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

Making prolog more expressive

AbstractThis paper introduces extended programs and extended goals for logic programming. A clause in an extended program can have an arbitrary first-order formula as its body. Similarly, an extended goal can have an arbitrary first-order formula as its body. The main results of the paper are the soundness of the negation as failure rule and SLDNF-resolution for extended programs and goals. We show how the increased expressibility of extended programs and goals can be easily implemented in any PROLOG system which has a sound implementation of the negation as failure rule. We also show how these ideas can be used to implement first-order logic as a query language in a deductive database system. An application to integrity constraints in deductive database systems is also given

A Denotational Semantics for First-Order Logic

In Apt and Bezem [AB99] (see cs.LO/9811017) we provided a computational interpretation of first-order formulas over arbitrary interpretations. Here we complement this work by introducing a denotational semantics for first-order logic. Additionally, by allowing an assignment of a non-ground term to a variable we introduce in this framework logical variables. The semantics combines a number of well-known ideas from the areas of semantics of imperative programming languages and logic programming. In the resulting computational view conjunction corresponds to sequential composition, disjunction to ``don't know'' nondeterminism, existential quantification to declaration of a local variable, and negation to the ``negation as finite failure'' rule. The soundness result shows correctness of the semantics with respect to the notion of truth. The proof resembles in some aspects the proof of the soundness of the SLDNF-resolution.Comment: 17 pages. Invited talk at the Computational Logic Conference (CL 2000). To appear in Springer-Verlag Lecture Notes in Computer Scienc

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

Logic Negation with Spiking Neural P Systems

Nowadays, the success of neural networks as reasoning systems is doubtless. Nonetheless, one of the drawbacks of such reasoning systems is that they work as black-boxes and the acquired knowledge is not human readable. In this paper, we present a new step in order to close the gap between connectionist and logic based reasoning systems. We show that two of the most used inference rules for obtaining negative information in rule based reasoning systems, the so-called Closed World Assumption and Negation as Finite Failure can be characterized by means of spiking neural P systems, a formal model of the third generation of neural networks born in the framework of membrane computing.Comment: 25 pages, 1 figur

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