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.

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

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Our aim in this article is to supplement the set of strong properties introduced in the preceding article ([Dix94]) with a set of weak principles in order to characterize semantics of logic programs. In [Dix94] we introduced our point of view: we observed that all semantics induce in a natural way a sceptical non-monotonic entailment relation SEM scept . We ask for the properties of these sceptical relations and use them to describe all possible semantics. We collect in this paper serious shortcomings of some semantics proposed recently. Their strange behaviour led us to formulate in a natural way certain principles to avoid these problems. We argue that any well-behaved semantics should satisfy these principles. The main results state that our list of weak principles is complete in the following sense: any well-behaved-semantics is an extension of the well-founded semantics WFS and coincides for stratified programs with Apt, Blair, and Walker's supported model M supp P . We also...

. Recently Brass and Dix have introduced the semantics D-WFS for general disjunctive logic programs. The interesting feature of this approach is that it is both semantically and proof-theoretically founded. Any program \Phi is associated a normalform res(\Phi), called the residual program, by a non-trivial bottom-up construction using least fixpoints of two monotonic operators. We show in this paper, that the original calculus, consisting of some simple transformations, has a very strong and appealing property: it is confluent and terminating. This means that all the transformations can be applied in any order: we always arrive at an irreducible program (no more transformation is applicable) and this program is already uniquely determined. Moreover, it coincides with the normalform res(\Phi) of the program we started with. The semantics D-WFS can be read off from res(\Phi) immediately. No proper subset of the calculus has these properties --- only when we restrict to certain subclasse...

Abstract. Knowledge representation formalisms used on the Semantic Web adhere to a strict open world assumption. Therefore, nonmonotonic reasoning techniques are often viewed with scepticism. Especially negation as failure, which intuitively adopts a closed world view, is often claimed to be unsuitable for the Web where knowledge is notoriously incomplete. Nonetheless, it was suggested in the ongoing discussions around rules extensions for languages like RDF(S) or OWL to allow at least restricted forms of negation as failure, as long as negation has an explicitly defined, finite scope. Yet clear definitions of such “scoped negation ” as well as formal semantics thereof are missing. We propose logic programs with contexts and scoped negation and discuss two possible semantics with desirable properties. We also argue that this class of logic programs can be viewed as a rule extension to a subset of RDF(S). 1

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Various semantics for logic programs with negation are described in terms of a dualized program together with additional axioms, some of which are second order formulas. The semantics of Clark, Fitting, and Kunen are characterized in this framework, and a nite rst-order presentation of Kunen's semantics is described. A new axiom to represent \common sense " reasoning is proposed for logic programs. It is shown that the well-founded semantics and stable models are de nable with this axiom. The roles of domain augmentation and domain closure are examined. A \domain foundation " axiom is proposed to replace the domain closure axiom. 1

We present the valid model semantics, a new approach to providing semantics for logic programs with negation, setterms and grouping. The valid model semantics is a threevalued semantics, and is defined in terms of a `normal form' computation. The valid model semantics also gives meaning to the generation and use of non-ground facts (i.e., facts with variables) in a computation. The formulation of the semantics in terms of a normal form computation offers important insight not only into the valid model semantics, but also into other semantics proposed earlier. We show that the valid model semantics extends the well-founded semantics in a natural manner, and has several advantages over it. The well-founded semantics can also be understood using a variant of the normal form computations that we use; the normal form computations used for valid semantics seem more natural than those used for well-founded semantics. We also show that the valid model semantics has several other desirable prop...

We inv estigate the declarative semantics of logic programs with negation. First, we propose the extended well-founded model semantics. Then we establish three important criteria of declarative semantics for logic programs. Finally we justify our extension through the comparison of different semantics. 1.