Profinite lambda-terms and parametricity

TL;DR

This paper introduces a new notion of profinite lambda-terms derived from Stone duality and Reynolds parametricity, generalizing automata theory concepts to lambda calculus and establishing a rich mathematical framework.

Contribution

It formulates a principled definition of profinite lambda-terms, shows their harmony with parametricity, and constructs a cartesian closed category including an embedding from standard lambda-terms.

Findings

Profinite lambda-terms form a cartesian closed category.

Embedding from lambda-terms to profinite lambda-terms is faithful.

Profinite words are homeomorphic to profinite lambda-terms via Church encoding.

Abstract

Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite lambda-term which, we show, generalizes at every type the traditional notion of profinite word coming from automata theory. We start by defining the Stone space of profinite lambda-terms as a projective limit of finite sets of usual lambda-terms, considered modulo a notion of equivalence based on the finite standard model. One main contribution of the paper is to establish that, somewhat surprisingly, the resulting notion of profinite lambda-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite lambda-term is compositional by constructing a cartesian closed category of profinite lambda-terms, and we establish that the embedding from lambda-terms modulo beta-eta-conversion to profinite lambda-terms is faithful using Statman's finite completeness theorem. Finally, we prove that the traditional Church encoding of finite words into lambda-terms can be extended to profinite words, and leads to a homeomorphism between the space of profinite words and the space of profinite lambda-terms of the corresponding Church type.