Skip to main content

Partial-Order Reduction for Parity Games with an Application on Parameterised Boolean Equation Systems

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12079)


Partial-order reduction (POR) is a well-established technique to combat the problem of state-space explosion. We propose POR techniques that are sound for parity games, a well-established formalism for solving a variety of decision problems. As a consequence, we obtain the first POR method that is sound for model checking for the full modal \(\mu \)-calculus. Our technique is applied to, and implemented for the fixed point logic called parameterised Boolean equation systems, which provides a high-level representation of parity games. Experiments indicate that substantial reductions can be achieved.


  1. Anderson, T.E.: The Performance of Spin Lock Alternatives for Shared-Memory Multiprocessors. IEEE Transactions on Parallel & Distributed Systems 1(1), 6–16 (1990).

    CrossRef  Google Scholar 

  2. Baier, C., Katoen, J.P.: Principles of model checking. MIT Press (2008)

    Google Scholar 

  3. Bønneland, F.M., Jensen, P.G., Larsen, K.G., Muñiz, M.: Partial Order Reduction for Reachability Games. In: CONCUR 2019. vol. 140, pp. 23:1–23:15 (2019).

  4. Bønneland, F.M., Jensen, P.G., Larsen, K.G., Mũniz, M., Srba, J.: Stubborn Set Reduction for Two-Player Reachability Games. arXiv:1912.09875 (2019)

  5. Bunte, O., Groote, J.F., Keiren, J.J.A., Laveaux, M., Neele, T., de Vink, E.P., Wesselink, J.W., Wijs, A.W., Willemse, T.A.C.: The mCRL2 Toolset for Analysing Concurrent Systems: Improvements in Expressivity and Usability. In: TACAS 2019. LNCS, vol. 11428, pp. 21–39 (2019).

  6. Cranen, S., Luttik, B., Willemse, T.A.C.: Proof graphs for parameterised Boolean equation systems. In: CONCUR 2013. LNCS, vol. 8052, pp. 470–484 (2013).

  7. Gerth, R., Kuiper, R., Peled, D., Penczek, W.: A Partial Order Approach to Branching Time Logic Model Checking. Information and Computation 150(2), 132–152 (1999).

    MathSciNet  CrossRef  MATH  Google Scholar 

  8. Godefroid, P.: Partial-Order Methods for the Verification of Concurrent Systems, LNCS, vol. 1032. Springer (1996).

  9. Groote, J.F., Sellink, M.P.A.: Confluence for process verification. Theoretical Computer Science 170(1-2), 47–81 (1996).

  10. Groote, J.F., Willemse, T.A.C.: Parameterised boolean equation systems. Theoretical Computer Science 343(3), 332–369 (2005).

    MathSciNet  CrossRef  MATH  Google Scholar 

  11. Heimbold, D., Luckham, D.: Debugging ada tasking programs. IEEE Software 2(2), 47–57 (1985).

    CrossRef  Google Scholar 

  12. Hesselink, W.H.: Invariants for the construction of a handshake register. Inf. Process. Lett. 68(4), 173–177 (1998).

    CrossRef  MATH  Google Scholar 

  13. Ip, C.N., Dill, D.L.: Better verification through symmetry. Formal Methods in System Design 9(1-2), 41–75 (1996).

  14. Kan, S., Huang, Z., Chen, Z., Li, W., Huang, Y.: Partial order reduction for checking LTL formulae with the next-time operator. Journal of Logic and Computation 27(4), 1095–1131 (2017).

    MathSciNet  CrossRef  MATH  Google Scholar 

  15. Keiren, J.J.A., Wesselink, J.W., Willemse, T.A.C.: Liveness Analysis for Parameterised Boolean Equation Systems. In: ATVA 2014. LNCS, vol. 8837, pp. 219–234 (2014).

