On the Community Structure of Bounded Model Checking SAT Problems

  • Guillaume Baud-Berthier
  • Jesús Giráldez-Cru
  • Laurent Simon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10491)


Following the impressive progress made in the quest for efficient SAT solving in the last years, a number of researches has focused on explaining performances observed on typical application problems. However, until now, tentative explanations were only partial, essentially because the semantic of the original problem was lost in the translation to SAT.

In this work, we study the behavior of so called “modern” SAT solvers under the prism of the first successful application of CDCL solvers, i.e., Bounded Model Checking. We trace the origin of each variable w.r.t. its unrolling depth, and show a surprising relationship between these time steps and the communities found in the CNF encoding. We also show how the VSIDS heuristic, the resolution engine, and the learning mechanism interact with the unrolling steps. Additionally, we show that the Literal Block Distance (LBD), used to identify good learnt clauses, is related to this measure.

Our work shows that communities identify strong dependencies among the variables of different time steps, revealing a structure that arises when unrolling the problem, and which seems to be caught by the LBD measure.


  1. 1.
    Ansótegui, C., Giráldez-Cru, J., Levy, J.: The community structure of SAT formulas. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 410–423. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-31612-8_31 CrossRefGoogle Scholar
  2. 2.
    Ansótegui, C., Giráldez-Cru, J., Levy, J., Simon, L.: using community structure to detect relevant learnt clauses. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 238–254. Springer, Cham (2015). doi: 10.1007/978-3-319-24318-4_18 CrossRefGoogle Scholar
  3. 3.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern sat solvers. In: Proceeding of IJCAI 2009, pp. 399–404 (2009)Google Scholar
  4. 4.
    Biere, A.: Splatz, Lingeling, Plingeling, Treengeling, YalSAT Entering the SAT Competition 2016. In: Proceeding of the SAT Competition 2016, Department of Computer Science Series of Publications B, vol. B-2016-1, pp. 44–45. University of Helsinki (2016)Google Scholar
  5. 5.
    Biere, A., Cimatti, A., Clarke, E., Zhu, Y.: Symbolic model checking without BDDs. In: Cleaveland, W.R. (ed.) TACAS 1999. LNCS, vol. 1579, pp. 193–207. Springer, Heidelberg (1999). doi: 10.1007/3-540-49059-0_14 CrossRefGoogle Scholar
  6. 6.
    Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theor. Exper. 2008(10), P10008 (2008)CrossRefGoogle Scholar
  7. 7.
    Darwiche, A., Pipatsrisawat, K.: Complete Algorithms, chap. 3, pp. 99–130. IOS Press (2009)Google Scholar
  8. 8.
    Eén, N., Biere, A.: Effective preprocessing in SAT through variable and clause elimination. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 61–75. Springer, Heidelberg (2005). doi: 10.1007/11499107_5 CrossRefGoogle Scholar
  9. 9.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004). doi: 10.1007/978-3-540-24605-3_37 CrossRefGoogle Scholar
  10. 10.
    Fortunato, S.: Community detection in graphs. Phys. Rep. 486(3–5), 75–174 (2010)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Giráldez-Cru, J., Levy, J.: A modularity-based random SAT instances generator. In: Proceeding of IJCAI 2015, pp. 1952–1958 (2015)Google Scholar
  12. 12.
    Giráldez-Cru, J., Levy, J.: Generating SAT instances with community structure. Artif. Intell. 238, 119–134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Giráldez-Cru, J., Levy, J.: Locality in random SAT instances. In: Proceeding of IJCAI 2017 (2017)Google Scholar
  14. 14.
    Gomes, C.P., Selman, B., Crato, N.: Heavy-tailed distributions in combinatorial search. In: Smolka, G. (ed.) CP 1997. LNCS, vol. 1330, pp. 121–135. Springer, Heidelberg (1997). doi: 10.1007/BFb0017434 CrossRefGoogle Scholar
  15. 15.
    Liang, J.H., Ganesh, V., Poupart, P., Czarnecki, K.: Learning rate based branching heuristic for SAT solvers. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 123–140. Springer, Cham (2016). doi: 10.1007/978-3-319-40970-2_9 Google Scholar
  16. 16.
    Luby, M., Sinclair, A., Zuckerman, D.: Optimal speedup of Las Vegas algorithms. Inf. Process. Lett. 47(4), 173–180 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Martins, R., Manquinho, V., Lynce, I.: Community-based partitioning for MaxSAT solving. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 182–191. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39071-5_14 CrossRefGoogle Scholar
  18. 18.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceeding of DAC 2001, pp. 530–535 (2001)Google Scholar
  19. 19.
    Neves, M., Martins, R., Janota, M., Lynce, I., Manquinho, V.: Exploiting resolution-based representations for MaxSAT solving. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 272–286. Springer, Cham (2015). doi: 10.1007/978-3-319-24318-4_20 CrossRefGoogle Scholar
  20. 20.
    Newman, M.E.J., Girvan, M.: Finding and evaluating community structure in networks. Phys. Rev. E 69(2), 026113 (2004)CrossRefGoogle Scholar
  21. 21.
    Newsham, Z., Ganesh, V., Fischmeister, S., Audemard, G., Simon, L.: Impact of community structure on SAT solver performance. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 252–268. Springer, Cham (2014). doi: 10.1007/978-3-319-09284-3_20 Google Scholar
  22. 22.
    Oh, C.: Between SAT and UNSAT: the fundamental difference in CDCL SAT. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 307–323. Springer, Cham (2015). doi: 10.1007/978-3-319-24318-4_23 CrossRefGoogle Scholar
  23. 23.
    Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175(2), 512–525 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Silva, J.P.M., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Katsirelos, G., Simon, L.: Eigenvector centrality in industrial SAT instances. In: Milano, M. (ed.) CP 2012. LNCS, pp. 348–356. Springer, Heidelberg (2012). doi: 10.1007/978-3-642-33558-7_27 CrossRefGoogle Scholar
  26. 26.
    Strichman, O.: Accelerating bounded model checking of safety properties. Form. Methods Syst. Des. 24(1), 5–24 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Guillaume Baud-Berthier
    • 1
    • 3
  • Jesús Giráldez-Cru
    • 2
  • Laurent Simon
    • 1
  1. 1.LaBRI, UMR 5800, University of BordeauxBordeauxFrance
  2. 2.KTH, Royal Institute of TechnologyStockholmSweden
  3. 3.SafeRiverParisFrance

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