How Hard Is Finding Shortest Counter-Example Lassos in Model Checking?

  • Rüdiger EhlersEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11800)


Modern model checkers help system engineers to pinpoint the reason for the faulty behavior of a system by providing counter-example traces. For finite-state systems and \(\omega \)-regular specifications, they come in the form of lassos. Lassos that are unnecessarily long should be avoided, as they make finding the cause for an error in a trace harder.

We give the first thorough characterization of the computational complexity of finding the shortest and approximately shortest counter-example lassos in model checking for the full class of \(\omega \)-regular specifications. We show how to build (potentially exponentially larger) tight automata for arbitrary \(\omega \)-regular specifications, which can be used to reduce finding shortest counter-example lassos for some finite-state system to finding a shortest accepting lasso in a (product) Büchi automaton. We then show that even approximating the size of the shortest counter-example lasso is an NP-hard problem for any polynomial approximation function, which demonstrates the hardness of obtaining short counter-examples in practical model checking. Minimizing only the length of the lasso cycle is however possible in polynomial time for a fixed but arbitrary upper limit on the size of strongly connected components in specification automata.


  1. 1.
    Adabala, K., Ehlers, R.: A fragment of linear temporal logic for universal very weak automata. In: Lahiri, S.K., Wang, C. (eds.) ATVA 2018. LNCS, vol. 11138, pp. 335–351. Springer, Cham (2018). Scholar
  2. 2.
    Calbrix, H., Nivat, M., Podelski, A.: Ultimately periodic words of rational w-languages. In: Brookes, S., Main, M., Melton, A., Mislove, M., Schmidt, D. (eds.) MFPS 1993. LNCS, vol. 802, pp. 554–566. Springer, Heidelberg (1994). Scholar
  3. 3.
    Clarke, E.M., Grumberg, O., McMillan, K.L., Zhao, X.: Efficient generation of counterexamples and witnesses in symbolic model checking. In: 32nd Conference on Design Automation (DAC), pp. 427–432. ACM Press (1995)Google Scholar
  4. 4.
    De Giacomo, G., Vardi, M.Y.: Linear temporal logic and linear dynamic logic on finite traces. In: 23rd International Joint Conference on Artificial Intelligence (IJCAI), pp. 854–860. IJCAI/AAAI (2013)Google Scholar
  5. 5.
    Edelkamp, S., Sulewski, D., Barnat, J., Brim, L., Simecek, P.: Flash memory efficient LTL model checking. Sci. Comput. Program. 76(2), 136–157 (2011)CrossRefGoogle Scholar
  6. 6.
    Ehlers, R.: Short witnesses and accepting lassos in \(\omega \)-Automata. In: Dediu, A.-H., Fernau, H., Martín-Vide, C. (eds.) LATA 2010. LNCS, vol. 6031, pp. 261–272. Springer, Heidelberg (2010). Scholar
  7. 7.
    Eisner, C., Fisman, D.: A Practical Introduction to PSL. Series on Integrated Circuits and Systems. Springer, Heidelberg (2006)Google Scholar
  8. 8.
    Etessami, K., Wilke, T., Schuller, R.A.: Fair simulation relations, parity games, and state space reduction for Büchi automata. SIAM J. Comput. 34(5), 1159–1175 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Farzan, A., Chen, Y.-F., Clarke, E.M., Tsay, Y.-K., Wang, B.-Y.: Extending automated compositional verification to the full class of omega-regular languages. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 2–17. Springer, Heidelberg (2008). Scholar
  10. 10.
    Gastin, P., Moro, P., Zeitoun, M.: Minimization of counterexamples in SPIN. In: Graf, S., Mounier, L. (eds.) SPIN 2004. LNCS, vol. 2989, pp. 92–108. Springer, Heidelberg (2004). Scholar
  11. 11.
    Groce, A., Visser, W.: What went wrong: explaining counterexamples. In: Ball, T., Rajamani, S.K. (eds.) SPIN 2003. LNCS, vol. 2648, pp. 121–136. Springer, Heidelberg (2003). Scholar
  12. 12.
    Holzmann, G.J.: The model checker SPIN. IEEE Trans. Softw. Eng. 23(5), 279–295 (1997)CrossRefGoogle Scholar
  13. 13.
    Kupferman, O., Sheinvald-Faragy, S.: Finding shortest witnesses to the nonemptiness of automata on infinite words. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 492–508. Springer, Heidelberg (2006). Scholar
  14. 14.
    Kupferman, O., Vardi, M.Y.: Model checking of safety properties. In: Halbwachs, N., Peled, D. (eds.) CAV 1999. LNCS, vol. 1633, pp. 172–183. Springer, Heidelberg (1999). Scholar
  15. 15.
    Maidl, M.: The common fragment of CTL and LTL. In: 41st Annual Symposium on Foundations of Computer Science (FOCS), pp. 643–652 (2000)Google Scholar
  16. 16.
    Schuppan, V., Biere, A.: Efficient reduction of finite state model checking to reachability analysis. STTT 5(2–3), 185–204 (2004)CrossRefGoogle Scholar
  17. 17.
    Schuppan, V., Biere, A.: Shortest counterexamples for symbolic model checking of LTL with past. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 493–509. Springer, Heidelberg (2005). Scholar
  18. 18.
    Schwoon, S., Esparza, J.: A note on on-the-fly verification algorithms. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 174–190. Springer, Heidelberg (2005). Scholar
  19. 19.
    Sebastiani, R., Tonetta, S.: “more deterministic” vs. “smaller” Büchi automata for efficient LTL model checking. In: 12th IFIP WG 10.5 Advanced Research Working Conference (CHARME), Correct Hardware Design and Verification Methods, pp. 126–140 (2003)Google Scholar
  20. 20.
    Tarjan, R.E.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Clausthal University of TechnologyClausthal-ZellerfeldGermany

Personalised recommendations