Z; Syntax-Free Developments

Author Vincent van Oostrom

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Vincent van Oostrom
  • Universität Innsbruck, Austria


Patrick Dehornoy introduced me to the main themes presented here, and indeed this paper was always intended to be a joint one. His work continues to be an inspiration. I want to thank Bertram Felgenhauer, Julian Nagele, and Christian Sternagel for discussions on their Isabelle formalisations of the Z-property.

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Vincent van Oostrom. Z; Syntax-Free Developments. In 6th International Conference on Formal Structures for Computation and Deduction (FSCD 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 195, pp. 24:1-24:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We present the Z-property and instantiate it to various rewrite systems: associativity, positive braids, self-distributivity, the lambda-calculus, lambda-calculi with explicit substitutions, orthogonal TRSs, .... The Z-property is proven equivalent to Takahashi’s angle property by means of a syntax-free notion of development. We show that several classical consequences of having developments such as confluence, normalisation, and recurrence, can be regained in a syntax-free way, and investigate how the notion corresponds to the classical syntactic notion of development in term rewriting.

Subject Classification

ACM Subject Classification
  • Theory of computation → Equational logic and rewriting
  • rewrite system
  • confluence
  • normalisation
  • recurrence


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