Negation In Logic Programming Research Articles

A fixed point theorem for non-monotonic functions

We present a fixed point theorem for a class of (potentially) non-monotonic functions over specially structured complete lattices. The theorem has as a special case the Knaster–Tarski fixed point theorem when restricted to the case of monotonic functions and Kleene's theorem when the functions are additionally continuous. From the practical side, the theorem has direct applications in the semantics of negation in logic programming. In particular, it leads to a more direct and elegant proof of the least fixed point result of [12]. Moreover, the theorem appears to have potential for possible applications outside the logic programming domain.

The infinite-valued semantics: overview, recent results and future directions

The infinite-valued semantics was introduced in Rondogiannis and Wadge (2005) as a purely logical way for capturing the meaning of well-founded negation in logic programming. The purpose of this paper is threefold: first, to give a non-technical introduction to the infinite-valued semantics; second, to discuss the applicability of the infinite-valued approach to syntactically richer extensions of logic programming; and third, to present the main open problems whose resolution would further enhance the applicability of the technique.

Well-founded semantics for Boolean grammars

Boolean grammars [A. Okhotin, Boolean grammars, Information and Computation 194 (1) (2004) 19–48] are a promising extension of context-free grammars that supports conjunction and negation in rule bodies. In this paper, we give a novel semantics for Boolean grammars which applies to all such grammars, independently of their syntax. The key idea of our proposal comes from the area of negation in logic programming, and in particular from the so-called well-founded semantics which is widely accepted in this area to be the “correct” approach to negation. We show that for every Boolean grammar there exists a distinguished (three-valued) interpretation of the non-terminal symbols, which satisfies all the rules of the grammar and at the same time is the least fixed-point of an operator associated with the grammar. Then, we demonstrate that every Boolean grammar can be transformed into an equivalent (under the new semantics) grammar in normal form. Based on this normal form, we propose an O ( n 3 ) algorithm for parsing that applies to any such normalized Boolean grammar. In summary, the main contribution of this paper is to provide a semantics which applies to all Boolean grammars while at the same time retaining the complexity of parsing associated with this type of grammars.

An infinite-game semantics for well-founded negation in logic programming

We present an infinite-game characterization of the well-founded semantics for function-free logic programs with negation. Our game is a simple generalization of the standard game for negation-less logic programs introduced by van Emden [M.H. van Emden, Quantitative deduction and its fixpoint theory, Journal of Logic Programming 3 (1) (1986) 37–53] in which two players, the Believer and the Doubter, compete by trying to prove (respectively disprove) a query. The standard game is equivalent to the minimum Herbrand model semantics of logic programming in the sense that a query succeeds in the minimum model semantics iff the Believer has a winning strategy for the game which begins with the Doubter doubting this query. The game for programs with negation that we propose follows the same rules as the standard one, except that the players swap roles every time the play “passes through” negation. We start our investigation by establishing the determinacy of the new game by using some classical tools from the theory of infinite-games. Our determinacy result immediately provides a novel and purely game-theoretic characterization of the semantics of negation in logic programming. We proceed to establish the connections of the game semantics to the existing semantic approaches for logic programming with negation. For this purpose, we first define a refined version of the game that uses degrees of winning and losing for the two players. We then demonstrate that this refined game corresponds exactly to the infinite-valued minimum model semantics of negation [P. Rondogiannis,W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 (2) (2005) 441–467]. This immediately implies that the unrefined game is equivalent to the well-founded semantics (since the infinite-valued semantics is a refinement of the well-founded semantics).

Epistemic foundation of stable model semantics

Stable model semantics has become a very popular approach for the management of negation in logic programming. This approach relies mainly on the closed world assumption to complete the available knowledge and its formulation has its basis in the so-called Gelfond–Lifschitz transformation. The primary goal of this work is to present an alternative and epistemic-based characterization of stable model semantics, to the Gelfond-Lifschitz transformation. In particular, we show that stable model semantics can be defined entirely as an extension of the Kripke-Kleene semantics. Indeed, we show that the closed world assumption can be seen as an additional source of ‘falsehood’ to be added cumulatively to the Kripke-Kleene semantics. Our approach is purely algebraic and can abstract from the particular formalism of choice as it is based on monotone operators (under the knowledge order) over bilattices only.

Failure and Equality in Functional Logic Programming

Constructive failure has been proposed recently as a programming construct useful for functional logic programming, playing a role similar to that of constructive negation in logic programming. On the other hand, almost any functional logic program requires the use of some kind of equality test between expressions. We face in this work in a rigorous way the interaction of failure and equality (even for non-ground expressions), which is a non trivial issue, requiring in particular the use of disequality conditions at the level of the operational mechanism of constructive failure. As an interesting side product, we develop a novel treatment of equality and disequality in functional logic programming, by giving them a functional status, which is better suited for practice than previous proposals.

