Advertisement

Extended Resolution Simulates DRAT

  • Benjamin Kiesl
  • Adrián Rebola-Pardo
  • Marijn J. H. Heule
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10900)

Abstract

We prove that extended resolution—a well-known proof system introduced by Tseitin—polynomially simulates \({\mathsf {DRAT}}\), the standard proof system in modern SAT solving. Our simulation procedure takes as input a \({\mathsf {DRAT}}\) proof and transforms it into an extended-resolution proof whose size is only polynomial with respect to the original proof. Based on our simulation, we implemented a tool that transforms \({\mathsf {DRAT}}\) proofs into extended-resolution proofs. We ran our tool on several benchmark formulas to estimate the increase in size caused by our simulation in practice. Finally, as a side note, we show how blocked-clause addition—a generalization of the extension rule from extended resolution—can be used to replace the addition of resolution asymmetric tautologies in \({\mathsf {DRAT}}\) without introducing new variables.

References

  1. 1.
    Baaz, M., Leitsch, A.: Methods of Cut-Elimination. Trends in Logic, vol. 3. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  2. 2.
    Biere, A.: Two pigeons per hole problem. In: Proceedings of SAT Competition 2013: Solver and Benchmark Descriptions, p. 103 (2013)Google Scholar
  3. 3.
    Chatalic, P., Simon, L.: Multi-resolution on compressed sets of clauses. In: Proceedings of the 12th IEEE International Conference on Tools with Artificial Intelligence (ICTAI 2000), pp. 2–10 (2000)Google Scholar
  4. 4.
    Cook, S.A.: A short proof of the pigeon hole principle using extended resolution. SIGACT News 8(4), 28–32 (1976)CrossRefGoogle Scholar
  5. 5.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Log. 44(1), 36–50 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39, 297–308 (1985)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Heule, M.J.H., Biere, A.: What a difference a variable makes. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 75–92. Springer, Cham (2018)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    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)CrossRefGoogle Scholar
  11. 11.
    Jussila, T., Sinz, C., Biere, A.: Extended resolution proofs for symbolic SAT solving with quantification. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 54–60. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Konev, B., Lisitsa, A.: Computer-aided proof of Erdős discrepancy properties. Artif. Intell. 224(C), 103–118 (2015)CrossRefGoogle Scholar
  13. 13.
    Kullmann, O.: On a generalization of extended resolution. Discret. Appl. Math. 96–97, 149–176 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lee, C.T.: A completeness theorem and a computer program for finding theorems derivable from given axioms. Ph.D. thesis (1967)Google Scholar
  15. 15.
    Philipp, T., Rebola-Pardo, A.: Towards a semantics of unsatisfiability proofs with inprocessing. In: Proceedings of the 21st International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR-21). EPiC Series in Computing, vol. 46, pp. 65–84. EasyChair (2017)Google 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)CrossRefGoogle Scholar
  17. 17.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. Stud. Math. Math. Log. 2, 115–125 (1968)Google Scholar
  18. 18.
    Urquhart, A.: The complexity of propositional proofs. Bull. Symb. Log. 1(4), 425–467 (1995)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Van Gelder, A.: Verifying RUP proofs of propositional unsatisfiability. In: Proceedings of the 10th International Symposium on Artificial Intelligence and Mathematics (ISAIM 2008) (2008)Google Scholar
  20. 20.
    Van Gelder, A.: Producing and verifying extremely large propositional refutations. Ann. Math. Artif. Intell. 65(4), 329–372 (2012)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wetzler, N., 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)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Benjamin Kiesl
    • 1
  • Adrián Rebola-Pardo
    • 1
  • Marijn J. H. Heule
    • 2
  1. 1.Institute of Logic and ComputationTU WienViennaAustria
  2. 2.Department of Computer ScienceThe University of TexasAustinUSA

Personalised recommendations