Advertisement

Local Soundness for QBF Calculi

  • Martin Suda
  • Bernhard GleissEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10929)

Abstract

We develop new semantics for resolution-based calculi for Quantified Boolean Formulas, covering both the CDCL-derived calculi and the expansion-derived ones. The semantics is centred around the notion of a partial strategy for the universal player and allows us to show in a local, inference-by-inference manner that these calculi are sound. It also helps us understand some less intuitive concepts, such as the role of tautologies in long-distance resolution or the meaning of the “star” in the annotations of IRM-calc. Furthermore, we show that a clause of any of these calculi can be, in the spirit of Curry-Howard correspondence, interpreted as a specification of the corresponding partial strategy. The strategy is total, i.e. winning, when specified by the empty clause.

Notes

Acknowledgements

We thank Olaf Beyersdorff, Leroy Chew, Uwe Egly, Mikoláš Janota, Adrián Rebola-Pardo, and Martina Seidl for interesting comments and inspiring discussions on the semantics of QBF.

References

  1. 1.
    Balabanov, V., Jiang, J.R., Janota, M., Widl, M.: Efficient extraction of QBF (counter)models from long-distance resolution proofs. In: Bonet, B., Koenig, S. (eds.) Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, Austin, Texas, USA, 25–30 January 2015, pp. 3694–3701. AAAI Press (2015)Google 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).  https://doi.org/10.1007/978-3-319-09284-3_12CrossRefzbMATHGoogle Scholar
  3. 3.
    Beyersdorff, O., Blinkhorn, J.: Dependency schemes in QBF calculi: semantics and soundness. In: Rueher, M. (ed.) CP 2016. LNCS, vol. 9892, pp. 96–112. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-44953-1_7CrossRefGoogle Scholar
  4. 4.
    Beyersdorff, O., Bonacina, I., Chew, L.: Lower bounds: from circuits to QBF proof systems. In: Proceedings of the ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016), pp. 249–260. ACM (2016)Google Scholar
  5. 5.
    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).  https://doi.org/10.1007/978-3-662-44465-8_8CrossRefzbMATHGoogle Scholar
  6. 6.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Proceedings of the STACS. LIPIcs, vol. 30, pp. 76–89. Schloss Dagstuhl (2015)Google Scholar
  7. 7.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Feasible interpolation for QBF resolution calculi. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9134, pp. 180–192. Springer, Heidelberg (2015).  https://doi.org/10.1007/978-3-662-47672-7_15CrossRefGoogle Scholar
  8. 8.
    Beyersdorff, O., Chew, L., Mahajan, M., Shukla, A.: Are short proofs narrow? QBF resolution is not simple. In: Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS 2016) (2016)Google Scholar
  9. 9.
    Beyersdorff, O., Chew, L., Schmidt, R.A., Suda, M.: Lifting QBF resolution calculi to DQBF. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 490–499. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_30CrossRefGoogle Scholar
  10. 10.
    Bjørner, N., Janota, M., Klieber, W.: On conflicts and strategies in QBF. In: Fehnker, A., McIver, A., Sutcliffe, G., Voronkov, A. (eds.) 20th International Conferences on Logic for Programming, Artificial Intelligence and Reasoning - Short Presentations, LPAR 2015, Suva, Fiji, 24–28 November 2015. EPiC Series in Computing, vol. 35, pp. 28–41. EasyChair (2015). http://www.easychair.org/publications/paper/255082
  11. 11.
    Bloem, R., Braud-Santoni, N., Hadzic, V.: QBF solving by counterexample-guided expansion. CoRR abs/1611.01553 (2016). http://arxiv.org/abs/1611.01553
  12. 12.
    Cimatti, A., Sebastiani, R. (eds.): SAT 2012. LNCS, vol. 7317. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31612-8CrossRefzbMATHGoogle Scholar
  13. 13.
    Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti and Sebastiani [12], pp. 100–113CrossRefGoogle Scholar
  14. 14.
    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).  https://doi.org/10.1007/978-3-319-40970-2_26CrossRefGoogle Scholar
  15. 15.
    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 2013. LNCS, vol. 8312, pp. 291–308. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-45221-5_21CrossRefzbMATHGoogle Scholar
  16. 16.
    Goultiaeva, A., Gelder, A.V., Bacchus, F.: A uniform approach for generating proofs and strategies for both true and false QBF formulas. In: Walsh, T. (ed.) Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI 2011, Barcelona, Catalonia, Spain, 16–22 July 2011, pp. 546–553. IJCAI/AAAI (2011),  https://doi.org/10.5591/978-1-57735-516-8/IJCAI11-099
  17. 17.
    Heule, M.J., Seidl, M., Biere, A.: Efficient extraction of Skolem functions from QRAT proofs. In: Formal Methods in Computer-Aided Design (FMCAD), pp. 107–114. IEEE (2014)Google Scholar
  18. 18.
    Heule, M.J.H., Seidl, M., Biere, A.: A unified proof system for QBF preprocessing. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS (LNAI), vol. 8562, pp. 91–106. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08587-6_7CrossRefGoogle Scholar
  19. 19.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. In: Cimatti and Sebastiani [12], pp. 114–128CrossRefGoogle Scholar
  20. 20.
    Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. JSAT 7(2–3), 71–76 (2010)Google Scholar
  23. 23.
    Rabe, M.N., Tentrup, L.: CAQE: a certifying QBF solver. In: Kaivola, R., Wahl, T. (eds.) Formal Methods in Computer-Aided Design, FMCAD 2015, Austin, Texas, USA, 27–30 September 2015, pp. 136–143. IEEE (2015)Google Scholar
  24. 24.
    Samulowitz, H., Bacchus, F.: Binary clause reasoning in QBF. In: Biere, A., Gomes, C.P. (eds.) SAT 2006. LNCS, vol. 4121, pp. 353–367. Springer, Heidelberg (2006).  https://doi.org/10.1007/11814948_33CrossRefGoogle Scholar
  25. 25.
    Seidl, M., Lonsing, F., Biere, A.: qbf2epr: a tool for generating EPR formulas from QBF. In: Proceedings of the PAAR-2012. EPiC, vol. 21, pp. 139–148. EasyChair (2013)Google Scholar
  26. 26.
    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).  https://doi.org/10.1007/978-3-319-09284-3_21CrossRefzbMATHGoogle Scholar
  27. 27.
    Slivovsky, F., Szeider, S.: Soundness of Q-resolution with dependency schemes. Theor. Comput. Sci. 612, 83–101 (2016).  https://doi.org/10.1016/j.tcs.2015.10.020MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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).  https://doi.org/10.1007/978-3-642-33558-7_47CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.TU WienViennaAustria

Personalised recommendations