skip to main content
research-article

New Resolution-Based QBF Calculi and Their Proof Complexity

Authors Info & Claims
Published:12 September 2019Publication History
Skip Abstract Section

Abstract

Modern QBF solvers typically use two different paradigms, conflict-driven clause learning (CDCL) solving or expansion solving. Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of these solvers, with Q-Resolution and its extensions relating to CDCL solving and ∀Exp+Res relating to expansion solving. This article defines two novel calculi, which are resolution-based and enable unification of some of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res.

However, the proof complexity of the QBF resolution proof systems is currently not well understood. In this article, we completely determine the relative power of the main QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi: proof systems for CDCL-based solvers (Q-resolution, universal, and long-distance Q-resolution) and proof systems for expansion-based solvers (∀Exp+Res and its generalizations IR-calc and IRM-calc defined here).

The most challenging part of this comparison is to exhibit hard formulas that underlie the exponential separations of the aforementioned proof systems. To this end, we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths-of-proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems such as long-distance Q-resolution and extensions.

With a completely different and novel counting argument, we show the hardness of the prominent formulas of Kleine Büning et al. [51] for the strong expansion-based calculus IR-calc.

References

  1. Sanjeev Arora and Boaz Barak. 2009. Computational Complexity -- A Modern Approach. Cambridge University Press. I--XXIV, 1--579 pages. Google ScholarGoogle ScholarDigital LibraryDigital Library
  2. Albert Atserias and Sergi Oliva. 2014. Bounded-width QBF is PSPACE-complete. J. Comput. Syst. Sci. 80, 7 (2014), 1415--1429.Google ScholarGoogle ScholarCross RefCross Ref
  3. Valeriy Balabanov and Jie-Hong R. Jiang. 2012. Unified QBF certification and its applications. Formal Methods in System Design 41, 1 (2012), 45--65. Google ScholarGoogle ScholarDigital LibraryDigital Library
  4. Valeriy Balabanov, Jie-Hong Roland Jiang, Mikoláš Janota, and Magdalena Widl. 2015. Efficient extraction of QBF (Counter)models from long-distance resolution proofs. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence. 3694--3701. Google ScholarGoogle ScholarDigital LibraryDigital Library
  5. Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. 2014. QBF Resolution systems and their proof complexities. In SAT. 154--169.Google ScholarGoogle Scholar
  6. Eli Ben-Sasson and Avi Wigderson. 2001. Short proofs are narrow - resolution made simple. J. ACM 48, 2 (2001), 149--169. Google ScholarGoogle ScholarDigital LibraryDigital Library
  7. Marco Benedetti. 2004. Evaluating QBFs via symbolic Skolemization. In LPAR, Franz Baader and Andrei Voronkov (Eds.), Vol. 3452. Springer, 285--300.Google ScholarGoogle Scholar
  8. Marco Benedetti and Hratch Mangassarian. 2008. QBF-based formal verification: Experience and perspectives. JSAT 5, 1--4 (2008), 133--191.Google ScholarGoogle Scholar
  9. Olaf Beyersdorff and Joshua Blinkhorn. 2016. Dependency schemes in QBF calculi: Semantics and soundness. In Principles and Practice of Constraint Programming - CP. 96--112.Google ScholarGoogle Scholar
  10. Olaf Beyersdorff, Ilario Bonacina, and Leroy Chew. 2016. Lower bounds: From circuits to QBF proof systems. In Proc. ACM Conference on Innovations in Theoretical Computer Science (ITCS). ACM, 249--260. Google ScholarGoogle ScholarDigital LibraryDigital Library
