Abstract not found

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

One of the most important and difficult problems in logic programming is the problem of finding a suitable declarative or intended semantics for logic programs. The importance of this problem stems from the declarative character of logic programming, whereas its difficulty can be largely attributed to the non-monotonic character of the negation operator used in logic programs. The problem can therefore be viewed as the problem of finding a suitable formalization of the type of non-monotonic reasoning used in logic programming. In this paper we introduce a semantics of logic programs based on the class PERF(P) of all, not necessarily Herbrand, perfect models of a program P and we show that the proposed semantics is not only natural but it also combines many of the desirable features of previous approaches, at the same time eliminating some of their drawbacks. For a positive program P, the class PERF(P) of perfect models coincides with the class MIN(P) of all minimal models of P. The per...

We introduce global SLS-resolution, a procedural semantics for well-founded negation as defined by Van Gelder, Ross and Schlipf. Global SLS-resolution extends Przymusinski 's SLS-resolution, and may be applied to all programs, whether locally stratified or not. 1 Global SLS-resolution is defined in terms of global trees, a new data structure representing the dependence of goals on derived negative subgoals. We prove that global SLS-resolution is sound with respect to the well-founded semantics, and complete for non-floundering queries. While not effective in general, global SLS-resolution is effective for classes of "acyclic" programs, and can be augmented with a memoing device to be effective for all function-free programs. This research was supported by the National Science Foundation under grant IRI-87-22886, by a grant from IBM Corporation, and by the United States Air Force Office of Scientific Research under contract AFOSR-88-0266. A preliminary version of this paper was presen...

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SLDNF-resolution is complete for allowed programs and allowed queries. But the condition of allowedness is very stringent and excludes many common Prolog constructs. We show that allowedness is a special case of a more general principle. We show that if the clauses of a normal program are correct with respect to an input/output specification then SLDNFresolution is complete for it. An input/output specification assigns to every predicate a set of positive and a set of negative mode specifications. A mode specification declares the arguments of predicates as input arguments, output arguments or normal arguments. Positive modes are used in positive calls and negative modes are used in negative calls. Definite programs together with definite goals, allowed programs together with allowed goals and many programs and goals used in practice are correct with respect to some input/output specification. Therefore our results imply that the three-valued Fitting/Kunen completion is the right declarative semantics for negation as failure. Keywords: Logic programming, negation as failure, SLDNF-resolution, completion of programs, three-valued logic. 1

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

The purpose of this paper is to compare three types of non-monotonic semantics: (a) proof-theoretic semantics based on the closed world assumption, (b) model-theoretic semantics based on the notion of a minimal model and (c) model-theoretic semantics based on the notion of a minimal Herbrand model. All of these semantics capture the non-monotonicity of common sense reasoning, i.e. the ability to withdraw conclusions after some new information is added to the original theories, and proved to be powerful enough to handle most examples of such reasoning presented in the literature. However, since these formalizations are based on different intuitions and often produce different results, the problem of understanding the relationship between them is especially important. In the first part of the paper we concentrate on the class of positive logic programs, also called definite theories. Although the three semantics usually differ for universal sentences, our main result shows that they alwa...

We propose a new negation rule for logic programming which derives existentially closed negative literals, and we define a version of completion for which this rule is sound and complete. The rule is called "Negation As Instantiation" (NAI). According to it, a negated atom succeeds whenever all the branches of the SLD-tree for the atom either fail or instantiate the atom. The set of the atoms whose negation is inferred by the NAI rule is proved equivalent to the complement of TC #!, where TC is the immediate consequence operator extended to nonground atoms (Falaschi et al., 1989). The NAI rule subsumes negation as failure on ground atoms, it excludes floundering and can be efficiently implemented. We formalize this way of handling negation in terms of SLDNI-resolution (SLD-resolution with Negation as Instantiation) . Finally, we amalgamate SLDNI-resolution and SLDNF-resolution, thus obtaining a new resolution procedure which is able to treat negative literals with both existentially qu...