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Proof Complexity of Fragments of Long-Distance Q-Resolution

  • Tomáš Peitl
  • Friedrich SlivovskyEmail author
  • Stefan Szeider
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11628)

Abstract

Q-resolution is perhaps the most well-studied proof system for Quantified Boolean Formulas (QBFs). Its proof complexity is by now well understood, and several general proof size lower bound techniques have been developed. The situation is quite different for long-distance Q-resolution (LDQ-resolution). While lower bounds on LDQ-resolution proof size have been established for specific families of formulas, we lack semantically grounded lower bound techniques for LDQ-resolution.

In this work, we study restrictions of LDQ-resolution. We show that a specific lower bound technique based on bounded-depth strategy extraction does not work even for reductionless Q-resolution by presenting short proofs of the QParity formulas. Reductionless Q-resolution is a variant of LDQ-resolution that admits merging but no universal reduction. We also prove a lower bound on the proof size of the completion principle formulas in reductionless Q-resolution. This shows that two natural fragments of LDQ-resolution are incomparable: Q-resolution, which allows universal reductions but no merging, and reductionless Q-resolution, which allows merging but no universal reductions. Finally, we develop semantically grounded lower bound techniques for fragments of LDQ-resolution, specifically tree-like LDQ-resolution and regular reductionless Q-resolution.

