Multi-core Nested Depth-First Search

  • Alfons Laarman
  • Rom Langerak
  • Jaco van de Pol
  • Michael Weber
  • Anton Wijs
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6996)


The LTL Model Checking problem is reducible to finding accepting cycles in a graph. The Nested Depth-First Search (Ndfs) algorithm detects accepting cycles efficiently: on-the-fly, with linear-time complexity and negligible memory overhead. The only downside of the algorithm is that it relies on an inherently-sequential, depth-first search. It has not been parallelized beyond running the independent nested search in a separate thread (dual core).

In this paper, we introduce, for the first time, a multi-core Ndfs algorithm that can scale beyond two threads, while maintaining exactly the same worst-case time complexity. We prove this algorithm correct, and present experimental results obtained with an implementation in the LTSmin tool set on the entire Beem benchmark database. We measured considerable speedups compared to the current state of the art in parallel cycle detection algorithms.


Model Check State Graph Linear Temporal Logic Input Graph Reachable State 
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.


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  1. 1.
    Baier, C., Katoen, J.P.: Principles of Model Checking. The MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  2. 2.
    Barnat, J., Brim, L., Ročkai, P.: Scalable Shared Memory LTL Model Checking. STTT 12(2), 139–153 (2010)CrossRefGoogle Scholar
  3. 3.
    Barnat, J., Brim, L., Češka, M., Ročkai, P.: DiVinE: Parallel Distributed Model Checker (Tool paper). In: Parallel and Distributed Methods in Verification and High Performance Computational Systems Biology (HiBi/PDMC 2010), pp. 4–7. IEEE, Los Alamitos (2010)Google Scholar
  4. 4.
    Barnat, J., Brim, L., Ročkai, P.: A Time-Optimal On-The-Fly Parallel Algorithm for Model Checking of Weak LTL Properties. In: Breitman, K., Cavalcanti, A. (eds.) ICFEM 2009. LNCS, vol. 5885, pp. 407–425. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Courcoubetis, C., Vardi, M.Y., Wolper, P., Yannakakis, M.: Memory-Efficient Algorithms for the Verification of Temporal Properties. Formal Methods in System Design 1(2/3), 275–288 (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dwyer, M.B., Elbaum, S.G., Person, S., Purandare, R.: Parallel Randomized State-Space Search. In: Proc. ICSE 2007, pp. 3–12. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  7. 7.
    Evangelista, S., Petrucci, L., Youcef, S.: Parallel nested depth-first searches for LTL model checking. In: Bultan, T., Hsiung, P.-A. (eds.) ATVA 2011. LNCS, vol. 6996, pp. 381–396. Springer, Heidelberg (2011)Google Scholar
  8. 8.
    Foster, I.: Designing and Building Parallel Programs. Addison-Wesley, Reading (1995)zbMATHGoogle Scholar
  9. 9.
    Gaiser, A., Schwoon, S.: Comparison of Algorithms for Checking Emptiness on Büchi Automata. CoRR, abs/0910.3766 (2009)Google Scholar
  10. 10.
    Geldenhuys, J., Valmari, A.: Tarjan’s Algorithm Makes On-the-Fly LTL Verification More Efficient. In: Jensen, K., Podelski, A. (eds.) TACAS 2004. LNCS, vol. 2988, pp. 205–219. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Holzmann, G.J., Bošnački, D.: The Design of a Multicore Extension of the SPIN Model Checker. IEEE Trans. On Software Engineering 33(10), 659–674 (2007)CrossRefGoogle Scholar
  12. 12.
    Holzmann, G.J., Bošnački, D.: The Design of a Multicore Extension of the SPIN Model Checker. IEEE Transactions on Software Engineering 33(10), 659–674 (2007)CrossRefGoogle Scholar
  13. 13.
    Holzmann, G.J., Joshi, R., Groce, A.: Swarm Verification. In: Proc. ASE 2008, pp. 1–6. IEEE Computer Society Press, Los Alamitos (2008)Google Scholar
  14. 14.
    Holzmann, G.J., Joshi, R., Groce, A.: Tackling Large Verification Problems with the Swarm Tool. In: Havelund, K., Majumdar, R. (eds.) SPIN 2008. LNCS, vol. 5156, pp. 134–143. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Holzmann, G.J., Peled, D., Yannakakis, M.: On Nested Depth First Search. In: The Spin Verification System, pp. 23–32. American Mathematical Society, Providence (1996)Google Scholar
  16. 16.
    Laarman, A.W., van de Pol, J.C., Weber, M.: Boosting Multi-Core Reachability Performance with Shared Hash Tables. In: Sharygina, N., Bloem, R. (eds.) Proceedings of the 10th International Conference on Formal Methods in Computer-Aided Design, Lugano, Swiss, USA. IEEE Computer Society, Los Alamitos (2010)Google Scholar
  17. 17.
    Laarman, A.W., van de Pol, J.C., Weber, M.: Multi-core LTSmin: Marrying modularity and scalability. In: Bobaru, M., Havelund, K., Holzmann, G., Joshi, R. (eds.) NFM 2011. LNCS, vol. 6617, pp. 506–511. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Moore, G.E.: Cramming more Components onto Integrated Circuits. Electronics 38(10), 114–117 (1965)Google Scholar
  19. 19.
    Pelánek, R.: BEEM: Benchmarks for Explicit Model Checkers. In: Bošnački, D., Edelkamp, S. (eds.) SPIN 2007. LNCS, vol. 4595, pp. 263–267. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Reif, J.H.: Depth-first Search is Inherently Sequential. Information Processing Letters 20(5), 229–234 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Schwoon, S., Esparza, J.: A Note on On-the-Fly Verification Algorithms. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 174–190. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  22. 22.
    Sivaraj, H., Gopalakrishnan, G.: Random Walk Based Heuristic Algorithms for Distributed Memory Model Checking. Electronic Notes in Theoretical Computer Science 89(1), 51–67 (2003)CrossRefzbMATHGoogle Scholar
  23. 23.
    Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification. In: Proc. 1st Symp. on Logic in Computer Science, Cambridge, pp. 332–344 (June 1986)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Alfons Laarman
    • 1
  • Rom Langerak
    • 1
  • Jaco van de Pol
    • 1
  • Michael Weber
    • 1
  • Anton Wijs
    • 2
  1. 1.Formal Methods and ToolsUniversity of TwenteThe Netherlands
  2. 2.Eindhoven University of TechnologyEindhovenThe Netherlands

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