skip to main content
research-article

Reasons for Hardness in QBF Proof Systems

Authors Info & Claims
Published:10 February 2020Publication History
Skip Abstract Section

Abstract

We aim to understand inherent reasons for lower bounds for QBF proof systems and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff and Pich, LICS’16). Here, we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) “genuine” QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ACM TOCT 2017). We prove a strong lower bound for relaxing QU-Res, which at the same time exhibits significant shortcomings of that model. Prompted by this, we introduce a hierarchy of new systems that improve Chen’s model and prove a strict separation for the complexity of proofs in this hierarchy. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction.

References

  1. Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. 2014. QBF resolution systems and their proof complexities. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’14). 154--169.Google ScholarGoogle Scholar
  2. Paul Beame, Richard M. Karp, Toniann Pitassi, and Michael E. Saks. 2002. The efficiency of resolution and Davis--Putnam procedures. SIAM J. Comput. 31, 4 (2002), 1048--1075.Google ScholarGoogle ScholarDigital LibraryDigital Library
  3. Paul Beame, Henry A. Kautz, and Ashish Sabharwal. 2004. Towards understanding and harnessing the potential of clause learning. J. Artif. Intell. Res. 22 (2004), 319--351.Google ScholarGoogle ScholarCross RefCross Ref
  4. Marco Benedetti and Hratch Mangassarian. 2008. QBF-based formal verification: Experience and perspectives. J. Sat. Boolean Model. Comput. 5, 1-4 (2008), 133--191.Google ScholarGoogle Scholar
  5. Olaf Beyersdorff and Joshua Blinkhorn. 2018. Genuine lower bounds for QBF expansion. In Proceedings of the 35th Symposium on Theoretical Aspects of Computer Science (STACS’18). 12:1--12:15.Google ScholarGoogle Scholar
  6. Olaf Beyersdorff, Joshua Blinkhorn, and Luke Hinde. 2019a. Size, cost, and capacity: A semantic technique for hard random QBFs. Logic. Methods Comput. Sci. 15, 1 (2019).Google ScholarGoogle Scholar
  7. Olaf Beyersdorff, Ilario Bonacina, and Leroy Chew. 2016. Lower bounds: From circuits to QBF proof systems. In Proceedings of the ACM Conference on Innovations in Theoretical Computer Science (ITCS’16). 249--260.Google ScholarGoogle ScholarDigital LibraryDigital Library
  8. Olaf Beyersdorff, Leroy Chew, Judith Clymo, and Meena Mahajan. 2019b. Short proofs in QBF expansion. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’19).Google ScholarGoogle ScholarCross RefCross Ref
  9. Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2014. On unification of QBF resolution-based calculi. In Proceedings of the Symposium on Mathematical Foundations of Computer Science (MFCS’14). 81--93.Google ScholarGoogle ScholarCross RefCross Ref
  10. Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2015. Proof complexity of resolution-based QBF calculi. In Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS’15). LIPIcs, 76--89.Google ScholarGoogle Scholar
  11. Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. 2017a. Feasible interpolation for QBF resolution calculi. Logic. Methods Comput. Sci. 13 (2017). Issue 2.Google ScholarGoogle Scholar
  12. Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. 2018. Understanding cutting planes for QBFs. Info. Comput. 262 (2018), 141--161.Google ScholarGoogle Scholar
  13. Olaf Beyersdorff and Luke Hinde. 2019. Characterising tree-like frege proofs for QBF. Info. Comput. 268 (2019).Google ScholarGoogle Scholar
  14. Olaf Beyersdorff, Luke Hinde, and Ján Pich. 2017b. Reasons for hardness in QBF proof systems. In Proceedings of the Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’17).Google ScholarGoogle Scholar
