Tight Bounds on the Descriptional Complexity of Regular Expressions

  • Hermann Gruber
  • Markus Holzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5583)

Abstract

We improve on some recent results on lower bounds for conversion problems for regular expressions. In particular we consider the conversion of planar deterministic finite automata to regular expressions, study the effect of the complementation operation on the descriptional complexity of regular expressions, and the conversion of regular expressions extended by adding intersection or interleaving to ordinary regular expressions. Almost all obtained lower bounds are optimal, and the presented examples are over a binary alphabet, which is best possible.

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References

  1. 1.
    Chandran, L.S., Kavitha, T.: The treewidth and pathwidth of hypercubes. Discrete Mathematics 306(3), 359–365 (2006)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Cohen, R.S.: Star height of certain families of regular events. Journal of Computer and System Sciences 4(3), 281–297 (1970)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Eggan, L.C.: Transition graphs and the star height of regular events. Michigan Mathematical Journal 10, 385–397 (1963)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ehrenfeucht, A., Zeiger, H.P.: Complexity measures for regular expressions. Journal of Computer and System Sciences 12(2), 134–146 (1976)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ellul, K., Krawetz, B., Shallit, J., Wang, M.-W.: Regular expressions: New results and open problems. Journal of Automata, Languages and Combinatorics 10(4), 407–437 (2005)MathSciNetMATHGoogle Scholar
  6. 6.
    Fürer, M.: The complexity of the inequivalence problem for regular expressions with intersection. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 234–245. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  7. 7.
    Gelade, W.: Succinctness of regular expressions with interleaving, intersection and counting. In: Ochmański, E., Tyszkiewicz, J. (eds.) MFCS 2008. LNCS, vol. 5162, pp. 363–374. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. In: Symposium on Theoretical Aspects of Computer Science, IBFI Schloss Dagstuhl, Germany. Dagstuhl Seminar Proceedings, vol. 08001, pp. 325–336 (2008)Google Scholar
  9. 9.
    Gruber, H., Holzer, M.: Finite automata, digraph connectivity, and regular expression size. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 39–50. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Gruber, H., Holzer, M.: Language operations with regular expressions of polynomial size. Theoretical Computer Science (accepted for publication) (2009)Google Scholar
  11. 11.
    Gruber, H., Holzer, M.: Provably shorter regular expressions from deterministic finite automata (Extended abstract). In: Ito, M., Toyama, M. (eds.) DLT 2008. LNCS, vol. 5257. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Gruber, H., Johannsen, J.: Optimal lower bounds on regular expression size using communication complexity. In: Amadio, R. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 273–286. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Hashiguchi, K.: Algorithms for determining relative star height and star height. Information and Computation 78(2), 124–169 (1988)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Hashiguchi, K., Honda, N.: Homomorphisms that preserve star height. Information and Control 30(3), 247–266 (1976)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley, Reading (1979)MATHGoogle Scholar
  16. 16.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82(1), 138–154 (2001)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Mayer, A.J., Stockmeyer, L.J.: Word problems - This time with interleaving. Information and Computation 115(2), 293–311 (1994)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McNaughton, R.: The loop complexity of pure-group events. Information and Control 11(1/2), 167–176 (1967)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Meyer, A.R., Fischer, M.J.: Economy of description by automata, grammars, and formal systems. In: IEEE Symposium on Switching and Automata Theory, pp. 188–191. IEEE Computer Society Press, Los Alamitos (1971)CrossRefGoogle Scholar
  20. 20.
    Warmuth, M.K., Haussler, D.: On the complexity of iterated shuffle. Journal of Computer and System Sciences 28(3), 345–358 (1984)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Hermann Gruber
    • 1
  • Markus Holzer
    • 1
  1. 1.Institut für InformatikUniversität GiessenGiessenGermany

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