Finitary Simulation of Infinitary $\beta$-Reduction via Taylor Expansion, and Applications

TL;DR

This paper extends Taylor expansion techniques to the infinitary lambda calculus, enabling finitary simulation of infinitary beta-reduction and simplifying proofs of normalization and confluence.

Contribution

It introduces a Taylor expansion for the infinitary lambda calculus and proves a generalized Commutation theorem, connecting normal forms to B"ohm trees.

Findings

Infinitary beta-reduction can be simulated via Taylor expansion.

The resource calculus reduction remains finite, confluent, and terminating.

Provides simplified proofs of normalization and confluence in infinitary lambda calculus.

Abstract

Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion of $\lambda$-terms has been broadly used as a tool to approximate the terms of several variants of the $\lambda$-calculus. Many results arise from a Commutation theorem relating the normal form of the Taylor expansion of a term to its B\"ohm tree. This led us to consider extending this formalism to the infinitary $\lambda$-calculus, since the $\Lambda_{\infty}^{001}$ version of this calculus has B\"ohm trees as normal forms and seems to be the ideal framework to reformulate the Commutation theorem. We give a (co-)inductive presentation of $\Lambda_{\infty}^{001}$. We define a Taylor expansion on this calculus, and state that the infinitary $\beta$-reduction can be simulated through this Taylor expansion. The target language is the usual resource calculus, and in particular the resource reduction remains finite, confluent and terminating. Finally, we state the generalised Commutation theorem and use our results to provide simple proofs of some normalisation and confluence properties in the infinitary $\lambda$-calculus.