Semantic Analysis of Normalisation by Evaluation for Typed Lambda Calculus

TL;DR

This paper offers a categorical and algebraic analysis of normalisation by evaluation for typed lambda calculus, unifying definability and normalisation, and implementing a normalisation program in a dependently-typed language.

Contribution

It introduces a unified categorical framework for normalisation by evaluation, combining extensional and intensional approaches, and provides a practical normalisation program.

Findings

Unified definability and normalisation results.

Normalisation function derived within a simply typed metatheory.

A dependently-typed normalisation-by-evaluation program.

Abstract

This paper studies normalisation by evaluation for typed lambda calculus from a categorical and algebraic viewpoint. The first part of the paper analyses the lambda definability result of Jung and Tiuryn via Kripke logical relations and shows how it can be adapted to unify definability and normalisation, yielding an extensional normalisation result. In the second part of the paper the analysis is refined further by considering intensional Kripke relations (in the form of Artin glueing) and shown to provide a function for normalising terms, casting normalisation by evaluation in the context of categorical glueing. The technical development includes an algebraic treatment of the syntax and semantics of the typed lambda calculus that allows the definition of the normalisation function to be given within a simply typed metatheory. A normalisation-by-evaluation program in a dependently-typed functional programming language is synthesised.