Non-parametric parametricity

Type abstraction and intensional type analysis are features seemingly at odds—type abstraction is intended to guarantee parametricity and representation independence, while type analysis is inherently non-parametric. Recently, however, several researchers have proposed and implemented “dynamic type generation” as a way to reconcile these features. The idea is that, when one defines an abstract type, one should also be able to generate at runtime a fresh type name, which may be used as a dynamic representative of the abstract type for purposes of type analysis. The question remains: in a language with non-parametric polymorphism, does dynamic type generation provide us with the same kinds of abstraction guarantees that we get from parametric polymorphism?Our goal is to provide a rigorous answer to this question. We define a step-indexed Kripke logical relation for a language with both non-parametric polymorphism (in the form of type-safe cast) and dynamic type generation. Our logical relation enables us to establish parametricity and representation independence results, even in a non-parametric setting, by attaching arbitrary relational interpretations to dynamically generated type names. In addition, we explore how programs that are provably equivalent in a more traditional parametric logical relation may be “wrapped” systematically to produce terms that are related by our non-parametric relation, and vice versa. This leads us to develop a “polarized” variant of our logical relation, which enables us to distinguish formally between positive and negative notions of parametricity.

A Note on a Parametric Relation of Variance Balanced Block Designs

The object of this communication is to prove the important parametric relation [Formula: see text] and hence [Formula: see text] for all connected Variance Balanced (VB) block designs, excepting the cquireplicatcd orthogonal designs which arc trivially VB.