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

Author:
Michele Basaldella

Independent Author, Cordenons, Italy

Independent Author, Cordenons, Italy
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ACM Transactions on Computational Logic Volume 22 Issue 1Article No.: 4pp 1–32https://doi.org/10.1145/3426471

Published:15 January 2021Publication History

ACM Transactions on Computational Logic

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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Index Terms

α β-Relations and the Actual Meaning of α-Renaming
1. Theory of computation

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Published in
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
School of Computer Science and Engineering, Hebrew University, Israel
Issue’s Table of Contents
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Publication History
- Published: 15 January 2021
- Accepted: 1 September 2020
- Revised: 1 May 2020
- Received: 1 June 2019
Published in tocl Volume 22, Issue 1

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Author Tags
α-renaming
λ-theory
Languages with binding operators
extensionality
lambda calculus
models of languages with binding operators
substitution
Qualifiers
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α β-Relations and the Actual Meaning of α-Renaming

ACM Transactions on Computational Logic

Abstract

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Cited By

Index Terms

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Extensional Higher-Order Paramodulation in Leo-III

Alpha-Structural Induction and Recursion for the Lambda Calculus in Constructive Type Theory