Semigroup Forum

, Volume 95, Issue 3, pp 475–508 | Cite as

The monoid of queue actions

  • Martin Huschenbett
  • Dietrich Kuske
  • Georg Zetzsche
Research Article


We model the behavior of a fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties such as conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is \({{\mathsf {N}}}{{\mathsf {L}}}\)-complete. Furthermore, we present an algebraic characterization of this monoid’s recognizable subsets. Finally, we prove that it is not Thurston-automatic.


Semi-Thue system Conjugacy problem Recognizable and rational subsets Automaticity 



G. Zetzsche supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD).


  1. 1.
    Berstel, J.: Transductions and Context-Free Languages. Teubner Studienbücher, Stuttgart (1979)CrossRefzbMATHGoogle Scholar
  2. 2.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: automata and interpretations. ACM Trans. Comput. Syst. 37(6), 641–674 (2004)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Book, R.V., Otto, F.: String-Rewriting Systems. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cannon, J.W., Epstein, D.B.A., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word Processing in Groups. Jones and Barlett, Boston (1992)zbMATHGoogle Scholar
  5. 5.
    Choffrut, Ch.: Conjugacy in free inverse monoids. Int. J. Algebra Comput. 3(2), 169–188 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Chrobak, M., Rytter, W.: Unique decipherability for partially commutative alphabet. In: Proceedings of Mathematical Foundations of Computer Science 1986, Bratislava, Czechoslovakia, August 25–29, 1996. Lecture Notes in Computer Science, vol. 233, pp. 256–263. Springer, Berlin (1986)Google Scholar
  7. 7.
    Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theor. Comput. Sci. 250(1–2), 365–391 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Duncan, A.J., Robertson, E.F., Ruškuc, N.: Automatic monoids and change of generators. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 127, pp. 403–409. Cambridge University Press, Cambridge (1999)Google Scholar
  9. 9.
    Duboc, C.: On some equations in free partially commutative monoids. Theor. Comput. Sci. 46, 159–174 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Frougny, Ch., Sakarovitch, J.: Synchronized rational relations of finite and infinite words. Theor. Comput. Sci. 108, 45–82 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Huschenbett, M., Kuske, D., Zetzsche, G.: The monoid of queue actions. In: MFCS’14. Lecture Notes in Computer Science, vol. 8634, pp. 340–351. Springer, Berlin (2014)Google Scholar
  12. 12.
    Kambites, M.: Formal languages and groups as memory. Commun. Algebra 37, 193–208 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kharlampovich, O., Khoussainov, B., Miasnikov, A.: From automatic structures to automatic groups. Preprint: arXiv:1107.3645 (2011)
  14. 14.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Logic and Computational Complexity. Lecture Notes in Computer Science, vol. 960, pp. 367–392. Springer, Berlin (1995)Google Scholar
  15. 15.
    Kuske, D., Prianychnykova, O.: The trace monoids in the queue monoid and in the direct product of two free monoids. In: DLT’16. Lecture Notes in Computer Science, vol. 9840, pp. 256–267. Springer, Berlin (2016)Google Scholar
  16. 16.
    Lohrey, M.: The rational subset membership problem for groups: a survey. In: Campbell, C.M., Quick, M.R., Robertson, E.F., Roney-Dougal, C.M. (eds.) Groups St Andrews 2013. London Mathematical Society Lecture Note Series, vol. 422, pp. 368–389. Cambridge University Press, Cambridge (2016.)Google Scholar
  17. 17.
    Lothaire, M.: Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 17. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  18. 18.
    Lentin, A., Schützenberger, M.P.: A combinatorial problem in the theory of free monoids. In: Bose, R.C., Dowling, T.A. (eds.) Combinatorial Mathematics and its Applications, pp. 128–144. North Carolina Press, Chapell Hill (1967)Google Scholar
  19. 19.
    Osipova, V.A.: On the conjugacy problem in semigroups. Proc. Steklov Inst. Math. 133, 169–182 (1973)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Otto, F.: Conjugacy in monoids with a special Church-Rosser presentation is decidable. Semigroup Forum 29, 223–240 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Zhang, L.: Conjugacy in special monoids. J. Algebra 143(2), 487–497 (1991)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Martin Huschenbett
    • 1
  • Dietrich Kuske
    • 2
  • Georg Zetzsche
    • 3
  1. 1.LondonUK
  2. 2.Institut für Theoretische InformatikTU IlmenauIlmenauGermany
  3. 3.LSV, CNRS & ENS CachanUniversité Paris-SaclayCachanFrance

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