PRuning Through Satisfaction

  • Marijn J. H. Heule
  • Benjamin Kiesl
  • Martina Seidl
  • Armin Biere
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10629)

Abstract

The classical approach to solving the satisfiability problem of propositional logic prunes unsatisfiable branches from the search space. We prune more agressively by also removing certain branches for which there exist other branches that are more satisfiable. This is achieved by extending the popular conflict-driven clause learning (CDCL) paradigm with so-called \(\mathsf {PR}\)-clause learning. We implemented our new paradigm, named satisfaction-driven clause learning (SDCL), in the SAT solver Lingeling. Experiments on the well-known pigeon hole formulas show that our method can automatically produce proofs of unsatisfiability whose size is cubic in the number of pigeons while plain CDCL solvers can only produce proofs of exponential size.

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References

  1. 1.
    Audemard, G., Katsirelos, G., Simon, L.: A restriction of extended resolution for clause learning SAT solvers. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI 2010), pp. 15–20. AAAI Press (2010)Google Scholar
  2. 2.
    Biere, A.: Splatz, lingeling, plingeling, treengeling, YalSAT entering the SAT Competition 2016. In: Proceedings of SAT competition 2016 – Solver and Benchmark Descriptions. Dep. of Computer Science Series of Publications B, vol. B-2016-1, pp. 44–45. University of Helsinki (2016)Google Scholar
  3. 3.
    Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)CrossRefGoogle Scholar
  4. 4.
    Haken, A.: The intractability of resolution. Theoretical Computer Science 39, 297–308 (1985)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Heule, M.J.H., Biere, A.: Proofs for satisfiability problems. In: All about Proofs, Proofs for All (APPA), Math. Logic and Foundations, vol. 55. College Pub (2015)Google Scholar
  6. 6.
    Heule, M.J.H., Kiesl, B., Biere, A.: Short proofs without new variables. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 130–147. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63046-5_9 CrossRefGoogle Scholar
  7. 7.
    Järvisalo, M., Heule, M.J.H., Biere, A.: Inprocessing rules. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 355–370. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31365-3_28 CrossRefGoogle Scholar
  8. 8.
    Kiesl, B., Seidl, M., Tompits, H., Biere, A.: Super-blocked clauses. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS (LNAI), vol. 9706, pp. 45–61. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_5 Google Scholar
  9. 9.
    Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Handbook of Satisfiability, pp. 339–401. IOS Press (2009)Google Scholar
  10. 10.
    Kullmann, O.: On a generalization of extended resolution. Discrete Applied Mathematics 96–97, 149–176 (1999)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Marques Silva, J.P., Sakallah, K.A.: GRASP: A search algorithm for propositional satisfiability. IEEE Trans. Computers 48(5), 506–521 (1999)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Monien, B., Speckenmeyer, E.: Solving satisfiability in less than \(2^n\) steps. Discrete Applied Mathematics 10(3), 287–295 (1985)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proceedings of the 38th Design Automation Conference (DAC 2001), pp. 530–535. ACM (2001)Google Scholar
  14. 14.
    Nordström, J.: On the interplay between proof complexity and SAT solving. SIGLOG News 2(3), 19–44 (2015)Google Scholar
  15. 15.
    Sinz, C.: Towards an optimal CNF encoding of boolean cardinality constraints. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 827–831. Springer, Heidelberg (2005). https://doi.org/10.1007/11564751_73 CrossRefGoogle Scholar
  16. 16.
    Sinz, C., Biere, A.: Extended resolution proofs for conjoining BDDs. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 600–611. Springer, Heidelberg (2006). https://doi.org/10.1007/11753728_60 CrossRefGoogle Scholar
  17. 17.
    Sörensson, N., Biere, A.: Minimizing learned clauses. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 237–243. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02777-2_23 CrossRefGoogle Scholar
  18. 18.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Automation of Reasoning: 2: Classical Papers on Computational Logic 1967–1970, pp. 466–483. Springer, Heidelberg (1983)Google Scholar
  19. 19.
    Urquhart, A.: The complexity of propositional proofs. In: Current Trends in Theoretical Computer Science, pp. 332–342. World Scientific (2001)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Marijn J. H. Heule
    • 1
  • Benjamin Kiesl
    • 2
  • Martina Seidl
    • 3
  • Armin Biere
    • 3
  1. 1.Department of Computer ScienceThe University of Texas at AustinAustinUSA
  2. 2.Institute of Information SystemsTU WienViennaAustria
  3. 3.Institute for Formal Models and VerificationJKU LinzLinzAustria

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