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Orbits of Abelian Automaton Groups

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Language and Automata Theory and Applications (LATA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11417))

Abstract

Automaton groups are a class of self-similar groups generated by invertible finite-state transducers [11]. Extending the results of Nekrashevych and Sidki [12], we describe a useful embedding of abelian automaton groups into a corresponding algebraic number field, and give a polynomial time algorithm to compute this embedding. We apply this technique to study iteration of transductions in abelian automaton groups. Specifically, properties of this number field lead to a polynomial-time algorithm for deciding when the orbits of a transduction are a rational relation. These algorithms were implemented in the SageMath computer algebra system and are available online [2].

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Acknowlegements

The authors would like to thank Eric Bach for his helpful feedback on a draft of this paper. We also thank Evan Bergeron and Chris Grossack for many helpful conversations on the results presented.

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Authors and Affiliations

  1. Department of Computer Sciences, University of Wisconsin-Madison, 1210 W. Dayton St., Madison, WI, 53706, USA

    Tim Becker

  2. Computer Science Department, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA, 15213, USA

    Klaus Sutner

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  1. Tim Becker
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  2. Klaus Sutner
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Correspondence to Tim Becker .

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Editors and Affiliations

  1. Rovira i Virgili University, Tarragona, Spain

    Carlos Martín-Vide

  2. Saint Petersburg State University, St. Petersburg, Russia

    Alexander Okhotin

  3. Ariel University, Ariel, Israel

    Dana Shapira

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Becker, T., Sutner, K. (2019). Orbits of Abelian Automaton Groups. In: Martín-Vide, C., Okhotin, A., Shapira, D. (eds) Language and Automata Theory and Applications. LATA 2019. Lecture Notes in Computer Science(), vol 11417. Springer, Cham. https://doi.org/10.1007/978-3-030-13435-8_5

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  • DOI: https://doi.org/10.1007/978-3-030-13435-8_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-13434-1

  • Online ISBN: 978-3-030-13435-8

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