On Isomorphism of "Functional" Intersection and Union Types

TL;DR

This paper investigates type isomorphism in systems with intersection and union types, providing a complex characterization and a main result based on isomorphism preserving reduction.

Contribution

It offers a novel characterization of type isomorphism in systems with intersection and union types, addressing the non-congruence issue.

Findings

A condition for type isomorphism is established.

Type isomorphism is characterized via interpretation as functions.

The approach uses Scott's inverse limit lambda-model.

Abstract

Type isomorphism is useful for retrieving library components, since a function in a library can have a type different from, but isomorphic to, the one expected by the user. Moreover type isomorphism gives for free the coercion required to include the function in the user program with the right type. The present paper faces the problem of type isomorphism in a system with intersection and union types. In the presence of intersection and union, isomorphism is not a congruence and cannot be characterised in an equational way. A characterisation can still be given, quite complicated by the interference between functional and non functional types. This drawback is faced in the paper by interpreting each atomic type as the set of functions mapping any argument into the interpretation of the type itself. This choice has been suggested by the initial projection of Scott's inverse limit lambda-model. The main result of this paper is a condition assuring type isomorphism, based on an isomorphism preserving reduction.