  11. Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2014. On unification of QBF resolution-based calculi. In International Symposium on Mathematical Foundations of Computer Science (MFCS). Springer, 81--93.Google ScholarGoogle ScholarCross RefCross Ref
  12. Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2015. Proof complexity of resolution-based QBF calculi. In Proc. Symposium on Theoretical Aspects of Computer Science (STACS). LIPIcs series, 76--89.Google ScholarGoogle Scholar
  13. Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2016. Extension variables in QBF resolution. In Beyond NP, Papers from the 2016 AAAI Workshop.Google ScholarGoogle Scholar
  14. Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. 2017. Feasible interpolation for QBF resolution calculi. Logical Methods in Computer Science 13 (2017). Issue 2.Google ScholarGoogle Scholar
  15. Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. 2018. Are short proofs narrow? QBF resolution is not so simple. ACM Transactions on Computational Logic 19 (2018). Issue 1. Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Olaf Beyersdorff, Leroy Chew, Renate A. Schmidt, and Martin Suda. 2016. Lifting QBF resolution calculi to DQBF. In SAT.Google ScholarGoogle Scholar
  17. Olaf Beyersdorff, Leroy Chew, and Karteek Sreenivasaiah. 2019. A game characterisation of tree-like Q-Resolution size. J. Comput. System Sci. 104 (2019), 82--101.Google ScholarGoogle ScholarCross RefCross Ref
  18. Olaf Beyersdorff, Nicola Galesi, and Massimo Lauria. 2010. A lower bound for the pigeonhole principle in tree-like resolution by asymmetric prover-delayer games. Inform. Process. Lett. 110, 23 (2010), 1074--1077. Google ScholarGoogle ScholarDigital LibraryDigital Library
  19. Olaf Beyersdorff, Nicola Galesi, and Massimo Lauria. 2013. A characterization of tree-like resolution size. Inform. Process. Lett. 113, 18 (2013), 666--671. Google ScholarGoogle ScholarDigital LibraryDigital Library
  20. Olaf Beyersdorff, Nicola Galesi, and Massimo Lauria. 2013. Parameterized complexity of DPLL search procedures. ACM Transactions on Computational Logic 14, 3 (2013). Google ScholarGoogle ScholarDigital LibraryDigital Library
  21. Olaf Beyersdorff and Ján Pich. 2016. Understanding Gentzen and Frege systems for QBF. In Proc. ACM/IEEE Symposium on Logic in Computer Science (LICS). Google ScholarGoogle ScholarDigital LibraryDigital Library
  22. Armin Biere. 2004. Resolve and expand. In SAT. 238--246.Google ScholarGoogle Scholar
  23. Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh (Eds.). 2009. Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, Vol. 185. IOS Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  24. Armin Biere, Florian Lonsing, and Martina Seidl. 2011. Blocked clause elimination for QBF. In International Conference on Automated Deduction CADE-23. 101--115. Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Maria Luisa Bonet, Juan Luis Esteban, Nicola Galesi, and Jan Johannsen. 2000. On the relative complexity of resolution refinements and cutting planes proof systems. SIAM J. Comput. 30, 5 (2000), 1462--1484. Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Uwe Bubeck and Hans Kleine Büning. 2007. Bounded universal expansion for preprocessing QBF. In Theory and Applications of Satisfiability Testing - SAT. 244--257. Google ScholarGoogle ScholarDigital LibraryDigital Library
  27. Samuel R. Buss. 2012. Towards NP-P via proof complexity and search. Ann. Pure Appl. Logic 163, 7 (2012), 906--917.Google ScholarGoogle ScholarCross RefCross Ref
  28. Hubie Chen. 2017. Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness. TOCT 9, 3 (2017), 15:1--15:20. Google ScholarGoogle ScholarDigital LibraryDigital Library
