Document Open Access Logo

Automatic Semigroups vs Automaton Semigroups (Track B: Automata, Logic, Semantics, and Theory of Programming)

Author Matthieu Picantin

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2019.124.pdf
  • Filesize: 0.62 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Rewrite systems
Keywords
  • Mealy machine
  • semigroup
  • rewriting system
  • automaticity
  • self-similarity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

We develop an effective and natural approach to interpret any semigroup admitting a special language of greedy normal forms as an automaton semigroup, namely the semigroup generated by a Mealy automaton encoding the behaviour of such a language of greedy normal forms under one-sided multiplication. The framework embraces many of the well-known classes of (automatic) semigroups: free semigroups, free commutative semigroups, trace or divisibility monoids, braid or Artin - Tits or Krammer or Garside monoids, Baumslag - Solitar semigroups, etc. Like plactic monoids or Chinese monoids, some neither left- nor right-cancellative automatic semigroups are also investigated, as well as some residually finite variations of the bicyclic monoid. It provides what appears to be the first known connection from a class of automatic semigroups to a class of automaton semigroups. It is worthwhile noting that, "being an automatic semigroup" and "being an automaton semigroup" become dual properties in a very automata-theoretical sense. Quadratic rewriting systems and associated tilings appear as the cornerstone of our construction.

Cite As Get BibTex

Matthieu Picantin. Automatic Semigroups vs Automaton Semigroups (Track B: Automata, Logic, Semantics, and Theory of Programming). In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 124:1-124:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.124

Author Details

Matthieu Picantin
  • IRIF UMR 8243 CNRS & Univ Paris Diderot, 75013 Paris, France

Funding

This work was partially supported by the French Agence Nationale pour la Recherche, through the project MealyM ANR-JS02-012-01.