References

  1. 1.
    Balabanov, V., Roland Jiang, J.-H.: Unified QBF certification and its applications. Formal Methods Syst. Des. 41(1), 45–65 (2012)CrossRefGoogle Scholar
  2. 2.
    Balabanov, V., Jiang, J.H.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, 25–30 January 2015, Austin, Texas, USA., pp. 3694–3701. AAAI Press (2015)Google 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, Cham (2014).  https://doi.org/10.1007/978-3-319-09284-3_12CrossRefzbMATHGoogle Scholar
  4. 4.
    Beyersdorff, O., Blinkhorn, J., Hinde, L.: Size, cost, and capacity: a semantic technique for hard random QBFs. Log. Methods Comput. Sci. 15(1), 13:1–13:39 (2019)Google Scholar
  5. 5.
    Beyersdorff, O., Blinkhorn, J., Mahajan, M.: Building strategies into QBF proofs. In: 36th International Symposium on Theoretical Aspects of Computer Science, STACS 2019, 13–16 March (2019), Berlin, Germany, pp. 14:1–14:18 (2019)Google Scholar
  6. 6.
    Beyersdorff, O., Chew, L., Janota, M.: Proof complexity of resolution-based QBF calculi. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science, STACS 2015, 4–7 March 2015, Garching, Germany, vol. 30 of LIPIcs, pp. 76–89. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2015)Google Scholar
  7. 7.
    Biere, A.: Resolve and expand. In: Proceedings of SAT 2004 Seventh International Conference on Theory and Applications of Satisfiability Testing, 10–13 May 2004, Vancouver, BC, Canada, pp. 59–70 (2004)Google Scholar
  8. 8.
    Biere, A.: Bounded model checking. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds) Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, pp. 457–481. IOS Press (2009)Google Scholar
  9. 9.
    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, EPiC Series in Computing, LPAR 2015, Suva, Fiji, 24–28 November 2015, vol. 35 pp. 28–41. EasyChair (2015)Google Scholar
  10. 10.
    Bollig, B., Wegener, I.: A very simple function that requires exponential size read-once branching programs. Inf. Process. Lett. 66(2), 53–57 (1998)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cadoli, M., Schaerf, M., Giovanardi, A., Giovanardi, M.: An algorithm to evaluate Quantified Boolean Formulae and its experimental evaluation. J. Automat. Reason. 28(2), 101–142 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chew, L.N.: QBF proof complexity. Ph.D. thesis, University of Leeds, UK (2017)Google Scholar
  13. 13.
    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
  14. 14.
    Giunchiglia, E., Narizzano, M., Tacchella, A.: Clause/term resolution and learning in the evaluation of quantified Boolean formulas. J. Artif. Intell. Res. 26, 371–416 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gomes, C.P., Kautz, H., Sabharwal, A., Selman, B.: Satisfiability solvers. In: Handbook of Knowledge Representation, volume 3 of Foundations of Artificial Intelligence, pp. 89–134. Elsevier (2008)Google Scholar
  16. 16.
    Håstad, J.: Computational Limitations of Small-depth Circuits. MIT Press, Cambridge, MA, USA (1987)Google Scholar
  17. 17.
    Janota, M., Klieber, W., Marques-Silva, J., Clarke, E.: Solving QBF with counterexample guided refinement. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 114–128. Springer, Heidelberg (2012).  https://doi.org/10.1007/978-3-642-31612-8_10CrossRefGoogle Scholar
  18. 18.
    Janota, M., Marques-Silva, J.: Solving QBF by clause selection. In: Yang, Q., Wooldridge, M. (eds.) Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, pp. 325–331. AAAI Press (2015)Google Scholar
  19. 19.
    Janota, M., Marques-Silva, J.: Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577, 25–42 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Janota, M.: On Q-resolution and CDCL QBF solving. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 402–418. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_25CrossRefzbMATHGoogle Scholar
  21. 21.
    Büning, H.K., Karpinski, M., Flögel, A.: Resolution for quantified Boolean formulas. Inf. Comput. 117(1), 12–18 (1995)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Klieber, W., Sapra, S., Gao, S., Clarke, E.: A non-prenex, non-clausal QBF solver with game-state learning. In: Strichman, O., Szeider, S. (eds.) SAT 2010. LNCS, vol. 6175, pp. 128–142. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-14186-7_12CrossRefGoogle Scholar
  23. 23.
    Lonsing, F., Egly, U., Van Gelder, A.: Efficient clause learning for quantified boolean formulas via QBF pseudo unit propagation. In: Järvisalo, M., Van Gelder, A. (eds.) SAT 2013. LNCS, vol. 7962, pp. 100–115. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39071-5_9CrossRefGoogle Scholar
  24. 24.
    Nechiporuk, I.: A Boolean function. Dokl. Akad. Nauk SSSR 169(4), 765–766 (1966)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Peitl, T., Slivovsky, F., Szeider, S.: Dependency learning for QBF. J. Artif. Intell. Res. 65, 181–208 (2019)CrossRefGoogle Scholar
  26. 26.
    Rabe, M.N., Tentrup, L.: CAQE: a certifying QBF solver. In: Kaivola, R., Wahl, R. (eds.) Formal Methods in Computer-Aided Design, FMCAD 2015, pp. 136–143. IEEE Computer Society (2015)Google Scholar
  27. 27.
    Rintanen, J.: Planning and SAT. In: Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, volume 185 of Frontiers in Artificial Intelligence and Applications, pp. 483–504. IOS Press (2009)Google Scholar
  28. 28.
    Ronald, L.: Rivest. Learning decision lists. Mach. Learn. 2(3), 229–246 (1987)Google Scholar
  29. 29.
    Stockmeyer, L.J., Meyer, A.R.: Word problems requiring exponential time. In: Proceedings of Theory of Computing, pp. 1–9. ACM (1973)Google Scholar
  30. 30.
    Tentrup, L.: Non-prenex QBF solving using abstraction. In: Creignou, N., Le Berre, D. (eds.) SAT 2016. LNCS, vol. 9710, pp. 393–401. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-40970-2_24CrossRefGoogle Scholar
  31. 31.
    Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Siekmann, J., Wrightson, G. (eds.) Automation of Reasoning. Classical Papers on Computer Science, pp. 466–483. Springer, Heidelberg (1983).  https://doi.org/10.1007/978-3-642-81955-1_28. Zap. Nauchn. Sem. Leningrad Otd. Mat. Inst. Akad. Nauk SSSR, 8:23–41, 1968. Russian. English translationCrossRefGoogle Scholar
  32. 32.
    van 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_47CrossRefzbMATHGoogle Scholar
  33. 33.
    Vizel, Y., Weissenbacher, G., Malik, S.: Boolean satisfiability solvers and their applications in model checking. Proc. IEEE 103(11), 2021–2035 (2015)CrossRefGoogle Scholar
  34. 34.
    Wegener, I.: Branching Programs and Binary Decision Diagrams. SIAM (2000)Google Scholar
  35. 35.
    Zhang, L., Malik, S.: Conflict driven learning in a quantified Boolean satisfiability solver. In: Pileggi, L.T., Kuehlmann, A. (eds.) Proceedings of the 2002 IEEE/ACM International Conference on Computer-aided Design, ICCAD 2002, San Jose, California, USA, 10–14 November 2002, pp. 442–449. ACM/IEEE Computer Society (2002)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Tomáš Peitl
    • 1
  • Friedrich Slivovsky
    • 1
    Email author
  • Stefan Szeider
    • 1
  1. 1.Algorithms and Complexity Group, TU WienViennaAustria

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