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International Conference on Theory and Applications of Satisfiability Testing

SAT 2018: Theory and Applications of Satisfiability Testing – SAT 2018 pp 217–234Cite as

Local Soundness for QBF Calculi

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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.

This work was supported by ERC Starting Grant 2014 SYMCAR 639270 and the Austrian research projects FWF S11403-N23 and S11409-N23.

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Notes

  1. 1.

    For technical reasons, we allow branching on universal variables.

  2. 2.

    Note that \(l_1\) may not be unique, but its index is (because of consistent branching).

  3. 3.

    In the tree perspective, decomposition basically just says that every non-empty tree has a root node labeled by some variable v and a left and right sub-tree.

  4. 4.

    The last is an arbitrary choice.

  5. 5.

    The proof of Lemma 3 is omitted due to lack of space.

  6. 6.

    There is also a simpler way of describing the Resolution rule for IR-calc, which does not rely on \({\textsf {inst}}\). However, the presentation in Fig. 5 is equivalent to it.

  7. 7.

    We actually do not need the usually stated assumption \(\mathsf{dom}(\mu ) = \mathsf{dom}(\sigma )\).

References

  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. 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_12

    Chapter  MATH  Google Scholar 

  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_7

    Chapter  Google Scholar 

  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. 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_8

    Chapter  MATH  Google Scholar 

  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. 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_15

    Chapter  Google Scholar 

  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. 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_30

    Chapter  Google Scholar 

  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. 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. Cimatti, A., Sebastiani, R. (eds.): SAT 2012. LNCS, vol. 7317. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31612-8

    Book  MATH  Google Scholar 

  13. Egly, U.: On sequent systems and resolution for QBFs. In: Cimatti and Sebastiani [12], pp. 100–113

    Chapter  Google Scholar 

  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_26

    Chapter  Google Scholar 

  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_21

    Chapter  MATH  Google Scholar 

  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. 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. 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_7

    Chapter  Google Scholar 

  19. Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.M.: Solving QBF with counterexample guided refinement. In: Cimatti and Sebastiani [12], pp. 114–128

    Chapter  Google Scholar 

  20. Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)

    Article  MathSciNet  Google Scholar 

  21. Kleine Büning, H., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)

    Article  MathSciNet  Google Scholar 

  22. Lonsing, F., Biere, A.: DepQBF: a dependency-aware QBF solver. JSAT 7(2–3), 71–76 (2010)

    Google Scholar 

  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. 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_33

    Chapter  Google Scholar 

  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. 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_21

    Chapter  MATH  Google Scholar 

  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.2015.10.020

    Article  MathSciNet  MATH  Google Scholar 

  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_47

    Chapter  Google Scholar 

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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.

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Authors and Affiliations

  1. TU Wien, Vienna, Austria

    Martin Suda & Bernhard Gleiss

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  1. Martin Suda
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Correspondence to Bernhard Gleiss .

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  1. Friedrich Schiller University Jena, Jena, Germany

    Olaf Beyersdorff

  2. Microsoft Research Ltd., Cambridge, United Kingdom

    Christoph M. Wintersteiger

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Suda, M., Gleiss, B. (2018). Local Soundness for QBF Calculi. In: Beyersdorff, O., Wintersteiger, C. (eds) Theory and Applications of Satisfiability Testing – SAT 2018. SAT 2018. Lecture Notes in Computer Science(), vol 10929. Springer, Cham. https://doi.org/10.1007/978-3-319-94144-8_14

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  • DOI: https://doi.org/10.1007/978-3-319-94144-8_14

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