Subtyping and Parametricity

Abstract

In this paper we study the interaction of subtyping and parametricity. We describe a logic for a programming language with parametric polymorphism and subtyping. The logic supports the formal definition and use of relational parametricity. We give two models for it, and compare it with other formal systems for the same language. In particular, we examine the "Penn interpretation" of subtyping as implicit coercion. Without subtyping, parametricity yields, for example, an encoding of abstract types and of initial algebras, with the corresponding proof principles of simulation and induction. With subtyping, we obtain partially abstract types and certain initial order-sorted algebras, and may derive proof principles for them. 1 Introduction A function is polymorphic if it works on inputs of several types. We may distinguish various notions of polymorphism, particularly parametric polymorphism (e.g. [Rey83]) and subtype polymorphism (e.g. [CW85]). These may exist in isolation, as in ML [MT..