Formal Parametric Polymorphism

A polymorphic function is parametric if its behavior does not depend on the type at which it is instantiated. Starting with Reynolds’s work, the study of parametricity is typically semantic. In this paper, we develop a syntactic approach to parametricity, and a formal system that embodies this approach, called system R . Girard’s system F deals with terms and types; R is an extension of F that deals also with relations between types. In R , it is possible to derive theorems about functions from their types, or “theorems for free”, as Wadler calls them. An easy “theorem for free” asserts that the type ∀(X)X→Bool contains only constant functions; this is not provable in F. There are many harder and more substantial examples. Various metatheorems can also be obtained, such as a syntactic version of Reynolds’s abstraction theorem.