‘Classical’ Negation in Nonmonotonic Reasoning and Logic Programming

Gelfond and Lifschitz were the first to point out the need for a symmetric negation in logic programming and they also proposed a specific semantics for such negation for logic programs with the stable semantics, which they called ‘classical’. Subsequently, several researchers proposed different, often incompatible, forms of symmetric negation for various semantics of logic programs and deductive databases. To the best of our knowledge, however, no systematic study of symmetric negation in non-monotonic reasoning was ever attempted in the past. In this paper we conduct such a systematic study of symmetric negation. We introduce and discuss two natural, yet different, definitions of symmetric negation: one is called strong negation and the other is called explicit negation. For logic programs with the stable semantics, both symmetric negations coincide with Gelfond–Lifschitz’ ‘classical negation’. We study properties of strong and explicit negation and their mutual relationship as well as their relationship to default negation ‘not’, and classical negation ‘¬’. We show how one can use symmetric negation to provide natural solutions to various knowledge representation problems, such as theory and interpretation update, and belief revision. Rather than to limit our discussion to some narrow class of nonmonotonic theories, such as the class of logic programs with some specific semantics, we conduct our study so that it is applicable to a broad class of non-monotonic formalisms. In order to achieve the desired level of generality, we define the notion of symmetric negation in the knowledge representation framework of AutoEpistemic logic of Beliefs, introduced by Przymusinski.

Negation and minimality in disjunctive databases

Two main approaches have been followed in the literature to give a semantics to non-Horn databases. The first one is based on considering the set of rules composing the programs as inference rules and interpreting the negation in the body as failure to prove. The other approach is based on the so-called closed-world assumption, and its objective is to define a stronger notion of consequence from a theory than the classical one where, very roughly, negative information can be inferred whenever its positive counterpart cannot be deduced from the theory. In this work, we generalize the semantics for negation in logic programs, putting together the constructive nature of the rule-based deductive databases with the syntax-independence of the closed-world reasoning rules. These generalized semantics are shown to be a well-motivated and well-founded alternative to closed-world assumptions since they enjoy nice semantic and computational properties.

Reasoning in inconsistent knowledge bases

Databases and knowledge bases could be inconsistent in many ways. For example, during the construction of an expert system, we may consult many different experts. Each expert may provide us with a group of rules and facts which are self-consistent. However, when we coalesce the facts and rules provided by these different experts, inconsistency may arise. Alternatively, knowledge bases may be inconsistent due to the presence of some erroneous information. Thus, a framework for reasoning about knowledge bases that contain inconsistent information is necessary. However, existing frameworks for reasoning with inconsistency do not support reasoning by cases and reasoning with the law of excluded middle (everything is either true or false). In this paper, we show how reasoning with cases, and reasoning with the law of excluded middle may be captured. We develop a declarative and operational semantics for knowledge bases that are possibly inconsistent. We compare and contrast our work with work on explicit and non-monotonic modes of negation in logic programs and suggest under what circumstances one framework may be preferred over another. >

What is failure? An approach to constructive negation

A standard approach to negation in logic programming is negation as failure. Its major drawback is that it cannot produce answer substitutions to negated queries. Approaches to overcoming this limitation are termed constructive negation. This work proposes an approach based on construction offailed trees for some instances of a negated query. For this purpose a generalization of the standard notion of a failed tree is needed. We show that a straightforward generalization leads to unsoundness and present a correct one.

Mixed integer programming methods for computing nonmonotonic deductive databases

Though the declarative semantics of both explicit and nonmonotonic negation in logic programs has been studied extensively, relatively little work has been done on computation and implementation of these semantics. In this paper, we study three different approaches to computing stable models of logic programs based on mixed integer linear programming methods for automated deduction introduced by R. Jeroslow. We subsequently discuss the relative efficiency of these algorithms. The results of experiments with a prototype compiler implemented by us tend to confirm our theoretical discussion. In contrast to resolution, the mixed integer programming methodology is both fully declarative and handles reuse of old computations gracefully. We also introduce, compare, implement, and experiment with linear constraints corresponding to four semantics for “explicit” negation in logic programs: the four-valued annotated semantics [Blair and Subrahmanian 1989], the Gelfond-Lifschitz semantics [1990], the over-determined models [Grant and Subrahmanian 1989], the Gelfond-Lifschitz semantics [1990], the over-determined models [Grant and Subrahmanian 1990], and the classical logic semantics. Gelfond and Lifschitz[1990] argue for simultaneous use of two modes of negation in logic programs, “classical” and “nonmonotonic,” so we give algorithms for computing “answer sets” for such logic programs too.

