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Term Rewriting Systems as Topological Dynamical Systems

Authors Soren Bjerg Andersen, Jakob Grue Simonsen



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Soren Bjerg Andersen
Jakob Grue Simonsen

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Soren Bjerg Andersen and Jakob Grue Simonsen. Term Rewriting Systems as Topological Dynamical Systems. In 23rd International Conference on Rewriting Techniques and Applications (RTA'12). Leibniz International Proceedings in Informatics (LIPIcs), Volume 15, pp. 53-68, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2012)
https://doi.org/10.4230/LIPIcs.RTA.2012.53

Abstract

Topological dynamics is, roughly, the study of phenomena related to iterations of continuous maps from a metric space to itself. We show how the rewrite relation in term rewriting gives rise to dynamical systems in two distinct, natural ways: (A) One in which any deterministic rewriting strategy induces a dynamical system on the set of finite and infinite terms endowed with the usual metric, and (B) one in which the unconstrained rewriting relation induces a dynamical system on sets of sets of terms, specifically the set of compact subsets of the set of finite and infinite terms endowed with the Hausdorff metric. For both approaches, we give sufficient criteria for the induced systems to be well-defined dynamical systems and for (A) we demonstrate how the classic topological invariant called topological entropy turns out to be much less useful in the setting of term rewriting systems than in symbolic dynamics.
Keywords
  • Term rewriting
  • dynamical systems
  • topology
  • symbolic dynamics

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