Relational Parametricity for Computational Effects

According to Strachey, a polymorphic program is parametric if it applies a uniform algorithm independently of the type instantiations at which it is applied. The notion of relational parametricity, introduced by Reynolds, is one possible mathematical formulation of this idea. Relational parametricity provides a powerful tool for establishing data abstraction properties, proving equivalences of datatypes, and establishing equalities of programs. Such properties have been well studied in a pure functional setting. Many programs, however, exhibit computational effects. In this paper, we develop a framework for extending the notion of relational parametricity to languages with effects.

1. Introduction

The theory of relational parametricity, proposed by Reynolds [21], provides a powerful framework for establishing properties of polymorphic programs and their types. Such properties include the “theorems for free” of Wadler [26], universal properties for datatype encodings, and representation independence properties for abstract datatypes. These results are well established, see e.g. [18], for the pure Girard/Reynolds second-order -calculus (a. k. a. system F) which provides a concise yet remarkably powerful calculus of typed total functions.