Synchronization Problems in Automata Without Non-trivial Cycles

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10329)

Abstract

We study the computational complexity of various problems related to synchronization of weakly acyclic automata, a subclass of widely studied aperiodic automata. We provide upper and lower bounds on the length of a shortest word synchronizing a weakly acyclic automaton or, more generally, a subset of its states, and show that the problem of approximating this length is hard. We also show inapproximability of the problem of computing the rank of a subset of states in a binary weakly acyclic automaton and prove that several problems related to recognizing a synchronizing subset of states in such automata are NP-complete.

Keywords

Synchronizing automata Computational complexity Weakly acyclic automata Subset rank 

References

  1. 1.
    Ananichev, D., Volkov, M.: Synchronizing monotonic automata. Theor. Comput. Sci. 327(3), 225–239 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Berlinkov, M.V.: On two algorithmic problems about synchronizing automata. In: Shur, A.M., Volkov, M.V. (eds.) DLT 2014. LNCS, vol. 8633, pp. 61–67. Springer, Cham (2014). doi:10.1007/978-3-319-09698-8_6 Google Scholar
  3. 3.
    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
  4. 4.
    Brzozowski, J., Fich, F.E.: Languages of R-trivial monoids. J. Comput. Syst. Sci. 20(1), 32–49 (1980)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cardoso, A.: The Černý Conjecture and Other Synchronization Problems. Ph.D. thesis. University of Porto, Portugal (2014)Google Scholar
  6. 6.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
  7. 7.
    Eppstein, D.: Reset sequences for monotonic automata. SIAM J. Comput. 19(3), 500–510 (1990)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gawrychowski, P., Straszak, D.: Strong inapproximability of the shortest reset word. In: Italiano, G.F., Pighizzini, G., Sannella, D.T. (eds.) MFCS 2015. LNCS, vol. 9234, pp. 243–255. Springer, Heidelberg (2015). doi:10.1007/978-3-662-48057-1_19 CrossRefGoogle Scholar
  9. 9.
    Gerencsér, B., Gusev, V.V., Jungers, R.M.: Primitive sets of nonnegative matrices and synchronizing automata. CoRR abs/1602.07556 (2016)Google Scholar
  10. 10.
    Jirásková, G., Masopust, T.: On the state and computational complexity of the reverse of acyclic minimal dfas. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 229–239. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31606-7_20 CrossRefGoogle Scholar
  11. 11.
    Kozen, D.: Lower bounds for natural proof systems. In: Proceedings of the 18th Annual Symposium on Foundations of Computer Science, pp. 254–266 (1977)Google Scholar
  12. 12.
    Martyugin, P.V.: Complexity of problems concerning carefully synchronizing words for PFA and directing words for NFA. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 288–302. Springer, Heidelberg (2010). doi:10.1007/978-3-642-13182-0_27 CrossRefGoogle Scholar
  13. 13.
    Mycielski, J.: Sur le coloriage des graphs. Colloquium Mathematicae 3(2), 161–162 (1955)MathSciNetMATHGoogle Scholar
  14. 14.
    Natarajan, B.K.: An algorithmic approach to the automated design of parts orienters. In: Proceedings of the 27th Annual Symposium on Foundations of Computer Science, pp. 132–142 (1986)Google Scholar
  15. 15.
    Pin, J.É.: On two combinatorial problems arising from automata theory. Ann. Discrete Math. 17, 535–548 (1983)MathSciNetMATHGoogle Scholar
  16. 16.
    Rystsov, I.K.: Rank of a finite automaton. Cybern. Syst. Anal. 28(3), 323–328 (1992)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Rystsov, I.K.: Polynomial complete problems in automata theory. Inform. Process. Lett. 16(3), 147–151 (1983)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rystsov, I.K.: Reset words for commutative and solvable automata. Theor. Comput. Sci. 172(1), 273–279 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ryzhikov, A.: Approximating the maximum number of synchronizing states in automata. CoRR abs/1608.00889 (2016)Google Scholar
  20. 20.
    Sandberg, S.: Homing and synchronizing sequences. In: Broy, M., Jonsson, B., Katoen, J.-P., Leucker, M., Pretschner, A. (eds.) Model-Based Testing of Reactive Systems. LNCS, vol. 3472, pp. 5–33. Springer, Heidelberg (2005). doi:10.1007/11498490_2 CrossRefGoogle Scholar
  21. 21.
    Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, 3rd edn. (2012)Google Scholar
  22. 22.
    Szykuła, M.: Improving the upper bound the length of the shortest reset words. CoRR abs/1702.05455 (2017)Google Scholar
  23. 23.
    Trahtman, A.N.: The Cerný conjecture for aperiodic automata. Discrete Math. Theor. Comput. Sci. 9(2), 3–10 (2007)MathSciNetMATHGoogle Scholar
  24. 24.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)Google Scholar
  25. 25.
    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
  26. 26.
    Vorel, V.: Subset synchronization and careful synchronization of binary finite automata. Int. J. Found. Comput. Sci. 27(5), 557–578 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3(6), 103–128 (2007)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Laboratoire G-SCOPUniversité Grenoble AlpesGrenobleFrance
  2. 2.United Institute of Informatics Problems of NASBMinskBelarus

Personalised recommendations