Dependent Pearl: Normalization by realizability

Abstract

For those of us who generally live in the world of syntax, semantic proof techniques such as reducibility, realizability or logical relations seem somewhat magical despite -- or perhaps due to -- their seemingly unreasonable effectiveness. Why do they work? At which point in the proof is "the real work" done? Hoping to build a programming intuition of these proofs, we implement a normalization argument for the simply-typed lambda-calculus with sums: instead of a proof, it is described as a program in a dependently-typed meta-language. The semantic technique we set out to study is Krivine's classical realizability, which amounts to a proof-relevant presentation of reducibility arguments -- unary logical relations. Reducibility assigns a predicate to each type, realizability assigns a set of realizers, which are abstract machines that extend lambda-terms with a first-class notion of contexts. Normalization is a direct consequence of an adequacy theorem or "fundamental lemma", which states that any well-typed term translates to a realizer of its type. We show that the adequacy theorem, when written as a dependent program, corresponds to an evaluation procedure. In particular, a weak normalization proof precisely computes a series of reduction from the input term to a normal form. Interestingly, the choices that we make when we define the reducibility predicates -- truth and falsity witnesses for each connective -- determine the evaluation order of the proof, with each datatype constructor behaving in a lazy or strict fashion. While most of the ideas in this presentation are folklore among specialists, our dependently-typed functional program provides an accessible presentation to a wider audience. In particular, our work provides a gentle introduction to abstract machine calculi which have recently been used as an effective research vehicle.