research-article

Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution

Authors:

Olaf Beyersdorff,

Joshua Blinkhorn,

Meena MahajanAuthors Info & Claims

LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

Pages 209 - 223

https://doi.org/10.1145/3373718.3394793

Published: 08 July 2020 Publication History

Abstract

We provide a tight characterisation of proof size in resolution for quantified Boolean formulas (QBF) by circuit complexity. Such a characterisation was previously obtained for a hierarchy of QBF Frege systems (Beyersdorff & Pich, LICS 2016), but leaving open the most important case of QBF resolution. Different from the Frege case, our characterisation uses a new version of decision lists as its circuit model, which is stronger than the CNFs the system works with. Our decision list model is well suited to compute countermodels for QBFs.

Our characterisation works for both Q-Resolution and QU-Resolution, which we show to be polynomially equivalent for QBFs of bounded quantifier alternation.

Using our characterisation we obtain a size-width relation for QBF resolution in the spirit of the celebrated result for propositional resolution (Ben-Sasson & Wigderson, J. ACM 2001). However, our result is not just a replication of the propositional relation --- intriguingly ruled out for QBF in previous research (Beyersdorff et al., ACM ToCL 2018) ---but shows a different dependence between size, width, and quantifier complexity.

We demonstrate that our new technique elegantly reproves known QBF hardness results and unifies previous lower-bound techniques in the QBF domain.

References

[1]

Yaroslav Alekseev, Dima Grigoriev, Edward A. Hirsch, and Iddo Tzameret. 2019. Semi-Algebraic Proofs, IPS Lower Bounds and the tau - Conjecture: Can a Natural Number be Negative? Electronic Colloquium on Computational Complexity 26 (2019), 142.

[2]

Valeriy Balabanov and Jie-Hong R. Jiang. 2012. Unified QBF Certification and its Applications. Formal Methods in System Design 41, 1 (2012), 45--65.

Digital Library

[3]

Paul Beame and Toniann Pitassi. 2001. Propositional Proof Complexity: Past, Present, and Future. In Current Trends in Theoretical Computer Science: Entering the 21st Century, G. Paun, G. Rozenberg, and A. Salomaa (Eds.). World Scientific Publishing, 42--70.

Digital Library

[4]

Eli Ben-Sasson and Avi Wigderson. 2001. Short proofs are narrow -resolution made simple. Journal of the ACM 48, 2 (2001), 149--169.

Digital Library

[5]

Olaf Beyersdorff. 2009. On the Correspondence Between Arithmetic Theories and Propositional Proof Systems - a Survey. Mathematical Logic Quarterly 55, 2 (2009), 116--137.

[6]

Olaf Beyersdorff, Joshua Blinkhorn, and Luke Hinde. 2019. Size, Cost, and Capacity: A Semantic Technique for Hard Random QBFs. Logical Methods in Computer Science 15, 1 (2019).

[7]

Olaf Beyersdorff, Ilario Bonacina, and Leroy Chew. 2016. Lower Bounds: From Circuits to QBF Proof Systems. In ACM Conference on Innovations in Theoretical Computer Science (ITCS), Madhu Sudan (Ed.). ACM, 249--260.

Digital Library

[8]

Olaf Beyersdorff, Leroy Chew, and Mikoláš Janota. 2015. Proof Complexity of Resolution-based QBF Calculi. In International Symposium on Theoretical Aspects of Computer Science (STACS) (Leibniz International Proceedings in Informatics (LIPIcs)), Ernst W. Mayr and Nicolas Ollinger (Eds.), Vol. 30. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 76--89.

[9]

Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. 2019. New Resolution-Based QBF Calculi and Their Proof Complexity. ACM Transactions on Computation Theory 11, 4 (2019), 26:1--26:42.

Digital Library

[10]

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.

[11]

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, 1 (2018), 1:1--1:26.

Digital Library

[12]

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.

[13]

Olaf Beyersdorff, Luke Hinde, and Ján Pich. 2020. Reasons for Hardness in QBF Proof Systems. ACM Transactions on Computation Theory 12, 2, Article 10 (2020), 27 pages.

Digital Library

[14]

Olaf Beyersdorff, Johannes Köbler, and Sebastian Müller. 2011. Proof Systems that Take Advice. Information and Computation 209, 3 (2011), 320--332.

Digital Library

[15]

Olaf Beyersdorff and Oliver Kullmann. 2014. Unified Characterisations of Resolution Hardness Measures. In International Conference on Theory and Practice of Satisfiability Testing (SAT) (Lecture Notes in Computer Science), Carsten Sinz and Uwe Egly (Eds.), Vol. 8561. Springer, 170--187.

[16]

Olaf Beyersdorff and Ján Pich. 2016. Understanding Gentzen and Frege Systems for QBF. In Symposium on Logic in Computer Science (LICS), Martin Grohe, Eric Koskinen, and Natarajan Shankar (Eds.). ACM, 146--155.

[17]

A. Blake. 1937. Canonical expressions in boolean algebra. Ph.D. Dissertation. University of Chicago.

[18]

Avrim Blum. 1992. Rank-r Decision Trees are a Subclass of r-Decision Lists. Inform. Process. Lett. 42, 4 (1992), 183--185.

