Tomorrow and All our Yesterdays: MTL Satisfiability over the Integers

  • Carlo A. Furia
  • Paola Spoletini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5160)


We investigate the satisfiability problem for metric temporal logic (MTL) with both past and future operators over linear discrete bi-infinite time models isomorphic to the integer numbers, where time is unbounded both in the future and in the past. We provide a technique to reduce satisfiability over the integers to satisfiability over the well-known mono-infinite time model of natural numbers, and we show how to implement the technique through an automata-theoretic approach. We also prove that MTL satisfiability over the integers is EXPSPACE-complete, hence the given algorithm is optimal in the worst case.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alur, R., Henzinger, T.A.: Real-time logics: Complexity and expressiveness. Information and Computation 104(1), 35–77 (1993)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bianculli, D., Morzenti, A., Pradella, M., San Pietro, P., Spoletini, P.: Trio2Promela: A model checker for temporal metric specifications. In: ICSE Companion, pp. 61–62 (2007)Google Scholar
  3. 3.
    Bouyer, P., Markey, N., Ouaknine, J., Worrell, J.: The cost of punctuality. In: Proceedings of LICS 2007, pp. 109–120. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  4. 4.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  5. 5.
    Coen-Porisini, A., Pradella, M., San Pietro, P.: A finite-domain semantics for testing temporal logic specifications. In: Ravn, A.P., Rischel, H. (eds.) FTRTFT 1998. LNCS, vol. 1486, pp. 41–54. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Demri, S., Schnoebelen, P.: The complexity of propositional linear temporal logics in simple cases. Information and Computation 174(1), 84–103 (2002)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Emerson, E.A.: Temporal and modal logic. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 996–1072. Elsevier Science, Amsterdam (1990)Google Scholar
  8. 8.
    Furia, C.A., Spoletini, P.: MTL satisfiability over the integers. Technical Report 2008.2, DEI, Politecnico di Milano (2008)Google Scholar
  9. 9.
    Gabbay, D.M., Hodkinson, I., Reynolds, M.: Temporal Logic: mathematical foundations and computational aspects, vol. 1. Oxford University Press, Oxford (1994)MATHGoogle Scholar
  10. 10.
    Holzmann, G.J.: The SPIN Model Checker: Primer and Reference Manual (2003)Google Scholar
  11. 11.
    Muller, D.E., Schupp, P.E., Saoudi, A.: On the decidability of the linear Z-temporal logic and the monadic second order theory. In: Proc. of ICCI 1992, pp. 2–5 (1992)Google Scholar
  12. 12.
    Perrin, D., Pin, J.-É.: Infinite Words. Pure and Applied Mathematics, vol. 141. Elsevier, Amsterdam (2004)MATHGoogle Scholar
  13. 13.
    Pradella, M., Morzenti, A., San Pietro, P.: The symmetry of the past and of the future. In: Proceedings of ESEC/FSE 2007, pp. 312–320 (2007)Google Scholar
  14. 14.
    Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. Journal of the ACM 32(3), 733–749 (1985)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Spoletini, P.: Verification of Temporal Logic Specification via Model Checking. PhD thesis, DEI, Politecnico di Milano (May 2005)Google Scholar
  16. 16.
    Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 133–164. Elsevier Science, Amsterdam (1990)Google Scholar
  17. 17.
    Vardi, M.Y.: Handbook of Modal Logic. In: Automata-Theoretic Techniques for Temporal Reasoning, pp. 971–990 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Carlo A. Furia
    • 1
  • Paola Spoletini
    • 2
  1. 1.DEIPolitecnico di MilanoMilanoItaly
  2. 2.DSCPIUniversità degli Studi dell’InsubriaComoItaly

Personalised recommendations