Programming in three-valued logic

Abstract

The aim of this paper is to propose a logical and algebraic theory which seems well-suited to logic programs with negation and deductive databases. This theory has similar properties to those of Prolog theory limited to programs with Horn clauses and thus can be considered as an extension of the usual theory. This parallel with logic programming without negation lies in the introduction of a third truth value (Indefinite) and of a new non-monotonic implication connective. Our proposition is different from the other ways of introducing a third truth value already used in Logic Programming and databases but it is somehow related to some of them, especially to Fitting's theory. We introduce a “consequence” operator associated with a logic program with negation which extends the operator of Apt and Van Emden. In the case of a consistent program, the post-fixpoints of this operator are the models of the program as they are usually. This operator is related to Fitting's one, the relation being obtained by completing the program. We finally give an operational semantics for a program with negation by the obtention of a three-valued interpreter from a bivalued one.