On the Interplay Between Babai and Černý’s Conjectures

  • François Gonze
  • Vladimir V. Gusev
  • Balázs Gerencsér
  • Raphaël M. Jungers
  • Mikhail V. Volkov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

Motivated by the Babai conjecture and the Černý conjecture, we study the reset thresholds of automata with the transition monoid equal to the full monoid of transformations of the state set. For automata with n states in this class, we prove that the reset thresholds are upper-bounded by \(2n^2-6n+5\) and can attain the value \(\tfrac{n(n-1)}{2}\). In addition, we study diameters of the pair digraphs of permutation automata and construct n-state permutation automata with diameter \(\tfrac{n^2}{4} + o(n^2)\).

References

  1. 1.
    Ananichev, D.S., Gusev, V.V., Volkov, M.V.: Primitive digraphs with large exponents and slowly synchronizing automata. J. Math. Sci. 192(3), 263–278 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ananichev, D.S., Volkov, M.V.: Some results on Černy type problems for transformation semigroups. In: Araújo, I.M., Branco, M.J.J., Fernandes, V.H., Gomes, G.M.S. (eds.) Semigroups and Languages, pp. 23–42. World Scientific (2004)Google Scholar
  3. 3.
    Araújo, J., Cameron, P.J., Steinberg, B.: Between primitive and 2-transitive: Synchronization and its friends. CoRR abs/1511.03184 (2015)Google Scholar
  4. 4.
    Babai, L., Seress, A.: On the diameter of permutation groups. Eur. J. Combin. 13(4), 231–243 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Berstel, J., Perrin, D., Reutenauer, C.: Codes and Automata. CUP, Cambridge (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Bondar, E.A., Volkov, M.V.: Completely reachable automata. In: Câmpeanu, C., Manea, F., Shallit, J. (eds.) DCFS 2016. LNCS, vol. 9777, pp. 1–17. Springer, Cham (2016). doi: 10.1007/978-3-319-41114-9_1 CrossRefGoogle Scholar
  7. 7.
    Brzozowski, J.A.: In search of most complex regular languages. Int. J. Found. Comput. Sci. 24(6), 691–708 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cameron, P.J.: Dixon’s theorem and random synchronization. Discrete Math. 313(11), 1233–1236 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Černý, J.: Poznámka k homogénnym experimentom s konečnými automatami. Mat.-fyz. Časopis Slovenskej Akadémie Vied 14(3), 208–216 (1964). in SlovakGoogle Scholar
  10. 10.
    Černý, J., Pirická, A., Rosenauerová, B.: On directable automata. Kybernetica 7, 289–298 (1971)MathSciNetMATHGoogle Scholar
  11. 11.
    Chevalier, P.Y., Hendrickx, J.M., Jungers, R.M.: Reachability of consensus and synchronizing automata. In: 54th IEEE Conference on Decision and Control (CDC), pp. 4139–4144. IEEE (2015)Google Scholar
  12. 12.
    Dixon, J.D.: The probability of generating the symmetric group. Math. Z. 110, 199–205 (1969)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Don, H.: The Černý conjecture and 1-contracting automata. Electr. J. Comb. 23(3), P3.12 (2016)Google Scholar
  14. 14.
    Dubuc, L.: Sur les automates circulaires et la conjecture de Černý. RAIRO Informatique Théorique et Applications 32, 21–34 (1998). in FrenchMathSciNetCrossRefGoogle Scholar
  15. 15.
    Frankl, P.: An extremal problem for two families of sets. Eur. J. Combin. 3, 125–127 (1982)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Frettlöh, D., Sing, B.: Computing modular coincidences for substitution tilings and point sets. Discrete Comput. Geom. 37, 381–407 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Friedman, J., Joux, A., Roichman, Y., Stern, J., Tillich, J.P.: The action of a few permutations on \(r\)-tuples is quickly transitive. Random Struct. Algorithms 12(4), 335–350 (1998)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ganyushkin, O., Mazorchuk, V.: Classical Finite Transformation Semigroups: An Introduction. Springer, London (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Gerencsér, B., Gusev, V.V., Jungers, R.M.: Primitive sets of nonnegative matrices and synchronizing automata. CoRR abs/1602.07556 (2016)Google Scholar
  20. 20.
