Model Checking Quantified Computation Tree Logic

  • Arend Rensink
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4137)


Propositional temporal logic is not suitable for expressing properties on the evolution of dynamically allocated entities over time. In particular, it is not possible to trace such entities through computation steps, since this requires the ability to freely mix quantification and temporal operators.

In this paper we study Quantified Computation Tree Logic (QCTL ), which extends the well-known propositional computation tree logic, PCTL, with first and (monadic) second order quantification. The semantics of QCTL is expressed on algebra automata, which are automata enriched with abstract algebras at each state, and with reallocations at each transition that express an injective renaming of the algebra elements from one state to the next. The reallocations enable minimization of the automata modulo bisimilarity, essentially through symmetry reduction. Our main result is to show that each combination of a QCTL formula and a finite algebra automaton can be transformed to an equivalent PCTL formula over an ordinary Kripke structure, while maintaining the symmetry reduction. The transformation is structure-preserving on the formulae. This gives rise to a method to lift any model checking technique for PCTL to QCTL.


Model Check Modal Logic Temporal Logic Propositional Formula Symmetry Reduction 
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.
    Baldan, P., Corradini, A., König, B., LLuch Lafuente, A.: A temporal graph logic for abstractions of graph rewrite systems. Draft (2005)Google Scholar
  2. 2.
    Basin, D.A., Matthews, S., Vigano, L.: Labelled modal logics: Quantifiers. Journal of Logic, Language and Information 7(3), 237–263 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bojanczyk, M., David, C., Muscholl, A., Schwentick, T., Segoufin, L.: Two-variable logic on words with data. Research Report 2005-004, LIAFA — Laboratoire d’Informatique Algorithmique: Fondements et Applications (2005)Google Scholar
  4. 4.
    Castellini, C., Smaill, A.: A modular, tactic-based approach for first-order temporal theorem proving. In: International Conference on Temporal Logic (ICTL) (2000)Google Scholar
  5. 5.
    Castellini, C., Smaill, A.: Proof planning for first-order temporal logic. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 235–249. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  6. 6.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  7. 7.
    Clarke, E.M., Emerson, E.A., Sistla, A.P.: Automatic verification of finite state concurrent systems using temporal logic specifications: A practical approach. In: Symposium on Principles of Programming Languages (POPL), pp. 117–126. ACM Press, New York (1983)Google Scholar
  8. 8.
    Corbett, J.C., Dwyer, M.B., Hatcliff, J., Robby: Expressing checkable properties of dynamic systems: the bandera specification language. International Journal on Software Tools for Technology 4(1), 34–56 (2002)CrossRefGoogle Scholar
  9. 9.
    Dam, M.: CTL⋆ and ECTL⋆ as fragments of the modal μ-calculus. Theoretical Comput. Sci. 126(1), 77–96 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Distefano, D., Katoen, J.-P., Rensink, A.: Who is pointing when to whom? on the automated verification of linked list structures. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 250–262. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Distefano, D., Rensink, A., Katoen, J.-P.: Model checking birth and death. In: Baeza-Yates, Montanari, Santoro (eds.) Foundations of Information Technology in the Era of Network and Mobile Computing. IFIP Conference Proceedings, vol. 223, pp. 435–447. Kluwer Academic Publishers, Dordrecht (2002)Google Scholar
  12. 12.
    Fitting, M.: Bertrand Russell, Herbrand’s Theorem, and the Assignment Statement. In: Calmet, J., Plaza, J. (eds.) AISC 1998. LNCS (LNAI), vol. 1476, p. 14. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  13. 13.
    Fitting, M.: On quantified modal logic. Fundamenta Informaticae 39(1), 5–121 (1999)MathSciNetGoogle Scholar
  14. 14.
    Garson, J.W.: Quantification in modal logic. In: Guenthner, F., Gabbay, D. (eds.) Handbook of Philosophical Logic, 2nd edn., vol. 3, pp. 267–323. Kluwer, Dordrecht (2001)Google Scholar
  15. 15.
    Holtzmann, G.J.: The Spin Model Checker: Primer and Reference Manual. Addison-Wesley, Reading (2003)Google Scholar
  16. 16.
    Montanari, U., Pistore, M.: History-dependent automata. Technical Report TR-11-98, Department of Computer Science, University of Pisa (1998)Google Scholar
  17. 17.
    Montanari, U., Pistore, M.: History-dependent automata: An introduction. In: Bernardo, M., Bogliolo, A. (eds.) SFM-Moby 2005. LNCS, vol. 3465, pp. 1–28. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Neven, F., Schwentick, T., Vianu, V.: Towards regular languages over infinite alphabets. In: Sgall, J., Pultr, A., Kolman, P. (eds.) MFCS 2001. LNCS, vol. 2136, pp. 560–572. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Rensink, A.: Towards model checking graph grammars. In: Leuschel, Gruner, Presti, (eds.), Workshop on Automated Verification of Critical Systems (AVoCS), Technical Report DSSE–TR–2003–2, pp. 150–160. University of Southampton (2003)Google Scholar
  20. 20.
    Rensink, A.: The GROOVE simulator: A tool for state space generation. In: Pfaltz, J.L., Nagl, M., Böhlen, B. (eds.) AGTIVE 2003. LNCS, vol. 3062, pp. 479–485. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  21. 21.
    Yahav, E., Reps, T., Sagiv, M., Wilhelm, R.: Verifying temporal heap properties specified via evolution logic. In: Degano, P. (ed.) ESOP 2003. LNCS, vol. 2618, pp. 204–222. Springer, Heidelberg (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Arend Rensink
    • 1
  1. 1.Department of Computer ScienceUniversity of TwenteThe Netherlands

Personalised recommendations