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α β-Relations and the Actual Meaning of α-Renaming

Published: 15 January 2021 Publication History

Abstract

In this work we provide an alternative, and equivalent, formulation of the concept of λ-theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We call α β-relations our alternative versions of λ-theories. We also clarify the actual role of α-renaming in the lambda calculus: it expresses a property of extensionality for a certain class of terms. To motivate the necessity of α-renaming, we construct an unusual denotational model of the lambda calculus that validates all structural and beta conditions but not α-renaming. The article also has a survey character.

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cover image ACM Transactions on Computational Logic
ACM Transactions on Computational Logic  Volume 22, Issue 1
January 2021
262 pages
ISSN:1529-3785
EISSN:1557-945X
DOI:10.1145/3436816
  • Editor:
  • Orna Kupferman
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 15 January 2021
Accepted: 01 September 2020
Revised: 01 May 2020
Received: 01 June 2019
Published in TOCL Volume 22, Issue 1

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Author Tags

  1. α-renaming
  2. λ-theory
  3. Languages with binding operators
  4. extensionality
  5. lambda calculus
  6. models of languages with binding operators
  7. substitution

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