Extended Resolution Simulates DRAT

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


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.


Extension Rule Proof System Tseitin Formulas Clause Deletion Pigeonhole Formulas 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  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
    Email author
  • 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