    Gonze, F., Jungers, R.M.: On the synchronizing probability function and the triple rendezvous time for synchronizing automata. SIAM J. Discrete Math. 30(2), 995–1014 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Grech, M., Kisielewicz, A.: The Černý conjecture for automata respecting intervals of a directed graph. Discrete Math. Theoret. Comput. Sci. 15(3), 61–72 (2013)MATHGoogle Scholar
  22. 22.
    Helfgott, H.A., Seress, A.: On the diameter of permutation groups. Ann. Math. 179(2), 611–658 (2014)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Helfgott, H.A.: Growth in groups: ideas and perspectives. Bull. Amer. Math. Soc. 52(3), 357–413 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Kari, J.: Synchronizing finite automata on Eulerian digraphs. Theoret. Comput. Sci. 295, 223–232 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Klyachko, A.A., Rystsov, I.K., Spivak, M.A.: An extremal combinatorial problem associated with the bound on the length of a synchronizing word in an automaton. Cybern. Syst. Anal. 23(2), 165–171 (1987)CrossRefMATHGoogle Scholar
  26. 26.
    Maslennikova, M.: Reset complexity of ideal languages. In: Bieliková, M. (ed.) SOFSEM 2012. Proceedings of the Institute of Computer Science Academy of Sciences of the Czech Republic, vol. II, pp. 33–44 (2012)Google Scholar
  27. 27.
    Natarajan, B.K.: An algorithmic approach to the automated design of parts orienters. In: 27th FOCS, pp. 132–142. IEEE (1986)Google Scholar
  28. 28.
    Natarajan, B.K.: Some paradigms for the automated design of parts feeders. Int. J. Robot. Res. 8(6), 89–109 (1989)CrossRefGoogle Scholar
  29. 29.
    Panteleev, P.: Preset distinguishing sequences and diameter of transformation semigroups. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 353–364. Springer, Cham (2015). doi: 10.1007/978-3-319-15579-1_27 Google Scholar
  30. 30.
    Pin, J.E.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)MathSciNetMATHGoogle Scholar
  31. 31.
    Rystsov, I.K.: Reset words for commutative and solvable automata. Theoret. Comput. Sci. 172(1), 273–279 (1997)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Rystsov, I.K.: Estimation of the length of reset words for automata with simple idempotents. Cybern. Syst. Anal. 36(3), 339–344 (2000)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Saloff-Coste, L.: Random walks on finite groups. In: Kesten, H. (ed.) Probability on Discrete Structures, pp. 263–346. Springer, Heidelberg (2004)Google Scholar
  34. 34.
    Salomaa, A.: Composition sequences for functions over a finite domain. Theoret. Comput. Sci. 292(1), 263–281 (2003)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Steinberg, B.: The averaging trick and the Černý conjecture. Int. J. Found. Comput. Sci. 22(7), 1697–1706 (2011)CrossRefMATHGoogle Scholar
  36. 36.
    Steinberg, B.: The Černý conjecture for one-cluster automata with prime length cycle. Theoret. Comput. Sci. 412(39), 5487–5491 (2011)MathSciNetCrossRefMATHGoogle Scholar
  37. 37.
    Szykuła, M.: Improving the upper bound the length of the shortest reset words. CoRR abs/1702.05455 (2017)Google Scholar
  38. 38.
    Volkov, M.V.: Synchronizing automata and the Černý conjecture. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 11–27. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-88282-4_4 CrossRefGoogle Scholar
  39. 39.
    Vorel, V.: Subset synchronization of transitive automata. In: Ésik, Z., Fülöp, Z. (eds.) AFL 2014. EPTCS, vol. 151, pp. 370–381 (2014)Google Scholar
  40. 40.
    Zubov, A.Y.: On the diameter of the group \(S_N\) with respect to a system of generators consisting of a complete cycle and a transposition. Tr. Diskretn. Mat. 2, 112–150 (1998). in RussianMathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • François Gonze
    • 1
  • Vladimir V. Gusev
    • 1
    • 2
  • Balázs Gerencsér
    • 3
  • Raphaël M. Jungers
    • 1
  • Mikhail V. Volkov
    • 2
  1. 1.ICTEAM InstituteUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Ural Federal UniversityEkaterinburgRussia
  3. 3.Alfréd Rényi Institute of MathematicsBudapestHungary

Personalised recommendations