Programming in three-valued logic

AbstractThe 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

Negation by default and unstratifiable logic programs

AbstractThe default approach to the theory of logic programs (and deductive databases) is based on the interpretation of negation by default rules. Default logic is a well-suited formalism to express the Closed World Assumption and to define the declarative semantics of stratifiable logic programs. The case of disjunctive consequences in rules is treated. General logic programs may not have a meaning with respect to default semantics. The contribution of the paper is to exhibit an interesting class of programs having a default semantics, called effectively stratifiable programs. This time, disjunctive consequences are not considered. Effective stratification is a weaker constraint than stratification, local stratification and weak stratification. Besides enlarging the class of stratifiable logic programs, the paper contributes to provide a constructive definition of well-founded models of logic programs. The class of effectively stratifiable logic programs matches the class of programs having a total well-founded model and in general, the default semantics extends the well-founded semantics