Evaluating functions as processes

TL;DR

This paper explores the relationship between lambda-calculus evaluation strategies and their simulation in the pi-calculus, revealing a tight correspondence with linear weak head reduction and its call-by-value variant.

Contribution

It establishes a precise connection between environment-based abstract machine evaluation and linear weak head reduction in the pi-calculus context.

Findings

Evaluation via abstract machines aligns with linear weak head reduction.

The correspondence extends to call-by-value evaluation strategies.

The relation between lambda-calculus and pi-calculus simulations is as tight as possible.

Abstract

A famous result by Milner is that the lambda-calculus can be simulated inside the pi-calculus. This simulation, however, holds only modulo strong bisimilarity on processes, i.e. there is a slight mismatch between beta-reduction and how it is simulated in the pi-calculus. The idea is that evaluating a lambda-term in the pi-calculus is like running an environment-based abstract machine, rather than applying ordinary beta-reduction. In this paper we show that such an abstract-machine evaluation corresponds to linear weak head reduction, a strategy arising from the representation of lambda-terms as linear logic proof nets, and that the relation between the two is as tight as it can be. The study is also smoothly rephrased in the call-by-value case, introducing a call-by-value analogous of linear weak head reduction.