Advertisement

Regular Expressions with Counting: Weak versus Strong Determinism

  • Wouter Gelade
  • Marc Gyssens
  • Wim Martens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5734)

Abstract

We study deterministic regular expressions extended with the counting operator. There exist two notions of determinism, strong and weak determinism, which almost coincide for standard regular expressions. This, however, changes dramatically in the presence of counting. In particular, we show that weakly deterministic expressions with counting are exponentially more succinct and strictly more expressive than strongly deterministic ones, even though they still do not capture all regular languages. In addition, we present a finite automaton model with counters, study its properties and investigate the natural extension of the Glushkov construction translating expressions with counting into such counting automata. This translation yields a deterministic automaton if and only if the expression is strongly deterministic. These results then also allow to derive upper bounds for decision problems for strongly deterministic expressions with counting.

Keywords

Regular Expression Regular Language Parse Tree Counter Variable Counting Operator 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Brüggemann-Klein, A.: Regular expressions into finite automata. Theor. Comput. Sci. 120(2), 197–213 (1993)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Brüggemann-Klein, A., Wood, D.: One-unambiguous regular languages. Information and Computation 142(2), 182–206 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Colazzo, D., Ghelli, G., Sartiani, C.: Efficient asymmetric inclusion between regular expression types. In: ICDT, pp. 174–182 (2009)Google Scholar
  4. 4.
    Dal-Zilio, S., Lugiez, D.: XML schema, tree logic and sheaves automata. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 246–263. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  5. 5.
    Esparza, J.: Decidability and complexity of Petri net problems – an introduction. In: Petri Nets, pp. 374–428 (1996)Google Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  7. 7.
    Gelade, W., Martens, W., Neven, F.: Optimizing schema languages for XML: Numerical constraints and interleaving. In: Schwentick, T., Suciu, D. (eds.) ICDT 2007. LNCS, vol. 4353, pp. 269–283. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    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
  9. 9.
    Hume, A.: A tale of two greps. Softw. Pract. and Exp. 18(11), 1063–1072 (1988)CrossRefGoogle Scholar
  10. 10.
    Kilpeläinen, P.: Inclusion of unambiguous #res is NP-hard (May 2004) (unpublished)Google Scholar
  11. 11.
    Kilpeläinen, P., Tuhkanen, R.: Regular expressions with numerical occurrence indicators — preliminary results. In: SPLST 2003, pp. 163–173 (2003)Google Scholar
  12. 12.
    Kilpeläinen, P., Tuhkanen, R.: Towards efficient implementation of XML schema content models. In: DOCENG 2004, pp. 239–241. ACM, New York (2004)Google Scholar
  13. 13.
    Kilpeläinen, P., Tuhkanen, R.: One-unambiguity of regular expressions with numeric occurrence indicators. Inform. Comput. 205(6), 890–916 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Koch, C., Scherzinger, S.: Attribute grammars for scalable query processing on XML streams. VLDB Journal 16(3), 317–342 (2007)CrossRefGoogle Scholar
  15. 15.
    Martens, W., Neven, F., Schwentick, T.: Complexity of decision problems for simple regular expressions. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 889–900. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  16. 16.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: FOCS, pp. 125–129 (1972)Google Scholar
  17. 17.
    Mount, D.W.: Bioinformatics: Sequence and Genome Analysis. Cold Spring Harbor Laboratory Press (September 2004)Google Scholar
  18. 18.
    Seidl, H., Schwentick, T., Muscholl, A., Habermehl, P.: Counting in trees for free. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 1136–1149. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and Jacobsthal’s function. Int. J. Found. Comp. Sc. 13(1), 145–159 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sperberg-McQueen, C.M.: Notes on finite state automata with counters (2004), http://www.w3.org/XML/2004/05/msm-cfa.html
  21. 21.
    Sperberg-McQueen, C.M., Thompson, H.: XML Schema (2005), http://www.w3.org/XML/Schema
  22. 22.
    Vardi, M.Y.: From monadic logic to PSL. Pillars of Computer Science, 656–681 (2008)Google Scholar
  23. 23.
    Wall, L., Christiansen, T., Orwant, J.: Programming Perl. O’Reilly, Sebastopol (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Wouter Gelade
    • 1
  • Marc Gyssens
    • 1
  • Wim Martens
    • 2
  1. 1.School for Information TechnologyHasselt University and Transnational University of LimburgBelgium
  2. 2.Technical University of DortmundGermany

Personalised recommendations