  29. Leroy Chew. 2017. QBF Proof Complexity. Ph.D. Dissertation. University of Leeds.Google ScholarGoogle Scholar
  30. Judith Clymo and Olaf Beyersdorff. 2018. Relating size and width in variants of Q-resolution. Inf. Process. Lett. 138 (2018), 1--6.Google ScholarGoogle ScholarCross RefCross Ref
  31. Stephen A. Cook and Phuong Nguyen. 2010. Logical Foundations of Proof Complexity. Cambridge University Press. Google ScholarGoogle ScholarDigital LibraryDigital Library
  32. Stephen A. Cook and Robert A. Reckhow. 1979. The relative efficiency of propositional proof systems. The Journal of Symbolic Logic 44, 1 (1979), 36--50.Google ScholarGoogle Scholar
  33. Uwe Egly. 2016. On stronger calculi for QBFs. In SAT.Google ScholarGoogle Scholar
  34. Uwe Egly, Martin Kronegger, Florian Lonsing, and Andreas Pfandler. 2017. Conformant planning as a case study of incremental QBF solving. Ann. Math. Artif. Intell. 80, 1 (2017), 21--45. Google ScholarGoogle ScholarDigital LibraryDigital Library
  35. Uwe Egly, Florian Lonsing, and Magdalena Widl. 2013. Long-distance resolution: Proof generation and strategy extraction in search-based QBF solving, See McMillan et al. {57}, 291--308.Google ScholarGoogle Scholar
  36. Merrick L. Furst, James B. Saxe, and Michael Sipser. 1984. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17, 1 (1984), 13--27.Google ScholarGoogle ScholarCross RefCross Ref
  37. Enrico Giunchiglia, Paolo Marin, and Massimo Narizzano. 2009. Reasoning with quantified Boolean formulas. See Biere et al. {23}, 761--780.Google ScholarGoogle Scholar
  38. Enrico Giunchiglia, Paolo Marin, and Massimo Narizzano. 2010. sQueezeBF: An effective preprocessor for QBFs based on equivalence reasoning. In SAT, Ofer Strichman and Stefan Szeider (Eds.), Vol. 6175. Springer, 85--98. Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. E. Giunchiglia, M. Narizzano, and A. Tacchella. 2006. Clause/term resolution and learning in the evaluation of quantified Boolean formulas. Journal of Artificial Intelligence Research 26, 1 (2006), 371--416. Google ScholarGoogle ScholarDigital LibraryDigital Library
  40. Alexandra Goultiaeva and Fahiem Bacchus. 2013. Recovering and utilizing partial duality in QBF. In Theory and Applications of Satisfiability Testing - SAT, M. Järvisalo and A. Van Gelder (Eds.). Springer, 83--99. Google ScholarGoogle ScholarDigital LibraryDigital Library
  41. Alexandra Goultiaeva, Martina Seidl, and Armin Biere. 2013. Bridging the gap between dual propagation and CNF-based QBF solving. In Design, Automation and Test in Europe, DATE. 811--814. Google ScholarGoogle ScholarDigital LibraryDigital Library
  42. Alexandra Goultiaeva, Allen Van Gelder, and Fahiem Bacchus. 2011. A uniform approach for generating proofs and strategies for both true and false QBF formulas. In IJCAI. 546--553. Google ScholarGoogle ScholarDigital LibraryDigital Library
  43. Johan Håstad. 1987. Computational Limitations of Small-depth Circuits. MIT Press, Cambridge, MA. Google ScholarGoogle ScholarDigital LibraryDigital Library
  44. Marijn J. H. Heule, Martina Seidl, and Armin Biere. 2017. Solution validation and extraction for QBF preprocessing. J. Autom. Reasoning 58, 1 (2017), 97--125. Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Mikoláš Janota. 2017. An Achilles’ heel of term-resolution. CoRR abs/1704.01071 (2017). https://arxiv.org/abs/1704.01071.Google ScholarGoogle Scholar
  46. Mikoláš Janota, Radu Grigore, and Joao Marques-Silva. 2013. On QBF proofs and preprocessing, See McMillan et al. {57}, 473--489.Google ScholarGoogle Scholar
