Efficiently Computing the Density of Regular Languages

  • Manuel Bodirsky
  • Tobias Gärtner
  • Timo von Oertzen
  • Jan Schwinghammer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2976)


A regular language L is called dense if the fraction f m of words of length m over some fixed signature that are contained in L tends to one if m tends to infinity. We present an algorithm that computes the number of accumulation points of (f m ) in polynomial time, if the regular language L is given by a finite deterministic automaton, and can then also efficiently check whether L is dense. Deciding whether the least accumulation point of (f m ) is greater than a given rational number, however, is coNP-complete. If the regular language is given by a non-deterministic automaton, checking whether L is dense becomes PSPACE-hard. We will formulate these problems as convergence problems of partially observable Markov chains, and reduce them to combinatorial problems for periodic sequences of rational numbers.


Markov Chain Rational Number Accumulation Point Regular Language Convergence Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and analysis of algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  2. 2.
    Berstel, J.: Sur la densité asymptotique de langages formels. In: ICALP, pp. 345–358 (1972)Google Scholar
  3. 3.
    Bhattacharya, R.N., Waymire, E.C.: Stochastic processes with applications, New York. Wiley Series in Probability and Mathematical Statistics (1990)Google Scholar
  4. 4.
    Bodirsky, M., Gärtner, T., von Oertzen, T., Schwinghammer, J.: Efficiently computing the density of regular languages, Full version, available under
  5. 5.
    Bürgisser, P., Clausen, M., Shokrollahi, M.: Algebraic Complexity Theory. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  6. 6.
    de Alfaro, L.: How to specify and verify the long-run average behavior of probabilistic systems. In: Proc. 13th IEEE Symp. on Logic in Computer Science. IEEE Computer Society Press, Los Alamitos (1998)Google Scholar
  7. 7.
    Gantmacher, F.R.: The Theory of Matrices. Chelsea Pub. Co., New York (1977)Google Scholar
  8. 8.
    Garey, M., Johnson, D.: A Guide to NP-completeness. CSLI Press, Stanford (1978)Google Scholar
  9. 9.
    Grandjean, E.: Complexity of the first-order theory of almost all finite structures. Information and Control 57, 180–204 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Mehlhorn, K., Näher, S.: LEDA. A platform for combinatorial and geometric computing. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  11. 11.
    Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Springer, Heidelberg (1978)zbMATHGoogle Scholar
  12. 12.
    Stockmeyer, L.J., Meyer, A.: Word problems requiring exponential time. In: Proc. 5th Ann. ACM Sypm. on Theory of Computing. Association of Computing Machinery, vol. 1–9 (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Manuel Bodirsky
    • 1
  • Tobias Gärtner
    • 2
  • Timo von Oertzen
    • 2
  • Jan Schwinghammer
    • 3
  1. 1.Humboldt-Universität zu BerlinGermany
  2. 2.Universität des SaarlandesGermany
  3. 3.University of SussexUK

Personalised recommendations