A Game Characterisation of Tree-like Q-resolution Size

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8977)


We provide a characterisation for the size of proofs in tree-like Q-Resolution by a Prover-Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution [10]. This gives the first successful transfer of one of the lower bound techniques for classical proof systems to QBF proof systems. We confirm our technique with two previously known hard examples. In particular, we give a proof of the hardness of the formulas of Kleine Büning et al. [20] for tree-like Q-Resolution.


Conjunctive Normal Form Partial Assignment Universal Variable Conjunctive Normal Form Formula Empty Clause 
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  1. 1.
    Atserias, A., Dalmau, V.: A combinatorial characterization of resolution width. Journal of Computer and System Sciences 74(3), 323–334 (2008)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHGoogle 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) CrossRefGoogle Scholar
  4. 4.
    Ben-Sasson, E., Harsha, P.,: Lower bounds for bounded depth Frege proofs via Buss-Pudlák games. ACM Trans. on Computational Logic 11(3) (2010)Google Scholar
  5. 5.
    Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. Journal of the ACM 48(2), 149–169 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Benedetti, M., Mangassarian, H.: QBF-based formal verification: Experience and perspectives. JSAT 5(1–4), 133–191 (2008)MathSciNetGoogle Scholar
  7. 7.
    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) CrossRefGoogle Scholar
  8. 8.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. ECCC 21, 120 (2014)Google Scholar
  9. 9.
    Beyersdorff, O., Galesi, N., Lauria, M.: A lower bound for the pigeonhole principle in tree-like resolution by asymmetric prover-delayer games. Information Processing Letters 110(23), 1074–1077 (2010)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Beyersdorff, O., Galesi, N., Lauria, M.: A characterization of tree-like resolution size. Information Processing Letters 113(18), 666–671 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Beyersdorff, O., Galesi, N., Lauria, M.,: Parameterized complexity of DPLL search procedures. ACM Trans. on Computational Logic 14(3) (2013)Google Scholar
  12. 12.
    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) CrossRefGoogle Scholar
  13. 13.
    Buss, S.R.: Towards NP-P via proof complexity and search. Ann. Pure Appl. Logic 163(7), 906–917 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Cook, S.A., Nguyen, P.: Logical Foundations of Proof Complexity. Cambridge University Press (2010)Google Scholar
  15. 15.
    Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 100–113. Springer, Heidelberg (2012) CrossRefGoogle Scholar
  16. 16.
    Egly, U., Lonsing, F., Widl, M.: Long-distance resolution: Proof generation and strategy extraction in search-based QBF solving. In: McMillan, K., Middeldorp, A., Voronkov, A. (eds.) LPAR-19 2013. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  17. 17.
    Esteban, J.L., Torán, J.: A combinatorial characterization of treelike resolution space. Information Processing Letters 87(6), 295–300 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Janota, M., Marques-Silva, J.: \(\forall \)Exp+Res does not p-simulate Q-resolution. International Workshop on Quantified Boolean Formulas (2013)Google Scholar
  19. 19.
    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) CrossRefGoogle Scholar
  20. 20.
    Büning, H.K., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)CrossRefzbMATHGoogle Scholar
  21. 21.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  22. 22.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. J. Symb. Log. 62(2), 457–486 (1997)CrossRefzbMATHGoogle Scholar
  23. 23.
    Pudlák, P.: Proofs as games. American Math. Monthly, pp. 541–550 (2000)Google Scholar
  24. 24.
    Pudlák, P., Impagliazzo, R.: A lower bound for DLL algorithms for SAT. In: Proc. 11th Symposium on Discrete Algorithms, pp. 128–136 (2000)Google Scholar
  25. 25.
    Rintanen, J.: Asymptotically optimal encodings of conformant planning in QBF. In: AAAI, pp. 1045–1050. AAAI Press, (2007)Google Scholar
  26. 26.
    Segerlind, N.: The complexity of propositional proofs. Bulletin of Symbolic Logic 13(4), 417–481 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Van Gelder, A.: Contributions to the theory of practical quantified Boolean formula solving. In: CP, pp. 647–663 (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.The Institute of Mathematical SciencesChennaiIndia

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