  16. Kozen, D.: Results on the propositional \(\mu \)-calculus. Theoretical Computer Science 27(3), 333–354 (1982).

    MathSciNet  CrossRef  MATH  Google Scholar 

  17. Laarman, A., Pater, E., van de Pol, J., Hansen, H.: Guard-based partial-order reduction. STTT 18(4), 427–448 (2016).

  18. Lann, G.L.: Distributed systems - towards a formal approach. IFIP 1977, 155–160 (1977)

    Google Scholar 

  19. Liebke, T., Wolf, K.: Taking Some Burden Off an Explicit CTL Model Checker. In: Petri Nets 2019. LNCS, vol. 11522, pp. 321–341 (2019).

  20. Milner, R.: A Calculus of Communicating Systems, LNCS, vol. 92. Springer (1980)

    Google Scholar 

  21. Neele, T., Valmari, A., Willemse, T.A.C.: The Inconsistent Labelling Problem of Stutter-Preserving Partial-Order Reduction. In: FoSSaCS 2020. LNCS, vol. 2077 (2020).

  22. Neele, T., Willemse, T.A.C., Groote, J.F.: Solving Parameterised Boolean Equation Systems with Infinite Data Through Quotienting. In: FACS 2018. LNCS, vol. 11222, pp. 216–236 (2018).

  23. Neele, T., Willemse, T.A.C., Groote, J.F.: Finding compact proofs for infinite-data parameterised Boolean equation systems. Science of Computer Programming 188, 102389 (2020).

    CrossRef  Google Scholar 

  24. Neele, T., Willemse, T.A.C., Wesselink, W.: Partial-Order Reduction for Parity Games with an Application on Parameterised Boolean Equation Systems (Technical Report). Tech. rep., Eindhoven University of Technology (2019)

    Google Scholar 

  25. Pelánek, R.: BEEM: Benchmarks for Explicit Model Checkers. In: SPIN 2007. LNCS, vol. 4595, pp. 263–267 (2007).

  26. Peled, D.: All from One, One for All: on Model Checking Using Representatives. In: CAV 1993. LNCS, vol. 697, pp. 409–423 (1993).

  27. Peled, D.: Combining partial order reductions with on-the-fly model-checking. FMSD 8(1), 39–64 (1996).

    CrossRef  MATH  Google Scholar 

  28. Ramakrishna, Y.S., Smolka, S.A.: Partial-Order Reduction in the Weak Modal Mu-Calculus. In: CONCUR 1997. LNCS, vol. 1243, pp. 5–24 (1997).

  29. Siegel, S.F.: What’s Wrong with On-the-Fly Partial Order Reduction. In: CAV 2019. LNCS, vol. 11562, pp. 478–495 (2019).

  30. Valmari, A.: A Stubborn Attack on State Explosion. Formal Methods in System Design 1(4), 297–322 (1992).

    CrossRef  MATH  Google Scholar 

  31. Valmari, A.: The state explosion problem. In: ACPN 1996. LNCS, vol. 1491, pp. 429–528 (1996).

  32. Valmari, A.: Stubborn Set Methods for Process Algebras. In: POMIV 1996. DIMACS, vol. 29, pp. 213–231 (1997).

  33. Valmari, A., Hansen, H.: Stubborn Set Intuition Explained. ToPNoC 10470(12), 140–165 (2017).

    CrossRef  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Thomas Neele .

Editor information

Editors and Affiliations

Rights and permissions

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Reprints and Permissions

Copyright information

© 2020 The Author(s)

About this paper

Verify currency and authenticity via CrossMark

Cite this paper

Neele, T., Willemse, T.A.C., Wesselink, W. (2020). Partial-Order Reduction for Parity Games with an Application on Parameterised Boolean Equation Systems. In: Biere, A., Parker, D. (eds) Tools and Algorithms for the Construction and Analysis of Systems. TACAS 2020. Lecture Notes in Computer Science(), vol 12079. Springer, Cham.

Download citation

  • DOI:

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-45236-0

  • Online ISBN: 978-3-030-45237-7

  • eBook Packages: Computer ScienceComputer Science (R0)