Branching Time? Pruning Time!

  • Markus Latte
  • Martin Lange
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)


The full branching time logic ctl* is a well-known specification logic for reactive systems. Its satisfiability and model checking problems are well understood. However, it is still lacking a satisfactory sound and complete axiomatisation. The only proof system known for ctl* is Reynolds’ which comes with an intricate and long completeness proof and, most of all, uses rules that do not possess the subformula property.

In this paper we consider a large fragment of ctl* which is characterised by disallowing certain nestings of temporal operators inside universal path quantifiers. This subsumes ctl  +  for instance. We present infinite satisfiability games for this fragment. Winning strategies for one of the players represent infinite tree models for satisfiable formulas. These can be pruned into finite trees using fixpoint strengthening and some simple combinatorial machinery such that the results represent proofs in a Hilbert-style axiom system for this fragment. Completeness of this axiomatisation is a simple consequence of soundness of the satisfiability games.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    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
  3. 3.
    Emerson, E.A., Jutla, C.S.: The complexity of tree automata and logics of programs. SIAM Journal on Computing 29(1), 132–158 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Friedmann, O., Latte, M., Lange, M.: A Decision Procedure for CTL* Based on Tableaux and Automata. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 331–345. Springer, Heidelberg (2010)Google Scholar
  5. 5.
    Gabbay, D., Pnueli, A., Shelah, S., Stavi, J.: The temporal analysis of fairness. In: Proc. 7th Symp. on Principles of Programming Languages, POPL 1980, pp. 163–173. ACM (1980)Google Scholar
  6. 6.
    Johannsen, J., Lange, M.: CTL +  is Complete for Double Exponential Time. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 767–775. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  7. 7.
    Kröger, F.: Temporal Logic of Programs. Springer (1987)Google Scholar
  8. 8.
    Kröger, F., Merz, S.: Temporal Logic and State Systems. Texts in Theoretical Computer Science. Springer (2008)Google Scholar
  9. 9.
    Lange, M., Stirling, C.: Focus games for satisfiability and completeness of temporal logic. In: Proc. 16th Symp. on Logic in Computer Science, LICS 2001, Boston, MA, USA. IEEE Computer Society Press (2001)Google Scholar
  10. 10.
    Lichtenstein, O., Pnueli, A.: Propositional temporal logics: Decidability and completeness. Logic Journal of the IGPL 8(1), 55–85 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems Specification. Springer (1992)Google Scholar
  12. 12.
    Martin, D.A.: Borel determinacy. Ann. Math. 102, 363–371 (1975)CrossRefzbMATHGoogle Scholar
  13. 13.
    Penczek, W.: Branching time and partial order in temporal logics. In: Bolc, L., Szałas, A. (eds.) Time and Logic – A Computational Approach, pp. 179–228. UCL Press, London (1995)Google Scholar
  14. 14.
    Prior, A.N.: Time and modality. Oxford University Press, Oxford (1957)zbMATHGoogle Scholar
  15. 15.
    Reynolds, M.: An axiomatization of full computation tree logic. Journal of Symbolic Logic 66(3), 1011–1057 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Reynolds, M.: An axiomatization of PCTL*. Information and Computation 201(1), 72–119 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Reynolds, M.: A tableau for bundled CTL*. Journal of Logic and Computation 17(1), 117–132 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Markus Latte
    • 1
  • Martin Lange
    • 2
  1. 1.Department of Computer ScienceUniversity of MunichGermany
  2. 2.School of Electrical Engineering and Computer ScienceUniversity of KasselGermany

Personalised recommendations