A finiteness structure on resource terms

Thomas Ehrhard (1)

1 PPS - Preuves, Programmes et Systèmes

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.

Mots clés: lambda-calculus; linear logic; lambda-calculus with resources; differential linear logic; finiteness spaces; topological vector spaces; system F

Domaines: Logique en informatique [cs.LO]

Thomas Ehrhard. A finiteness structure on resource terms. 25th Annual IEEE Symposium on Logic in Computer Science, LICS 2010, Jul 2010, Edinburgh, United Kingdom. pp.402-410, ⟨10.1109/LICS.2010.38⟩. ⟨hal-00448431⟩