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Automaton Semigroups: The Two-state Case

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Abstract

We prove that semigroups generated by reversible two-state Mealy automata have remarkable growth properties: they are either finite or free. We give an effective procedure to decide finiteness or freeness of such semigroups when the generating automaton is also invertible.

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Notes

  1. Timings obtained on an Intel Xeon computer with clock speed 2.13GHz; programs written in GAP [12].

References

  1. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The design and analysis of computer algorithms. Addison-Wesley (1974)

  2. Akhavi, A., Klimann, I., Lombardy, S., Mairesse, J., Picantin, M.: On the finiteness problem for automaton (semi)groups. Int. J. Algebra Comput. 22(6), 26 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alešin, S.V.: Finite automata and the Burnside problem for periodic groups. Mat. Zametki 11, 319–328 (1972)

    MathSciNet  Google Scholar 

  4. Antonenko, A.S.: On transition functions of Mealy automata of finite growth. Matematychni Studii 29(1), 3–17 (2008)

    MathSciNet  MATH  Google Scholar 

  5. Bartholdi, L.: FR Functionally recursive groups – a GAP package, v.1.2.4.2 (2011)

  6. Bartholdi, L., Reznykov, I.I., Sushchanskiı̆, V.I.: The smallest Mealy automaton of intermediate growth. J. Algebra 295(2), 387–414 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cain, A.J.: Automaton semigroups. Theor. Comput. Sci. 410(47-49), 5022–5038 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. D’Angeli, D., Rodaro, E.: Groups and semigroups defined by colorings of synchronizing automata. Int. J. Algebra Comput., 21 (2014)

  11. de la Harpe, P.: Topics in geometric group theory. University of Chicago Press (2000)

  12. The GAP Group: GAP – groups, algorithms, and programming, v.4.4.12 (2008)

  13. Gillibert, P.: The finiteness problem for automaton semigroups is undecidable. Int. J. Algebra Comput. 24(01), 1–9 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Glasner, Y., Mozes, Sh.: Automata and square complexes. Geom. Dedicata 111(1), 43–6 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Grigorchuk, R., żuk, A.: The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata 87, 209–244 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grigorchuk, R.I.: On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen. 14(1), 53–54 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  17. Grigorchuk, R.I., Nekrashevich, V.V., Sushchanskiı̆, V.I.: Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova 231, 134–214 (2000)

    MathSciNet  Google Scholar 

  18. Klimann, I., Mairesse, J., Picantin, M.: Implementing computations in automaton (semi)groups. In. Proc. 17th CIAA, volume 7381 of LNCS, pp 240–252 (2012)

  19. Klimann, I.: The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable. In Proc. 30th STACS, vol. 20 of LIPIcs, pp 502–513 (2013)

  20. Maltcev, V.: Cayley automaton semigroups. Int. J. Algebra Comput. 19(1), 79–95 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mintz, A.: On the Cayley semigroup of a finite aperiodic semigroup. Int. J. Algebra Comput. 19(6), 723–746 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Muntyan, Y., Savchuk, D.: sutomgrp Automata Groups – a GAP package, v.1.1.4.1 (2008)

  23. Nekrashevych, V.: Self-similar groups, volume 117 of mathematical surveys and monographs. American mathematical society, providence, RI (2005)

  24. Savchuk, D.M., Vorobets, Y.: Automata generating free products of groups of order 2. J. Algebra 336(1), 53–66 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sidki, S.N.: Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (NewYork) 100(1), 1925–1943 (2000). Algebra, 12

    Article  MathSciNet  MATH  Google Scholar 

  26. Silva, P.V., Steinberg, B.: On a class of automata groups generalizing lamplighter groups. Int. J. Algebra Comput. 15(5-6), 1213–1234 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Steinberg, B., Vorobets, M., Vorobets, Y.: Automata over a binary alphabet generating free groups of even rank. Int. J. Algebra Comput. 21(1-2), 329–354 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Vorobets, M., Vorobets, Y.: On a free group of transformations defined by an automaton. Geom. Dedicata 124, 237–249 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  29. Vorobets, M., Vorobets, Y.: On a series of finite automata defining free transformation groups. Groups Geom Dyn. 4, 377–405 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I would like to thank Jean Mairesse and Matthieu Picantin for numerous discussions around this topic.

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Correspondence to Ines Klimann.

Additional information

An extended abstract was presented at STACS’13 [19].

The author is partially supported by ANR Project MealyM ANR-JCJC-12-JS02-012-01.

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Klimann, I. Automaton Semigroups: The Two-state Case. Theory Comput Syst 58, 664–680 (2016). https://doi.org/10.1007/s00224-014-9594-0

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