A finiteness structure on resource terms

TL;DR

This paper introduces a finiteness structure on resource lambda-terms that ensures a well-behaved linear topological vector space framework, preventing infinite coefficients during reductions in typed lambda-calculus extensions.

Contribution

It proposes a novel finiteness structure on resource terms that maintains coherence properties and controls infinite coefficients in algebraic lambda-calculus extensions.

Findings

Finiteness structure induces a linearly topologized vector space.

Prevents infinite coefficients during reduction.

Maintains coherence properties in typed settings.

Abstract

In our paper "Uniformity and the Taylor expansion of ordinary lambda-terms" (with Laurent Regnier), we studied a translation of lambda-terms as infinite linear combinations of resource lambda-terms, from a calculus similar to Boudol's lambda-calculus with resources and based on ideas coming from differential linear logic and differential lambda-calculus. The good properties of this translation wrt. beta-reduction were guaranteed by a coherence relation on resource terms: normalization is "linear and stable" (in the sense of the coherence space semantics of linear logic) wrt. this coherence relation. Such coherence properties are lost when one considers non-deterministic or algebraic extensions of the lambda-calculus (the algebraic lambda-calculus is an extension of the lambda-calculus where terms can be linearly combined). We introduce a "finiteness structure" on resource terms which induces a linearly topologized vector space structure on terms and prevents the appearance of infinite coefficients during reduction, in typed settings.