On the State and Computational Complexity of the Reverse of Acyclic Minimal DFAs

  • Galina Jirásková
  • Tomáš Masopust
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7381)


We study the state complexity of the reverse of acyclic minimal deterministic finite automata, and the computational complexity of the following problem: Given an acyclic minimal DFA, is the minimal DFA for the reverse also acyclic? Note that we allow self-loops in acyclic automata. We show that there exists a language accepted by an acyclic minimal DFA such that the minimal DFA for its reverse is exponential with respect to the number of states, and we establish a tight bound on the state complexity of the reverse of acyclic DFAs. We also give a direct proof of the fact that the minimal DFA for the reverse is acyclic if and only if the original acyclic minimal DFA satisfies a certain structural property, which can be tested in quadratic time.


State Complexity Regular Language Maximal State Reachable State Quadratic Time 
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  1. 1.
    Boutin, O., Komenda, J., Masopust, T., Schmidt, K., van Schuppen, J.H.: Hierarchical control with partial observations: Sufficient conditions. In: Proc. of IEEE Conference on Decision and Control and European Control Conference (CDC-ECC 2011), Orlando, Florida, USA, pp. 1817–1822 (2011)Google Scholar
  2. 2.
    Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Proc. of the Symposium on Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Institute of Brooklyn, New York (1963)Google Scholar
  3. 3.
    Champarnaud, J.M., Khorsi, A., Paranthoën, T.: Split and join for minimizing: Brzozowski’s algorithm,
  4. 4.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)Google Scholar
  5. 5.
    Jirásková, G., Šebej, J.: Note on Reversal of Binary Regular Languages. In: Holzer, M., Kutrib, M., Pighizzini, G. (eds.) DCFS 2011. LNCS, vol. 6808, pp. 212–221. Springer, Heidelberg (2011)Google Scholar
  6. 6.
    Jirásková, G., Masopust, T.: On a structural property in the state complexity of projected regular languages. Theoretical Computer Science (in press, 2012), doi:10.1016/j.tcs.2012.04.009Google Scholar
  7. 7.
    Klíma, O., Polák, L.: On biautomata. In: Proc. of NCMA 2011., vol. 282, pp. 153–164. Austrian Computer Society (2011)Google Scholar
  8. 8.
    Komenda, J., Masopust, T., van Schuppen, J.H.: Synthesis of controllable and normal sublanguages for discrete-event systems using a coordinator. Systems & Control Letters 60(7), 492–502 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Komenda, J., Masopust, T., van Schuppen, J.H.: Supervisory control synthesis of discrete-event systems using a coordination scheme. Automatica 48(2), 247–254 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Leiss, E.: Succinct representation of regular languages by boolean automata. Theoretical Computer Science 13, 323–330 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mirkin, B.G.: On dual automata. Kibernetika 2, 7–10 (1966) (in Russian); English translation: Cybernetics 2, 6–9 (1966)MathSciNetGoogle Scholar
  12. 12.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoretical Computer Science 320, 315–329 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Simon, I.: Hierarchies of Events with Dot-Depth One. Ph.D. thesis, Dep. of Applied Analysis and Computer Science, University of Waterloo, Canada (1972)Google Scholar
  14. 14.
    Simon, I.: Piecewise Testable Events. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 214–222. Springer, Heidelberg (1975)Google Scholar
  15. 15.
    Stern, J.: Characterizations of some classes of regular events. Theoretical Computer Science 35, 17–42 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Stern, J.: Complexity of some problems from the theory of automata. Information and Control 66(3), 163–176 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Trahtman, A.N.: A Package TESTAS for Checking Some Kinds of Testability. In: Champarnaud, J.-M., Maurel, D. (eds.) CIAA 2002. LNCS, vol. 2608, pp. 228–232. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Trahtman, A.N.: Piecewise and Local Threshold Testability of DFA. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 347–358. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexities of some basic operations on regular languages. Theoretical Computer Science 125(2), 315–328 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Galina Jirásková
    • 1
  • Tomáš Masopust
    • 2
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovak Republic
  2. 2.Institute of MathematicsAcademy of Sciences of the Czech RepublicBrnoCzech Republic

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