Acta Informatica

, Volume 55, Issue 3, pp 191–212 | Cite as

The complexity of counting models of linear-time temporal logic

Original Article

Abstract

We determine the complexity of counting models of bounded size of specifications expressed in linear-time temporal logic. Counting word-models is #P-complete, if the bound is given in unary, and as hard as counting accepting runs of nondeterministic polynomial space Turing machines, if the bound is given in binary. Counting tree-models is as hard as counting accepting runs of nondeterministic exponential time Turing machines, if the bound is given in unary. For a binary encoding of the bound, the problem is at least as hard as counting accepting runs of nondeterministic exponential space Turing machines, and not harder than counting accepting runs of nondeterministic doubly-exponential time Turing machines. Finally, counting arbitrary transition systems satisfying a formula is #P-hard and not harder than counting accepting runs of nondeterministic polynomial time Turing machines with a PSPACE oracle, if the bound is given in unary. If the bound is given in binary, then counting arbitrary models is as hard as counting accepting runs of nondeterministic exponential time Turing machines.

Notes

Acknowledgements

We would like to thank Markus Lohrey and an anonymous reviewer for bringing Ladner’s work on polynomial space counting [12] to our attention.

References

  1. 1.
    Arora, S., Barak, B.: Computational Complexity: A Modern Approach, 1st edn. Cambridge University Press, New York (2009)CrossRefMATHGoogle Scholar
  2. 2.
    Bertoni, A. Mauri, G., Sabadini, N.: A characterization of the class of functions computable in polynomial time on random access machines. In: STOC 1981, pp 168–176. ACM (1981)Google Scholar
  3. 3.
    Biere, A.: Bounded model checking. In: Biere, A., Heule, M., Van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 457–481. IOS Press (2009)Google Scholar
  4. 4.
    Bloem, R., Gamauf, H.-J., Hofferek, G., Könighofer, B., Könighofer, R.: Synthesizing robust systems with RATSY. In: Peled, D., Schewe, S. (eds.) SYNT 2012, Volume 84 of EPTCS, pp. 47–53. Open Publishing Association (2012)Google Scholar
  5. 5.
    Bohy, A., Bruyère, V., Filiot, E., Jin, N., Raskin, J.-F.: Acacia+, a tool for LTL synthesis. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012, Volume 7358 of LNCS, pp. 652–657. Springer, New York (2012)Google Scholar
  6. 6.
    Burch, J.R., Clarke, E.M., McMillan, K.L., Dill, D.L., Hwang, L.J.: Symbolic model checking: \(10^{20}\) states and beyond. Inf. Comput. 98(2), 142–170 (1992)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ehlers, R.: Unbeast: symbolic bounded synthesis. In: Abdulla, P.A., Rustan, K., Leino, M. (eds.) TACAS 2011, Volume 6605 of LNCS, pp. 272–275. Springer, New York (2011)Google Scholar
  8. 8.
    Finkbeiner, B., Schewe, S.: Bounded synthesis. Int. J. Softw. Tools Technol. Transf. 15(5–6), 519–539 (2013)CrossRefMATHGoogle 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, Volume 8370 of LNCS, pp. 360–371. Springer, New York (2014)Google Scholar
  10. 10.
    Hemaspaandra, L.A., Vollmer, H.: The satanic notations: counting classes beyond #P and other definitional adventures. SIGACT News 26(1), 2–13 (1995)CrossRefGoogle Scholar
  11. 11.
    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, Volume 5556 of LNCS, pp. 235–246. Springer, New York (2009)Google Scholar
  12. 12.
    Ladner, R.E.: Polynomial space counting problems. SIAM J. Comput. 18(6), 1087–1097 (1989)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Liśkiewicz, M., Ogihara, M., Toda, S.: The complexity of counting self-avoiding walks in subgraphs of two-dimensional grids and hypercubes. Theor. Comput. Sci. 1–3(304), 129–156 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Littman, M.L., Majercik, S.M., Pitassi, T.: Stochastic boolean satisfiability. J. Autom. Reason. 27, 2001 (2000)MathSciNetMATHGoogle Scholar
  15. 15.
    Lohrey, M., Schmidt-Schauß, M.: Processing succinct matrices and vectors. In: Hirsch, E.A., Kuznetsov, S.O., Pin, J.-É., Vereshchagin, N.K. (eds.) CSR 2014, Volume 8476 of LNCS, pp. 245–258. New York, Springer (2014)Google Scholar
  16. 16.
    Morwood, D., Bryce, D.: Evaluating temporal plans in incomplete domains. In: Hoffmann, J., Selman, B. (eds.) AAAI 2012. AAAI Press, Menlo Park (2012)Google Scholar
  17. 17.
    Pnueli, A.: The temporal logic of programs. In: FOCS 1977, pp. 46–57. IEEE Computer Society (1977)Google Scholar
  18. 18.
    Prasad Sistla, A., Clarke, E.M.: The complexity of propositional linear temporal logics. J. ACM 32(3), 733–749 (1985)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Prasad Sistla, A.: Safety, liveness and fairness in temporal logic. Form. Asp. Comput. 6(5), 495–511 (1994)CrossRefMATHGoogle Scholar
  20. 20.
    Torfah, H., Zimmermann, M.: The complexity of counting models of linear-time temporal logic. In: Raman, V., Suresh, S.P. (eds.) FSTTCS 2014, Volume 29 of LIPIcs, pp. 241–252. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, Wadern (2014)Google Scholar
  21. 21.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Williams, R.: A counting class based on PSPACE. http://web.stanford.edu/~rrwill/sharp-p-pspace.pdf (1999)

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Reactive Systems GroupSaarland UniversitySaarbrückenGermany

Personalised recommendations