Cyclic Shift on Prefix-Free Languages

  • Jozef Jirásek
  • Galina Jirásková
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7913)


We prove that the cyclic shift of a prefix-free language represented by a minimal complete n-state deterministic finite automaton is recognized by a deterministic automaton of at most (2n − 3) n − 2 states. We also show that this bound is tight in the quaternary case, and that it cannot be met by using any smaller alphabet. In the ternary and binary cases, we still get exponential lower bounds.


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  1. 1.
    Birget, J.-C.: Intersection and union of regular languages and state complexity. Inform. Process. Letters 43, 185–190 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Gruber, H., Holzer, M.: Language operations with regular expressions of polynomial size. Theoret. Comput. Sci. 410, 3281–3289 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Han, Y.-S., Salomaa, K., Wood, D.: Operational state complexity of prefix-free regular languages. In: Automata, Formal Languages, and Related Topics, pp. 99–115. University of Szeged, Hungary (2009)Google Scholar
  4. 4.
    Han, Y.-S., Salomaa, K., Wood, D.: Nondeterministic state complexity of basic operations for prefix-free regular languages. Fund. Inform. 90, 93–106 (2009)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Han, Y.-S., Salomaa, K., Yu, S.: State complexity of combined operations for prefix-free regular languages. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 398–409. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Jirásková, G., Krausová, M.: Complexity in prefix-free regular languages. In: McQuillan, I., Pighizzini, G., Trost, B. (eds.) Proc. 12th DCFS, pp. 236–244. University of Saskatchewan, Saskatoon (2010)Google Scholar
  7. 7.
    Jirásková, G., Okhotin, A.: State complexity of cyclic shift. Theor. Inform. Appl. 42, 335–360 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Krausová, M.: Prefix-free regular languages: Closure properties, difference, and left quotient. In: Kotásek, Z., Bouda, J., Černá, I., Sekanina, L., Vojnar, T., Antoš, D. (eds.) MEMICS 2011. LNCS, vol. 7119, pp. 114–122. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  9. 9.
    Maslov, A.N.: Estimates of the number of states of finite automata. Soviet Math. Dokl. 11, 1373–1375 (1970)zbMATHGoogle Scholar
  10. 10.
    Maslov, A.N.: The cyclic shift of languages. Problemy Peredači Informacii 9, 81–87 (1973) (Russian)Google Scholar
  11. 11.
    Oshiba, T.: Closure property of the family of context-free languages under the cyclic shift operation. Electron. Commun. Japan 55, 119–122 (1972)MathSciNetGoogle Scholar
  12. 12.
    Sipser, M.: Introduction to the theory of computation. PWS Publishing Company, Boston (1997)zbMATHGoogle Scholar
  13. 13.
    Yu, S.: Regular languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. I, ch. 2, pp. 41–110. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  14. 14.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125, 315–328 (1994)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jozef Jirásek
    • 1
  • Galina Jirásková
    • 2
  1. 1.Institute of Computer Science, Faculty of ScienceP.J. Šafárik UniversityKošiceSlovakia
  2. 2.Mathematical InstituteSlovak Academy of SciencesKošiceSlovakia

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