Note on Reversal of Binary Regular Languages

  • Galina Jirásková
  • Juraj Šebej
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6808)


We present binary deterministic finite automata of n states that meet the upper bound 2n on the state complexity of reversal. The automata have a single final state and are one-cycle-free-path, thus the witness languages are deterministic union-free. This result allows us to describe a binary language such that the nondeterministic state complexity of the language and of its complement is n and n + 1, respectively, while the state complexity of the language is 2n. We also show that there is no regular language with state complexity 2n such that both the language and its complement have nondeterministic state complexity n.


Regular languages reversal state complexity nondeterministic state complexity deterministic union-free languages 


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  1. 1.
    Brzozowski, J.A.: Canonical Regular Expressions and Minimal State Graphs for Definite Events. In: Proceedings of the Symposium on Mathematical Theory of Automata, New York, NY, April 24-26 (1962); Fox, J. (ed.) MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press of the Polytechnic Institute of Brooklyn, Brooklyn, NY (1963)Google Scholar
  2. 2.
    Brzozowski, J., Jirásková, G., Li, B.: Quotient complexity of ideal languages. In: López-Ortiz, A. (ed.) LATIN 2010. LNCS, vol. 6034, pp. 208–221. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Brzozowski, J., Jirásková, G., Zou, C.: Quotient complexity of closed languages. In: Ablayev, F., Mayr, E.W. (eds.) CSR 2010. LNCS, vol. 6072, pp. 84–95. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  4. 4.
    Champarnaud, J.-M., Khorsi, A., Paranthoën, T.: Split and join for minimizing: Brzozowski’s algorithm,
  5. 5.
    Jirásková, G., Masopust, T.: Complexity in union-free regular languages. In: Gao, Y., Lu, H., Seki, S., Yu, S. (eds.) DLT 2010. LNCS, vol. 6224, pp. 255–266. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Leiss, E.: Succinct representation of regular languages by Boolean automata. Theoret. Comput. Sci. 13, 323–330 (1981)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Lupanov, U.I.: A comparison of two types of finite automata. Problemy Kibernetiki 9, 321–326 (1963) (in Russian)Google Scholar
  8. 8.
    Mera, F., Pighizzini, G.: Complementing unary nondeterministic automata. Theor. Comput. Sci. 330, 349–360 (2005)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Mirkin, B.G.: On dual automata. Kibernetika (Kiev) 2, 7–10 (1966) (in Russian); English translation: Cybernetics 2, 6–9 (1966)MathSciNetGoogle Scholar
  10. 10.
    Rabin, M., Scott, D.: Finite automata and their decision problems. IBM Res. Develop. 3, 114–129 (1959)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoret. Comput. Sci. 320, 315–329 (2004)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Sipser, M.: Introduction to the theory of computation. PWS Publishing Company, Boston (1997)MATHGoogle Scholar
  13. 13.
    Šebej, J.: Reversal of regular languages and state complexity. In: Pardubská, D. (ed.) Proc. 10th ITAT, pp. 47–54. Šafárik University, Košice (2010)Google Scholar
  14. 14.
    Yu, S.: Chapter 2: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, pp. 41–110. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Galina Jirásková
    • 1
  • Juraj Šebej
    • 2
  1. 1.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia
  2. 2.Institute of Computer ScienceP.J. Šafárik UniversityKošiceSlovakia

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