  47. Mikoláš Janota, William Klieber, Joao Marques-Silva, and Edmund Clarke. 2016. Solving QBF with counterexample guided refinement. Artificial Intelligence 234 (2016), 1--25. Google ScholarGoogle ScholarDigital LibraryDigital Library
  48. Mikoláš Janota and Joao Marques-Silva. 2015. Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci. 577 (2015), 25--42. Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Toni Jussila, Armin Biere, Carsten Sinz, Daniel Kröning, and Christoph M. Wintersteiger. 2007. A first step towards a unified proof checker for QBF. In Theory and Applications of Satisfiability Testing - SAT. 201--214. Google ScholarGoogle ScholarDigital LibraryDigital Library
  50. Hans Kleine Büning and Uwe Bubeck. 2009. Theory of quantified Boolean formulas. See Biere et al. {23}, 735--760.Google ScholarGoogle Scholar
  51. Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. 1995. Resolution for quantified Boolean formulas. Inf. Comput. 117, 1 (1995), 12--18. Google ScholarGoogle ScholarDigital LibraryDigital Library
  52. Hans Kleine Büning, K. Subramani, and Xishun Zhao. 2007. Boolean functions as models for quantified Boolean formulas. J. Autom. Reasoning 39, 1 (2007), 49--75. Google ScholarGoogle ScholarDigital LibraryDigital Library
  53. William Klieber, Samir Sapra, Sicun Gao, and Edmund M. Clarke. 2010. A non-prenex, non-clausal QBF solver with game-state learning. In Theory and Applications of Satisfiability Testing - SAT. 128--142. Google ScholarGoogle ScholarDigital LibraryDigital Library
  54. Jan Krajíček. 1997. Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. The Journal of Symbolic Logic 62, 2 (1997), 457--486.Google ScholarGoogle ScholarCross RefCross Ref
  55. Florian Lonsing and Armin Biere. 2010. Integrating dependency schemes in search-based QBF solvers. In Theory and Applications of Satisfiability Testing - SAT. 158--171. Google ScholarGoogle ScholarDigital LibraryDigital Library
  56. Florian Lonsing, Uwe Egly, and Allen Van Gelder. 2013. Efficient clause learning for quantified Boolean formulas via QBF pseudo unit propagation. In Theory and Applications of Satisfiability Testing - SAT. 100--115. Google ScholarGoogle ScholarDigital LibraryDigital Library
  57. Kenneth L. McMillan, Aart Middeldorp, and Andrei Voronkov (Eds.). 2013. Logic for Programming, Artificial Intelligence, and Reasoning, LPAR. Springer.Google ScholarGoogle Scholar
  58. Christos H. Papadimitriou. 1994. Computational Complexity. Addison-Wesley.Google ScholarGoogle Scholar
  59. Tomás Peitl, Friedrich Slivovsky, and Stefan Szeider. 2016. Long distance Q-resolution with dependency schemes. In SAT. 500--518.Google ScholarGoogle Scholar
  60. Pavel Pudlák. 1997. Lower bounds for resolution and cutting planes proofs and monotone computations. The Journal of Symbolic Logic 62, 3 (1997), 981--998.Google ScholarGoogle ScholarCross RefCross Ref
  61. Pavel Pudlák and Russell Impagliazzo. 2000. A lower bound for DLL algorithms for SAT. In Proc. 11th Symposium on Discrete Algorithms. 128--136. Google ScholarGoogle ScholarDigital LibraryDigital Library
  62. Jussi Rintanen. 2007. Asymptotically optimal encodings of conformant planning in QBF. In AAAI. AAAI Press, 1045--1050. Google ScholarGoogle ScholarDigital LibraryDigital Library
  63. Ronald L. Rivest. 1987. Learning decision lists. Machine Learning 2, 3 (1987), 229--246. Google ScholarGoogle ScholarDigital LibraryDigital Library
  64. John Alan Robinson. 1965. A machine-oriented logic based on the resolution principle. J. ACM 12, 1 (1965), 23--41. Google ScholarGoogle ScholarDigital LibraryDigital Library
  65. Marko Samer. 2008. Variable dependencies of quantified CSPs. In Logic for Programming, Artificial Intelligence, and Reasoning - LPAR. 512--527. Google ScholarGoogle ScholarDigital LibraryDigital Library
