Game Semantics and Normalization by Evaluation

Abstract

International audienceWe show that Hyland and Ong's game semantics for PCF can be presented using normalization by evaluation (nbe). We use the bijective correspondence between innocent well-bracketed strategies and PCF Böhm trees, and show how operations on PCF Böhm trees, such as composition, can be computed lazily and simply by nbe. The usual equations characteristic of games follow from the nbe construction without reference to low-level game-theoretic machinery. As an illustration, we give a Haskell program computing the application of innocent strategies

A Compositional Cost Model for the λ-calculus

“…One possibility is normalization by evaluation [25]: computing the denotation of a term as the element of a datatype in a high-level language, which can then be reified back to its normal form. Implementing this lazily allows infinitary normal forms (Böhm trees) to be computed [26], [27], and has been used to describe the composition of innocent strategies [28]. This is potentially a form of complete laziness, because higher-order functions are represented as shareable elements of lazy datatypes.…”