Extended Dependency Graphs and Efficient Distributed Fixed-Point Computation

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10258)


Equivalence and model checking problems can be encoded into computing fixed points on dependency graphs. Dependency graphs represent causal dependencies among the nodes of the graph by means of hyper-edges. We suggest to extend the model of dependency graphs with so-called negation edges in order to increase their applicability. The graphs (as well as the verification problems) suffer from the state space explosion problem. To combat this issue, we design an on-the-fly algorithm for efficiently computing fixed points on extended dependency graphs. Our algorithm supplements previous approaches with the possibility to back-propagate, in certain scenarios, the domain value 0, in addition to the standard back-propagation of the value 1. Finally, we design a distributed version of the algorithm, implement it in an open-source tool, and demonstrate the efficiency of our general approach on the benchmark of Petri net models and CTL queries from the Model Checking Contest 2016.


Model Check Dependency Graph Local Algorithm Computation Tree Logic Negation Edge 
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 would like to thank to Frederik Boenneland, Jakob Dyhr, Mads Johannsen and Torsten Liebke for their help with running LoLA experiments. The work was funded by Sino-Danish Basic Research Center IDEA4CPS, Innovation Fund Denmark center DiCyPS and ERC Advanced Grant LASSO. The last author is partially affiliated with FI MU in Brno.


