Well-Typed Logic Programs Are not Wrong

TL;DR

This paper explores the type systems of logic programs, proposing a more general condition than the traditional head condition to ensure subject reduction, and discusses its implications and relation to polymorphic recursion.

Contribution

It introduces a generalized condition for type preservation in logic programs, relaxing the traditional head condition and linking it to polymorphic recursion.

Findings

Proposes a more general condition for subject reduction in logic programs.

Establishes a connection between the head condition and polymorphic recursion.

Discusses implications for the design of type systems in logic programming.

Abstract

We consider prescriptive type systems for logic programs (as in Goedel 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.