On solving equations and disequations

We are interested in the problem of solving a system 〈s l = t l : 1 ≤ i ≤ n, p j ≠ q j : 1 ≤ j ≤ m 〉 of equations and disequations, also known as disunification . Solutions to disunification problems are substitutions for the variables of the problem that make the two terms of each equation equal, but leave those of the disequations different. We investigate this in both algebraic and logical contexts where equality is defined by an equational theory and more generally by a definitive clause equality theory E. We show how E-disunification can be reduced to E-unification, that is, solving equations only, and give a disunification algorithm for theories given a unification algorithm. In fact, this result shows that for theories in which the solutions of all unification problems can also be represented finitely. We sketch how disunification can be applied to handle negation in logic programming with equality in a similar style to Colmerauer's logic programming with rational trees, and to represent many solutions to AC-unification problems by a few solutions to ACI-disunification problems.

Logic programming and negation: A survey

We survey here various approaches which were proposed to incorporate negation in logic programs. We concentrate on the proof-theoretic and model-theoretic issues and the relationships between them.

A procedural semantics for well-founded negation in logic programs

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. 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 nonfloundering queries. Although not effective in general, global SLS-resolution is effective for classes of “acrylic” programs and can be augmented with a memoing device to be effective for all function-free programs.

A Modal Semantics of Negation in Logic Programming

The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

Classical negation in logic programs and disjunctive databases

An important limitation of traditional logic programming as a knowledge representation tool, in comparison with classical logic, is that logic programming does not allow us to deal directly with incomplete information. In order to overcome this limitation, we extend the class of general logic programs by including classical negation, in addition to negation-as-failure. The semantics of such extended programs is based on the method of stable models. The concept of a disjunctive database can be extended in a similar way. We show that some facts of commonsense knowledge can be represented by logic programs and disjunctive databases more easily when classical negation is available. Computationally, classical negation can be eliminated from extended programs by a simple preprocessor. Extended programs are identical to a special case of default theories in the sense of Reiter.

The expressive power of stratified logic programs

The study of negation in logic programming has been the topic of substantial research activity during the past several years, starting with the negation as failure semantics in Clark (1978), and Apt and van Emden (1982). More recently, a major direction of research has focused on the class of stratified logic programs, in which no predicate is defined recursively in terms of its own negation and which can be given natural semantics in terms of iterated fixpoints. Stratified logic programs were introduced and studied first by Chandra and Hare1 (1985), but soon attracted the interest of researchers from both database theory and artificial intelligence. Recent research work on stratified logic programs and their generalizations includes the papers by Apt, Blair, and Walker (1988), Van Gelder (1986) Lifschitz (1988), Przymusinski (1988), Apt and Pugin (1987) and others. At the same time, stratified logic programs became the choice for the treatment of negation in the NAIL ! system developed at Stanford University by Ullman and his co-workers (cf. Morris

RECOGNIZING NON-FLOUNDERING LOGIC PROGRAMS AND GOALS

The introduction of negation in Logic Programming using the Negation as Failure Rule causes some problems regarding the completeness of the SLDNF-Resolution proof procedure. One of the causes of incompleteness arises when evaluating a non-ground negative literal. This is solved by forbidding these evaluations. Obviously, there is the possibility of having only non-ground negative literals in the goal (the floundering of the goal). There is a class of programs and goals (allowed) that has been proved to have the non-floundering property. In this paper an algorithm is proposed which recognizes a wider class of programs with this property and which is based on abstract interpretation techniques.

A transformational approach to negation in logic programming

A transformation technique is introduced which, given the Horn-clause definition of a set of predicates p i , synthesizes the definitions of new predicate p̃ i which can be used, under a suitable refutation procedure, to compute the finite failure set of p i . This technique exhibits some computational advantages, such as the possibility of computing nonground negative goals still preserving the capability of producing answers. The refutation procedure, named SLDN refutation, is proved sound and complete with respect to the completed program.

Negation in logic programming

We define a semantics for negation as failure in logic programming. Our semantics may be viewed as a cross between the approaches of Clark [5] and Fitting [7]. As does [7], our semantics corresponds well with real PROLOG in the standard examples used in the literature to illustrate problems with [5]. Also, PROLOG and the common variants of it are sound but not complete for our semantics. Unlike [7], our semantics is constructive, in that the set of supported queries is recursively enumerable. Thus, a complete interpreter exists in theory, although we point out that there are serious difficulties in building one that works well in practice.