Feasible Interpolation for QBF Resolution Calculi

  • Olaf Beyersdorff
  • Leroy Chew
  • Meena Mahajan
  • Anil Shukla
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. We establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF calculi as well as largely extends the scope of classical feasible interpolation. We apply our technique to obtain new exponential lower bounds to all resolution-based QBF systems for a new class of QBF formulas based on the clique problem. Finally, we show how feasible interpolation relates to the recently established lower bound method based on strategy extraction [7].


Proof System Conjunctive Normal Form Strategy Extraction Winning Strategy Interpolation Theorem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Boppana, R.B.: The monotone circuit complexity of boolean functions. Combinatorica 7(1), 1–22 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Balabanov, V., Jiang, J.-H.R.: Unified QBF certification and its applications. Formal Methods in System Design 41(1), 45–65 (2012)zbMATHCrossRefGoogle Scholar
  3. 3.
    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, Heidelberg (2014) Google Scholar
  4. 4.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. Journal of the ACM 48(2), 149–169 (2001)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Benedetti, M., Mangassarian, H.: QBF-based formal verification: Experience and perspectives. JSAT 5(1–4), 133–191 (2008)MathSciNetGoogle Scholar
  6. 6.
    Beyersdorff, O., Chew, L., Janota, M.: On unification of QBF resolution-based calculi. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 81–93. Springer, Heidelberg (2014) Google Scholar
  7. 7.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In : STACS, pp. 76–89 (2015)Google Scholar
  8. 8.
    Beyersdorff, O., Chew, L., Sreenivasaiah, K.: A game characterisation of tree-like Q-resolution size. In: Dediu, A.-H., Formenti, E., Martín-Vide, C., Truthe, B. (eds.) LATA 2015. LNCS, vol. 8977, pp. 486–498. Springer, Heidelberg (2015) Google Scholar
  9. 9.
    Beyersdorff, O., Kullmann, O.: Unified characterisations of resolution hardness measures. In: Sinz, C., Egly, U. (eds.) SAT 2014. LNCS, vol. 8561, pp. 170–187. Springer, Heidelberg (2014) Google Scholar
  10. 10.
    Bonet, M.L., Domingo, C., Gavaldà, R., Maciel, A., Pitassi, T.: Non-automatizability of bounded-depth Frege proofs. Computational Complexity 13(1–2), 47–68 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Maria Luisa Bonet: Toniann Pitassi, and Ran Raz. On interpolation and automatization for Frege systems. SIAM Journal on Computing 29(6), 1939–1967 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Craig, W.: Three uses of the Herbrand-Gentzen theorem in relating model theory and proof theory. The Journal of Symbolic Logic 22(3), 269–285 (1957)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Egly, U., Kronegger, M., Lonsing, F., Pfandler, A.: Conformant planning as a case study of incremental QBF solving (2014). CoRR, abs/1405.7253Google Scholar
  14. 14.
    Goultiaeva, A., Van Gelder, A., Bacchus, F.: A uniform approach for generating proofs and strategies for both true and false QBF formulas. In: IJCAI, pp. 546–553 (2011)Google Scholar
  15. 15.
    Hrubeš, P.: On lengths of proofs in non-classical logics. Annals of Pure and Applied Logic 157(2–3), 194–205 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Hans Kleine Büning: Marek Karpinski, and Andreas Flögel. Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)zbMATHCrossRefGoogle Scholar
  18. 18.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. J. Symb. Log. 62(2), 457–486 (1997)zbMATHCrossRefGoogle Scholar
  19. 19.
    Krajíček, J.: Forcing with random variables and proof complexity, vol. 382, Lecture Note Series. London Mathematical Society (2011)Google Scholar
  20. 20.
    Krajíček, J., Pudlák, P.: Some consequences of cryptographical conjectures for \(S^1_2\) and \(EF\). Information and Computation 140(1), 82–94 (1998)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Mundici, D.: Tautologies with a unique Craig interpolant, uniform vs. nonuniform complexity. Annals of Pure and Applied Logic 27, 265–273 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  22. 22.
    Pudlák, P.: Lower bounds for resolution and cutting planes proofs and monotone computations. The Journal of Symbolic Logic 62(3), 981–998 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  23. 23.
    Pudlák, P.: Proofs as games. American Math. Monthly, pp. 541–550 (2000)Google Scholar
  24. 24.
    Rintanen, J.: Asymptotically optimal encodings of conformant planning in QBF. In: AAAI, pp. 1045–1050. AAAI Press (2007)Google Scholar
  25. 25.
    Van Gelder, A.: Contributions to the theory of practical quantified boolean formula solving. In: Milano, M. (ed.) CP 2012. LNCS, vol. 7514, pp. 647–663. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  26. 26.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: ICCAD, pp. 442–449 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Olaf Beyersdorff
    • 1
  • Leroy Chew
    • 1
  • Meena Mahajan
    • 2
  • Anil Shukla
    • 2
  1. 1.School of ComputingUniversity of LeeddsLeeddsUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations