Efficient CNF Simplification Based on Binary Implication Graphs

  • Marijn J. H. Heule
  • Matti Järvisalo
  • Armin Biere
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6695)


This paper develops techniques for efficiently detecting redundancies in CNF formulas. We introduce the concept of hidden literals, resulting in the novel technique of hidden literal elimination. We develop a practical simplification algorithm that enables “Unhiding” various redundancies in a unified framework. Based on time stamping literals in the binary implication graph, the algorithm applies various binary clause based simplifications, including techniques that, when run repeatedly until fixpoint, can be too costly. Unhiding can also be applied during search, taking learnt clauses into account. We show that Unhiding gives performance improvements on real-world SAT competition benchmarks.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bacchus, F.: Enhancing Davis Putnam with extended binary clause reasoning. In: Proc. AAAI, pp. 613–619. AAAI Press, Menlo Park (2002)Google Scholar
  2. 2.
    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
  3. 3.
    Gershman, R., Strichman, O.: Cost-effective hyper-resolution for preprocessing CNF formulas. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 423–429. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Han, H., Somenzi, F.: Alembic: An efficient algorithm for CNF preprocessing. In: Proc. DAC, pp. 582–587. IEEE, Los Alamitos (2007)Google Scholar
  5. 5.
    Järvisalo, M., Biere, A., Heule, M.J.H.: Blocked clause elimination. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 129–144. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  6. 6.
    Heule, M.J.H., Järvisalo, M., Biere, A.: Clause elimination procedures for CNF formulas. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 357–371. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  7. 7.
    Marques-Silva, J.P.: Algebraic simplification techniques for propositional satisfiability. In: Dechter, R. (ed.) CP 2000. LNCS, vol. 1894, pp. 537–542. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Van Gelder, A.: Toward leaner binary-clause reasoning in a satisfiability solver. Annals of Mathematics and Artificial Intelligence 43(1), 239–253 (2005)CrossRefzbMATHGoogle Scholar
  9. 9.
    Li, C.M.: Integrating equivalency reasoning into Davis-Putnam procedure. In: Proc. AAAI, pp. 291–296 (2000)Google Scholar
  10. 10.
    Brafman, R.: A simplifier for propositional formulas with many binary clauses. IEEE Transactions on Systems, Man, and Cybernetics, Part B 34(1), 52–59 (2004)CrossRefGoogle Scholar
  11. 11.
    Aho, A., Garey, M., Ullman, J.: The transitive reduction of a directed graph. SIAM Journal on Computing 1(2), 131–137 (1972)CrossRefzbMATHGoogle Scholar
  12. 12.
    Biere, A.: Lingeling, Plingeling, PicoSAT and PrecoSAT at SAT Race 2010. FMV Report Series Technical Report 10/1, Johannes Kepler University, Linz, Austria (2010)Google Scholar
  13. 13.
    del Val, Á.: Simplifying binary propositional theories into connected components twice as fast. In: Nieuwenhuis, R., Voronkov, A. (eds.) LPAR 2001. LNCS (LNAI), vol. 2250, pp. 392–406. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  14. 14.
    Soos, M.: Cryptominisat 2.5.0, sat race 2010 solver description (2010)Google Scholar
  15. 15.
    Korovin, K.: iProver – an instantiation-based theorem prover for first-order logic (System description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 292–298. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Groote, J.F., Warners, J.P.: The propositional formula checker HeerHugo. J. Autom. Reasoning 24(1/2), 101–125 (2000)CrossRefzbMATHGoogle Scholar
  17. 17.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Computing 1(2) (1972)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Marijn J. H. Heule
    • 1
  • Matti Järvisalo
    • 2
  • Armin Biere
    • 3
  1. 1.Department of Software TechnologyDelft University of TechnologyThe Netherlands
  2. 2.Department of Computer ScienceUniversity of HelsinkiFinland
  3. 3.Institute for Formal Models and VerificationJohannes Kepler University LinzAustria

Personalised recommendations