A Theory of Parametric Polymorphism and an Application

Abstract: This thesis revisits the well-known notion of parametric polymorphism in the light of modern developments in type-theory. Additionally, applications of parametric polymorphism are also presented. The first part of the thesis presents a theoretical investigation of the semantics of parametric polymorphism of and within type-theories with dependent types. It is shown how the meaning of polymorphic, possibly dependent, types can be reflected within type-theory itself, via a simple syntactic transformation. This self-referential property opens the door to internalise the transformation in type-theory, and we study one possible way to do so. We also examine how the translation relates to various specific features of type-theory, such as proof irrelevance and realizability. The second part is concerned with applications of parametric polymorphism relevant to software engineers. Two applications are presented. First, we present a schema to reduce polymorphic properties to equivalent monomorphic properties, for the purpose of testing. Our proof uses parametricity and properties of initial algebras. Second, we compare Haskell type classes with their counterpart in C++. We contribute to an improved understanding of language support for generic programming, and hope to facilitate discussion between mainstream and functional programming language communities.