Recursive types are not conservative over F≤

Abstract

F≤ is a type system used to study the integration of inclusion and parametric polymorphism. F≤ does not include a notion of recursive types, but extensions of F≤ with recursive types are widely used as a basis for foundational studies about the type systems of functional and object-oriented languages. In this paper we show that adding recursive types results in a non conservative extension of the system. This means that the algorithm for F≤ subtyping (the kernel of the algorithm for F≤ typing) is no longer complete for the extended system, even when it is applied only to judgements where no recursive type appears, and that most of the proofs of known properties of F≤ do not hold for the extended system; this is the case, for example, for Pierce's proof of undecidability of F≤. However, we prove that this non conservativity is limited to a very special class of subtyping judgements, the “diverging judgements” introduced in [Ghe]. This last result implies that the extension of F≤ with recursive types could be still useful for practical purposes.

This work was carried out with the partial support of E.E.C., Esprit Basic Research Action 6309 FIDE2 and of “Progetto Finalizzato Sistemi Informatici e Calcolo Parallelo” of the Italian National Research Council under grant No.91.00877.PF69.