On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

  • Rafaela Bastos
  • Sabine Broda
  • António Machiavelo
  • Nelma MoreiraEmail author
  • Rogério Reis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9777)


Extended regular expressions (with complement and intersection) are used in many applications due to their succinctness. In particular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard regular expressions or equivalent nondeterministic finite automata (NFA). For practical purposes it is important to study the average behaviour of conversions between these models. In this paper, we focus on the conversion of regular expressions with intersection to nondeterministic finite automata, using partial derivatives and the notion of support. First, we give a tight upper bound of \(2^{O(n)}\) for the worst-case number of states of the resulting partial derivative automaton, where n is the size of the expression. Using the framework of analytic combinatorics, we then establish an upper bound of \((1.056 +o(1))^n\) for its asymptotic average-state complexity, which is significantly smaller than the one for the worst case.


Regular Expression Real Zero Analytic Combinatorics Positive Zero Average Complexity 
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.


  1. 1.
    Antimirov, V.: Partial derivatives of regular expressions and finite automaton constructions. Theoret. Comput. Sci. 155(2), 291–319 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antimirov, V.M., Mosses, P.D.: Rewriting extended regular expressions. In: Rozenberg, G., Salomaa, A. (eds.) 1st DLT. pp. 195–209. World Scientific (1994)Google Scholar
  3. 3.
    Bastos, R.: Manipulation of Extended Regular Expressions with Derivatives. Master’s thesis, Faculdade de Ciências da Universidade do Porto (2015)Google Scholar
  4. 4.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average state complexity of partial derivative automata. Int. J. Found. Comput. Sci. 22(7), 1593–1606 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: On the average size of Glushkov and partial derivative automata. Int. J. Found. Comput. Sci. 23(5), 969–984 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: A Hitchhiker’s guide to descriptional complexity through analytic combinatorics. Theoret. Comput. Sci. 528, 85–100 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Broda, S., Machiavelo, A., Moreira, N., Reis, R.: Partial derivative automaton for regular expressions with shuffle. In: Shallit, J., Okhotin, A. (eds.) DCFS 2015. LNCS, vol. 9118, pp. 21–32. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  8. 8.
    Brzozowski, J.A.: Derivatives of regular expressions. JACM 11(4), 481–494 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caron, P., Champarnaud, J.-M., Mignot, L.: Partial derivatives of an extended regular expression. In: Dediu, A.-H., Inenaga, S., Martín-Vide, C. (eds.) LATA 2011. LNCS, vol. 6638, pp. 179–191. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Caron, P., Champarnaud, J., Mignot, L.: A general framework for the derivation of regular expressions. RAIRO - Theor. Inf. Appl. 48(3), 281–305 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Champarnaud, J.M., Ziadi, D.: From Mirkin’s prebases to Antimirov’s word partial derivatives. Fundam. Inform. 45(3), 195–205 (2001)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Christiansen, T., Foy, B.D., Wall, L., Orwant, J.: Programming Perl, 4th edn. O’Reilly Media, Sebastopol (2012)zbMATHGoogle Scholar
  13. 13.
    Flajolet, P., Sedgewick, R.: Analytic Combinatorics. CUP, Cambridge (2008)zbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Gelade, W.: Succinctness of regular expressions with interleaving, intersection and counting. Theoret. Comput. Sci. 411(31–33), 2987–2998 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gelade, W., Neven, F.: Succinctness of the complement and intersection of regular expressions. In: Albers, S., Weil, P. (eds.) 25th STACS. LIPIcs, vol. 1, pp. 325–336. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany (2008)Google Scholar
  17. 17.
    Gruber, H.: On the descriptional and algorithmic complexity of regular languages. Ph.D. thesis, Justus Liebig University Giessen (2010)Google Scholar
  18. 18.
    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
  19. 19.
    Jiang, T., Ravikumar, B.: A note on the space complexity of some decision problems for finite automata. Inf. Process. Lett. 40(1), 25–31 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Mirkin, B.G.: An algorithm for constructing a base in a language of regular expressions. Eng. Cybern. 5, 51–57 (1966)Google Scholar
  21. 21.
    Petersen, H.: The membership problem for regular expressions with intersection is complete in LOGCFL. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 513–522. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  22. 22.
    Sen, K., Rosu, G.: Generating optimal monitors for extended regular expressions. Electr. Notes Theor. Comput. Sci. 89, 231–250 (2003)Google Scholar
  23. 23.
    van der Vlist, E.: RELAX NG. O’Reilly Media, Cambridge (2003)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  • Rafaela Bastos
    • 1
  • Sabine Broda
    • 1
  • António Machiavelo
    • 1
  • Nelma Moreira
    • 1
    Email author
  • Rogério Reis
    • 1
  1. 1.CMUP and Faculdade de Ciências da Universidade do PortoPortoPortugal

Personalised recommendations