Boundary Points and Resolution

  • Eugene Goldberg
  • Panagiotis Manolios


We use the notion of boundary points to study resolution proofs. Given a CNF formula F, an l(x)-boundary point is a complete assignment falsifying only clauses of F having the same literal l(x) of variable x. An l(x)-boundary point p mandates a resolution on variable x. Adding the resolvent of this resolution to F eliminates p as an l(x)-boundary point. Any resolution proof has to eventually eliminate all boundary points of F. Hence one can study resolution proofs from the viewpoint of boundary point elimination. We use equivalence checking formulas to compare proofs of their unsatisfiability built by a conflict-driven SAT-solver and very short proofs tailored to these formulas. We show experimentally that in contrast to proofs generated by this SAT-solver, almost every resolution of a specialized proof eliminates a boundary point. This implies that one may use the share of resolutions eliminating boundary points as a metric of proof quality. We argue that obtaining proofs with a high value of this metric requires taking into account the formula structure. We show that for any unsatisfiable CNF formula there always exists a proof consisting only of resolutions eliminating cut boundary points (which are a relaxation of the notion of boundary points). This result enables building resolution SAT-solvers that are driven by elimination of cut boundary points.

This chapter is an extended version of the conference paper [9].


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alekhnovich, M., Razborov, A.: Resolution is not automatizable unless w[p] is tractable. SIAM Journal on Computing 38(4), 1347–1363 (2008)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: A. Robinson, A. Voronkov (eds.) Handbook of Automated Reasoning, vol. I,  chap. 2, pp. 19–99. North-Holland Elsevier Science (2001)
  3. 3.
    Biere, A.: Picosat essentials. Journal of Substance Abuse Treatment 4(2–4), 75–97 (2008)MATHGoogle Scholar
  4. 4.
    Bonet, L., Pitassi, T., Raz, R.: On interpolation and automatization for Frege systems. SIAM Journal of Computing 29(6), 1939–1967 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Babic, D., Bingham, J., Hu, A.: Efficient sat solving: Beyond supercubes. In: Design Automation Conference, pp. 744–749. Anaheim, California, USA (2005)Google Scholar
  6. 6.
    Eén, N., Sörensson, N.: An extensible sat-solver. In: Proceedings of SAT, pp. 502–518. Santa Margherita Ligure, Italy (2003)Google Scholar
  7. 7.
    Goldberg, E.: Proving unsatisfiability of CNFs locally. Journal of Automated Reasoning 28(4), 417–434 (2002)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Goldberg, E.: A decision-making procedure for resolution-based sat-solvers. In: Proceedings of SAT-08, pp. 119–132. Guangzhou, China (2008)Google Scholar
  9. 9.
    Goldberg, E.: Boundary points and resolution. In: Proceedings of SAT-09, pp. 147–160. Swansea, Wales, United Kingdom (2009)Google Scholar
  10. 10.
    Goldberg, E., Novikov, Y.: Berkmin: A fast and robust sat-solver. Discrete Applied Mathematics 155(12), 1549–1561 (2007). Doi:
  11. 11.
    Goldberg, E., Prasad, M., Brayton, R.: Using problem symmetry in search based satisfiability algorithms. In: Proceedings of DATE ’02, pp. 134–141. Paris, France (2002)Google Scholar
  12. 12.
    Marques-Silva, J., Sakallah, K.: Grasp–-A new search algorithm for satisfiability. In: Proceedings of ICCAD-96, pp. 220–227. Washington, DC (1996)Google Scholar
  13. 13.
    Moskewicz, M., Madigan, C., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient sat solver. In: Design Automation Conference-01, pp. 530–535. New York, NY (2001). Doi:
  14. 14.
    Nadel, A.: Backtrack search algorithms for propositional logic satisfiability: Review and innovations. Master’s thesis, The Hebrew University (2002)Google Scholar
  15. 15.
    Zhang, H.: Sato: An efficient propositional prover. In: Proceedings of CADE-97, pp. 272–275. Springer, London (1997)Google Scholar
  16. 16.
    Ryan, L.: The siege sat-solver.∼cl/software/siege (2010)
  17. 17.

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Northeastern UniversityBastonUSA

Personalised recommendations