  66. Marko Samer and Stefan Szeider. 2009. Backdoor sets of quantified Boolean formulas. J. Autom. Reasoning 42, 1 (2009), 77--97. Google ScholarGoogle ScholarDigital LibraryDigital Library
  67. Horst Samulowitz and Fahiem Bacchus. 2006. Binary clause reasoning in QBF. In SAT, Armin Biere and Carla P. Gomes (Eds.), Vol. 4121. Springer, 353--367. Google ScholarGoogle ScholarDigital LibraryDigital Library
  68. Nathan Segerlind. 2007. The complexity of propositional proofs. Bulletin of Symbolic Logic 13, 4 (2007), 417--481.Google ScholarGoogle ScholarCross RefCross Ref
  69. Friedrich Slivovsky and Stefan Szeider. 2016. Soundness of Q-resolution with dependency schemes. Theor. Comput. Sci. 612 (2016), 83--101. Google ScholarGoogle ScholarDigital LibraryDigital Library
  70. Allen Van Gelder. 2011. Variable independence and resolution paths for quantified Boolean formulas. In Principles and Practice of Constraint Programming - CP. 789--803. Google ScholarGoogle ScholarDigital LibraryDigital Library
  71. Allen Van Gelder. 2012. Contributions to the theory of practical quantified Boolean formula solving. In CP, Michela Milano (Ed.), Vol. 7514. Springer, 647--663.Google ScholarGoogle Scholar
  72. Allen Van Gelder. 2013. Primal and dual encoding from applications into quantified Boolean formulas. In CP. 694--707.Google ScholarGoogle Scholar
  73. H. Vollmer. 1999. Introduction to Circuit Complexity -- A Uniform Approach. Springer Verlag, Berlin. Google ScholarGoogle ScholarDigital LibraryDigital Library
  74. Lintao Zhang and Sharad Malik. 2002. Conflict driven learning in a quantified Boolean satisfiability solver. In ICCAD. 442--449. Google ScholarGoogle ScholarDigital LibraryDigital Library
  75. Lintao Zhang and Sharad Malik. 2002. Towards a symmetric treatment of satisfaction and conflicts in quantified boolean formula evaluation. In Principles and Practice of Constraint Programming - CP. 200--215. http://link.springer.de/link/service/series/0558/bibs/2470/24700200.htm. Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. New Resolution-Based QBF Calculi and Their Proof Complexity

        Recommendations

        Comments

        Login options

        Check if you have access through your login credentials or your institution to get full access on this article.

        Sign in

        Full Access

        • Published in

          cover image ACM Transactions on Computation Theory
          ACM Transactions on Computation Theory  Volume 11, Issue 4
          December 2019
          252 pages
          ISSN:1942-3454
          EISSN:1942-3462
          DOI:10.1145/3331049
          Issue’s Table of Contents

          Copyright © 2019 ACM

          Publisher

          Association for Computing Machinery

          New York, NY, United States

          Publication History

          • Published: 12 September 2019
          • Accepted: 1 June 2019
          • Revised: 1 April 2019
          • Received: 1 September 2018
          Published in toct Volume 11, Issue 4

          Permissions

          Request permissions about this article.

          Request Permissions

          Check for updates

          Qualifiers

          • research-article
          • Research
          • Refereed

        PDF Format

        View or Download as a PDF file.

        PDF

        eReader

        View online with eReader.

        eReader

        HTML Format

        View this article in HTML Format .

        View HTML Format
        About Cookies On This Site

        We use cookies to ensure that we give you the best experience on our website.

        Learn more

        Got it!