On Normal Forms and Equivalence for Logic Programs

It is known that larger classes of formulae than Horn clauses may be used as logic programming languages. One such class of formulae is hereditary Harrop formulae, for which an operational notion of provability has been studied, and it is known that operational provability corresponds to prov-ability in intuitionistic logic. In this paper we discuss the notion of a normal form for this class of formulae, and show how this may be given by removing disjunctions and existential quantiications from programs. Whilst the normal form of the program preserves operational provability, there are operationally equivalent programs which are not intuitionistically equivalent. As it is known that classical logic is too strong to precisely capture operational provability for larger classes of programs than Horn clauses, the appropriate logic in which to study questions of equivalence is an intermediate logic. We explore the nature of the required logic, and show that this may be obtained by the addition of the Independence of Premise axioms to intuitionistic logic. We show how equivalence in this logic captures the notion of operational provability, in that logically equivalent programs are operationally equivalent. This result suggests that the natural logic in which to study logic programming is a slightly stronger constructive logic than intuitionistic logic.