State Complexity of Basic Operations on Finite Languages

  • C. Câmpeanu
  • K. CulikII
  • Kai Salomaa
  • Sheng Yu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2214)

Abstract

The state complexity of basic operations on regular languages has been studied in [9],[10],[11]. Here we focus on finite languages. We show that the catenation of two finite languages accepted by an mstate and an n-state DFA, respectively, with m > n is accepted by a DFA of (mn + 3)2n−2 − 1 states in the two-letter alphabet case, and this bound is shown to be reachable. We also show that the tight upperbounds for the number of states of a DFA that accepts the star of an n-state finite language is 2n−3 + 2n−4 in the two-letter alphabet case. The same bound for reversal is 3 · 2p−1 − 1 when n is even and 2p − 1 when n is odd. Results for alphabets of an arbitrary size are also obtained. These upper-bounds for finite languages are strictly lower than the corresponding ones for general regular languages.

References

  1. 1.
    J.A. Brzozowski, “Canonical regular expressions and minimal state graphs for definite events”, Mathematical Theory of Automata, vol. 12 of MRI Symposia Series, Polytechnic Press, NY, 1962, 529–561.Google Scholar
  2. 2.
    C. Câmpeanu, “Regular languages and programming languages”, Revue Roumaine de Linguistique-CLTA, 23 (1986), 7–10.Google Scholar
  3. 3.
    J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages, and Computation, Addison Wesley (1979), Reading, Mass.MATHGoogle Scholar
  4. 4.
    E. Leiss, “Succinct representation of regular languages by boolean automata”, Theoretical Computer Science 13 (1981) 323–330.CrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Salomaa, Theory of Automata, Pergamon Press (1969), Oxford.MATHGoogle Scholar
  6. 6.
    K. Salomaa and S. Yu, “NFA to DFA Transformation for Finite Languages over Arbitrary Alphabets”, Journal of Automata, Languages and Combinatorics, 2 (1997) 3, 177–186.MathSciNetGoogle Scholar
  7. 7.
    B.W. Watson, Taxonomies and Toolkits of Regular Language Algorithms, PhD Dissertation, Department of Mathematics and Computing Science, Eindhoven University of Technology, 1995.Google Scholar
  8. 8.
    D. Wood, Theory of Computation, John Wiley and Sons, 1987.Google Scholar
  9. 9.
    S. Yu, Q. Zhuang, K. Salomaa, “The state complexities of some basic operations on regular languages”, Theoretical Computer Science 125 (1994) 315–328.CrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Yu, Q. Zhuang, “On the State Complexity of Intersection of Regular Languages”, ACM SIGACT News, vol. 22, no. 3, (1991) 52–54.CrossRefGoogle Scholar
  11. 11.
    S. Yu, Regular Languages, in: Handbook of Formal Languages, Vol. I, G. Rozenberg and A. Salomaa eds., Springer Verlag, 1997, pp. 41–110.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • C. Câmpeanu
    • 1
  • K. CulikII
    • 2
  • Kai Salomaa
    • 3
  • Sheng Yu
    • 3
  1. 1.Fundamentals of Computer Science DepartmentFaculty of Mathematics University of BucharestRomania
  2. 2.Department of Computer ScienceUniversity of South CarolinaColumbiaUSA
  3. 3.Department of Computer ScienceThe University of Western OntarioLondonCanada

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