well-founded semantics and the stable semantics. We compare them especially to two older semantics, the two-valued and three-valued program completion semantics. We identify the expressive power of the stable semantics, and in fairly general circumstances that of the well-founded semantics. In particular, over infinite Herbrand universes, the four semantics all have the same expressive power. We discuss a feature of certain logic programming semantics, which we call the Principle of Stratification, a feature allowing a program to be built easily in modules. The three-valued program completion and well-founded semantics satisfy this principle. Over infinite Herbrand models, we consider a notion of translatability between the three-valued program completion and well-founded semantics which is in a sense uniform in the strata. In this sense of uniform translatability we show the well-founded semantics to be more expressive than the three-valued program completion. The proof is a corollary of our result that over non-Herbrand infinite models, the well-founded semantics is more expressive than the three-valued program completion semantics. 1

this paper. In particular, the paper reports on a very significant progress made recently in this area. It also presents some results which have not yet appeared in print. The paper is organized as follows. In the next two sections we define deductive databases and logic programs. Subsequently, in Sections 4 and 5, we discuss model theory and fixed points, which play a crucial role in the definition of semantics. Section 6 is the main section of the paper and is entirely devoted to a systematic exposition and comparison of various proposed semantics. In Section 7 we discuss the relationship between declarative semantics of deductive databases and logic programs and non-monotonic reasoning. Section 8 contains concluding remarks. 2 Deductive Databases

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...

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The present prolegomena consist, as all indeed do, in a critical discussion serving to introduce and interpret the extended works that follow in this book. As a result, the book is not a mere collection of excellent papers in their own specialty, but provides also the basics of the motivation, background history, important themes, bridges to other areas, and a common technical platform of the principal formalisms and approaches, augmented with examples. In the

We try to assess to what extent declarative programming can be realized in Prolog and which aspects of correctness of Prolog programs can be dealt with by means of declarative interpretation. More specifically, we shall discuss termination of Prolog programs, partial correctness, absence of errors and the safe use of negation. 1991 Mathematics Subject Classification: 68Q40, 68T15. CR Categories: F.3.2., F.4.1, H.3.3, I.2.3. Keywords and Phrases: declarative programming, Prolog programs, verification. Notes. This research was partly supported by the ESPRIT Basic Research Action 6810 (Compulog 2). This paper will appear as invited lecture in: Proc. of International Logic Programming Symposium (ILPS '93), The MIT Press, D. Miller (editor). It also appeared as a Technical Report No CT-93-06 in the ILLC Prepublication Series of the University of Amsterdam. 1