Logical Relations as Types: Proof-Relevant Parametricity for Program Modules

Abstract

The theory of program modules is of interest to language designers not only for its practical importance to programming, but also because it lies at the nexus of three fundamental concerns in language design: the phase distinction , computational effects , and type abstraction . We contribute a fresh “synthetic” take on program modules that treats modules as the fundamental constructs, in which the usual suspects of prior module calculi (kinds, constructors, dynamic programs) are rendered as derived notions in terms of a modal type-theoretic account of the phase distinction. We simplify the account of type abstraction (embodied in the generativity of module functors) through a lax modality that encapsulates computational effects, placing projectibility of module expressions on a type-theoretic basis. Our main result is a (significant) proof-relevant and phase-sensitive generalization of the Reynolds abstraction theorem for a calculus of program modules, based on a new kind of logical relation called a parametricity structure . Parametricity structures generalize the proof-irrelevant relations of classical parametricity to proof- relevant families, where there may be non-trivial evidence witnessing the relatedness of two programs—simplifying the metatheory of strong sums over the collection of types, for although there can be no “relation classifying relations,” one easily accommodates a “family classifying small families.” Using the insight that logical relations/parametricity is itself a form of phase distinction between the syntactic and the semantic, we contribute a new synthetic approach to phase separated parametricity based on the slogan logical relations as types , by iterating our modal account of the phase distinction. We axiomatize a dependent type theory of parametricity structures using two pairs of complementary modalities (syntactic, semantic) and (static, dynamic), substantiated using the topos theoretic Artin gluing construction. Then, to construct a simulation between two implementations of an abstract type, one simply programs a third implementation whose type component carries the representation invariant.

Strict universes for Grothendieck topoi

“…Artin gluing is used by computer scientists to prove metatheorems for type theories and programming languages such as normalization, canonicity, decidability, parametricity, conservativity, and computational adequacy. Sterling and Harper [SH21] have introduced synthetic Tait computability as an abstraction for working in the internal language of glued topoi, taking the realignment law (U8) in its internal form (see Section 5) as a basic axiom.…”

“…It is of practical interest to employ Martin-Löf type theory (MLTT) as an internal language for a variety of categories. In addition to the standard applications of internal methods to mathematics, the existence of models of MLTT in topoi is a critical ingredient for a number of recent results in type theory and programming languages, including the generalized abstraction theorem of Sterling and Harper [SH21] and the proofs of normalization for cubical type theory and multi-modal dependent type theory [Gra21;SA21].…”

Normalization for Cubical Type Theory

“…In this paper, we combine ideas of Orton and Pitts with the synthetic Tait computability (STC) theory of Sterling and Harper [55], which factors out bureaucratic aspects of syntactic metatheory. In STC, one considers an extensional type theory whose types are (proof-relevant) logical relations; the underlying syntax is exposed via a proof-irrelevant proposition syn under which the syntactic part of a logical relation is projected.…”

“…The past several years have witnessed an explosion in semantic computability techniques for establishing syntactic metatheorems [4,20,22,26,38,51,55,56,61]. What makes semantic computability different from "free-hand" computability is that it is expressed as a gluing model, parameterized in the generic model of the type theory; hence one is always working with typed terms up to judgmental equality.…”

“…Gratzer et al [29] used gluing to prove canonicity for a general multi-modal dependent type theory. Sterling and Harper [55] employ a different gluing argument to establish a proof-relevant generalization of the Reynolds Abstraction Theorem for a calculus of ML modules. Gratzer and Sterling [28] used gluing to establish the conservativity of higher-order judgments for dependent type theories.…”