MUS Extraction Using Clausal Proofs

  • Anton Belov
  • Marijn J. H. Heule
  • Joao Marques-Silva
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8561)


Recent work introduced an effective method for extraction of reduced unsatisfiable cores of CNF formulas as a by-product of validation of clausal proofs emitted by conflict-driven clause learning SAT solvers. In this paper, we demonstrate that this method for trimming CNF formulas can also benefit state-of-the-art tools for the computation of a Minimal Unsatisfiable Subformula (MUS). Furthermore, we propose a number of techniques that improve the quality of trimming, and demonstrate a significant positive impact on the performance of MUS extractors from the improved trimming.


Unit Clause Resolution Graph Input Formula Resolution Proof Easy Instance 
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.
    Marques-Silva, J., Janota, M., Belov, A.: Minimal sets over monotone predicates in boolean formulae. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 592–607. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  2. 2.
    Belov, A., Lynce, I., Marques-Silva, J.: Towards efficient MUS extraction. AI Communications 25(2), 97–116 (2012)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Nadel, A., Ryvchin, V., Strichman, O.: Efficient MUS extraction with resolution. In: [24], pp.197–200Google Scholar
  4. 4.
    Audemard, G., Lagniez, J.M., Simon, L.: Improving glucose for incremental SAT solving with assumptions: Application to MUS extraction, In: [25], pp. 309–317Google Scholar
  5. 5.
    Lagniez, J.M., Biere, A.: Factoring out assumptions to speed up MUS extraction. In: [25], pp. 276–292Google Scholar
  6. 6.
    Dershowitz, N., Hanna, Z., Nadel, A.: A scalable algorithm for minimal unsatisfiable core extraction. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 36–41. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Zhang, L., Malik, S.: Validating SAT solvers using an independent resolution-based checker: Practical implementations and other applications. In: DATE, pp. 10880–10885. IEEE Computer Society (2003)Google Scholar
  8. 8.
    Belov, A., Marques-Silva, J.: MUSer2: An efficient MUS extractor. Journal of Satisfiability 8, 123–128 (2012)Google Scholar
  9. 9.
    Wieringa, S., Heljanko, K.: Asynchronous multi-core incremental SAT solving. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 139–153. Springer, Heidelberg (2013)Google Scholar
  10. 10.
    Goldberg, E.I., Novikov, Y.: Verification of proofs of unsatisfiability for CNF formulas. In: DATE, pp. 10886–10891 (2003)Google Scholar
  11. 11.
    Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.: Trimming while checking clausal proofs, In: [24], pp. 181–188Google Scholar
  12. 12.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. J. ACM 7(3), 201–215 (1960)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    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)CrossRefGoogle Scholar
  14. 14.
    Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)Google Scholar
  15. 15.
    Van Gelder, A.: Verifying RUP proofs of propositional unsatisfiability. In: ISAIM (2008)Google Scholar
  16. 16.
    Liffiton, M.H., Sakallah, K.A.: Algorithms for computing minimal unsatisfiable subsets of constraints. J. Autom. Reasoning 40(1), 1–33 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Nadel, A.: Boosting minimal unsatisfiable core extraction. In: FMCAD, pp. 121–128 (October 2010)Google Scholar
  18. 18.
    Kullmann, O., Lynce, I., Marques-Silva, J.: Categorisation of clauses in conjunctive normal forms: Minimally unsatisfiable sub-clause-sets and the lean kernel. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 22–35. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Marques-Silva, J., Lynce, I.: On improving MUS extraction algorithms. In: Sakallah, K.A., Simon, L. (eds.) SAT 2011. LNCS, vol. 6695, pp. 159–173. Springer, Heidelberg (2011)Google Scholar
  20. 20.
    Marques-Silva, J.: Minimal unsatisfiability: Models, algorithms and applications. In: ISMVL, pp. 9–14 (2010)Google Scholar
  21. 21.
    Eén, N., Sörensson, N.: Temporal induction by incremental SAT solving. Electr. Notes Theor. Comput. Sci. 89(4), 543–560 (2003)CrossRefGoogle Scholar
  22. 22.
    Audemard, G., Simon, L.: Predicting learnt clauses quality in modern SAT solvers. In: Boutilier, C. (ed.) IJCAI, pp. 399–404 (2009)Google Scholar
  23. 23.
    Jussila, T., Sinz, C., Biere, A.: Extended resolution proofs for symbolic SAT solving with quantification. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 54–60. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Formal Methods in Computer-Aided Design, FMCAD 2013, Portland, OR, USA, October 20-23. IEEE (2013)Google Scholar
  25. 25.
    Järvisalo, M., Van Gelder, A. (eds.): SAT 2013. LNCS, vol. 7962. Springer, Heidelberg (2013)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Anton Belov
    • 1
  • Marijn J. H. Heule
    • 2
  • Joao Marques-Silva
    • 1
    • 3
  1. 1.Complex and Adaptive Systems LaboratoryUniversity College DublinIreland
  2. 2.The University of Texas at AustinUSA
  3. 3.IST/INESC-IDTechnical University of LisbonPortugal

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