Advertisement

Parameterised Boolean Equation Systems

  • Jan Friso Groote
  • Tim Willemse
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3170)

Abstract

Boolean equation system are a useful tool for verifying formulas from modal mu-calculus on transition systems (see [9] for an excellent treatment). We are interested in an extension of boolean equation systems with data. This allows to formulate and prove a substantially wider range of properties on much larger and even infinite state systems. In previous works [4,6] it has been outlined how to transform a modal formula and a process, both containing data, to a so-called parameterised boolean equation system, or equation system for short. In this article we focus on techniques to solve such equation systems.

We introduce a new equivalence between equation systems, because existing equivalences are not compositional. We present techniques similar to Gauß elimination as outlined in [9] that allow to solve each equation system provided a single equation can be solved. We give several techniques for solving single equations, such as approximation (known), patterns (new) and invariants (new). Finally, we provide several small but illustrative examples of verifications of modal mu-calculus formulas on concrete processes to show the use of the techniques.

Keywords

Model Check Equation System Input Stream Modal Formula Predicate Variable 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bradfield, J., Stirling, C.: Modal logics and mu-calculi: An introduction. In: Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.) Handbook of process algebra, pp. 293–330. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  2. 2.
    Cousot, P.: Semantic foundations of program analysis, ch. 10. In: Muchnick, S.S., Jones, N.D. (eds.) Program Flow Analysis: Theory and Applications, New Jersey, USA, pp. 303–342. Prentice-Hall, Inc., Englewood Cliffs (1981)Google Scholar
  3. 3.
    Emerson, E.A., Lei, C.-L.: Efficient model checking in fragments of the propositional mu-calculus. In: First IEEE Symposium on Logic in Computer Science, LICS 1986, pp. 267–278. IEEE Computer Society Press, Los Alamitos (1986)Google Scholar
  4. 4.
    Groote, J.F., Mateescu, R.: Verification of temporal properties of processes in a setting with data. In: Haeberer, A.M. (ed.) AMAST 1998. LNCS, vol. 1548, pp. 74–90. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Groote, J.F., Reniers, M.A.: Algebraic process verification. In: Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.) Handbook of Process Algebra, ch. 17, pp. 1151–1208. Elsevier, North-Holland (2001)CrossRefGoogle Scholar
  6. 6.
    Groote, J.F., Willemse, T.A.C.: A checker for modal formulas for processes with data. Technical Report CSR 02-16, Eindhoven University of Technology, Department of Mathematics and Computer Science (2002)Google Scholar
  7. 7.
    Groote, J.F., Willemse, T.A.C.: Parameterised boolean equation systems. Computer Science Report 04/09, Department of Mathematics and Computer Science, Eindhoven University of Technology (2004)Google Scholar
  8. 8.
    Kozen, D.: Results on the propositional mu-calculus. Theoretical Computer Science 27, 333–354 (1983)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Mader, A.: Modal μ-calculus, model checking and gaußelimination. In: Brinksma, E., Steffen, B., Cleaveland, W.R., Larsen, K.G., Margaria, T. (eds.) TACAS 1995. LNCS, vol. 1019, pp. 72–88. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  10. 10.
    Mader, A.: Verification of Modal Properties Using Boolean Equation Systems. PhD thesis, Technical University of Munich (1997)Google Scholar
  11. 11.
    Vergauwen, B., Lewi, J.: Efficient local correctness checking for single and alternating boolean equation systems. In: Shamir, E., Abiteboul, S. (eds.) ICALP 1994. LNCS, vol. 820, pp. 302–315. Springer, Heidelberg (1994)Google Scholar
  12. 12.
    Willemse, T.A.C.: Semantics and Verification in Process Algebras with Data and Timing. PhD thesis, Eindhoven University of Technology (February 2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Jan Friso Groote
    • 1
  • Tim Willemse
    • 1
    • 2
  1. 1.Department of Mathematics and Computer ScienceEindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.Faculty of Science, Mathematics and Computing ScienceUniversity of NijmegenNijmegenThe Netherlands

Personalised recommendations