Timed Vacuity

  • Hana Chockler
  • Shibashis GuhaEmail author
  • Orna Kupferman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10951)


Vacuity is a leading sanity check in model-checking, applied when the system is found to satisfy the specification. The check detects situations where the specification passes in a trivial way, say when a specification that requires every request to be followed by a grant is satisfied in a system with no requests. Such situations typically reveal problems in the modelling of the system or the specification, and indeed vacuity detection is a part of most industrial model-checking tools.

Existing research and tools for vacuity concern discrete-time systems and specification formalisms. We introduce real-time vacuity, which aims to detect problems with real-time modelling. Real-time logics are used for the specification and verification of systems with a continuous-time behavior. We study vacuity for the branching real-time logic TCTL, and focus on vacuity with respect to the time constraints in the specification. Specifically, the logic TCTL includes the temporal operator \(U^J\), which specifies real-time eventualities in real-time systems: the parameter Open image in new window is an interval with integral boundaries that bounds the time in which the eventuality should hold. We define several tightenings for the \(U^J\) operator. These tightenings require the eventuality to hold within a strict subset of J. We prove that vacuity detection for TCTL is PSPACE-complete, thus it does not increase the complexity of model-checking of TCTL. Our contribution involves an extension, termed TCTL\(^+\), of TCTL, which allows the interval J not to be continuous, and for which model checking stays in PSPACE. Finally, we describe a method for ranking vacuity results according to their significance.