References

  1. Ali Akhavi, Ines Klimann, Sylvain Lombardy, Jean Mairesse, and Matthieu Picantin. On the finiteness problem for automaton (semi)groups. Internat. J. Algebra Comput., 22(6):1-26, 2012. Google Scholar
  2. Stanislas V. Alëšin. Finite automata and the Burnside problem for periodic groups. Mat. Zametki, 11:319-328, 1972. Google Scholar
  3. Stanislav V. Alëšin. A free group of finite automata. Vestnik Moskov. Univ. Ser. I Mat. Mekh., 4:12-14, 1983. Google Scholar
  4. Algirdas Avižienis. Signed-digit number representations for fast parallel arithmetic. IRE Trans. Electronic Computers, 10(3):389-400, 1961. Google Scholar
  5. Laurent Bartholdi. FR - GAP package "Computations with functionally recursive groups", Version 2.4.5, 2018. URL: http://www.gap-system.org/Packages/fr.html.
  6. Laurent Bartholdi, Thibault Godin, Ines Klimann, and Matthieu Picantin. A New Hierarchy for Automaton Semigroups. In 23rd International Conference on Implementation and Applications of Automata (CIAA 2018), volume 10977 of LNCS, pages 71-83, 2018. Google Scholar
  7. Laurent Bartholdi and Pedro V. Silva. Groups defined by automata. In J.-É. Pin, editor, AutoMathA Handbook. Europ. Math. Soc., 2010. (arXiv version: https://arxiv.org/abs/1012.1531).
  8. Laurent Bartholdi and Pedro V. Silva. Rational subsets of groups. In J.-É. Pin, editor, AutoMathA Handbook. Europ. Math. Soc., 2010. (arXiv version: https://arxiv.org/abs/1012.1532).
  9. Laurent Bartholdi and Zoran Šuniḱ. Some solvable automaton groups. In Topological and asymptotic aspects of group theory, volume 394 of Contemp. Math., pages 11-29. Amer. Math. Soc., Providence, RI, 2006. Google Scholar
  10. Ievgen V. Bondarenko, Natalia V. Bondarenko, Saïd N. Sidki, and Flavia R. Zapata. 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. Jungers. Google Scholar
  11. Tara Brough and Alan J. Cain. Automaton semigroup constructions. Semigroup Forum, 90(3):763-774, 2015. Google Scholar
  12. Tara Brough and Alan J. Cain. Automaton semigroups: new constructions results and examples of non-automaton semigroups. Theoret. Comput. Sci., 674:1-15, 2017. Google Scholar
  13. Kai-Uwe Bux et al. Selfsimilar groups and conformal dynamics - Problem List. AIM workshop 2006. URL: http://www.aimath.org/WWN/selfsimgroups/selfsimgroups.pdf.
  14. Alan J. Cain. Automaton semigroups. Theoret. Comput. Sci., 410(47-49):5022-5038, 2009. Google Scholar
  15. Alan J. Cain. Personal communication, 2016. Google Scholar
  16. Alan J. Cain, Robert D. Gray, and António Malheiro. Rewriting systems and biautomatic structures for Chinese, hypoplactic, and Sylvester monoids. Internat. J. Algebra Comput., 25(1-2):51-80, 2015. Google Scholar
  17. Colin M. Campbell, Edmund F. Robertson, Nikola Ruškuc, and Richard M. Thomas. Automatic semigroups. Theoret. Comput. Sci., 250(1-2):365-391, 2001. Google Scholar
  18. Augustin-Louis Cauchy. Sur les moyens d'éviter les erreurs dans les calculs numériques, volume 5 of Cambridge Library Collection - Mathematics, pages 431-442. Cambridge University Press, 2009. Google Scholar
  19. Daniele D'Angeli, Thibault Godin, Ines Klimann, Matthieu Picantin, and Emanuele Rodaro. Boundary action of automaton groups without singular points and Wang tilings. Submitted, 2016. URL: http://arxiv.org/abs/1604.07736.
  20. Patrick Dehornoy. Garside and quadratic normalisation: a survey. In 19th International Conference on Developments in Language Theory (DLT 2015), volume 9168 of LNCS, pages 14-45, 2015. Google Scholar
  21. Patrick Dehornoy et al. Foundations of Garside theory. Europ. Math. Soc. Tracts in Mathematics, volume 22, 2015. URL: http://www.math.unicaen.fr/~garside/Garside.pdf.
  22. Patrick Dehornoy and Yves Guiraud. Quadratic normalization in monoids. Internat. J. Algebra Comput., 26(5):935-972, 2016. Google Scholar
  23. Murray Elder. Automaticity, almost convexity and falsification by fellow traveler properties of some finitely presented groups. PhD thesis, Univ Melbourne, 2000. Google Scholar
  24. David B. A. Epstein, James W. Cannon, Derek F. Holt, Silvio V. F. Levy, Michael S. Paterson, and William P. Thurston. Word processing in groups. Jones and Bartlett Publishers, Boston, MA, 1992. Google Scholar
  25. Pierre Gillibert. The finiteness problem for automaton semigroups is undecidable. Internat. J. Algebra Comput., 24(1):1-9, 2014. Google Scholar
  26. Thibault Godin, Ines Klimann, and Matthieu Picantin. On torsion-free semigroups generated by invertible reversible Mealy automata. In 9th International Conference on Language and Automata Theory and Applications (LATA 2015), pages 328-339, 2015. Google Scholar
  27. Rostislav I. Grigorchuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53-54, 1980. Google Scholar
  28. Rostislav I. Grigorchuk. Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat., 48(5):939-985, 1984. Google Scholar
  29. Yves Guiraud and Matthieu Picantin. Resolutions by differential graded polygraphs. In preparation, 2019. Google Scholar
