Negation as failure. II

AbstractThe use of the negation as failure rule in logic programming is often considered to be tantamount to reasoning from Clark's “completed data base” [2]. Continuing the investigations of Clark and Shepherdson [2,7], we show that this is not fully equivalent to negation as failure either using classical logic or the more appropriate intuitionistic logic. We doubt whether there is any simple and useful logical meaning of negation as failure in the general case, and study in detail some special kinds of data base where the relationship of the completed data base to negation as failure is closer, e.g. where the data base is definite Horn or hierarchic

Reasoning about Minimal Belief and Negation as Failure

We investigate the problem of reasoning in the propositional fragment of MBNF, the logic of minimal belief and negation as failure introduced by Lifschitz, which can be considered as a unifying framework for several nonmonotonic formalisms, including default logic, autoepistemic logic, circumscription, epistemic queries, and logic programming. We characterize the complexity and provide algorithms for reasoning in propositional MBNF. In particular, we show that entailment in propositional MBNF lies at the third level of the polynomial hierarchy, hence it is harder than reasoning in all the above mentioned propositional formalisms for nonmonotonic reasoning. We also prove the exact correspondence between negation as failure in MBNF and negative introspection in Moore's autoepistemic logic

Negation-as-failure considered harmful

In logic programs, negation-as-failure has been used both for representing negative information and for providing default nonmonotonic inference. In this paper we argue that this twofold role is not only unnecessary for the expressiveness of the language, but it also plays against declarative programming, especially if further negation symbols such as strong negation are also available. We therefore propose a new logic programming approach in which negation and default inference are independent, orthogonal concepts. Semantical characterization of this approach is given in the style of answer sets, but other approaches are also possible. Finally, we compare them with the semantics for logic programs with two kinds of negation.Red de Universidades con Carreras en Informática (RedUNCI

The consistency of negation as failure

AbstractClark's attempt [1] to validate negation as failure in first order logic is shown to contain some fundamental errors. In particular, we show that the motivation for the completed database, the definition of the completed database, and the attempt to validate negation as failure in terms of it are illogical, that the completed database cannot be regarded as the intended meaning of the database, and that the closed world assumption is generally absurd and, in any case, irrelevant. A validation is given using a consistent first order extension of the database and hence in the only terms which appear to make any sense, namely, consistency with the database. However, it seems that the query evaluation process, with negation interpreted as failure, is of no practical use as a theorem prover

A linear axiomatization of negation as failure

AbstractThis paper is concerned with the axiomatization of success and failure in propositional logic programming. It deals with the actual implementation of SLDNF in PROLOG, as opposed to the general nondeterministic SLDNF evaluation method. Given any propositional program P, a linear theory LTP is defined (the linear translation of P) and the following results are proved for any literal A: soundness of PROLOG evaluation (if the goal A PROLOG-succeeds on P, then LTP⊢lin A, and if A PROLOG-fails on P, then LTP⊢lin A⊥), and completeness of PROLOG evaluation (if LTP⊢lin A, then the goal A PROLOG-succeeds on P, and if LTP⊢lin A⊥, then A PROLOG-fails on P). Here ⊢lin means provability in linear logic, and A⊥ is the linear negation of A

Learning logic programs with negation as failure

Normal logic programs are usually shorter and easier to write and understand than definite logic programs. As a consequence, it is worth investigating their learnability, if Inductive Logic Program- ming is to be proposed as an alternative tool for software development and Software Engineering at large. In this paper we present an exten- sion of the ILP system TRACY, called TRACY-not, able to learn normal logic programs. The method is proved to be sound, in the sense that it outputs a program which is complete and consistent w.r.t.the ex- amples, and complete, in the sense that it does find a solution when it exists. Compared to learning systems based on extensionality,TRACY and TRACY not are less dependent on the kind and number of training examples, which is due to the intensional evaluation of the hypothe- ses and, for TRACY-not, to the possibility to have restricted hypothesis spaces through the use of negation

Belnap's epistemic states and negation-as-failure

Generalizing Belnap's system of epistemic states [Bel77] we obtain the system of disjunctive factbases which is the paradigm for all other kinds of disjunctive knowledge bases. Disjunctive factbases capture the nonmonotonic reasoning based on paraminimal models. In the schema of a disjunctive factbase, certain predicates of the resp. domain are declared to be exact, i.e. two-valued, and in turn some of these exact predicates are declared to be subject to the Closed-World Assumption (CWA). Thus, we distinguish between three kinds of predicates: inexact predicates, exact predicates subject to the CWA, and exact predicates not subject to the CWA

A Polynomial Translation of Logic Programs with Nested Expressions into Disjunctive Logic Programs: Preliminary Report

Nested logic programs have recently been introduced in order to allow for arbitrarily nested formulas in the heads and the bodies of logic program rules under the answer sets semantics. Nested expressions can be formed using conjunction, disjunction, as well as the negation as failure operator in an unrestricted fashion. This provides a very flexible and compact framework for knowledge representation and reasoning. Previous results show that nested logic programs can be transformed into standard (unnested) disjunctive logic programs in an elementary way, applying the negation as failure operator to body literals only. This is of great practical relevance since it allows us to evaluate nested logic programs by means of off-the-shelf disjunctive logic programming systems, like DLV. However, it turns out that this straightforward transformation results in an exponential blow-up in the worst-case, despite the fact that complexity results indicate that there is a polynomial translation among both formalisms. In this paper, we take up this challenge and provide a polynomial translation of logic programs with nested expressions into disjunctive logic programs. Moreover, we show that this translation is modular and (strongly) faithful. We have implemented both the straightforward as well as our advanced transformation; the resulting compiler serves as a front-end to DLV and is publicly available on the Web.Comment: 10 pages; published in Proceedings of the 9th International Workshop on Non-Monotonic Reasonin