  1. 1.
    Alur, R., Courcoubetis, C., Dill, D.: Model-checking in dense real-time. Inf. Comput. 104(1), 2–34 (1993)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Alur, R., Dill, D.: A theory of timed automata. Theoret. Comput. Sci. 126(2), 183–236 (1994)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alur, R., Feder, T., Henzinger, T.A.: The benefits of relaxing punctuality. J. ACM 43(1), 116–146 (1996)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alur, R., Henzinger, T.A.: Logics and models of real time: a survey. In: de Bakker, J.W., Huizing, C., de Roever, W.P., Rozenberg, G. (eds.) REX 1991. LNCS, vol. 600, pp. 74–106. Springer, Heidelberg (1992). Scholar
  5. 5.
    Armoni, R., et al.: Enhanced vacuity detection in linear temporal logic. In: Hunt, W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 368–380. Springer, Heidelberg (2003). Scholar
  6. 6.
    Beer, I., Ben-David, S., Eisner, C., Rodeh, Y.: Efficient detection of vacuity in ACTL formulas. Formal Methods Syst. Des. 18(2), 141–162 (2001)CrossRefGoogle Scholar
  7. 7.
    Bustan, D., Flaisher, A., Grumberg, O., Kupferman, O., Vardi, M.Y.: Regular vacuity. In: Borrione, D., Paul, W. (eds.) CHARME 2005. LNCS, vol. 3725, pp. 191–206. Springer, Heidelberg (2005). Scholar
  8. 8.
    Chechik, M., Gheorghiu, M., Gurfinkel, A.: Finding environment guarantees. In: Dwyer, M.B., Lopes, A. (eds.) FASE 2007. LNCS, vol. 4422, pp. 352–367. Springer, Heidelberg (2007). Scholar
  9. 9.
    Chockler, H., Gurfinkel, A., Strichman, O.: Beyond vacuity: towards the strongest passing formula. In: Proceedings of the 8th International Conference on Formal Methods in Computer-Aided Design, pp. 1–8 (2008)Google Scholar
  10. 10.
    Chockler, H., Halpern, J.Y.: Responsibility and blame: a structural-model approach. In: Proceedings of the 19th International Joint Conference on Artificial Intelligence, pp. 147–153 (2003)Google Scholar
  11. 11.
    Chockler, H., Strichman, O.: Before and after vacuity. Formal Methods Syst. Des. 34(1), 37–58 (2009)CrossRefGoogle Scholar
  12. 12.
    Clarke, E., Grumberg, O., Long, D.: Verification tools for finite-state concurrent systems. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) REX 1993. LNCS, vol. 803, pp. 124–175. Springer, Heidelberg (1994). Scholar
  13. 13.
    Clarke, E.M., Grumberg, O., McMillan, K.L., Zhao, X.: Efficient generation of counterexamples and witnesses in symbolic model checking. In: Proceedings of the 32st Design Automation Conference, pp. 427–432. IEEE Computer Society (1995)Google Scholar
  14. 14.
    Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. J. ACM 42, 857–907 (1995)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Ben-David, S., Kupferman, O.: A framework for ranking vacuity results. In: Van Hung, D., Ogawa, M. (eds.) ATVA 2013. LNCS, vol. 8172, pp. 148–162. Springer, Cham (2013). Scholar
  16. 16.
    Dokhanchi, A., Hoxha, B., Fainekos, G.E.: Formal requirement elicitation and debugging for testing and verification of cyber-physical systems. CoRR, abs/1607.02549 (2016)Google Scholar
  17. 17.
    Emerson, E.A., Halpern, J.Y.: Sometimes and not never revisited: on branching versus linear time. J. ACM 33(1), 151–178 (1986)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Fisman, D., Kupferman, O., Sheinvald-Faragy, S., Vardi, M.Y.: A framework for inherent vacuity. In: Chockler, H., Hu, A.J. (eds.) HVC 2008. LNCS, vol. 5394, pp. 7–22. Springer, Heidelberg (2009). Scholar
  19. 19.
    Gurfinkel, A., Chechik, M.: Extending extended vacuity. In: Hu, A.J., Martin, A.K. (eds.) FMCAD 2004. LNCS, vol. 3312, pp. 306–321. Springer, Heidelberg (2004). Scholar
  20. 20.
    Gurfinkel, A., Chechik, M.: How vacuous is vacuous? In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 451–466. Springer, Heidelberg (2004). Scholar
  21. 21.
    Henzinger, T.A., Manna, Z., Pnueli, A.: Timed transition systems. In: de Bakker, J.W., Huizing, C., de Roever, W.P., Rozenberg, G. (eds.) REX 1991. LNCS, vol. 600, pp. 226–251. Springer, Heidelberg (1992). Scholar
  22. 22.
    Kupferman, O.: Sanity checks in formal verification. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 37–51. Springer, Heidelberg (2006). Scholar
  23. 23.
    Kupferman, O., Li, W., Seshia, S.A.: A theory of mutations with applications to vacuity, coverage, and fault tolerance. In: Proceedings of the 8th International Conference on Formal Methods in Computer-Aided Design, pp. 1–9 (2008)Google Scholar
  24. 24.
    Kupferman, O., Vardi, M.Y.: Vacuity detection in temporal model checking. Softw. Tools Technol. Transf. 4(2), 224–233 (2003)CrossRefGoogle Scholar
  25. 25.
    Namjoshi, K.S.: An efficiently checkable, proof-based formulation of vacuity in model checking. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 57–69. Springer, Heidelberg (2004). Scholar
  26. 26.
    Purandare, M., Somenzi, F.: Vacuum cleaning CTL formulae. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 485–499. Springer, Heidelberg (2002). Scholar
  27. 27.
    Purandare, M., Wahl, T., Kroening, D.: Strengthening properties using abstraction refinement. In: Proceedings of Design, Automation and Test in Europe (DATE), pp. 1692–1697. IEEE (2009)Google Scholar
  28. 28.
    Stockmeyer, L.J.: On the combinational complexity of certain symmetric boolean functions. Math. Syst. Theory 10, 323–336 (1977)MathSciNetCrossRefGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.King’s College LondonLondonUK
  2. 2.Université Libre de BruxellesBrusselsBelgium
  3. 3.The Hebrew UniversityJerusalemIsrael

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