Advertisement

The Ranges of Accepting State Complexities of Languages Resulting From Some Operations

  • Michal Hospodár
  • Markus Holzer
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10977)

Abstract

We examine the accepting state complexity, i.e., the minimal number of accepting states of deterministic finite automata (DFAs) for languages resulting from unary and binary operations on languages with accepting state complexity given as a parameter. This is continuation of the work of [J. Dassow: On the number of accepting states of finite automata, J. Autom., Lang. Comb., 21, 2016]. We solve most of the open problems mentioned thereof. In particular, we consider the operations of intersection, symmetric difference, right and left quotients, reversal, and permutation (on finite languages), where we obtain precise ranges of the accepting state complexities.

References

  1. 1.
    Cho, D.-J., Goč, D., Han, Y.-S., Ko, S.-K., Palioudakis, A., Salomaa, K.: State complexity of permutation on finite languages over a binary alphabet. Theoret. Comput. Sci. 682, 67–78 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chrobak, M.: Finite automata and unary languages. Theoret. Comput. Sci. 47, 149–158 (1986)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Dassow, J.: On the number of accepting states of finite automata. J. Autom. Lang. Comb. 21(1–2), 55–67 (2016)Google Scholar
  4. 4.
    Dassow, J.: Descriptional complexity and operations – two non-classical cases. In: Pighizzini, G., Câmpeanu, C. (eds.) DCFS 2017. LNCS, vol. 10316, pp. 33–44. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-60252-3_3CrossRefzbMATHGoogle Scholar
  5. 5.
    Geffert, V.: Magic numbers in the state hierarchy of finite automata. Inf. Comput. 205(11), 1652–1670 (2007)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Harrison, M.A.: Introduction to Formal Language Theory. Addison-Wesley, Boston (1978)zbMATHGoogle Scholar
  7. 7.
    Hricko, M., Jirásková, G., Szabari, A.: Union and intersection of regular languages and descriptional complexity. In: DCFS 2005, pp. 170–181 (2005)Google Scholar
  8. 8.
    Iwama, K., Kambayashi, Y., Takaki, K.: Tight bounds on the number of states of DFAs that are equivalent to \(n\)-state NFAs. Theoret. Comput. Sci. 237(1–2), 485–494 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Iwama, K., Matsuura, A., Paterson, M.: A family of NFAs which need \(2^n-\alpha \) deterministic states. Theoret. Comput. Sci. 301(1–3), 451–462 (2003)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Jirásková, G.: Magic numbers and ternary alphabet. Internat. J. Found. Comput. Sci. 22(2), 331–344 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Jirásková, G.: Personal communication (2017)Google Scholar
  12. 12.
    Jirásková, G., Šebej, J.: Reversal of binary regular languages. Theoret. Comput. Sci. 449, 85–92 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Leiss, E.: Succinct representation of regular languages by Boolean automata. Theoret. Comput. Sci. 13, 323–330 (1981)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Šebej, J.: Reversal on regular languages and descriptional complexity. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 265–276. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39310-5_25CrossRefzbMATHGoogle Scholar
  15. 15.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 41–110. Springer, Heidelberg (1997).  https://doi.org/10.1007/978-3-642-59136-5CrossRefGoogle Scholar
  16. 16.
    Yu, S., Zhuang, Q.: On the state complexity of intersection of regular languages. SIGACT News 22(3), 52–54 (1991)CrossRefGoogle Scholar
  17. 17.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia
  2. 2.Institut für InformatikUniversität GiessenGiessenGermany

Personalised recommendations