Short Proofs Without New Variables

  • Marijn J. H. Heule
  • Benjamin Kiesl
  • Armin Biere
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)

Abstract

Adding and removing redundant clauses is at the core of state-of-the-art SAT solving. Crucial is the ability to add short clauses whose redundancy can be determined in polynomial time. We present a characterization of the strongest notion of clause redundancy (i.e., addition of the clause preserves satisfiability) in terms of an implication relationship. By using a polynomial-time decidable implication relation based on unit propagation, we thus obtain an efficiently checkable redundancy notion. A proof system based on this notion is surprisingly strong, even without the introduction of new variables—the key component of short proofs presented in the proof complexity literature. We demonstrate this strength on the famous pigeon hole formulas by providing short clausal proofs without new variables.

References

  1. 1.
    Clarke, E.M., Biere, A., Raimi, R., Zhu, Y.: Bounded model checking using satisfiability solving. Formal Meth. Syst. Des. 19(1), 7–34 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Ivančić, F., Yang, Z., Ganai, M.K., Gupta, A., Ashar, P.: Efficient SAT-based bounded model checking for software verification. Theor. Comput. Sci. 404(3), 256–274 (2008)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Konev, B., Lisitsa, A.: Computer-aided proof of Erdős discrepancy properties. Artif. Intell. 224(C), 103–118 (2015)CrossRefMATHGoogle Scholar
  4. 4.
    Heule, M.J.H., Kullmann, O., Marek, V.W.: Solving and verifying the boolean pythagorean triples problem via cube-and-conquer. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 228–245. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_15 Google Scholar
  5. 5.
    Wetzler, N.D., Heule, M.J.H., Hunt Jr., W.A.: DRAT-trim: efficient checking and trimming using expressive clausal proofs. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 422–429. Springer, Cham (2014). doi:10.1007/978-3-319-09284-3_31 Google Scholar
  6. 6.
    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). doi:10.1007/978-3-642-31365-3_28 CrossRefGoogle Scholar
  7. 7.
    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). doi:10.1007/978-3-319-40229-1_5 Google Scholar
  8. 8.
    Kleine Büning, H., Kullmann, O.: Minimal unsatisfiability and autarkies. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. IOS Press, pp. 339–401 (2009)Google Scholar
  9. 9.
    Weaver, S., Franco, J.V., Schlipf, J.S.: Extending existential quantification in conjunctions of BDDs. JSAT 1(2), 89–110 (2006)MATHGoogle Scholar
  10. 10.
    Andersson, G., Bjesse, P., Cook, B., Hanna, Z.: A proof engine approach to solving combinational design automation problems. In: Proceedings of the 39th Annual Design Automation Conference (DAC 2002). ACM, pp. 725–730 (2002)Google Scholar
  11. 11.
    Crawford, J., Ginsberg, M., Luks, E., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning (KR 1996). Morgan Kaufmann, pp. 148–159 (1996)Google Scholar
  12. 12.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Stud. Math. Math. Logic 2, 115–125 (1968)Google Scholar
  13. 13.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39, 297–308 (1985)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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). AAAI Press (2010)Google Scholar
  15. 15.
    Manthey, N., Heule, M.J.H., Biere, A.: Automated reencoding of boolean formulas. In: Biere, A., Nahir, A., Vos, T. (eds.) HVC 2012. LNCS, vol. 7857, pp. 102–117. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39611-3_14 CrossRefGoogle Scholar
  16. 16.
    Devriendt, J., Bogaerts, B., Bruynooghe, M., Denecker, M.: Improved static symmetry breaking for SAT. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 104–122. Springer, Cham (2016). doi:10.1007/978-3-319-40970-2_8 Google Scholar
  17. 17.
    Balyo, T., Heule, M.J.H., Järvisalo, M.: SAT competition 2016: recent developments. In: Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI 2017). AAAI Press (2017, to appear)Google Scholar
  18. 18.
    Goldberg, E.I., Novikov, Y.: Verification of proofs of unsatisfiability for CNF formulas. In: Proceedings of the Conference on Design, Automation and Test in Europe (DATE 2003). IEEE Computer Society, pp. 10886–10891 (2003)Google Scholar
  19. 19.
    Van Gelder, A.: Producing and verifying extremely large propositional refutations. Ann. Math. Artif. Intell. 65(4), 329–372 (2012)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kullmann, O.: On a generalization of extended resolution. Discrete Appl. Math. 96–97, 149–176 (1999)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Järvisalo, M., Biere, A., Heule, M.J.H.: Simulating circuit-level simplifications on CNF. J. Autom. Reasoning 49(4), 583–619 (2012)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)CrossRefGoogle Scholar
  23. 23.
    Heule, M.J.H., Hunt Jr., W.A., Wetzler, N.D.: Expressing symmetry breaking in DRAT proofs. In: Felty, A.P., Middeldorp, A. (eds.) CADE 2015. LNCS (LNAI), vol. 9195, pp. 591–606. Springer, Cham (2015). doi:10.1007/978-3-319-21401-6_40 CrossRefGoogle Scholar
  24. 24.
    Urquhart, A.: The complexity of propositional proofs. Bull. Symbolic Logic 1(4), 425–467 (1995)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Nordström, J.: A simplified way of proving trade-off results for resolution. Inf. Process. Lett. 109(18), 1030–1035 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

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

Personalised recommendations