Generalized Model Checking: Reasoning about Partial State Spaces

  • Glenn Bruns
  • Patrice Godefroid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1877)


We discuss the problem of model checking temporal properties on partial Kripke structures, which were used in [BG99] to represent incomplete state spaces. We first extend the results of [BG99] by showing that the model-checking problem for any 3-valued temporal logic can be reduced to two model-checking problems for the corresponding 2-valued temporal logic. We then introduce a new semantics for 3-valued temporal logics that can give more definite answers than the previous one. With this semantics, the evaluation of a formula φ on a partial Kripke structure M returns the third truth value ⊥ (read “unknown”) only if there exist Kripke structures M 1 and M 2 that both complete M and such that M 1 satisfies φ while M 2 violates φ, hence making the value of φ on M truly unknown. The partial Kripke structure M can thus be viewed as a partial solution to the satisfiability problem which reduces the solution space to complete Kripke structures that are more complete than M with respect to a completeness preorder. This generalized model-checking problem is thus a generalization of both satisfiability (all Kripke structures are potential solutions) and model checking (a single Kripke structure needs to be checked). We present algorithms and complexity bounds for the generalized model-checking problem for various temporal logics.


Model Check Modal Logic Temporal Logic Kripke Structure Tree Automaton 
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. ALW89.
    Martín Abadi, Leslie Lamport, and Pierre Wolper. Realizable and unrealizable concurrent program specifications. In Proc. 16th Int. Colloquium on Automata, Languages and Programming, volume 372 of Lecture Notes in Computer Science, pages 1–17. Springer-Verlag, July 1989.CrossRefGoogle Scholar
  2. BG99.
    Glenn Bruns and Patrice Godefroid. Model checking partial state spaces with 3-valued temporal logics. In N. Halbwachs and D. Peled, editors, Proceedings of CAV’ 99, LNCS 1633, pages 274–287, 1999.Google Scholar
  3. BVW94.
    Orna Bernholtz, Moshe Y. Vardi, and Pierre Wolper. An automata-theoretic approach to branching-time model checking. In Computer Aided Verification, Proc. 6th Int. Workshop, volume 818 of Lecture Notes in Computer Science, pages 142–155, Stanford, California, June 1994. Springer-Verlag.Google Scholar
  4. CE81.
    E. M. Clarke and E. A. Emerson. Design and Synthesis of Synchronization Skeletons using Branching-Time Temporal Logic. In D. Kozen, editor, Proceedings of the Workshop on Logic of Programs, Yorktown Heights, volume 131 of Lecture Notes in Computer Science, pages 52–71. Springer-Verlag, 1981.CrossRefGoogle Scholar
  5. CES86.
    E.M. Clarke, E.A. Emerson, and A.P. Sistla. Automatic verification of finite-state concurrent systems using temporal logic specifications. ACM Transactions on Programming Languages and Systems, 8(2):244–263, January 1986.Google Scholar
  6. Eme90.
    E. A. Emerson. Temporal and modal logic. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science. Elsevier/MIT Press, Amsterdam/Cambridge, 1990.Google Scholar
  7. Fit92a.
    Melvin Fitting. Many-valued modal logics I. Fundamenta Informaticae, 15:235–254, 1992.MathSciNetGoogle Scholar
  8. Fit92b.
    Melvin Fitting. Many-valued modal logics II. Fundamenta Informaticae, 17:55–73, 1992.zbMATHMathSciNetGoogle Scholar
  9. HM85.
    M. Hennessy and R. Milner. Algebraic laws for nondeterminism and concurrency. Journal of the ACM, 32(1):137–161, 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Kle87.
    Stephen Cole Kleene. Introduction to Metamathematics. North Holland, 1987.Google Scholar
  11. Koz83.
    D. Kozen. Results on the Propositional Mu-Calculus. Theoretical Computer Science, 27:333–354, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  12. LT88.
    Kim G. Larsen and Bent Thomsen. A modal process logic. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science, pages 203–210. IEEE Computer Society Press, 1988.Google Scholar
  13. Mil89.
    R. Milner. Communication and Concurrency. Prentice Hall, 1989.Google Scholar
  14. Mor89.
    Osamu Morikawa. Some modal logics based on a three-valued logic. Notre Dame Journal of Formal Logic, 30(1):130–137, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  15. MP92.
    Z. Manna and A. Pnueli. The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer-Verlag, 1992.Google Scholar
  16. Par81.
    D. M. R. Park. Concurrency and automata on infinite sequences. In P. Deussen, editor, 5 th GI Conference, volume 104 of Lecture Notes in Computer Science, pages 167–183. Springer-Verlag, 1981.Google Scholar
  17. PR89a.
    A. Pnueli and R. Rosner. On the synthesis of a reactive module. In Proc. of the Sixteenth Symposim on Principles of Programming Languages, Austin, January 1989.Google Scholar
  18. PR89b.
    A. Pnueli and R. Rosner. On the synthesis of an asynchronous reactive module. In Proceedings of ICALP’89, Stresa, July 1989.Google Scholar
  19. Seg67.
    Krister Segerberg. Some modal logics based on a three-valued logic. Theoria, 33:53–71, 1967.zbMATHMathSciNetCrossRefGoogle Scholar
  20. SRW99.
    Mooly Sagiv, Thomas Reps, and Reinhard Wilhelm. Parametric shape analysis via 3-valued logic. In Proceedings of the 26th Annual ACM Symposium on Principles of Programming Languages, 1999.Google Scholar
  21. Sti87.
    Colin Stirling. Modal logics for communicating systems. Theoretical Computer Science, 49:331–347, 1987.CrossRefMathSciNetGoogle Scholar
  22. SW90.
    C. Stirling and D. Walker. CCS, liveness and local model checking in the linear-time mu-calculus. In Proc. First International Workshop on Automatic Verification Methods for Finite State Systems, volume 407 of Lecture Notes in Computer Science, pages 166–178. Springer-Verlag, 1990.Google Scholar
  23. Tar55.
    A. Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific J. of Maths, 5:285–309, 1955.zbMATHMathSciNetGoogle Scholar
  24. Var97a.
    M.Y. Vardi. Alternating automata: Checking truth and validity for temporal logics. In Proceedings of CADE’97, 1997.Google Scholar
  25. Var97b.
    M.Y. Vardi. Why is modal logic so robustly decidable? In Proceedings of DIMACS Workshop on Descriptive Complexity and Finite Models. AMS, 1997.Google Scholar
  26. VW86.
    M.Y. Vardi and P. Wolper. Automata-theoretic techniques for modal logics of programs. Journal of Computer and System Science, 32(2):183–221, April 1986.Google Scholar
  27. Wal88.
    D. J. Walker. Bisimulations and divergence. In Proceedings of the 3rd Annual Symposium on Logic in Computer Science. IEEE Computer Society Press, 1988.Google Scholar
  28. Wol83.
    Pierre Wolper. Temporal logic can be more expressive. Information and Control, 56(1–2):72–99, 1983.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Glenn Bruns
    • 1
  • Patrice Godefroid
    • 1
  1. 1.Bell LaboratoriesLucent TechnologiesNapervilleUSA

Personalised recommendations