Computation tree logic CTL* and path quantifiers in the monadic theory of the binary tree

  • Thilo Hafer
  • Wolfgang Thomas
Temporal Logic, Concurrent Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 267)


In this paper we study the expressive power of the full branching time logic CTL* (of Clarke, Emerson, Halpern and Sistla), by comparing it with fragments of monadic second-order logic over the binary tree. We show that over binary tree models the system CTL* has the same expressive power as monadic second-order logic in which set quantification is restricted to infinite paths (in particular, the full strength of first-order logic is captured here by CTL*). This generalizes Kamp's theorem on the equivalence between propositional linear time logic and first-order logic over the ordering of the natural numbers. For the proof an extension of CTL* by past operators is introduced. The transition from monadic formulas to CTL* rests on a model-theoretic decomposition lemma for trees that is justified by an application of the Ehrenfeucht-Fraissé game. The paper concludes by discussing variants of the main result, dealing for example with trees of higher branching index and the extended branching time logic ECTL*.


Binary Tree Kripke Structure Tree Language Computation Tree Logic Linear Time Temporal Logic 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B86]
    Browne, M.C., An improved algorithm for the automatic verification of finite state systems using temporal logic, 1st IEEE Sympos. Logic in Computer Science, 1986, pp. 260–266.Google Scholar
  2. [CE82]
    Clarke, E.M., Emerson, E.A., Using branching time temporal logic to synthesize synchronization skeletons, Science of Computer Pogramming 2 (1982), 241–266.CrossRefGoogle Scholar
  3. [CES83]
    Clarke, E.M., Emerson, E.A., Sistla, A.P., Automatic verification of finite-state concurrent systems using temporal logic specifications: A practical approach, Proc. 10th Ann. ACM Sympos. on Principles of Programming Languages, 1983, pp. 117–126.Google Scholar
  4. [EH86]
    Emerson, E.A., Halpern, J.Y., 'sometimes’ and ‘notnever’ revisited: On branching versus linear time temporal logic, Journal of the ACM 33 (1986), 151–178.CrossRefGoogle Scholar
  5. [ES84]
    Emerson, E.A., Sistla, A.P., Deciding (full) branching time logic, Information and Control 61 (1984), 175–201.CrossRefGoogle Scholar
  6. [GPSS80]
    Gabbay, D., Pnueli, A., Shelah, A., Stavi, J., The temporal analysis of fairness, Proc. 7th Ann. ACM Sympos. on Principles of Programming Languages, 1980, pp. 163–173.Google Scholar
  7. [GS83]
    Gurevich, Y., Shelah, S., Rabin's uniformization problem, Journal of Symbolic Logic 48 (1983), 1105–1119.Google Scholar
  8. [Ha86]
    Hafer, Th., On the expressive completeness of CTL*, Schriften zur Informatik und angewandten Mathematik 123, RWTH Aachen, 1986.Google Scholar
  9. [Heu86]
    Heuter, U., Baumsprachen in der Logik erster Stufe: Eine Charakterisierung durch reguläre Ausdrücke, Schriften zur Informatik und angewandten Mathematik 122, RWTH Aachen, 1986.Google Scholar
  10. [HKP82]
    Harel, D., Kozen, D., Parikh, R., Process logic: expressiveness, decidability and completeness, Journal of Computer and System Sciences 18 (1982), 144–170.Google Scholar
  11. [Ka68]
    Kamp, H.W., Tense logic and the theory of linear order, Ph.D. Thesis, University of California, Los Angeles, 1968.Google Scholar
  12. [KP83]
    Kozen, D., Parikh, R., A decision procedure for the propositional μ-calculus, Second Workshop on Logics of Programs, Lecture Notes in Computer Science 164 (1983), 313–325.Google Scholar
  13. [LPZ85]
    Lichtenstein, O., Pnueli, A., Zuck, L., The glory of the past, Proc. Logics of Programs, New York, Lecture Notes in Computer Science 193 (1985), 196–218.Google Scholar
  14. [Ra69]
    Rabin, M.O., Decidability of second-order theories and automata on infinite trees, Trans. Amer. Math. Soc. 141 (1969), 1–35.Google Scholar
  15. [Ra70]
    Rabin, M.O., Weakly definable relations and special automata, in "Math. Logic and Found. of Set Theory", Y. Bar-Hillel, Ed., North-Holland, 1970, pp. 1–23.Google Scholar
  16. [Ro82]
    Rosenstein, J.G., Linear orderings, Academic Press, New York, 1982.Google Scholar
  17. [Th84]
    Thomas, W., Logical aspects in the study of tree languages, in "9th Coll. on Trees in Algebra and Programming", B. Courcelle, Ed., Cambridge University Press, 1984, pp. 31–49.Google Scholar
  18. [Th87]
    Thomas, W., On chain logic, path logic, and first-order logic over infinite trees, 2nd IEEE Sympos. Logic in Computer Science, 1987.Google Scholar
  19. [Wo83]
    Wolper, P., Temporal logic can be more expressive, Information and Control 56 (1983), 72–99.CrossRefGoogle Scholar
  20. [VS85]
    Vardi, M.Y., Stockmeyer, L., Improved upper and lower bounds for modal logics of programs: Preliminary report, Proc. 17th ACM Sympos. on Theory of Computing, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • Thilo Hafer
    • 1
  • Wolfgang Thomas
    • 1
  1. 1.Lehrstuhl für Informatik IIRWTH AachenAachenWest-Germany

Personalised recommendations