Normalization proofs for the un-typed μμ'-calculus

A long-standing open problem of Parigot has been solved by David and Nour, namely, they gave a syntactical and arithmetical proof of the strong normalization of the untyped μμ'-reduction. In connection with this, we present in this paper a proof of the weak normalization of the μ and μ'-ru...

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Published in:AIMS mathematics Vol. 5; no. 4; pp. 3702 - 3713
Main Authors: Battyányi, Péter, Nour, Karim
Format: Journal Article
Language:English
Published: AIMS Press 2020
Subjects:
normalization algorithm
strong normalization
weak normalization
λμ-calculus
μμ'-calculus
ISSN:2473-6988, 2473-6988
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Abstract A long-standing open problem of Parigot has been solved by David and Nour, namely, they gave a syntactical and arithmetical proof of the strong normalization of the untyped μμ'-reduction. In connection with this, we present in this paper a proof of the weak normalization of the μ and μ'-rules. The algorithm works by induction on the complexity of the term. Our algorithm does not necessarily give a unique normal form: sometimes we may get different normal forms depending on our choice. We also give a simpler proof of the strong normalization of the same reduction. Our proof is based on a notion of a norm defined on the terms.
AbstractList A long-standing open problem of Parigot has been solved by David and Nour, namely, they gave a syntactical and arithmetical proof of the strong normalization of the untyped μμ'-reduction. In connection with this, we present in this paper a proof of the weak normalization of the μ and μ'-rules. The algorithm works by induction on the complexity of the term. Our algorithm does not necessarily give a unique normal form: sometimes we may get different normal forms depending on our choice. We also give a simpler proof of the strong normalization of the same reduction. Our proof is based on a notion of a norm defined on the terms.
Author Nour, Karim
Battyányi, Péter
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Cites_doi 10.1007/s001530050076
10.1017/S0960129598002667
10.1016/j.tcs.2012.02.027
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CorporateAuthor 2 Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS, LAMA, LIMD, 73000 Chambéry, France
1 Department of Computer Science, Faculty of Informatics, University of Debrecen, Kassai út 26, 4028 Debrecen, Hungary
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References A. Saurin (6)
8
1
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P. de Groote (4)
K. Nour (3)
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  article-title: i>Arithmetical proofs of strong normalization results for symmetric lambda calculi</i
– ident: 1
  article-title: M. Parigot, Free Deduction: An Analysis of "Computations" in Classical Logic, In: A. Voronkov, editors. Logic Programming Lecture Notes in Artificial Intelligence 592, Berlin,
– ident: 2
  article-title: M. Parigot, λμ-calculus: An algorithmic interpretation of classical natural deduction, In: A. Voronkov, editors. Logic Programming and Automated Reasoning, LPAR 1992, Lecture Notes in Artificial Intelligence 624, Berlin,
– ident: 3
  article-title: i>La valeur d'un entier classique en λμ-calcul</i
  publication-title: Archive Math. Logic
  doi: 10.1007/s001530050076
– ident: 4
  article-title: i>An environment machine for the λμ-calculus</i
  publication-title: Math. Struct. Comput. Sci.
  doi: 10.1017/S0960129598002667
– ident: 6
  article-title: i>Böhm theorem and Böhm trees for the Λμ-calculus</i
  publication-title: Theor. Comput. Sci.
  doi: 10.1016/j.tcs.2012.02.027
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SubjectTerms normalization algorithm
strong normalization
weak normalization
λμ-calculus
μμ'-calculus
Title Normalization proofs for the un-typed μμ'-calculus
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