SMT-Based Induction Methods for Timed Systems

  • Roland Kindermann
  • Tommi Junttila
  • Ilkka Niemelä
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7595)


Modeling time-related aspects is important in many applications of verification methods. For precise results, it is necessary to interpret time as a dense domain, e.g. using timed automata as a formalism, even though the system’s resulting infinite state space is challenging for verification methods. Furthermore, fully symbolic treatment of both timing related and non-timing related elements of the state space seems to offer an attractive approach to model checking timed systems with a large amount of non-determinism. This paper presents an SMT-based timed system extension to the IC3 algorithm, a SAT-based novel, highly efficient, complete verification method for untimed systems. Handling of the infinite state spaces of timed system in the extended IC3 algorithm is based on suitably adapting the well-known region abstraction for timed systems. Additionally, k-induction, another symbolic verification method for discrete time systems, is extended in a similar fashion to support timed systems. Both methods are evaluated and experimentally compared to a booleanization-based verification approach that uses the original discrete time IC3 algorithm.


Model Check Proof Obligation Bound Model Check Linear Arithmetic Random Property 
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. Theor. Comput. Sci. 126(2), 183–235 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Alur, R.: Timed Automata. In: Halbwachs, N., Peled, D.A. (eds.) CAV 1999. LNCS, vol. 1633, pp. 8–22. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  3. 3.
    Bengtsson, J.E., Yi, W.: Timed Automata: Semantics, Algorithms and Tools. In: Desel, J., Reisig, W., Rozenberg, G. (eds.) ACPN 2003. LNCS, vol. 3098, pp. 87–124. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Behrmann, G., David, A., Larsen, K.G.: A Tutorial on Uppaal. In: Bernardo, M., Corradini, F. (eds.) SFM-RT 2004. LNCS, vol. 3185, pp. 200–236. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  5. 5.
    Wang, F.: Efficient verification of timed automata with BDD-like data structures. International Journal on Software Tools for Technology Transfer 6(1), 77–97 (2004)CrossRefGoogle Scholar
  6. 6.
    Gruhn, P., Cheddie, H.L.: Safety Instrumented Systems: Design, Analysis, and Justification. ISA (2006)Google Scholar
  7. 7.
    Kindermann, R., Junttila, T., Niemelä, I.: Modeling for symbolic analysis of safety instrumented systems with clocks. In: Proc. ACSD 2011, pp. 185–194. IEEE (2011)Google Scholar
  8. 8.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer (1992)Google Scholar
  9. 9.
    Kindermann, R., Junttila, T., Niemelä, I.: Beyond Lassos: Complete SMT-Based Bounded Model Checking for Timed Automata. In: Giese, H., Rosu, G. (eds.) FORTE 2012 and FMOODS 2012. LNCS, vol. 7273, pp. 84–100. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Audemard, G., Cimatti, A., Kornilowicz, A., Sebastiani, R.: Bounded Model Checking for Timed Systems. In: Peled, D.A., Vardi, M.Y. (eds.) FORTE 2002. LNCS, vol. 2529, pp. 243–259. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Sorea, M.: Bounded model checking for timed automata. Electronic Notes in Theoretical Computer Science 68(5), 116–134 (2002)CrossRefGoogle Scholar
  12. 12.
    Sheeran, M., Singh, S., Stålmarck, G.: Checking Safety Properties Using Induction and a SAT-Solver. In: Johnson, S.D., Hunt Jr., W.A. (eds.) FMCAD 2000. LNCS, vol. 1954, pp. 108–125. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  13. 13.
    Eén, N., Sörensson, N.: Temporal induction by incremental SAT solving. Electronic Notes in Theoretical Computer Science 89(4), 543–560 (2003)CrossRefGoogle Scholar
  14. 14.
    Bradley, A.R.: SAT-Based Model Checking without Unrolling. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 70–87. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Behrmann, G., Bouyer, P., Fleury, E., Larsen, K.G.: Static Guard Analysis in Timed Automata Verification. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 254–270. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  16. 16.
    Morbé, G., Pigorsch, F., Scholl, C.: Fully Symbolic Model Checking for Timed Automata. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 616–632. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Satisfiability, pp. 825–885. IOS Press (2009)Google Scholar
  18. 18.
    Björkmann, K., Frits, J., Valkonen, J., Heljanko, K., Niemelä, I.: Model-based analysis of a stepwise shutdown logic. VTT Working Papers 115. VTT Technical Research Centre of Finland, Espoo (2009)Google Scholar
  19. 19.
    de Moura, L., Rueß, H., Sorea, M.: Bounded Model Checking and Induction: From Refutation to Verification. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 14–26. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Dutertre, B., de Moura, L.: A Fast Linear-Arithmetic Solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Een, N., Mishchenko, A., Brayton, R.: Efficient implementation of property directed reachability. In: Proceedings of IWLS, IEEE/ACM (2011)Google Scholar
  22. 22.
    Nieuwenhuis, R., Oliveras, A.: DPLL(T) with Exhaustive Theory Propagation and Its Application to Difference Logic. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 321–334. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Cotton, S., Maler, O.: Fast and Flexible Difference Constraint Propagation for DPLL(T). In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 170–183. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Kindermann, R., Junttila, T., Niemelä, I.: SMT-based induction methods for timed systems. document arXiv:1204.5639v1(cs.LO) (2012)Google Scholar
  25. 25.
    Cimatti, A., Clarke, E., Giunchiglia, E., Giunchiglia, F., Pistore, M., Roveri, M., Sebastiani, R., Tacchella, A.: NuSMV 2: An OpenSource Tool for Symbolic Model Checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 359–364. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  26. 26.
    Lahtinen, J., Björkman, K., Valkonen, J., Frits, J., Niemelä, I.: Analysis of an emergency diesel generator control system by compositional model checking. VTT Working Papers 156. VTT Technical Research Centre of Finland, Espoo (2010)Google Scholar
  27. 27.
    Hoder, K., Bjørner, N.: Generalized Property Directed Reachability. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 157–171. Springer, Heidelberg (2012)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Roland Kindermann
    • 1
  • Tommi Junttila
    • 1
  • Ilkka Niemelä
    • 1
  1. 1.Department of Information and Computer ScienceAalto UniversityAaltoFinland

Personalised recommendations