Extended Computation Tree Logic

  • Roland Axelsson
  • Matthew Hague
  • Stephan Kreutzer
  • Martin Lange
  • Markus Latte
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6397)


We introduce a generic extension of the popular branching-time logic CTL which refines the temporal until and release operators with formal languages. For instance, a language may determine the moments along a path that an until property may be fulfilled. We consider several classes of languages leading to logics with different expressive power and complexity, whose importance is motivated by their use in model checking, synthesis, abstract interpretation, etc. We show that even with context-free languages on the until operator the logic still allows for polynomial time model-checking despite the significant increase in expressive power. This makes the logic a promising candidate for applications in verification. In addition, we analyse the complexity of satisfiability and compare the expressive power of these logics to CTL* and extensions of PDL.


Model Check Temporal Logic Expressive Power Label Transition System Proof Obligation 
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.
    Inc. Accellera Organization. Formal semantics of Accellera property specification language. In: Appendix B (2004),
  2. 2.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. 36th Ann. ACM Symp. on Theory of Computing, STOC 2004, pp. 202–211 (2004)Google Scholar
  3. 3.
    Armoni, R., Fix, L., Flaisher, A., Gerth, R., Ginsburg, B., Kanza, T., Landver, A., Mador-Haim, S., Singerman, E., Tiemeyer, A., Vardi, M.Y., Zbar, Y.: The ForSpec temporal logic: A new temporal property specification language. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, pp. 296–311. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Arnold, A., Vincent, A., Walukiewicz, I.: Games for synthesis of controllers with partial observation. Theor. Comput. Sci. 303(1), 7–34 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beer, I., Ben-David, S., Landver, A.: On-the-fly model checking of RCTL formulas. In: Y. Vardi, M. (ed.) CAV 1998. LNCS, vol. 1427, pp. 184–194. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  7. 7.
    Brázdil, T., Cerná, I.: Model checking of regCTL. Computers and Artificial Intelligence 25(1) (2006)Google Scholar
  8. 8.
    Chandra, A.K., Kozen, D.C., Stockmeyer, L.J.: Alternation. Journal of the ACM 28(1), 114–133 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Clarke, E.M., Emerson, E.A.: Synthesis of synchronization skeletons for branching time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  10. 10.
    Clarke, E.M., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Counterexample-guided abstraction refinement for symbolic model checking. Journal of the ACM 50(5), 752–794 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dawar, A., Grädel, E., Kreutzer, S.: Inflationary fixed points in modal logics. ACM Transactions on Computational Logic 5(2), 282–315 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Emerson, E.A., Halpern, J.Y.: Decision procedures and expressiveness in the temporal logic of branching time. Journal of Computer and System Sciences 30, 1–24 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Emerson, E.A., Halpern, J.Y.: “Sometimes” and “not never” revisited: On branching versus linear time temporal logic. Journal of the ACM 33(1), 151–178 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM Journal on Computing 29(1), 132–158 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. In: Annual IEEE Symposium on Foundations of Computer Science, pp. 328–337 (1988)Google Scholar
  16. 16.
    Esparza, J.: Decidability of model-checking for infinite-state concurrent systems. Acta Informatica 34, 85–107 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. Journal of Computer and System Sciences 18(2), 194–211 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Harel, D., Kozen, D., Tiuryn, J.: Dynamic Logic. MIT Press, Cambridge (2000)zbMATHGoogle Scholar
  19. 19.
    Harel, D., Pnueli, A., Stavi, J.: Propositional dynamic logic of nonregular programs. Journal of Computer and System Sciences 26(2), 222–243 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Henriksen, J.G., Thiagarajan, P.S.: Dynamic linear time temporal logic. Annals of Pure and Applied Logic 96(1-3), 187–207 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Walukiewicz, I.: Model checking CTL properties of pushdown systems. In: Kapoor, S., Prasad, S. (eds.) FSTTCS 2000. LNCS, vol. 1974, pp. 127–138. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  22. 22.
    Kozen, D.: Results on the propositional μ-calculus. TCS 27, 333–354 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kupferman, O., Piterman, N., Vardi, M.Y.: Extended temporal logic revisited. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 519–535. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  24. 24.
    Bozzelli, L.: Complexity results on branching-time pushdown model checking. Theor. Comput. Sci. 379(1-2), 286–297 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lange, M., Latte, M.: A CTL-based logic for program abstractions. In: de Queiroz, R. (ed.) WoLLIC 2010. LNCS (LNAI), vol. 6188, pp. 19–33. Springer, Heidelberg (2010)Google Scholar
  26. 26.
    Löding, C., Lutz, C., Serre, O.: Propositional dynamic logic with recursive programs. J. Log. Algebr. Program. 73(1-2), 51–69 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mateescu, R., Monteiro, P.T., Dumas, E., de Jong, H.: Computation tree regular logic for genetic regulatory networks. In: Cha, S(S.), Choi, J.-Y., Kim, M., Lee, I., Viswanathan, M. (eds.) ATVA 2008. LNCS, vol. 5311, pp. 48–63. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  28. 28.
    Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal 2(3), 115–125 (1959)zbMATHGoogle Scholar
  29. 29.
    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4, 177–192 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Sistla, A.P., Clarke, E.M.: The complexity of propositional linear temporal logics. Journal of the Association for Computing Machinery 32(3), 733–749 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Streett, R.S.: Propositional dynamic logic of looping and converse is elementarily decidable. Information and Control 54(1/2), 121–141 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    van Emde Boas, P.: The convenience of tilings. In: Sorbi, A. (ed.) Complexity, Logic, and Recursion Theory. Lecture notes in pure and applied mathematics, vol. 187, pp. 331–363. Marcel Dekker, Inc., New York (1997)Google Scholar
  33. 33.
    Vardi, M.Y., Stockmeyer, L.: Improved upper and lower bounds for modal logics of programs. In: Proc. 17th Symp. on Theory of Computing, STOC 1985, Baltimore, USA, pp. 240–251. ACM, New York (1985)Google Scholar
  34. 34.
    Vardi, M.Y., Wolper, P.: Reasoning about infinite computations. Information and Computation 115(1), 1–37 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Viswanathan, M., Viswanathan, R.: A higher order modal fixed point logic. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 512–528. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  36. 36.
    Walukiewicz, I.: Pushdown processes: Games and model-checking. Information and Computation 164(2), 234–263 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Wolper, P.: Temporal logic can be more expressive. In: SFCS 1981: Proceedings of the 22nd Annual Symposium on Foundations of Computer Science, Washington, DC, USA, pp. 340–348. IEEE Computer Society, Los Alamitos (1981)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Roland Axelsson
    • 1
  • Matthew Hague
    • 2
  • Stephan Kreutzer
    • 2
  • Martin Lange
    • 3
  • Markus Latte
    • 1
  1. 1.Department of Computer ScienceLudwig-Maximilians-Universität MunichGermany
  2. 2.Oxford University Computing LaboratoryUK
  3. 3.Department of Elect. Engineering and Computer ScienceUniversity of KasselGermany

Personalised recommendations