Programming in Three-valued Logic

The aim of this paper is to propose a logical and algebraic theory which seems well-suited to logic programs with negatior, and deductive databases. This theory has similar properties to those of Prolog theory limited to programs with Horn clauses and tbns can be considered as sn 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 the. " , espeklly to Fitting's theor,l. We introduce a " consequencs " 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 uf this opera:or are the models of the program as they are usually. This operator is related 10 Fittine's one, the relation being obtained by completing the program. We ficslly $ve an operational semnntics for B program with negation by the abtention ofa three-valued interpreter from a bivalued one. When treating negation in logic programming, many problems must be faced and attempts to solve them may be sorted out more or less according to their faithfulness to negation as failure, which is the negation really used in Prolog [4, 10, 14, 22, 231. It does not seem possible to stay close to Prolog while having an easy axiomatical semantics: we are not sure to find a sound and complete semantics for negation as faiiure. A negation other than negation as failure is also used in expert systems working in forwards chaining. The theory we propose here is quite general and can be applied to both Logic Programming and this negation. It has good theoretical and algebraic properties similar to those obtained in logic programming with negation. The parallel lies in the introduction of a third truth value, indefinite and a new non-monotonic implication connective denoted by +. We thus obtain a very simple way of treating negation