Digital Library

[19]

Nader H. Bshouty. 1996. A Subexponential Exact Learning Algorithm for DNF Using Equivalence Queries. Inform. Process. Lett. 59, 1 (1996), 37--39.

Digital Library

[20]

Samuel R. Buss. 2012. Towards NP-P via proof complexity and search. Annals of Pure and Applied Logic 163, 7 (2012), 906--917.

[21]

Hubie Chen. 2017. Proof Complexity Modulo the Polynomial Hierarchy: Understanding Alternation as a Source of Hardness. ACM Transactions on Computation Theory 9, 3 (2017), 15:1--15:20.

Digital Library

[22]

Judith Clymo. [n.d.]. Ph.D. Dissertation. School of Computing, University of Leeds. in preparation.

[23]

Judith Clymo and Olaf Beyersdorff. 2018. Relating size and width in variants of Q-resolution. Inform. Process. Lett. 138 (2018), 1--6.

[24]

Stephen A. Cook and Phuong Nguyen. 2010. Logical Foundations of Proof Complexity. Cambridge University Press, Cambridge.

[25]

Stephen A. Cook and Robert A. Reckhow. 1979. The Relative Efficiency of Propositional Proof Systems. Journal of Symbolic Logic 44, 1 (1979), 36--50.

[26]

Nadia Creignou and Daniel Le Berre (Eds.). 2016. International Conference on Theory and Practice of Satisfiability Testing (SAT). Lecture Notes in Computer Science, Vol. 9710. Springer.

[27]

Uwe Egly, Martin Kronegger, Florian Lonsing, and Andreas Pfandler. 2017. Conformant planning as a case study of incremental QBF solving. Annals of Mathematics and Artificial Intelligence 80, 1 (2017), 21--45.

Digital Library

[28]

Allen Van Gelder. 2012. Contributions to the Theory of Practical Quantified Boolean Formula Solving. In International Conference on Principles and Practice of Constraint Programming (CP) (Lecture Notes in Computer Science), Michela Milano (Ed.), Vol. 7514. Springer, 647--663.

[29]

Joshua A. Grochow and Toniann Pitassi. 2018. Circuit Complexity, Proof Complexity, and Polynomial Identity Testing: The Ideal Proof System. Journal of the ACM 65, 6 (2018), 37:1--37:59.

Digital Library

[30]

Armin Haken. 1985. The intractability of Resolution. Theoretical Computer Science 39 (1985), 297--308.

[31]

J. Håstad. 1987. Computational Limitations of Small Depth Circuits. MIT Press, Cambridge.

[32]

Mikolás Janota. 2016. On Q-Resolution and CDCL QBF Solving, See [26], 402--418.

[33]

Mikoláš Janota and João Marques-Silva. 2015. Expansion-based QBF solving versus Q-resolution. Theoretical Computer Science 577 (2015), 25--42.

Digital Library

[34]

Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. 1995. Resolution for Quantified Boolean Formulas. Information and Computation 117, 1 (1995), 12--18.

Digital Library

[35]

Roman Kontchakov, Luca Pulina, Ulrike Sattler, Thomas Schneider, Petra Selmer, Frank Wolter, and Michael Zakharyaschev. 2009. Minimal Module Extraction from DL-Lite Ontologies Using QBF Solvers. In International Joint Conference on Artificial Intelligence (IJCAI), Craig Boutilier (Ed.). AAAI Press, 836--841.

[36]

Jan Krajíček. 1995. Bounded Arithmetic, Propositional Logic, and Complexity Theory. Encyclopedia of Mathematics and Its Applications, Vol. 60. Cambridge University Press, Cambridge.

Digital Library

[37]

Matthias Krause. 2006. On the computational power of Boolean decision lists. Computational Complexity 14, 4 (2006), 362--375.

Digital Library

[38]

Oliver Kullmann. 2004. Upper and Lower Bounds on the Complexity of Generalised Resolution and Generalised Constraint Satisfaction Problems. Annals of Mathematics and Artificial Intelligence 40, 3-4 (2004), 303--352.

Digital Library

[39]

Florian Lonsing and Uwe Egly. 2018. Evaluating QBF Solvers: Quantifier Alternations Matter. In International Conference on Principles and Practice of Constraint Programming (CP) (Lecture Notes in Computer Science), John N. Hooker (Ed.), Vol. 11008. Springer, 276--294.

[40]

Florian Lonsing, Uwe Egly, and Allen Van Gelder. 2013. Efficient Clause Learning for Quantified Boolean Formulas via QBF Pseudo Unit Propagation. In International Conference on Theory and Applications of Satisfiability Testing (SAT) (Lecture Notes in Computer Science), Matti Järvisalo and Allen Van Gelder (Eds.), Vol. 7962. Springer, 100--115.

Digital Library

[41]

Florian Lonsing, Uwe Egly, and Martina Seidl. 2016. Q-Resolution with Generalized Axioms, See [26], 435--452.

