A Refined Resolution Calculus for CTL

  • Lan Zhang
  • Ullrich Hustadt
  • Clare Dixon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5663)


In this paper, we present a refined resolution-based calculus for Computation Tree Logic (CTL). The calculus requires a polynomial time computable transformation of an arbitrary CTL formula to an equi-satisfiable clausal normal form formulated in an extension of CTL with indexed existential path quantifiers. The calculus itself consists of a set of resolution rules which can be used as the basis for an EXPTIME decision procedure for the satisfiability problem of CTL. We prove soundness and completeness of the calculus. In addition, we introduce CTL-RP, our implementation of the calculus as well as some experimental results.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abate, P., Goré, R.: The Tableaux Workbench. In: Cialdea Mayer, M., Pirri, F. (eds.) TABLEAUX 2003. LNCS, vol. 2796, pp. 230–236. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Abate, P., Goré, R., Widmann, F.: One-Pass Tableaux for Computation Tree Logic. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS, vol. 4790, pp. 32–46. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Alur, R., Henzinger, T.A., Kupferman, O.: Alternating-time temporal logic. J. ACM 49(5), 672–713 (2002)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Handbook of Automated Reasoning, vol. 1, pp. 19–99. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  5. 5.
    Bolotov, A.: Clausal Resolution for Branching-Time Temporal Logic. PhD thesis, Manchester Metropolitan University (2000)Google Scholar
  6. 6.
    Bolotov, A., Dixon, C.: Resolution for Branching Time Temporal Logics: Applying the Temporal Resolution Rule. In: Proc. TIME 2000, pp. 163–172. IEEE, Los Alamitos (2000)Google Scholar
  7. 7.
    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
  8. 8.
    Clarke, E.M., Grumberg, O., Peled, D.A.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  9. 9.
    Dixon, C.: Temporal Resolution Using a Breadth-First Search Algorithm. Annals of Mathematics and Artificial Intelligence 22(1-2), 87–115 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Emerson, E.A.: Temporal and Modal Logic. In: Handbook of Theoretical Computer Science, pp. 996–1072. Elsevier, Amsterdam (1990)Google Scholar
  11. 11.
    Emerson, E.A., Halpern, J.Y.: Decision Procedures and Expressiveness in the Temporal Logic of Branching Time. J. Comput. Syst. Sci. 30(1), 1–24 (1985)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Fisher, M., Dixon, C., Peim, M.: Clausal Temporal Resolution. ACM Transactions on Computational Logic 2(1), 12–56 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hustadt, U., Konev, B.: TRP++: A Temporal Resolution Prover. In: Collegium Logicum, pp. 65–79. Kurt Gödel Society (2004)Google Scholar
  14. 14.
    Huth, M., Ryan, M.: Logic in Computer Science: Modelling and Reasoning about Systems. Cambridge University Press, Cambridge (2004)CrossRefMATHGoogle Scholar
  15. 15.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems. Springer, Heidelberg (1992)CrossRefMATHGoogle Scholar
  16. 16.
    Max-Planck-Institut Informatik: SPASS: An Automated Theorem Prover for First- Order Logic with Equality, http://www.spass-prover.org/
  17. 17.
    Vardi, M.Y., Wolper, P.: Automata-Theoretic Techniques for Modal Logics of Programs. J. Comput. Syst. Sci. 32(2), 183–221 (1986)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Weidenbach, C., Schmidt, R.A., Hillenbrand, T., Rusev, R., Topic, D.: System description: Spass version 3.0. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 514–520. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Zhang, L., Hustadt, U., Dixon, C.: First-order Resolution for CTL. Technical Report ULCS-08-010, Department of Computer Science, University of Liverpool (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Lan Zhang
    • 1
  • Ullrich Hustadt
    • 1
  • Clare Dixon
    • 1
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK

Personalised recommendations