Spectral Gap in Timed Automata

  • Eugene Asarin
  • Nicolas Basset
  • Aldric Degorre
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8053)


Various problems about probabilistic and non-probabilistic timed automata (computing probability density, language volume or entropy) can be naturally phrased as iteration of linear operators in Banach spaces. Convergence of such iterations is guaranteed whenever the operator’s spectrum has a gap. In this article, for operators used in entropy computation, we use the theory of positive operators to establish the existence of such a gap. This allows to devise simple numeric algorithms for computing the entropy and prove their exponential convergence.


Spectral Radius Positive Operator Regular Language Entry Region Periodic Component 
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.
    Alur, R., Dill, D.L.: A theory of timed automata. TCS 126, 183–235 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Asarin, E., Basset, N., Béal, M.-P., Degorre, A., Perrin, D.: Toward a timed theory of channel coding. In: Jurdziński, M., Ničković, D. (eds.) FORMATS 2012. LNCS, vol. 7595, pp. 27–42. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Asarin, E., Basset, N., Degorre, A., Perrin, D.: Generating functions of timed languages. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 124–135. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Asarin, E., Degorre, A.: Volume and entropy of regular timed languages: Analytic approach. In: Ouaknine, J., Vaandrager, F.W. (eds.) FORMATS 2009. LNCS, vol. 5813, pp. 13–27. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Asarin, E., Degorre, A.: Volume and entropy of regular timed languages: Discretization approach. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 69–83. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Asarin, E., Degorre, A.: Two size measures for timed languages. In: FSTTCS. LIPIcs, vol. 8, pp. 376–387 (2010)Google Scholar
  7. 7.
    Basset, N.: A maximal entropy stochastic process for a timed automaton. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part II. LNCS, vol. 7966, pp. 61–73. Springer, Heidelberg (2013)Google Scholar
  8. 8.
    Basset, N., Asarin, E.: Thin and thick timed regular languages. In: Fahrenberg, U., Tripakis, S. (eds.) FORMATS 2011. LNCS, vol. 6919, pp. 113–128. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  9. 9.
    Bertrand, N., Bouyer, P., Brihaye, T., Markey, N.: Quantitative model-checking of one-clock timed automata under probabilistic semantics. In: QEST, pp. 55–64 (2008)Google Scholar
  10. 10.
    Chomsky, N., Miller, G.A.: Finite state languages. Information and Control 1(2), 91–112 (1958)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Denardo, E.: Periods of connected networks and powers of nonnegative matrices. Mathematics of Operations Research 2(1), 20–24 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Katō, T.: Perturbation Theory for Linear Operators. Springer (1995)Google Scholar
  13. 13.
    Krasnosel’skij, M., Lifshits, E., Sobolev, A.: Positive Linear Systems: The method of positive operators. Heldermann Verlag, Berlin (1989)zbMATHGoogle Scholar
  14. 14.
    Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press (1995)Google Scholar
  15. 15.
    Maler, O., Larsen, K.G., Krogh, B.H.: On zone-based analysis of duration probabilistic automata. In: INFINITY. EPTCS, vol. 39, pp. 33–46 (2010)Google Scholar
  16. 16.
    Sassoli, L., Vicario, E.: Close form derivation of state-density functions over DBM domains in the analysis of non-Markovian models. In: QEST, pp. 59–68 (2007)Google Scholar
  17. 17.
    Seneta, E.: Non-Negative Matrices and Markov Chains. Springer (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eugene Asarin
    • 1
  • Nicolas Basset
    • 1
    • 2
  • Aldric Degorre
    • 1
  1. 1.LIAFAUniversity Paris Diderot and CNRSFrance
  2. 2.LIGMUniversity Paris-Est Marne-la-Vallée and CNRSFrance

Personalised recommendations