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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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References

  1. Bartholdi, L.: Book review: combinatorial algebra: syntax and semantics. Bull. AMS 54(4), 681–686 (2017)

    Article  Google Scholar 

  2. Becker, T.: Embeddings and orbits of abelian automaton groups (2018). https://github.com/tim-becker/thesis-code

  3. Bondarenko, I., et al.: Classification of groups generated by 3-state automata over a 2-letter alphabet. Algebra Discrete Math. 1, April 2008

    Google Scholar 

  4. Brough, M., Khoussainov, B., Nelson, P.: Sequential automatic algebras. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds.) CiE 2008. LNCS, vol. 5028, pp. 84–93. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-69407-6_9

    Chapter  Google Scholar 

  5. Faugère, J.C., El Din, M.S., Spaenlehauer, P.J.: Gröbner bases of bihomogeneous ideals generated by polynomials of bidegree (1, 1): algorithms and complexity. J. Symbolic Comput. 46(4), 406–437 (2011)

    Article  MathSciNet  Google Scholar 

  6. Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskii, V.I.: Automata, dynamical systems and groups. Proc. Steklov Inst. Math. 231, 128–203 (2000)

    MathSciNet  MATH  Google Scholar 

  7. Hungerford, T.W.: Algebra. Springer, New York (1974). https://doi.org/10.1007/978-1-4612-6101-8

    Book  MATH  Google Scholar 

  8. Ireland, K., Rosen, M., Rosen, M.: A Classical Introduction to Modern Number Theory. Graduate Texts in Mathematics. Springer, New York (1990). https://doi.org/10.1007/978-1-4757-2103-4

    Book  MATH  Google Scholar 

  9. Latimer, C.G., MacDuffee, C.C.: A correspondence between classes of ideals and classes of matrices. Ann. Math. 34(2), 313–316 (1933). http://www.jstor.org/stable/1968204

    Article  MathSciNet  Google Scholar 

  10. Lazard, D.: Solving zero-dimensional algebraic systems. J. Symbolic Comput. 13(2), 117–131 (1992). https://doi.org/10.1016/S0747-7171(08)80086-7. http://www.sciencedirect.com/science/article/pii/S0747717108800867

    Article  MathSciNet  MATH  Google Scholar 

  11. Nekrashevych, V.: Self-similar groups. Mathematical Surveys and Monographs. American Mathematical Society, Providence (2014)

    MATH  Google Scholar 

  12. Nekrashevych, V., Sidki, S.: Automorphisms of the binary tree: state-closed subgroups and dynamics of 1/2-endomorphisms. London Math. Soc. Lect. Note Ser. 311, 375–404 (2004)

    MathSciNet  MATH  Google Scholar 

  13. Okano, T.: Invertible binary transducers and automorphisms of the binary tree. Master’s thesis, Carnegie Mellon University (2015)

    Google Scholar 

  14. Raney, G.N.: Sequential functions. J. ACM (JACM) 5(2), 177–180 (1958)

    Article  MathSciNet  Google Scholar 

  15. Robinson, R.M.: Conjugate algebraic integers on a circle. Mathematische Zeitschrift 110(1), 41–51 (1969). https://doi.org/10.1007/BF01114639

    Article  MathSciNet  MATH  Google Scholar 

  16. Stein, W.: Algebraic Number Theory, a Computational Approach. Harvard, Massachusetts (2012)

    Google Scholar 

  17. Sutner, K.: Abelian invertible automata. In: Adamatzky, A. (ed.) Reversibility and Universality. ECC, vol. 30, pp. 37–59. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73216-9_3

    Chapter  Google Scholar 

  18. Sutner, K., Lewi, K.: Iterating inverse binary transducers. J. Automata, Lang. Comb. 17(2–4), 293–313 (2012)

    MathSciNet  MATH  Google Scholar 

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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

Authors
  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

  • eBook Packages: Computer ScienceComputer Science (R0)

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