Provably Shorter Regular Expressions from Deterministic Finite Automata

(Extended Abstract)
  • Hermann Gruber
  • Markus Holzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5257)


We study the problem of finding good elimination orderings for the state elimination algorithm, which is one of the most popular algorithms for the conversion of finite automata into equivalent regular expressions. Based on graph separator techniques we are able to describe elimination strategies that remove states in large induced subgraphs that are “simple” like, e.g., independent sets or subgraphs of bounded treewidth, of the underlying automaton, that lead to regular expressions of moderate size. In particular, we show that there is an elimination ordering such that every language over a binary alphabet accepted by an n-state deterministic finite automaton has alphabetic width at most O(1.742 n ), which is, to our knowledge, the algorithm with currently the best known performance guarantee. Finally, we apply our technique to the question on the effect of language operations on regular expression size. In case of the intersection operation we prove an upper bound which matches, up to a small factor, a lower bound recently obtained in [9,10], and thus settles an open problem stated in [7].


Regular Expression Regular Language Internal Vertex Performance Guarantee Input Alphabet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Brzozowski, J.A.: Derivatives of regular expressions. Journal of the ACM 11(4), 481–494 (1964)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chung, F.R.K.: Spectral Graph Theory. In: CBMS Regional Conference Series in Mathematics, vol. 92. American Mathematical Society (1997)Google Scholar
  3. 3.
    Delgado, M., Morais, J.: Approximation to the smallest regular expression for a given regular language. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 312–314. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Domaratzki, M.: State complexity of proportional removals. Journal of Automata, Languages and Combinatorics 7(4), 455–468 (2002)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Edwards, K., Farr, G.E.: Planarization and fragmentability of some classes of graphs. Discrete Mathematics 308(12), 2396–2406 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Ehrenfeucht, A., Zeiger, H.P.: Complexity measures for regular expressions. Journal of Computer and System Sciences 12(2), 134–146 (1976)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Ellul, K., Krawetz, B., Shallit, J., Wang, M.: Regular expressions: New results and open problems. Journal of Automata, Languages and Combinatorics 10(4), 407–437 (2005)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gelade, W.: Succinctness of regular expressions with interleaving, intersection and counting. In: Proceedings of the 33rd International Symposium on Mathematical Foundations of Computer Science, Turoń, Poland, August 2008. LNCS. Springer, Heidelberg (to appear, 2008)Google Scholar
  9. 9.
    Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. In: Albers, S., Weil, P. (eds.) Proceedings of the 25th Symposium on Theoretical Aspects of Computer Science, Bordeaux, France, February 2008. Dagstuhl Seminar Proceedings, vol. 08001, pp. 325–336. Internationales Begegnungs- und Forschungszentrum fuer Informatik (IBFI), Schloss Dagstuhl, Germany (2008)Google Scholar
  10. 10.
    Gruber, H., Holzer, M.: Finite automata, digraph connectivity, and regular expression size. In: Aceto, L., Damgaard, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walkuwiewicz, I. (eds.) Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Reykjavik, Iceland, July 2008. Springer, Heidelberg (2008)Google Scholar
  11. 11.
    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
  12. 12.
    Han, Y.-S., Wood, D.: Obtaining shorter regular expressions from finite-state automata. Theoretical Computer Science 370(1-3), 110–120 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ilie, L., Yu, S.: Follow automata. Information and Computation 186(1), 140–162 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kleene, S.C.: Representation of events in nerve nets and finite automata. In: Shannon, C.E., McCarthy, J. (eds.) Automata Studies, Annals of Mathematics Studies, pp. 3–42. Princeton University Press, Princeton (1956)Google Scholar
  15. 15.
    Lipton, R.J., Tarjan, R.E.: A separator theorem for planar graphs. SIAM Journal on Applied Mathematics 36(2), 177–189 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    McIntosh, H.V.: REEX: A CONVERT program to realize the McNaughton-Yamada analysis algorithm. Technical Report AIM-153, MIT Artificial Intelligence Laboratory (January 1968)Google Scholar
  17. 17.
    McNaughton, R., Yamada, H.: Regular expressions and state graphs for automata. IRA Transactions on Electronic Computers 9(1), 39–47 (1960)CrossRefGoogle Scholar
  18. 18.
    Morais, J.J., Moreira, N., Reis, R.: Acyclic automata with easy-to-find short regular expressions. In: Farré, J., Litovsky, I., Schmitz, S. (eds.) CIAA 2005. LNCS, vol. 3845, pp. 349–350. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms 7(3), 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Turán, P.: On an extremal problem in graph theory (in Hungarian). Matematicko Fizicki Lapok 48, 436–452 (1941)zbMATHGoogle Scholar
  21. 21.
    Wood, D.: Theory of Computation. John Wilet & Sons (1987)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hermann Gruber
    • 1
  • Markus Holzer
    • 2
  1. 1.Institut für InformatikLudwig-Maximilians-Universität MünchenMünchenGermany
  2. 2.Institut für InformatikTechnische Universität MünchenGarching bei MünchenGermany

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