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Abstract

Quantified Boolean formulas generalize propositional formulas by admitting quantifications over propositional variables. We compare proof systems with different quantifier handling paradigms for quantified Boolean formulas (QBFs) with respect to their ability to allow succinct proofs. We analyze cut-free sequent systems extended by different quantifier rules and show that some rules are better than some others.

Q-resolution is an elegant extension of propositional resolution to QBFs and is applicable to formulas in prenex conjunctive normal form. In Q-resolution, there is no explicit handling of quantifiers by specific rules. Instead the forall reduction rule which operates on single clauses inspects the global quantifier prefix. We show that there are classes of formulas for which there are short cut-free tree proofs in a sequent system, but any Q-resolution refutation of the negation of the formula is exponential.

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References

  1. 1.
    Balabanov, V., Jiang, J.-H.R.: Resolution Proofs and Skolem Functions in QBF Evaluation and Applications. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 149–164. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Buss, S.: An introduction to proof theory. In: Handbook of Proof Theory, pp. 1–78. Elsevier, Amsterdam (1998)CrossRefGoogle Scholar
  3. 3.
    Clote, P., Kranakis, E.: Boolean Functions and Models of Computation. Springer, Heidelberg (2002)Google Scholar
  4. 4.
    Cook, S.A., Morioka, T.: Quantified propositional calculus and a second-order theory for NC1. Arch. Math. Log. 44(6), 711–749 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Cook, S.A., Reckhow, R.A.: On the lengths of proofs in the propositional calculus (preliminary version). In: STOC, pp. 135–148 (1974)Google Scholar
  6. 6.
    Egly, U., Seidl, M., Woltran, S.: A solver for QBFs in negation normal form. Constraints 14(1), 38–79 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gentzen, G.: Untersuchungen über das logische Schließen. Mathematische Zeitschrift 39, 176–210, 405–431 (1935)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Haken, A.: The intractability of resolution. Theor. Comput. Sci. 39, 297–308 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Jussila, T., Biere, A., Sinz, C., Kroning, 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)CrossRefGoogle Scholar
  10. 10.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)zbMATHCrossRefGoogle Scholar
  11. 11.
    Krajíček, J.: Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and its Application, vol. 60. Cambridge University Press (1995)Google Scholar
  12. 12.
    Plaisted, D.A., Greenbaum, S.: A structure-preserving clause form translation. J. Symb. Comput. 2(3), 293–304 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Tseitin, G.S.: On the Complexity of Derivation in Propositional Calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, Part II. Seminars in Mathematics, vol. 8, pp. 234–259. Steklov Mathematical Institute, Leningrad (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Uwe Egly
    • 1
  1. 1.Institut für Informationssysteme 184/3Technische Universität WienViennaAustria

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