Consistent Consequence for Boolean Equation Systems

  • Maciej W. Gazda
  • Tim A. C. Willemse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)


Inspired by the concept of a consistent correlation for Boolean equation systems, we introduce and study a novel relation, called consistent consequence. We show that it can be used as an approximation of the solution to an equation system. For the closed, simple and recursive fragment of equation systems we prove that it coincides with direct simulation for parity games. In addition, we show that deciding both consistent consequence and consistent correlations are coNP-complete problems, and we provide a sound and complete proof system for consistent consequence. As an application, we define a novel abstraction mechanism for parameterised Boolean equation systems and we establish its correctness using our theory.


Equation System Consistent Consequence Proof System Proposition Variable Label Transition System 
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. 1.
    Arnold, A., Crubille, P.: A linear algorithm to solve fixed-point equations on transition systems. Information Processing Letters 20(1), 57–66 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Trans. Program. Lang. Syst. 16(5), 1512–1542 (1994)CrossRefGoogle Scholar
  3. 3.
    Delgrande, J.P., Gupta, A.: Two results in negation-free logic. Applied Mathematics Letters 6(6), 79–83 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Etessami, K., Wilke, T., Schuller, R.A.: Fair simulation relations, parity games, and state space reduction for büchi automata. SIAM J. Comput. 34(5), 1159–1175 (2005)CrossRefzbMATHGoogle Scholar
  5. 5.
    Fritz, C., Wilke, T.: Simulation Relations for Alternating Parity Automata and Parity Games. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 59–70. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Groote, J.F., Willemse, T.A.C.: Parameterised boolean equation systems. Theor. Comput. Sci. 343(3), 332–369 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Keinänen, M.: Techniques for Solving Boolean Equation Systems. PhD thesis, Helsinki University of Technology (2006)Google Scholar
  8. 8.
    Loiseaux, C., Graf, S., Sifakis, J., Bouajjani, A., Bensalem, S.: Property preserving abstractions for the verification of concurrent systems. Formal Methods in System Design 6(1), 11–44 (1995)CrossRefzbMATHGoogle Scholar
  9. 9.
    Mader, A.: Verification of Modal Properties Using Boolean Equation Systems. PhD thesis, Technische Universität München (1997)Google Scholar
  10. 10.
    Mateescu, R.: Vérification des propriétés temporelles des programmes parallèles. PhD thesis, Institut National Polytechnique de Grenoble (1998)Google Scholar
  11. 11.
    Mateescu, R.: A Generic On-the-Fly Solver for Alternation-Free Boolean Equation Systems. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 81–96. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 12.
    McNaughton, R.: Infinite games played on finite graphs. Annals of Pure and Applied Logic 65(2), 149–184 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Orzan, S., Wesselink, J.W., Willemse, T.A.C.: Static Analysis Techniques for Parameterised Boolean Equation Systems. In: Kowalewski, S., Philippou, A. (eds.) TACAS 2009. LNCS, vol. 5505, pp. 230–245. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Willemse, T.A.C.: Consistent Correlations for Parameterised Boolean Equation Systems with Applications in Correctness Proofs for Manipulations. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 584–598. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  15. 15.
    Zhang, D., Cleaveland, R.: Fast Generic Model-Checking for Data-Based Systems. In: Wang, F. (ed.) FORTE 2005. LNCS, vol. 3731, pp. 83–97. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Maciej W. Gazda
    • 1
  • Tim A. C. Willemse
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations