Profinite lambda-terms and parametricity

Abstract

Combining ideas coming from Stone duality and Reynolds parametricity, we formulate in a clean and principled way a notion of profinite λ-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 λ-terms as a projective limit of finite sets of usual λterms, 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 λ-term coming from Stone duality lives in perfect harmony with the principles of Reynolds parametricity. In addition, we show that the notion of profinite λ-term is compositional by constructing a cartesian closed category of profinite λ-terms, and we establish that the embedding from λ-terms modulo βηconversion to profinite λ-terms is faithful using Statman's finite completeness theorem. Finally, we prove a parametricity theorem for Church encodings of word and ranked tree languages, which states that every parametric family of elements in the finite standard model is the interpretation of a profinite λ-term. This result shows that our notion of profinite λ-term conservatively extends the existing notion of profinite word and provides a natural framework for defining and studying profinite trees.