  15. Olaf Beyersdorff, Johannes Köbler, and Sebastian Müller. 2011. Proof systems that take advice. Info. Comput. 209, 3 (2011), 320--332.Google ScholarGoogle ScholarDigital LibraryDigital Library
  16. Olaf Beyersdorff and Ján Pich. 2016. Understanding gentzen and frege systems for QBF. In Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS’16). 146--155.Google ScholarGoogle ScholarDigital LibraryDigital Library
  17. A. Blake. 1937. Canonical Expressions in Boolean Algebra. Ph.D. Dissertation. University of Chicago.Google ScholarGoogle Scholar
  18. 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
  19. 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
  20. Hubie Chen. 2017. Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness. Trans. Comput. Theory 9, 3 (2017), 15:1--15:20. DOI:https://doi.org/10.1145/3087534Google ScholarGoogle Scholar
  21. Stephen A. Cook and Tsuyoshi Morioka. 2005. Quantified propositional calculus and a second-order theory for NC1. Arch. Math. Log. 44, 6 (2005), 711--749.Google ScholarGoogle ScholarCross RefCross Ref
  22. Stephen A. Cook and Phuong Nguyen. 2010. Logical Foundations of Proof Complexity. Cambridge University Press.Google ScholarGoogle Scholar
  23. Stephen A. Cook and Robert A. Reckhow. 1979. The relative efficiency of propositional proof systems. J. Symbol. Log. 44, 1 (1979), 36--50.Google ScholarGoogle ScholarCross RefCross Ref
  24. Martin Davis and Hilary Putnam. 1960. A computing procedure for quantification theory. J. ACM 7 (1960), 210--215.Google ScholarGoogle ScholarDigital LibraryDigital Library
  25. Uwe Egly. 2012. On sequent systems and resolution for QBFs. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’12). 100--113.Google ScholarGoogle ScholarDigital LibraryDigital Library
  26. Uwe Egly. 2016. On stronger calculi for QBFs. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’16). 419--434.Google ScholarGoogle ScholarCross RefCross Ref
  27. 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. DOI:https://doi.org/10.1007/s10472-016-9501-2Google ScholarGoogle ScholarDigital LibraryDigital Library
  28. Merrick L. Furst, James B. Saxe, and Michael Sipser. 1984. Parity, circuits, and the polynomial-time hierarchy. Math. Syst. Theory 17, 1 (1984), 13--27.Google ScholarGoogle Scholar
  29. Enrico Giunchiglia, Paolo Marin, and Massimo Narizzano. 2009. Reasoning with quantified Boolean formulas. In Handbook of Satisfiability. IOS Press, 761--780.Google ScholarGoogle Scholar
  30. Armin Haken. 1985. The intractability of resolution. Theor. Comput. Sci. 39 (1985), 297--308.Google ScholarGoogle ScholarCross RefCross Ref
  31. Johan Håstad. 1989. Almost optimal lower bounds for small depth circuits. In Randomness and Computation, Advances in Computing Reasearch, Vol 5, S. Micali (Ed.). JAI Press, 143--170.Google ScholarGoogle Scholar
  32. Mikolás Janota, William Klieber, Joao Marques-Silva, and Edmund M. Clarke. 2016. Solving QBF with counterexample guided refinement. J. Artific. Intell. 234 (2016), 1--25.Google ScholarGoogle ScholarDigital LibraryDigital Library
  33. Mikolás 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
  34. Emil Jerábek. 2004. Dual weak pigeonhole principle, Boolean complexity, and derandomization. Ann. Pure Appl. Logic 129, 1–3 (2004), 1--37.Google ScholarGoogle ScholarCross RefCross Ref
  35. Benjamin Kiesl, Marijn J. H. Heule, and Martina Seidl. 2017. A little blocked literal goes a long way. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’17). 281--297.Google ScholarGoogle ScholarCross RefCross Ref
  36. Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. 1995. Resolution for quantified Boolean formulas. Info. Comput. 117, 1 (1995), 12--18.Google ScholarGoogle ScholarDigital LibraryDigital Library