  1. 1.
    Barnat J. et al.: DiVinE 3.0 – an explicit-state model checker for multithreaded C & C++ programs. In: Sharygina N., Veith H. (eds) CAV 2013. LNCS, vol 8044, pp. 863–868. Springer, Heidelberg (2013)Google Scholar
  2. 2.
    Bellettini, C., Camilli, M., Capra, L., Monga, M.: Distributed CTL model checking in the cloud. arXiv preprint arXiv:1310.6670 (2013)
  3. 3.
    Bollig, B., Leucker, M., Weber, M.: Local parallel model checking for the alternation-free \(\mu \)-calculus. In: Bošnački, D., Leue, S. (eds.) SPIN 2002. LNCS, vol. 2318, pp. 128–147. Springer, Heidelberg (2002). doi: 10.1007/3-540-46017-9_11 CrossRefGoogle Scholar
  4. 4.
    Brim, L., Crhova, J., Yorav, K.: Using assumptions to distribute CTL model checking. ENTCS 68(4), 559–574 (2002)zbMATHGoogle Scholar
  5. 5.
    Cassez, F., David, A., Fleury, E., Larsen, K.G., Lime, D.: Efficient on-the-fly algorithms for the analysis of timed games. In: Abadi, M., Alfaro, L. (eds.) CONCUR 2005. LNCS, vol. 3653, pp. 66–80. Springer, Heidelberg (2005). doi: 10.1007/11539452_9 CrossRefGoogle Scholar
  6. 6.
    Christoffersen, P., Hansen, M., Mariegaard, A., Ringsmose, J.T., Larsen, K.G., Mardare, R.: Parametric verification of weighted systems. In: SynCoP 2015, vol. 44, pp. 77–90. OASIcs, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  7. 7.
    Clarke, E., Grumberg, O., Jha, S., Lu, Y., Veith, H.: Progress on the state explosion problem in model checking. In: Wilhelm, R. (ed.) Informatics. LNCS, vol. 2000, pp. 176–194. Springer, Heidelberg (2001). doi: 10.1007/3-540-44577-3_12 CrossRefGoogle Scholar
  8. 8.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching time temporal logic. In: Kozen, D. (ed.) Logic of Programs 1981. LNCS, vol. 131, pp. 52–71. Springer, Heidelberg (1982). doi: 10.1007/BFb0025774 CrossRefGoogle Scholar
  9. 9.
    Clarke, E.M., Emerson, E.A., Sifakis, J.: Model checking: algorithmic verification and debugging. Commun. ACM 52(11), 74–84 (2009)CrossRefGoogle Scholar
  10. 10.
    Dalsgaard, A.E., Enevoldsen, S., Larsen, K.G., Srba, J.: Distributed computation of fixed points on dependency graphs. In: Fränzle, M., Kapur, D., Zhan, N. (eds.) SETTA 2016. LNCS, vol. 9984, pp. 197–212. Springer, Cham (2016). doi: 10.1007/978-3-319-47677-3_13 CrossRefGoogle Scholar
  11. 11.
    David, A., Jacobsen, L., Jacobsen, M., Jørgensen, K.Y., Møller, M.H., Srba, J.: TAPAAL 2.0: integrated development environment for timed-arc Petri nets. In: Flanagan, C., König, B. (eds.) TACAS 2012. LNCS, vol. 7214, pp. 492–497. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-28756-5_36 CrossRefGoogle Scholar
  12. 12.
    Esparza, J.: Decidability of model checking for infinite-state concurrent systems. Acta Informatica 34(2), 85–107 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Garavel, H., Lang, F., Mateescu, R., Serwe, W.: CADP 2011: a toolbox for the construction and analysis of distributed processes. STTT 15(2), 89–107 (2013)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gibson-Robinson, T., Armstrong, P., Boulgakov, A., Roscoe, A.W.: FDR3 — a modern refinement checker for CSP. In: Ábrahám, E., Havelund, K. (eds.) TACAS 2014. LNCS, vol. 8413, pp. 187–201. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54862-8_13 CrossRefGoogle Scholar
  15. 15.
    Greenlaw, R., Hoover, H.J., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory, vol. 200. Oxford University Press Inc, New York (1995)zbMATHGoogle Scholar
  16. 16.
    Groote, J., Mousavi, M.: Modeling and Analysis of Communicating Systems. The MIT Press, Cambridge (2014)zbMATHGoogle Scholar
  17. 17.
    Grumberg, O., Heyman, T., Schuster, A.: Distributed symbolic model checking for \(\mu \)-calculus. Formal Methods Syst. Des. 26(2), 197–219 (2005)CrossRefzbMATHGoogle Scholar
  18. 18.
    Heiner, M., Rohr, C., Schwarick, M.: MARCIE – model checking and reachability analysis done efficiently. In: Colom, J.-M., Desel, J. (eds.) PETRI NETS 2013. LNCS, vol. 7927, pp. 389–399. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38697-8_21 CrossRefGoogle Scholar
  19. 19.
    Holzmann, G.: Spin Model Checker, the: Primer and Reference Manual. Addison-Wesley Professional, Boston (2003)Google Scholar
  20. 20.
    Jensen, J., Larsen, K., Srba, J., Oestergaard, L.: Efficient model checking of weighted CTL with upper-bound constraints. STTT 18(4), 409–426 (2016)CrossRefGoogle Scholar
  21. 21.
    Jensen, J.F., Nielsen, T., Oestergaard, L.K., Srba, J.: TAPAAL and reachability analysis of P/T nets. In: Koutny, M., Desel, J., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency XI. LNCS, vol. 9930, pp. 307–318. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-53401-4_16 CrossRefGoogle Scholar
  22. 22.
    Joubert, C., Mateescu, R.: Distributed on-the-fly model checking and test case generation. In: Valmari, A. (ed.) SPIN 2006. LNCS, vol. 3925, pp. 126–145. Springer, Heidelberg (2006). doi: 10.1007/11691617_8 CrossRefGoogle Scholar
  23. 23.
    Kant, G., Laarman, A., Meijer, J., Pol, J., Blom, S., Dijk, T.: LTSmin: high-performance language-independent model checking. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 692–707. Springer, Heidelberg (2015). doi: 10.1007/978-3-662-46681-0_61 Google Scholar
  24. 24.
    Keinänen, M.: Techniques for solving boolean equation systems. Research Report A105, Doctoral dissertation, Laboratory for Theoretical Computer Science, Helsinki University of Technology, pp. xii+95 (2006)Google Scholar
  25. 25.
    Keiren, J.J.A.: Advanced reduction techniques for model checking. Ph.D. thesis, Eindhoven University of Technology (2013)Google Scholar
  26. 26.
    Kordon, F., Garavel, H., Hillah, L.M., Hulin-Hubard, F., Chiardo, G., Hamez, A., Jezequel, L., Miner, A., Meijer, J., Paviot-Adet, E., Racordon, D., Rodriguez, C., Rohr, C., Srba, J., Thierry-Mieg, Y., Trinh, G., Wolf, K.: Complete results for the 2016th edition of the model checking contest.
  27. 27.
    Kordon, F., Garavel, H., Hillah, L.M., Hulin-Hubard, F., Linard, A., Beccuti, M., Hamez, A., Lopez-Bobeda, E., Jezequel, L., Meijer, J., Paviot-Adet, E., Rodriguez, C., Rohr, C., Srba, J., Thierry-Mieg, Y., Wolf, K.: Complete results for the 2015th edition of the model checking contest (2015)Google Scholar
  28. 28.
    Kozen, D.: Results on the propositional \(\mu \)-calculus. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 348–359. Springer, Heidelberg (1982). doi: 10.1007/BFb0012782 CrossRefGoogle Scholar
  29. 29.
    Liu, X., Smolka, S.A.: Simple linear-time algorithms for minimal fixed points. In: ICALP 1998. LNCS, vol. 1443, pp. 53–66. Springer, Heidelberg (1998)Google Scholar
  30. 30.
    Tan, L., Cleaveland, R.: Evidence-based model checking. In: Brinksma, E., Larsen, K.G. (eds.) CAV 2002. LNCS, vol. 2404, pp. 455–470. Springer, Heidelberg (2002). doi: 10.1007/3-540-45657-0_37 CrossRefGoogle Scholar
  31. 31.
    Wolf, K.: Running LoLA 2.0 in a model checking competition. In: Koutny, M., Desel, J., Kleijn, J. (eds.) Transactions on Petri Nets and Other Models of Concurrency XI. LNCS, vol. 9930, pp. 274–285. Springer, Heidelberg (2016). doi: 10.1007/978-3-662-53401-4_13 CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Computer ScienceAalborg UniversityAalborg EastDenmark
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

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