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A New Hierarchy for Automaton Semigroups

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Implementation and Application of Automata (CIAA 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10977))

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Abstract

We define a new strict and computable hierarchy for the family of automaton semigroups, which reflects the various asymptotic behaviors of the state-activity growth. This hierarchy extends that given by Sidki for automaton groups, and also gives new insights into the latter. Its exponential part coincides with a notion of entropy for some associated automata.

We prove that the Order Problem is decidable when the state-activity is bounded. The Order Problem remains open for the next level of this hierarchy, that is, when the state-activity is linear. Gillibert showed that it is undecidable in the whole family.

The former results are implemented and will be available in the GAP package FR developed by the first author.

T. Godin—supported by the Academy of Finland grant 296018.

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Author information

Authors and Affiliations

  1. École Normale Supérieure, Paris, France

    Laurent Bartholdi

  2. Georg-August-Universität zu Göttingen, Göttingen, Germany

    Laurent Bartholdi

  3. University of Turku, Turku, Finland

    Thibault Godin

  4. IRIF, UMR 8243 CNRS & Université Paris Diderot, Paris, France

    Ines Klimann & Matthieu Picantin

Authors
  1. Laurent Bartholdi
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  2. Thibault Godin
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  3. Ines Klimann
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  4. Matthieu Picantin
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Corresponding author

Correspondence to Matthieu Picantin .

Editor information

Editors and Affiliations

  1. University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada

    Cezar Câmpeanu

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Bartholdi, L., Godin, T., Klimann, I., Picantin, M. (2018). A New Hierarchy for Automaton Semigroups. In: Câmpeanu, C. (eds) Implementation and Application of Automata. CIAA 2018. Lecture Notes in Computer Science(), vol 10977. Springer, Cham. https://doi.org/10.1007/978-3-319-94812-6_7

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  • DOI: https://doi.org/10.1007/978-3-319-94812-6_7

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94811-9

  • Online ISBN: 978-3-319-94812-6

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