Completeness of a Procedure to Calculate Logic Programs by Replacement with the Bodies of Rules and a Proof Procedure with Respect to a General Axiom

Abstract

In this paper, a logic program is considered as the union of a set of rules and an axiom which defines basic predicates and functions, and a procedure to calculate logic programs is considered as the combination of a replacement procedure with the bodies of the rules and a proof procedure with respect to the axiom. It is proved that a goal is a logical consequence of Δ ∪ Γ if and only if there exists n such that the logical formula obtained by replacing atomic formulae in the goal n times is a logical consequence of Δ, where Γ is the set of the rules, and Δ is the axiom. Moreover, conditions concerning Γ and Δ are described.