QRAT Polynomially Simulates \(\forall \text {-Exp+Res}\)

  • Benjamin KieslEmail author
  • Martina Seidl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11628)


The proof system \(\forall \text {-Exp+Res}\) formally captures expansion-based solving of quantified Boolean formulas (QBFs) whereas the \(\mathsf {QRAT}\) proof system captures QBF preprocessing. From previous work it is known that certain families of formulas have short proofs in \(\mathsf {QRAT}\) but not in \(\forall \text {-Exp+Res}\). However, it was not known if the two proof systems were incomparable (i.e., if there also existed QBFs with short \(\forall \text {-Exp+Res}\) proofs but without short \(\mathsf {QRAT}\) proofs), or if \(\mathsf {QRAT}\) polynomially simulates \(\forall \text {-Exp+Res}\). We close this gap of the QBF-proof-complexity landscape by presenting a polynomial simulation of \(\forall \text {-Exp+Res}\) in \(\mathsf {QRAT}\). Our simulation shows how definition introduction combined with extended-universal reduction can mimic the concept of universal expansion.


  1. 1.
    Balabanov, V., Jiang, J.R.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefGoogle Scholar
  2. 2.
    Balabanov, V., Widl, M., Jiang, J.-H.R.: QBF resolution systems and their proof complexities. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 154–169. Springer, Cham (2014). Scholar
  3. 3.
    Beyersdorff, O., Bonacina, I., Chew, L.: Lower bounds: from circuits to QBF proof systems. In: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016), pp. 249–260. ACM (2016)Google Scholar
  4. 4.
    Beyersdorff, O., Chew, L., Janota, M.: On unification of QBF resolution-based calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014. LNCS, vol. 8635, pp. 81–93. Springer, Heidelberg (2014). Scholar
  5. 5.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). LIPIcs, vol. 30, pp. 76–89. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2015)Google Scholar
  6. 6.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Are short proofs narrow? QBF resolution is not simple. In: Proceedings of the 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). LIPIcs, vol. 47, pp. 15:1–15:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  7. 7.
    Bloem, R., Braud-Santoni, N., Hadzic, V., Egly, U., Lonsing, F., Seidl, M.: Expansion-based QBF solving without recursion. In: Proceedings of the International Conference on Formal Methods in Computer Aided Design (FMCAD 2018), pp. 1–10. IEEE (2018)Google Scholar
  8. 8.
    Chen, H.: Proof complexity modulo the polynomial hierarchy: understanding alternation as a source of hardness. In: Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). LIPIcs, vol. 55, pp. 94:1–94:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2016)Google Scholar
  9. 9.
    Cook, S.A., Reckhow, R.A.: The relative efficiency of propositional proof systems. J. Symb. Logic 44(1), 36–50 (1979)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Egly, U.: On stronger calculi for QBFs. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 419–434. Springer, Cham (2016). Scholar
  11. 11.
    Gelder, A.: Variable independence and resolution paths for quantified boolean formulas. In: Lee, J. (ed.) CP 2011. LNCS, vol. 6876, pp. 789–803. Springer, Heidelberg (2011). Scholar
  12. 12.
    Heule, M.J.H., Seidl, M., Biere, A.: Solution validation and extraction for QBF preprocessing. J. Autom. Reasoning 58(1), 1–29 (2016)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Janota, M.: On Q-resolution and CDCL QBF solving. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 402–418. Springer, Cham (2016). Scholar
  14. 14.
    Janota, M., Grigore, R., Marques-Silva, J.: On QBF proofs and preprocessing. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR 2013. LNCS, vol. 8312, pp. 473–489. Springer, Heidelberg (2013). Scholar
  15. 15.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. Artif. Intell. 234, 1–25 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Janota, M., Marques-Silva, J.: On propositional QBF expansions and Q-resolution. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 67–82. Springer, Heidelberg (2013). Scholar
  17. 17.
    Jussila, T., Biere, A., Sinz, C., Kröning, D., Wintersteiger, C.M.: A first step towards a unified proof checker for QBF. In: Marques-Silva, J., Sakallah, K.A. (eds.) SAT 2007. LNCS, vol. 4501, pp. 201–214. Springer, Heidelberg (2007). Scholar
  18. 18.
    Kiesl, B., Heule, M.J.H., Seidl, M.: A little blocked literal goes a long way. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 281–297. Springer, Cham (2017). Scholar
  19. 19.
    Kiesl, B., Rebola-Pardo, A., Heule, M.J.H.: Extended resolution simulates DRAT. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS (LNAI), vol. 10900, pp. 516–531. Springer, Cham (2018). Scholar
  20. 20.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Lonsing, F., Egly, U.: DepQBF 6.0: a search-based QBF solver beyond traditional QCDCL. In: de Moura, L. (ed.) CADE 2017. LNCS (LNAI), vol. 10395, pp. 371–384. Springer, Cham (2017). Scholar
  22. 22.
    Peitl, T., Slivovsky, F., Szeider, S.: Dependency learning for QBF. In: Gaspers, S., Walsh, T. (eds.) SAT 2017. LNCS, vol. 10491, pp. 298–313. Springer, Cham (2017). Scholar
  23. 23.
    Slivovsky, F., Szeider, S.: Variable dependencies and Q-resolution. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 269–284. Springer, Cham (2014). Scholar
  24. 24.
    Slivovsky, F., Szeider, S.: Soundness of Q-resolution with dependency schemes. Theor. Comput. Sci. 612, 83–101 (2016)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Tentrup, L.: On expansion and resolution in CEGAR based QBF solving. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10427, pp. 475–494. Springer, Cham (2017). Scholar
  26. 26.
    Gelder, A.: Contributions to the theory of practical quantified boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, pp. 647–663. Springer, Heidelberg (2012). Scholar
  27. 27.
    Wetzler, N., Heule, M.J.H., Hunt, 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). Scholar
  28. 28.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified boolean satisfiability solver. In: Proceedings of the 2002 IEEE/ACM International Conference on Computer-aided Design (ICCAD 2002), pp. 442–449. ACM/IEEE Computer Society (2002)Google Scholar

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Authors and Affiliations

  1. 1.Institute of Logic and ComputationTU WienViennaAustria
  2. 2.CISPA Helmholtz Center for Information SecuritySaarbrückenGermany
  3. 3.Institute for Formal Models and VerificationJKU LinzLinzAustria

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