  30. Alexander Hess. Factorable monoids: resolutions and homology via discrete Morse theory. PhD thesis, Univ Bonn, 2012. URL: http://hss.ulb.uni-bonn.de/2012/2932/2932.pdf.
  31. Alexander Hess and Viktoriya Ozornova. Factorability, string rewriting and discrete Morse theory. Submitted. URL: http://arxiv.org/abs/1412.3025.
  32. Michael Hoffmann. Automatic Semigroups. PhD thesis, Univ Leicester, 2001. Google Scholar
  33. Michael Hoffmann and Richard M. Thomas. Biautomatic semigroups. In 15th International Symposium on Fundamentals of Computation Theory (FCT 2005), volume 3623 of LNCS, pages 56-67, 2005. Google Scholar
  34. Ines Klimann. The finiteness of a group generated by a 2-letter invertible-reversible Mealy automaton is decidable. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013), volume 20 of LIPIcs, pages 502-513, 2013. Google Scholar
  35. Ines Klimann, Jean Mairesse, and Matthieu Picantin. Implementing Computations in Automaton (Semi)groups. In 17th International Conference on Implementation and Applications of Automata (CIAA 2012), volume 7381 of LNCS, pages 240-252, 2012. Google Scholar
  36. Ines Klimann and Matthieu Picantin. Automaton (semi)groups: Wang tilings and Schreier tries. In Valérie Berthé and Michel Rigo, editors, Sequences, Groups, and Number Theory. Trends in Mathematics, 2018. Google Scholar
  37. Ines Klimann, Matthieu Picantin, and Dmytro Savchuk. A Connected 3-State Reversible Mealy Automaton Cannot Generate an Infinite Burnside Group. In 19th International Conference on Developments in Language Theory (DLT 2015), volume 9168 of LNCS, pages 313-325, 2015. Google Scholar
  38. Ines Klimann, Matthieu Picantin, and Dmytro Savchuk. Orbit automata as a new tool to attack the order problem in automaton groups. J. Algebra, 445:433-457, 2016. Google Scholar
  39. Daan Krammer. An asymmetric generalisation of Artin monoids. Groups Complex. Cryptol., 5:141-168, 2013. Google Scholar
  40. Yaroslav Lavrenyuk, Volodymyr Mazorchuk, Andriy Oliynyk, and Vitaliy Sushchansky. Faithful group actions on rooted trees induced by actions of quotients. Comm. Algebra, 35(11):3759-3775, 2007. Google Scholar
  41. Anatoly I. Malcev. On the immersion of an algebraic ring into a field. Math. Ann., 113(1):686-691, 1937. Google Scholar
  42. Anatoly I. Malcev. Über die Einbettung von assoziativen Systemen in Gruppen. Rec. Math. [Mat. Sbornik] N.S., 6 (48):331-336, 1939. Google Scholar
  43. Anatoly I. Malcev. Über die Einbettung von assoziativen Systemen in Gruppen. II. Rec. Math. [Mat. Sbornik] N.S., 8 (50):251-264, 1940. Google Scholar
  44. Victor D. Mazurov and Evgeny I. Khukhro. Unsolved problems in group theory. The Kourovka Notebook. No 19. URL: https://kourovka-notebook.org/.
  45. David McCune. Groups and Semigroups Generated by Automata. PhD thesis, Univ Nebraska-Lincoln, 2011. Google Scholar
  46. Yevgen Muntyan and Dmytro Savchuk. AutomGrp - GAP package for computations in self-similar groups and semigroups, Version 1.3, 2016. URL: http://www.gap-system.org/Packages/automgrp.html.
  47. Volodymyr V. Nekrashevych. Self-similar groups, volume 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. Google Scholar
  48. Viktoriya Ozornova. Factorability, discrete Morse theory, and a reformularion of K(π, 1)-conjecture. PhD thesis, Univ Bonn, 2013. URL: http://hss.ulb.uni-bonn.de/2013/3117/3117.pdf.
  49. Matthieu Picantin. Finite transducers for divisibility monoids. Theoret. Comput. Sci., 362(1-3):207-221, 2006. Google Scholar
  50. Matthieu Picantin. Tree products of cyclic groups and HNN extensions. Preprint, 2015. URL: http://arxiv.org/abs/1306.5724v4.
  51. Matthieu Picantin. Automates, (semi)groupes, dualités. Habilitation à diriger des recherches, Univ Paris Diderot, 2017. URL: https://www.irif.fr/~picantin/papers/hdr_memoire.pdf and URL: https://www.irif.fr/~picantin/papers/hdr_soutenance.pdf.
  52. Matthieu Picantin. Automatic semigroups vs automaton semigroups. Full version of the current paper, 2019. URL: http://arxiv.org/abs/1609.09364v5.
  53. Jacques Sakarovitch. Elements of Automata Theory. Cambridge University Press, New York, NY, USA, 2009. Google Scholar
  54. Marcel-Paul Schützenberger. Pour le monoïde plaxique. Math. Inform. Sci. Humaines, 140:5-10, 1997. Google Scholar
  55. Pedro V. Silva. Groups and Automata: A Perfect Match. In 14th International Workshop on Descriptional Complexity of Formal Systems (DCFS 2012), volume 7386 of LNCS, pages 50-63, 2012. Google Scholar
  56. Pedro V. Silva and Benjamin Steinberg. On a class of automata groups generalizing lamplighter groups. Internat. J. Algebra Comput., 15(5-6):1213-1234, 2005. Google Scholar
  57. Bartosz Tarnawski. Automatic groups as groups defined by transducers. Master’s thesis, Univ Warsaw, Faculty of Mathematics, Informatics and Mechanics, Poland, 2017. Google Scholar
  58. Daniel T. Wise. A non-Hopfian automatic group. J. Algebra, 180(3):845-847, 1996. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact