When Boolean Satisfiability Meets Gaussian Elimination in a Simplex Way

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7358)


Recent research on Boolean satisfiability (SAT) reveals modern solvers’ inability to handle formulae in the abundance of parity (xor) constraints. Although xor-handling in SAT solving has attracted much attention, challenges remain to completely deduce xor-inferred implications and conflicts, to effectively reduce expensive overhead, and to directly generate compact interpolants. This paper integrates SAT solving tightly with Gaussian elimination in the style of Dantzig’s simplex method. It yields a powerful tool overcoming these challenges. Experiments show promising performance improvements and efficient derivation of compact interpolants, which are otherwise unobtainable.


Gaussian Elimination Conjunctive Normal Form Conjunctive Normal Form Formula Nonbasic Variable Resolution Refutation 
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.


  1. 1.
  2. 2.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. IOS Press (2009)Google Scholar
  3. 3.
    Berkeley Logic Synthesis and Verification Group. ABC: A System for Sequential Synthesis and Verification,
  4. 4.
    Baumgartner, P., Massacci, F.: The taming of the (X)OR. In: Proc. Int’l Conf. on Computational Logic, pp. 508–522 (2000)Google Scholar
  5. 5.
    Cimatti, A., Griggio, A., Sebastiani, R.: Efficient generation of Craig interpolants in satisfiability modulo theories. ACM Trans. on Computational Logic 12(1), 7 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chen, J.-C.: XORSAT: An efficient algorithm for the DIMACS 32-bit parity problem, CoRR abs/cs/0703006 (2007)Google Scholar
  7. 7.
    Chen, J.-C.: Building a Hybrid SAT Solver via Conflict-Driven, Look-Ahead and XOR Reasoning Techniques. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 298–311. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  8. 8.
    Craig, W.: Linear reasoning: A new form of the Herbrand-Gentzen theorem. J. Symbolic Logic 22(3), 250–268 (1957)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Dantzig, G.: Maximization of linear function of variables subject to linear inequalities. Activity Analysis of Production and Allocation, 339–347 (1951)Google Scholar
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Gomes, C., Sabharwal, A., Selman, B.: Model counting: A new strategy for obtaining good bounds. In: Proc. National Conf. on Artificial Intelligence (AAAI), pp. 54–61 (2006)Google Scholar
  12. 12.
    Haanpaa, H., Jarvisalo, M., Kaski, P., Niemela, I.: Hard satisfiable clause sets for benchmarking equivalence reasoning techniques. Journal on Satisfiability, Boolean Modeling and Computation 2(1-4), 27–46 (2006)Google Scholar
  13. 13.
    Jiang, J.-H.R., Lee, C.-C., Mishchenko, A., Huang, C.-Y.: To SAT or not to SAT: Scalable exploration of functional dependency. IEEE Trans. on Computers 59(4), 457–467 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Liu, H.-Y., Chou, Y.-C., Lin, C.-H., Jiang, J.-H.R.: Towards Completely Automatic Decoder Synthesis. In: Proc. Int’l Conf. on Computer Aided Design (ICCAD), pp. 389–395 (2011)Google Scholar
  15. 15.
    Li, C.M.: Integrating equivalency reasoning into Davis-Putnam procedure. In: Proc. National Conf. on Artificial Intelligence (AAAI), pp. 291–296 (2000)Google Scholar
  16. 16.
    Laitinen, T., Junttila, T., Niemela, I.: Extending clause learning DPLL with parity reasoning. In: Proc. European Conference on Artificial Intelligence (ECAI), pp. 21–26 (2010)Google Scholar
  17. 17.
    Laitinen, T., Junttila, T., Niemela, I.: Equivalence class based parity reasoning with DPLL(XOR). In: Proc. Int’l Conf. on Tools with Artificial Intelligence (ICTAI), pp. 649–658 (2011)Google Scholar
  18. 18.
    Mishchenko, A., Chatterjee, S., Brayton, R., Een, N.: Improvements to combinational equivalence checking. In: Proc. Int’l Conf. on Computer-Aided Design (ICCAD), pp. 836–843 (2006)Google Scholar
  19. 19.
    McMillan, K.L.: Interpolation and SAT-Based Model Checking. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 1–13. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    McMillan, K.L.: An interpolating theorem prover. Theoretical Computer Science 345(1), 101–121 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Massacci, F., Marraro, L.: Logical cryptanalysis as a SAT-problem: Encoding and analysis. Journal of Automated Reasoning 24(1-2), 165–203 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Moskewicz, M., Madigan, C., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proc. Design Automation Conf. (DAC), pp. 530–535 (2001)Google Scholar
  23. 23.
    Marques-Silva, J., Sakallah, K.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. on Computeres 48(5), 506–521 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: From an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). Journal of the ACM 53(6), 937–977 (2006)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Soos, M., Nohl, K., Castelluccia, C.: Extending SAT Solvers to Cryptographic Problems. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 244–257. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  26. 26.
    Soos, M.: Enhanced Gaussian elimination in DPLL-based SAT solvers. In: Proc. Pragmatics of SAT (2010)Google Scholar
  27. 27.
    Warners, J., van Maaren, H.: A two-phase algorithm for solving a class of hard satisfiability problems. Operations Research Letters 23(3-5), 81–88 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Yorsh, G., Musuvathi, M.: A Combination Method for Generating Interpolants. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 353–368. Springer, Heidelberg (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Electrical Engineering / Graduate Institute of Electronics EngineeringNational Taiwan UniversityTaipeiTaiwan

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