The Density of Linear-Time Properties

  • Bernd Finkbeiner
  • Hazem TorfahEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10482)


Finding models for linear-time properties is a central problem in verification and planning. We study the distribution of linear-time models by investigating the density of linear-time properties over the space of ultimately periodic words. The density of a property over a bound n is the ratio of the number of lasso-shaped words of length n, that satisfy the property, to the total number of lasso-shaped words of length n. We investigate the problem of computing the density for both linear-time properties in general and for the important special case of \(\omega \)-regular properties. For general linear-time properties, the density is not necessarily convergent and can oscillates indefinitely for certain properties. However, we show that the oscillation is bounded by the growth of the sets of bad- and good-prefix of the property. For \(\omega \)-regular properties, we show that the density is always convergent and provide a general algorithm for computing the density of \(\omega \)-regular properties as well as more specialized algorithms for certain sub-classes and their combinations.


  1. 1.
    Asarin, E., Blockelet, M., Degorre, A., Dima, C., Mu, C.: Asymptotic behaviour in temporal logic. In: LICS 2014. ACM, New York (2014)Google Scholar
  2. 2.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. Representation and Mind Series. The MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  3. 3.
    Bodirsky, M., Gärtner, T., von Oertzen, T., Schwinghammer, J.: Efficiently computing the density of regular languages. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 262–270. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24698-5_30 CrossRefGoogle Scholar
  4. 4.
    Chomsky, N., Miller, G.A.: Finite state languages. Inf. Control 1(2), 91–112 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Clarke, E., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Form. Methods Syst. Des. 19(1), 7–34 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    Demaine, E.D., López-Ortiz, A., Munro, J.I.: On universally easy classes for NP-complete problems. Theoret. Comput. Sci. 304, 471–476 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eisman, G., Ravikumar, B.: Approximate recognition of non-regular languages by finite automata. In: Proceedings of the 28th Australasian Conference on Computer Science, ACSC 2005, Darlinghurst, Australia, vol. 38 (2005)Google Scholar
  8. 8.
    Faran, R., Kupferman, O.: Spanning the spectrum from safety to liveness. In: Finkbeiner, B., Pu, G., Zhang, L. (eds.) ATVA 2015. LNCS, vol. 9364, pp. 183–200. Springer, Cham (2015). doi: 10.1007/978-3-319-24953-7_13 CrossRefGoogle Scholar
  9. 9.
    Finkbeiner, B., Torfah, H.: Counting models of linear-time temporal logic. In: Dediu, A.-H., Martín-Vide, C., Sierra-Rodríguez, J.-L., Truthe, B. (eds.) LATA 2014. LNCS, vol. 8370, pp. 360–371. Springer, Cham (2014). doi: 10.1007/978-3-319-04921-2_29 CrossRefGoogle Scholar
  10. 10.
    Flajolet, P.: Analytic models and ambiguity of context-free languages. Theoret. Comput. Sci. 49(23), 283–309 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grosu, R., Smolka, S.A.: Monte Carlo model checking. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 271–286. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31980-1_18 CrossRefGoogle Scholar
  12. 12.
    Hartwig, M.: On the density of regular and context-free languages. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 318–327. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-14031-0_35 CrossRefGoogle Scholar
  13. 13.
    Kuhtz, L., Finkbeiner, B.: LTL path checking is efficiently parallelizable. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 235–246. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02930-1_20 CrossRefGoogle Scholar
  14. 14.
    Markey, N., Schnoebelen, P.: Model checking a path. In: Amadio, R., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 251–265. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45187-7_17 CrossRefGoogle Scholar
  15. 15.
    Meyer, A.R., Stockmeyer, L.J.: The equivalence problem for regular expressions with squaring requires exponential space. In: 13th Annual Symposium on Switching and Automata Theory, SWAT 1972. IEEE, Washington, DC (1972)Google Scholar
  16. 16.
    Patrizi, F., Lipovetzky, N., De Giacomo, G., Geffner, H.: Computing infinite plans for LTL goals using a classical planner. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, Barcelona, Catalonia, Spain, 16–22 July 2011. IJCAI/AAAI (2011)Google Scholar
  17. 17.
    Piterman, N.: From nondeterministic büchi and streett automata to deterministic parity automata. In: 21st Annual IEEE Symposium on Logic in Computer Science (LICS 2006), pp. 255–264 (2006)Google Scholar
  18. 18.
    Pnueli, A.: The temporal logic of programs. In: 18th Annual Symposium on Foundations of Computer Science, pp. 46–57. IEEE Computer Society (1977)Google Scholar
  19. 19.
    Rozier, K.Y., Vardi, M.Y.: LTL satisfiability checking. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 149–167. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-73370-6_11 CrossRefGoogle Scholar
  20. 20.
    Schewe, S.: Synthesis for probabilistic environments. In: Graf, S., Zhang, W. (eds.) ATVA 2006. LNCS, vol. 4218, pp. 245–259. Springer, Heidelberg (2006). doi: 10.1007/11901914_20 CrossRefGoogle Scholar
  21. 21.
    Szilard, A., Yu, S., Zhang, K., Shallit, J.: Characterizing regular languages with polynomial densities. In: Havel, I.M., Koubek, V. (eds.) MFCS 1992. LNCS, vol. 629, pp. 494–503. Springer, Heidelberg (1992). doi: 10.1007/3-540-55808-X_48 CrossRefGoogle Scholar
  22. 22.
    Thomas, W.: Facets of synthesis: revisiting church’s problem. In: de Alfaro, L. (ed.) FoSSaCS 2009. LNCS, vol. 5504, pp. 1–14. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-00596-1_1 CrossRefGoogle Scholar
  23. 23.
    Torfah, H., Zimmermann, M.: The complexity of counting models of linear-time temporal logic. Acta Inform. pp. 1–22 (2016)Google Scholar
  24. 24.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Saarland UniversitySaarbrückenGermany

Personalised recommendations