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

Authors: Matthieu Picantin

Published in: LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)


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.

Cite as

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)


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@InProceedings{picantin:LIPIcs.ICALP.2019.124,
  author =	{Picantin, Matthieu},
  title =	{{Automatic Semigroups vs Automaton Semigroups}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{124:1--124:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.124},
  URN =		{urn:nbn:de:0030-drops-107004},
  doi =		{10.4230/LIPIcs.ICALP.2019.124},
  annote =	{Keywords: Mealy machine, semigroup, rewriting system, automaticity, self-similarity}
}
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