On Torsion-Free Semigroups Generated by Invertible Reversible Mealy Automata

  • Thibault Godin
  • Ines Klimann
  • Matthieu PicantinEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8977)


This paper addresses the torsion problem for a class of automaton semigroups, defined as semigroups of transformations induced by Mealy automata, aka letter-by-letter transducers with the same input and output alphabet. The torsion problem is undecidable for automaton semigroups in general, but is known to be solvable within the well-studied class of (semi)groups generated by invertible bounded Mealy automata. We focus on the somehow antipodal class of invertible reversible Mealy automata and prove that for a wide subclass the generated semigroup is torsion-free.


Automaton semigroup Reversible mealy automaton   Labeled orbit tree Torsion-free semigroup 


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  1. 1.
    Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi) groups. Internat. J. Algebra Comput. 22(6), 26 (2012)Google Scholar
  2. 2.
    Alešin, S.V.: Finite automata and the Burnside problem for periodic groups. Mat. Zametki 11, 319–328 (1972)MathSciNetGoogle Scholar
  3. 3.
    Antonenko, A.S.: On transition functions of Mealy automata of finite growth. Matematychni Studii. 29(1), 3–17 (2008)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Antonenko, A.S., Berkovich, E.L.: Groups and semigroups defined by some classes of Mealy automata. Acta Cybernetica 18(1), 23–46 (2007)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Bartholdi, L., Kaimanovich, V.A., Nekrashevych, V.V.: On amenability of automata groups. Duke Math. J. 154(3), 575–598 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Bartholdi, L., Reznykov, I.I., Sushchanskiĭ, V.I.: The smallest Mealy automaton of intermediate growth. J. Algebra 295(2), 387–414 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Bartholdi, L., Silva, P.V.: Groups defined by automata. In: Handbook AutoMathA, ArXiv:cs.FL/1012.1531, ch. 24 (2010)Google Scholar
  8. 8.
    Baumslag, G., Boone, W.W., Neumann, B.H.: Some unsolvable problems about elements and subgroups of groups. Math. Scand. 7, 191–201 (1959)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Bondarenko, I., Grigorchuk, R.I., Kravchenko, R., Muntyan, Y., Nekrashevych, V., Savchuk, D., Šunić, Z.: On classification of groups generated by 3-state automata over a 2-letter alphabet. Algebra Discrete Math. (1), 1–163 (2008)Google Scholar
  10. 10.
    Bondarenko, I.V., Bondarenko, N.V., Sidki, S.N., Zapata, F.R.: On the conjugacy problem for finite-state automorphisms of regular rooted trees. Groups Geom. Dyn. 7(2), 323–355 (2013). with an appendix by Raphaël M. JungersGoogle Scholar
  11. 11.
    Cain, A.J.: Automaton semigroups. Theor. Comput. Sci. 410, 5022–5038 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Eilenberg, S.: Automata, languages, and machines, vol. A. Academic Press (A subsidiary of Harcourt Brace Jovanovich, Publishers), New York (1974)Google Scholar
  13. 13.
    Gawron, P.W., Nekrashevych, V.V., Sushchansky, V.I.: Conjugation in tree automorphism groups. Internat. J. Algebra Comput. 11(5), 529–547 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. Internat. J. Algebra Comput. 24(1), 1–9 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Grigorchuk, R.I.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Grigorchuk, R.I.: On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR 271(1), 30–33 (1983)MathSciNetGoogle Scholar
  17. 17.
    Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskiĭ, V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231, 134–214 (2000)Google Scholar
  18. 18.
    Klimann, I.: The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable. In: Proc. 30th STACS. LIPIcs, vol. 20, pp. 502–513 (2013)Google Scholar
  19. 19.
    Klimann, I., Mairesse, J., Picantin, M.: Implementing computations in automaton (semi) groups. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 240–252. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  20. 20.
    Klimann, I., Picantin, M., Savchuk, D.: A connected 3-state reversible Mealy automaton cannot generate an infinite Burnside group. arXiv:1409.6142 (2014)
  21. 21.
    Macedonska, O., Nekrashevych, V.V., Sushchansky, V.I.: Commensurators of groups and reversible automata. Dopov. Nats. Akad. Nauk Ukr., Mat. Pryr. Tekh. Nauky (12), 36–39 (2000)Google Scholar
  22. 22.
    Maltcev, V.: Cayley automaton semigroups. Internat. J. Algebra Comput. 19(1), 79–95 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Mintz, A.: On the Cayley semigroup of a finite aperiodic semigroup. Internat. J. Algebra Comput. 19(6), 723–746 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Nekrashevych, V.: Self-similar groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence (2005) zbMATHGoogle Scholar
  25. 25.
    Russyev, A.: Finite groups as groups of automata with no cycles with exit. Algebra and Discrete Mathematics 9(1), 86–102 (2010)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Sakarovitch, J.: Elements of Automata Theory. Cambridge University Press (2009)Google Scholar
  27. 27.
    Sidki, S.N.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (New York) 100(1), 1925–1943 (2000). algebra, 12Google Scholar
  28. 28.
    Silva, P.V., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Internat. J. Algebra Comput. 15(5–6), 1213–1234 (2005)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Thibault Godin
    • 1
  • Ines Klimann
    • 1
  • Matthieu Picantin
    • 1
    Email author
  1. 1.Université Paris Diderot, Sorbonne Paris Cité, LIAFA, UMR 7089 CNRSParisFrance

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