Sequence Types and Infinitary Semantics

Vial, Pierre

doi:10.48550/arxiv.2102.07515

Cited by 1 publication

(1 citation statement)

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“…Even if the "plain" infinitary λ-calculus does not enjoy confluence, several results of confluence and of normalisation modulo "meaningless" terms have been established [Ken+97;Cza14;Cza20], as well as a standardisation theorem using coinductive techniques [EP13]. Some normalisation properties have also been characterised using non-idempotent intersection types [Via17;Via21].…”

Section: Introductionmentioning

confidence: 99%

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

Cerda,

Auclair

2023

Logical Methods in Computer Science

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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.

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