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International Conference on Rewriting Techniques and Applications

RTA 2002: Rewriting Techniques and Applications pp 24–50Cite as

Axiomatic Rewriting Theory VI: Residual Theory Revisited

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2378))

Abstract

Residual theory is the algebraic theory of confluence for the λ-calculus, and more generally conflict-free rewriting systems (=without critical pairs). The theory took its modern shape in Lévy’s PhD thesis, after Church, Rosser and Curry’s seminal steps. There, Lévy introduces a permutation equivalence between rewriting paths, and establishes that among all confluence diagrams P → N ← Q completing a span P ← M → Q, there exists a minimum such one, modulo permutation equivalence. Categorically, the diagram is called a pushout.

In this article, we extend Lévy’s residual theory, in order to enscope “border-line” rewriting systems, which admit critical pairs but enjoy a strong Church-Rosser property (=existence of pushouts.) Typical examples are the associativity rule and the positive braid rewriting systems. Finally, we show that the resulting theory reformulates and clarifies Lévy’s optimality theory for the λ-calculus, and its so-called “extraction procedure”.

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Authors and Affiliations

  1. Equipe PPS, CNRS, Université Paris 7, France

    Paul-André Melliès

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  1. Paul-André Melliès
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Editors and Affiliations

  1. LIFL Bâtiment M3, Université de Lille 1, Cité Scientifique, 59655, Villeneuve d’Ascq cedex, France

    Sophie Tison

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Melliès, PA. (2002). Axiomatic Rewriting Theory VI: Residual Theory Revisited. In: Tison, S. (eds) Rewriting Techniques and Applications. RTA 2002. Lecture Notes in Computer Science, vol 2378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45610-4_4

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  • DOI: https://doi.org/10.1007/3-540-45610-4_4

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  • Print ISBN: 978-3-540-43916-5

  • Online ISBN: 978-3-540-45610-0

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