An Abstract Interpretation Perspective on Linear vs. Branching Time

  • Francesco Ranzato
  • Francesco Tapparo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3780)


It is known that the branching time language ACTL and the linear time language ∀ LTL of universally quantified formulae of LTL have incomparable expressive powers, i.e., Sem(ACTL) and Sem(∀LTL) are incomparable sets. Within a standard abstract interpretation framework, ACTL can be viewed as an abstract interpretation LTL ∀  of LTL where the universal path quantifier ∀ abstracts each linear temporal operator of LTL to a corresponding branching state temporal operator of ACTL. In abstract interpretation terms, it turns out that the universal path quantifier abstraction of LTL is incomplete. In this paper we reason on a generic abstraction α over a domain A of a generic linear time language L. This approach induces both a language αL of α-abstracted formulae of L and an abstract language L α whose operators are the best correct abstractions in A of the linear operators of L. When the abstraction α is complete for the operators in L it turns out that α L and L α have the same expressive power, so that trace-based model checking of α L can be reduced with no lack of precision to A-based model checking of L α . This abstract interpretation-based approach allows to compare temporal languages at different levels of abstraction and to view the standard linear vs. branching time comparison as a particular instance.


Model Check Complete Lattice Expressive Power Abstract Interpretation Atomic Proposition 
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  1. 1.
    Browne, M.C., Clarke, E.M., Grumberg, O.: Characterizing finite Kripke structures in propositional temporal logic. Theor. Comp. Sci. 59, 115–131 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Clarke, E.M., Draghicescu, I.A.: Expressibility results for linear time and branching time logics. In: de Bakker, J.W., de Roever, W.-P., Rozenberg, G. (eds.) Linear Time, Branching Time and Partial Order in Logics and Models for Concurrency. LNCS, vol. 354, pp. 428–437. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  3. 3.
    Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: Proc. 4th ACM POPL, pp. 238–252 (1977)Google Scholar
  4. 4.
    Cousot, P., Cousot, R.: Systematic design of program analysis frameworks. In: Proc. 6th ACM POPL, pp. 269–282 (1979)Google Scholar
  5. 5.
    Cousot, P., Cousot, R.: Temporal abstract interpretation. In: Proc. 27th ACM POPL, pp. 12–25 (2000)Google Scholar
  6. 6.
    Dams, D.: Abstract Interpretation and Partition Refinement for Model Checking. Ph.D. Thesis, Univ. Eindhoven (1996)Google Scholar
  7. 7.
    Dams, D.: Flat fragments of CTL and CTL*: separating the expressive and distinguishing powers. Logic J. of the IGPL 7(1), 55–78 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Emerson, E.A., Halpern, J.Y.: “Sometimes” and “Not Never” revisited: on branching versus linear time temporal logic. J. ACM 33(1), 151–178 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Giacobazzi, R., Ranzato, F., Scozzari, F.: Making abstract interpretations complete. J 47(2), 361–416 (2000)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Groote, J.F., Vaandrager, F.: An efficient algorithm for branching bisimulation and stuttering equivalence. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 626–638. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  11. 11.
    Kupferman, O., Vardi, M.: Relating linear and branching model checking. In: Proc. IFIP PROCOMET, pp. 304–326. Chapman & Hall, Boca Raton (1998)Google Scholar
  12. 12.
    Kupferman, O., Vardi, M.: Freedom, weakness and determinism: from linear-time to branching-time. In: Proc. 13th LICS, pp. 81–92. IEEE Press, Los Alamitos (1998)Google Scholar
  13. 13.
    Lamport, L.: Sometimes is sometimes “not never” – on the temporal logic of programs. In: Proc. 7th ACM POPL, pp. 174–185 (1980)Google Scholar
  14. 14.
    Maidl, M.: The common fragment of CTL and LTL. In: Proc. 41st FOCS, pp. 643–652 (2000)Google Scholar
  15. 15.
    Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Ranzato, F.: On the completeness of model checking. In: Sands, D. (ed.) ESOP 2001. LNCS, vol. 2028, pp. 137–154. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  17. 17.
    Ranzato, F., Tapparo, F.: Strong preservation as completeness in abstract interpretation. In: Schmidt, D. (ed.) ESOP 2004. LNCS, vol. 2986, pp. 18–32. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Ranzato, F., Tapparo, F.: An abstract interpretation-based refinement algorithm for strong preservation. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 140–156. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Vardi, M.: Sometimes and not never re-revisited: on branching vs. linear time. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 1–17. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  20. 20.
    Vardi, M.: Branching vs. Linear time: Final showdown. In: Margaria, T., Yi, W. (eds.) TACAS 2001. LNCS, vol. 2031, pp. 1–22. Springer, Heidelberg (2001)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Francesco Ranzato
    • 1
  • Francesco Tapparo
    • 1
  1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità di PadovaItaly

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