Model checking and boolean graphs

  • Henrik Reif Andersen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 582)


This paper describes a method for translating a satisfaction problem of the modal μ-calculus into a problem of finding a certain marking of a boolean graph. By giving algorithms to solve the graph problem, we present a global model checking algorithm for the modal μ-calculus of alternation depth one, which has time-complexity ¦A¦¦T¦, where ¦A¦ is the size of the assertion and ¦T¦ is the size of the model (a labelled transition system). This algorithm extends to an algorithm for the full modal μ-calculus which runs in time (¦A¦¦T¦)ad, where ad is the alternation depth, improving on earlier presented algorithms. Moreover, a local algorithm is presented for alternation depth one, which runs in time ¦A¦¦T¦log(¦A¦¦T¦), improving on the earlier published algorithms that are all at least exponential.


  1. [AC88]
    André Arnold and Paul Crubille. A linear algorithm to solve fixed-point equations on transitions systems. Information Processing Letters, 29:57–66, 1988.Google Scholar
  2. [AW91]
    Henrik Reif Andersen and Glynn Winskel. Compositional checking of satisfaction. In Larsen and Skou [LS91]. To appear.Google Scholar
  3. [Bek84]
    H. Bekić. Definable operations in general algebras, and the theory of automata and flow charts. Lecture Notes in Computer Science, 177, 1984.Google Scholar
  4. [Cle90]
    Rance Cleaveland. Tableau-based model checking in the propositional mucalculus. Acta Informatica, 27:725–747, 1990.Google Scholar
  5. [CLR90]
    Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest. Introduction to Algorithms. McGraw-Hill, 1990.Google Scholar
  6. [CPS89]
    Rance Cleaveland, Joachim Parrow, and Bernhard Steffen. The Concurrency Workbench: A semantics based tool for the verification of concurrent systems. Technical Report ECS-LFCS-89-83, Laboratory for Foundations of Computer Science, Uni. of Edinburgh, August 1989.Google Scholar
  7. [CS91]
    Rance Cleaveland and Bernhard Steffen. A linear-time model-checking algorithm for the alternation-free modal mu-calculus. In Larsen and Skou [LS91]. To appear.Google Scholar
  8. [Dam90]
    Mads Dam. Translating CTL* into the modal μ-calculus. Technical Report ECS-LFCS-90-123, Laboratory for Foundations of Computer Science, Uni. of Edinburgh, November 1990.Google Scholar
  9. [DG84]
    William F. Dowling and Jean H. Gallier. Linear-time algorithms for testing the satisfiability of propositional Horn formulae. Journal of Logic Programming, 1 (3):267–284, 1984.Google Scholar
  10. [EL86]
    E. Allen Emerson and Chin-Luang Lei. Efficient model checking in fragments of the propositional mu-calculus. In Symposium on Logic in Computer Science, Proceedings, pages 267–278. IEEE, 1986.Google Scholar
  11. [Koz83]
    Dexter Kozen. Results on the prepositional mu-calculus. Theoretical Computer Science, 27, 1983.Google Scholar
  12. [Lax88]
    Kim G. Laxsen. Proof systems for Hennessy-Milner logic with recursion. In Proceedings of CAAP, 1988.Google Scholar
  13. [LGZ89]
    Kim G. Larsen, J.C. Godskesen, and M. Zeeberg. TAV-Tools for Automatic Verification. Technical Report R 89-19, Aalborg Universitetscenter, 1989.Google Scholar
  14. [LS91]
    Kim G. Larsen and Arne Skou, editors. Proceedings of the 3rd Workshop on Computer Aided Verification, Aalborg, LNCS. Springer-Verlag, July 1991. To appear.Google Scholar
  15. [LX90]
    Kim G. Larsen and Liu Xinxin. Compositionality through an operational semantics of contexts. In M.S. Paterson, editor, Proceedings of ICALP, volume 443 of LNCS, pages 526–539. Springer-Verlag, 1990.Google Scholar
  16. [SW89]
    Colin Stirling and David Walker. Local model checking in the modal mucalculus. In Proceedings of TAPSOFT, 1989.Google Scholar
  17. [Tar55]
    A. Tarski. A lattice-theoretical fixpoint theorem and its applications. Pacific Journal of Mathematics, 5, 1955.Google Scholar
  18. [Win89]
    Glynn Winskel. A note on model checking the modal v-calculus. In Ausiello, Dezani-Ciancaglini, and Rocca, editors, Proceedings of ICALP, volume 372 of LNCS, 1989.Google Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Henrik Reif Andersen
    • 1
  1. 1.Department of Computer ScienceAarhus UniversityAarhus CDenmark

Personalised recommendations