Using SMT for Solving Fragments of Parameterised Boolean Equation Systems

  • Ruud P. J. Koolen
  • Tim A. C. WillemseEmail author
  • Hans Zantema
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9364)


Fixpoint logics such as parameterised Boolean equation systems (PBESs) provide a unifying framework in which a number of practical decision problems can be encoded. Efficient evaluation methods (solving methods in the terminology of PBESs) are needed to solve the encoded decision problems. We present a sound pseudo-decision procedure that uses SMT solvers for solving conjunctive and disjunctive PBESs. These are important fragments, allowing to encode typical verification problems and planning problems. Our experiments, conducted with a prototype implementation, show that the new solving procedure is complementary to existing techniques for solving PBESs.


Model Check Satisfiability Modulo Theory Predicate Variable Bound Model Check Model Check Problem 
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.



We thank Jan Friso Groote (TU/e) for helpful discussions and feedback on our ideas, and the referees for their constructive feedback.


  1. 1.
    Alpuente, M., Feliú, M.A., Joubert, C., Villanueva, A.: Datalog-based program analysis with BES and RWL. In: de Moor, O., Gottlob, G., Furche, T., Sellers, A. (eds.) Datalog 2010. LNCS, vol. 6702, pp. 1–20. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  2. 2.
    Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 171–177. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  3. 3.
    Cavada, R., et al.: The nuXmv Symbolic Model Checker. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 334–342. Springer, Heidelberg (2014) Google Scholar
  4. 4.
    Clarke, E., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Form. Methods Syst. Des. 19(1), 7–34 (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cranen, S., Groote, J.F., Keiren, J.J.A., Stappers, F.P.M., de Vink, E.P., Wesselink, W., Willemse, T.A.C.: An overview of the mCRL2 toolset and its recent advances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013 (ETAPS 2013). LNCS, vol. 7795, pp. 199–213. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  6. 6.
    Cranen, S., Luttik, B., Willemse, T.A.C.: Proof graphs for parameterised boolean equation systems. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013 – Concurrency Theory. LNCS, vol. 8052, pp. 470–484. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  7. 7.
    Dutertre, B., De Moura, L.: The yices SMT solver. Technical report (2006)Google Scholar
  8. 8.
    Garavel, H., Lang, F., Mateescu, R., Serwe, W.: CADP 2010: a toolbox for the construction and analysis of distributed processes. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 372–387. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  9. 9.
    Groote, J.F., Willemse, T.A.C.: Model-checking processes with data. Sci. Comput. Program 56(3), 251–273 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Groote, J.F., Willemse, T.A.C.: Parameterised boolean equation systems. Theor. Comput. Sci. 343(3), 332–369 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kant, G., van de Pol, J.C.: Generating and solving symbolic parity games. In: GRAPHITE. EPTCS, vol. 159 , pp. 2–14 (2014)Google Scholar
  12. 12.
    Koolen, R.P.J.: Solving conjunctive and disjunctive parameterised Boolean equation systems using SMT solvers. Master’s thesis, Eindhoven University of Technology (2014)Google Scholar
  13. 13.
    Orzan, S., Wesselink, 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.
    Oshman, R., Grumberg, O.: A new approach to bounded model checking for branching time logics. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 410–424. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  15. 15.
    Ploeger, B., Wesselink, W., Willemse, T.A.C.: Verification of reactive systems via instantiation of parameterised Boolean equation systems. Inf. Comput. 209(4), 637–663 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ruud P. J. Koolen
    • 1
  • Tim A. C. Willemse
    • 1
    Email author
  • Hans Zantema
    • 1
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations