Well-Typed Logic Programs Are not Wrong

Abstract : We consider prescriptive type systems for logic programs (as in Gödel or Mercury). In such systems, the typing is static, but it guarantees an operational property: if a program is «well-typed», then all derivations starting in a «well-typed» query are again «well-typed». This property has been called subject reduction. We show that this property can also be phrased as a property of the proof-theoretic semantics of logic programs, thus abstracting from the usual operational (top-down) semantics. This proof-theoretic view leads us to questioning a condition which is usually considered necessary for subject reduction, namely the head condition. It states that the head of each clause must have a type which is a variant (and not a proper instance) of the declared type. We provide a more general condition, thus reestablishing a certain symmetry between heads and body atoms. The condition ensures that in a derivation, the types of two unified terms are themselves unifiable. We discuss possible implications of this result. We also discuss the relationship between the head condition and polymorphic recursion, a concept known in functional programming.