Advertisement

DFAs and PFAs with Long Shortest Synchronizing Word Length

  • Michiel de Bondt
  • Henk Don
  • Hans ZantemaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10396)

Abstract

It was conjectured by Černý in 1964, that a synchronizing DFA on n states always has a synchronizing word of length at most \((n-1)^2\), and he gave a sequence of DFAs for which this bound is reached. Until now a full analysis of all DFAs reaching this bound was only given for \(n \le 4\), and with bounds on the number of symbols for \(n \le 10\). Here we give the full analysis for \(n \le 6\), without bounds on the number of symbols.

For PFAs on \(n\le 6\) states we do a similar analysis as for DFAs and find the maximal shortest synchronizing word lengths, exceeding \((n-1)^2\) for \(n =4,5,6\). For arbitrary n we use rewrite systems to construct a PFA on three symbols with exponential shortest synchronizing word length, giving significantly better bounds than earlier exponential constructions. We give a transformation of this PFA to a PFA on two symbols keeping exponential shortest synchronizing word length, yielding a better bound than applying a similar known transformation.

References

  1. 1.
    de Bondt, M., Don, H., Zantema, H.: DFAs and PFAs with long shortest synchronizing word length (2017). https://arxiv.org/abs/1703.07618
  2. 2.
    Černy, J.: Poznámka k homogénnym experimentom s konečnými automatmi. Matematicko-fyzikálny časopis, Slovensk. Akad. Vied 14(3), 208–216 (1964)Google Scholar
  3. 3.
    Černy, J., Piricka, A., Rosenauerova, B.: On directable automata. Kybernetika 7(4), 289–298 (1971)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Don, H., Zantema, H.: Finding DFAs with maximal shortest synchronizing word length. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 249–260. Springer, Cham (2017). doi: 10.1007/978-3-319-53733-7_18 CrossRefGoogle Scholar
  5. 5.
    Gerencsér, B., Gusev, V.V., Jungers, R.M.: Primitive sets of nonnegative matrices and synchronizing automata (2016). https://arxiv.org/abs/1602.07556
  6. 6.
    Kari, J.: A counterexample to a conjecture concerning synchronizing word in finite automata. EATCS Bull. 73, 146–147 (2001)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Martyugin, P.V.: A lower bound for the length of the shortest carefully synchronizing words. Russ. Math. (Iz. VUZ) 54(1), 46–54 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Pin, J.E.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Roman, A.: A note on Černý conjecture for automata with 3-letter alphabet. J. Autom. Lang. Comb. 13(2), 141–143 (2008)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Rystsov, I.: Asymptotic estimate of the length of a diagnostic word for a finite automaton. Cybernetics 16(2), 194–198 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Szykuła, M.: Improving the upper bound the length of the shortest reset word (2017). https://arxiv.org/abs/1702.05455
  12. 12.
    Trahtman, A.N.: An efficient algorithm finds noticeable trends and examples concerning the Černy conjecture. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 789–800. Springer, Heidelberg (2006). doi: 10.1007/11821069_68 CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Vorel, V.: Subset synchronization and careful synchronization of binary finite automata. Int. J. Found. Comput. Sci. 27(5), 557–578 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceRadboud University NijmegenNijmegenThe Netherlands
  2. 2.Department of MathematicsFree UniversityAmsterdamThe Netherlands
  3. 3.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands

Personalised recommendations