  37. Jan Krajícek, Pavel Pudlák, and Alan R. Woods. 1995. An exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle. Random Struct. Algor. 7, 1 (1995), 15--40.Google ScholarGoogle ScholarDigital LibraryDigital Library
  38. Jan Krajíček. 1995. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and Its Applications, Vol. 60. Cambridge University Press, Cambridge.Google ScholarGoogle ScholarDigital LibraryDigital Library
  39. Jan Krajíček and Pavel Pudlák. 1990. Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 36 (1990), 29--46.Google ScholarGoogle ScholarCross RefCross Ref
  40. Florian Lonsing and Uwe Egly. 2018a. Evaluating QBF solvers: Quantifier alternations matter. In Proceedings of the Conference on Principles and Practice of Constraint Programming (CP’18). 276--294.Google ScholarGoogle ScholarCross RefCross Ref
  41. Florian Lonsing and Uwe Egly. 2018b. QRAT+: Generalizing QRAT by a more powerful QBF redundancy property. In Proceedings of the 9th International Joint Conference on Automated Reasoning (IJCAR’18). 161--177.Google ScholarGoogle ScholarCross RefCross Ref
  42. Florian Lonsing, Uwe Egly, and Martina Seidl. 2016. Q-resolution with generalized axioms. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’16). 435--452.Google ScholarGoogle ScholarCross RefCross Ref
  43. Tomás Peitl, Friedrich Slivovsky, and Stefan Szeider. 2016. Long distance q-resolution with dependency schemes. In Proceedings of the Conference on Theory and Applications of Satisfiability Testing (SAT’16). 500--518.Google ScholarGoogle ScholarCross RefCross Ref
  44. Knot Pipatsrisawat and Adnan Darwiche. 2011. On the power of clause-learning SAT solvers as resolution engines. Artif. Intell. 175, 2 (2011), 512--525.Google ScholarGoogle ScholarDigital LibraryDigital Library
  45. Toniann Pitassi, Paul Beame, and Russell Impagliazzo. 1993. Exponential lower bounds for the pigeonhole principle. Comput. Complex. 3 (1993), 97--140.Google ScholarGoogle ScholarDigital LibraryDigital Library
  46. Jussi Rintanen. 2007. Asymptotically optimal encodings of conformant planning in QBF. In Proceedings of the 22nd AAAI Conference on Artificial Intelligence. 1045--1050.Google ScholarGoogle Scholar
  47. 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
  48. Horst Samulowitz and Fahiem Bacchus. 2005. Using SAT in QBF. In Proceedings of the Conference on Principles and Practice of Constraint Programming (CP’05). 578--592.Google ScholarGoogle ScholarDigital LibraryDigital Library
  49. Nathan Segerlind. 2007. The complexity of propositional proofs. Bull. Symbol. Logic 13, 4 (2007), 417--481.Google ScholarGoogle ScholarCross RefCross Ref
  50. 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
  51. Allen Van Gelder. 2012. Contributions to the theory of practical quantified Boolean formula solving. In Proceedings of the Conference on Principles and Practice of Constraint Programming (CP’12). 647--663.Google ScholarGoogle ScholarCross RefCross Ref
  52. Lintao Zhang and Sharad Malik. 2002. Conflict driven learning in a quantified Boolean Satisfiability solver. In Proceedings of the IEEE/ACM International Conference on Computer-aided Design (ICCAD’02). 442--449.Google ScholarGoogle ScholarDigital LibraryDigital Library

Index Terms

  1. Reasons for Hardness in QBF Proof Systems

      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 12, Issue 2
        June 2020
        138 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3382781
        Issue’s Table of Contents

        Copyright © 2020 ACM

        Publisher

        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 10 February 2020
        • Accepted: 1 November 2019
        • Revised: 1 October 2019
        • Received: 1 December 2017
        Published in toct Volume 12, Issue 2

        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!