[42]

Hratch Mangassarian, Andreas G. Veneris, and Marco Benedetti. 2010. Robust QBF Encodings for Sequential Circuits with Applications to Verification, Debug, and Test. IEEE Trans. Comput. 59, 7 (2010), 981--994.

Digital Library

[43]

Jakob Nordström. 2015. On the interplay between proof complexity and SAT solving. SIGLOG News 2, 3 (2015), 19--44.

Digital Library

[44]

Luca Pulina and Martina Seidl. 2019. The 2016 and 2017 QBF solvers evaluations (QBFEVAL'16 and QBFEVAL'17). Artificial Intelligence 274 (2019), 224--248.

[45]

Alexander A. Razborov. 1987. Lower bounds for the size of circuits of bounded depth with basis (∧, &oplus;}. Mathematical Notes 41, 4 (1987), 333--338.

[46]

Ronald L. Rivest. 1987. Learning Decision Lists. Machine Learning 2, 3 (1987), 229--246.

Digital Library

[47]

John Alan Robinson. 1965. A Machine-Oriented Logic Based on the Resolution Principle. Journal of the ACM 12, 1 (1965), 23--41.

Digital Library

[48]

Nathan Segerlind. 2007. The Complexity of Propositional Proofs. Bulletin of Symbolic Logic 13, 4 (2007), 417--481.

[49]

R. Smolensky. 1987. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In ACM Symposium on Theory of Computing (STOC), Alfred V. Aho (Ed.). ACM, 77--82.

Digital Library

[50]

Raymond M. Smullyan. 1995. First-order Logic. Dover Publications.

[51]

Moshe Y. Vardi. 2014. Boolean Satisfiability: Theory and Engineering. Commun. ACM 57, 3 (2014), 5.

Digital Library

Cited By

Böhm BBeyersdorff O(2023)Lower Bounds for QCDCL via Formula GaugeJournal of Automated Reasoning10.1007/s10817-023-09683-167:4Online publication date: 27-Sep-2023
https://doi.org/10.1007/s10817-023-09683-1
Beyersdorff OBlinkhorn JMahajan MPeitl T(2022)Hardness Characterisations and Size-width Lower Bounds for QBF ResolutionACM Transactions on Computational Logic10.1145/356528624:2(1-30)Online publication date: 28-Sep-2022
https://dl.acm.org/doi/10.1145/3565286
Beyersdorff OBlinkhorn J(2021)A simple proof of QBF hardnessInformation Processing Letters10.1016/j.ipl.2021.106093(106093)Online publication date: Jan-2021
https://doi.org/10.1016/j.ipl.2021.106093
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Index Terms

Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution
1. Theory of computation
  1. Computational complexity and cryptography
    1. Circuit complexity
    2. Proof complexity

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cover image ACM Conferences

LICS '20: Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

July 2020

986 pages

ISBN:9781450371049

DOI:10.1145/3373718

Conference Chairs:
Holger Hermanns,
Lijun Zhang,
Naoki Kobayashi,
General Chair:
Dale Miller

Copyright © 2020 ACM.

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Published: 08 July 2020

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Carl Zeiss Foundation
John Templeton Foundation

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LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science

July 8 - 11, 2020

Saarbrücken, Germany

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LICS '20 Paper Acceptance Rate 69 of 174 submissions, 40%;

Overall Acceptance Rate 215 of 622 submissions, 35%

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Cited By

Böhm BBeyersdorff O(2023)Lower Bounds for QCDCL via Formula GaugeJournal of Automated Reasoning10.1007/s10817-023-09683-167:4Online publication date: 27-Sep-2023
https://doi.org/10.1007/s10817-023-09683-1
Beyersdorff OBlinkhorn JMahajan MPeitl T(2022)Hardness Characterisations and Size-width Lower Bounds for QBF ResolutionACM Transactions on Computational Logic10.1145/356528624:2(1-30)Online publication date: 28-Sep-2022
https://dl.acm.org/doi/10.1145/3565286
Beyersdorff OBlinkhorn J(2021)A simple proof of QBF hardnessInformation Processing Letters10.1016/j.ipl.2021.106093(106093)Online publication date: Jan-2021
https://doi.org/10.1016/j.ipl.2021.106093
Sigley SBeyersdorff O(2021)Proof Complexity of Modal ResolutionJournal of Automated Reasoning10.1007/s10817-021-09609-9Online publication date: 13-Oct-2021
https://doi.org/10.1007/s10817-021-09609-9
Böhm BBeyersdorff O(2021)Lower Bounds for QCDCL via Formula GaugeTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_5(47-63)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_5
Beyersdorff OPulina LSeidl MShukla A(2021)QBFFam: A Tool for Generating QBF Families from Proof ComplexityTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_3(21-29)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_3
Mengel SSlivovsky F(2021)Proof Complexity of Symbolic QBF ReasoningTheory and Applications of Satisfiability Testing – SAT 202110.1007/978-3-030-80223-3_28(399-416)Online publication date: 2-Jul-2021
https://doi.org/10.1007/978-3-030-80223-3_28

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