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Automatic Semigroups vs Automaton Semigroups (Track B: Automata, Logic, Semantics, and Theory of Programming)

Author Matthieu Picantin

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Matthieu Picantin
  • IRIF UMR 8243 CNRS & Univ Paris Diderot, 75013 Paris, France

Funding

This work was partially supported by the French Agence Nationale pour la Recherche, through the project MealyM ANR-JS02-012-01.

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Matthieu Picantin. Automatic Semigroups vs Automaton Semigroups (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 124:1-124:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.124

Abstract

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups, free commutative semigroups, trace or divisibility monoids, braid or Artin - Tits or Krammer or Garside monoids, Baumslag - Solitar semigroups, etc. Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroups to a class of automaton semigroups. It is worthwhile noting that, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as the cornerstone of our construction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Rewrite systems
Keywords
  • Mealy machine
  • semigroup
  • rewriting system
  • automaticity
  • self-similarity

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