research-article

Lower Bounds: From Circuits to QBF Proof Systems

Authors:

Olaf Beyersdorff,

Ilario Bonacina,

Chew LeroyAuthors Info & Claims

ITCS '16: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

Pages 249 - 260

https://doi.org/10.1145/2840728.2840740

Published: 14 January 2016 Publication History

Abstract

A general and long-standing belief in the proof complexity community asserts that there is a close connection between progress in lower bounds for Boolean circuits and progress in proof size lower bounds for strong propositional proof systems. Although there are famous examples where a transfer from ideas and techniques from circuit complexity to proof complexity has been effective, a formal connection between the two areas has never been established so far. Here we provide such a formal relation between lower bounds for circuit classes and lower bounds for Frege systems for quantified Boolean formulas (QBF).

Starting from a propositional proof system P we exhibit a general method how to obtain a QBF proof system P+∀red{P}, which is inspired by the transition from resolution to Q-resolution. For us the most important case is a new and natural hierarchy of QBF Frege systems C-Frege+∀red that parallels the well-studied propositional hierarchy of C-Frege systems, where lines in proofs are restricted to belong to a circuit class C.

Building on earlier work for resolution [Beyersdorff, Chew and Janota, 2015a] we establish a lower bound technique via strategy extraction that transfers arbitrary lower bounds for the circuit class C to lower bounds in C-Frege+∀red.

By using the full spectrum of state-of-the-art circuit lower bounds, our new lower bound method leads to very strong lower bounds for QBF \FREGE systems:

exponential lower bounds and separations for the QBF proof system AC^o[p]-Frege+∀red for all primes p;

an exponential separation of AC^o[p]-Frege+∀red from TCo/d-Frege+∀red;

an exponential separation of the hierarchy of constant-depth systems ACo/d-Frege+∀red by formulas of depth independent of d.

In the propositional case, all these results correspond to major open problems.

References

[1]

Miklós Ajtai. The complexity of the pigeonhole-principle. Combinatorica, 14 (4): 417--433, 1994.

[2]

Sanjeev Arora and Boaz Barak. Computational Complexity - A Modern Approach. Cambridge University Press, 2009.

Digital Library

[3]

Valeriy Balabanov and Jie-Hong R. Jiang. Unified QBF certification and its applications. Form. Methods Syst. Des., 41 (1): 45--65, August 2012.

Digital Library

[4]

Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. QBF resolution systems and their proof complexities. In SAT, pages 154--169, 2014.

[5]

Paul Beame and Toniann Pitassi. Propositional proof complexity: Past, present, and future. In G. Paun, G. Rozenberg, and A. Salomaa, editors, Current Trends in Theoretical Computer Science: Entering the 21st Century, pages 42--70. World Scientific Publishing, 2001.

Digital Library

[6]

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

Digital Library

[7]

Marco Benedetti and Hratch Mangassarian. QBF-based formal verification: Experience and perspectives. JSAT, 5 (1-4): 133--191, 2008.

[8]

Olaf Beyersdorff and Oliver Kullmann. Unified characterisations of resolution hardness measures. In SAT, pages 170--187, 2014.

[9]

Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. On unification of QBF resolution-based calculi. In MFCS, II, pages 81--93, 2014.

[10]

Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. Proof complexity of resolution-based QBF calculi. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), pages 76--89, 2015.

[11]

Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. Feasible interpolation for QBF resolution calculi. In ICALP. Springer, 2015.

[12]

Archie Blake. Canonical Expressions in Boolean Algebra. PhD thesis, 1937. University of Chicago.

[13]

Maria Luisa Bonet, Toniann Pitassi, and Ran Raz. On interpolation and automatization for Frege systems. SIAM Journal on Computing, 29 (6): 1939--1967, 2000.

Digital Library

[14]

Maria Luisa Bonet, Carlos Domingo, Ricard Gavaldà, Alexis Maciel, and Toniann Pitassi. Non-automatizability of bounded-depth Frege proofs. Computational Complexity, 13 (1-2): 47--68, 2004.

Digital Library

[15]

Ravi B. Boppana and Michael Sipser. Handbook of theoretical computer science (vol. A). chapter The Complexity of Finite Functions, pages 757--804. MIT Press, Cambridge, MA, USA, 1990.

Digital Library

[16]

Samuel R. Buss. Towards NP-P via proof complexity and search. Ann. Pure Appl. Logic, 163 (7): 906--917, 2012.

[17]

Samuel R. Buss and Peter Clote. Cutting planes, connectivity, and threshold logic. Archive for Mathematical Logic, 35 (1): 33--62, 1996.

[18]

Matthew Clegg, Jeff Edmonds, and Russell Impagliazzo. Using the Groebner basis algorithm to find proofs of unsatisfiability. In Proc. 28th ACM Symposium on Theory of Computing, pages 174--183, 1996.

Digital Library

[19]

Stephen A. Cook and Tsuyoshi Morioka. Quantified propositional calculus and a second-order theory for NC1. Arch. Math. Log., 44 (6): 711--749, 2005.

[20]

Stephen A. Cook and Robert A. Reckhow. The relative efficiency of propositional proof systems. Journal of Symbolic Logic, 6: 169--184, 1979.

[21]

William Cook, Collette R. Coullard, and György Turán. On the complexity of cutting-plane proofs. Discrete Applied Mathematics, 18 (1): 25--38, 1987.

Digital Library

[22]

Uwe Egly. On sequent systems and resolution for qbfs. In Theory and Applications of Satisfiability Testing - SAT 2012, pages 100--113, 2012.

Digital Library

[23]

Uwe Egly, Martin Kronegger, Florian Lonsing, and Andreas Pfandler. Conformant planning as a case study of incremental QBF solving. In Artificial Intelligence and Symbolic Computation AISC 2014, pages 120--131, 2014.

[24]

Alexandra Goultiaeva, Allen Van Gelder, and Fahiem Bacchus. A uniform approach for generating proofs and strategies for both true and false QBF formulas. In IJCAI, pages 546--553, 2011.

Digital Library

[25]

Amin Haken. The intractability of resolution. Theoretical Computer Science, 39: 297--308, 1985.

[26]

Johan Håstad. Almost optimal lower bounds for small depth circuits. In Proc. 18th STOC, pages 6--20. ACM Press, 1986.

Digital Library

[27]

Marijn Heule, Martina Seidl, and Armin Biere. A unified proof system for QBF preprocessing. In IJCAR, pages 91--106, 2014.

[28]

Pavel Hrubeš. On lengths of proofs in non-classical logics. Annals of Pure and Applied Logic, 157 (2-3): 194--205, 2009.

[29]

Mikolás Janota and Joao Marques-Silva. Expansion-based QBF solving versus Q-resolution. Theor. Comput. Sci., 577: 25--42, 2015.

Digital Library

[30]

Emil Jeřábek. Weak pigeonhole principle, and randomized computation. PhD thesis, Faculty of Mathematics and Physics, Charles University, Prague, 2005.

[31]

Emil Jeřábek. Substitution Frege and extended Frege proof systems in non-classical logics. Annals of Pure and Applied Logic, 159 (1-2): 1--48, 2009.

[32]

Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. Resolution for quantified Boolean formulas. Inf. Comput., 117 (1): 12--18, 1995.

Digital Library

[33]

Jan Krajíček. Bounded Arithmetic, Propositional Logic and Complexity Theory. Cambridge University Press, 1995.

Digital Library

[34]

Jan Krajíček. Interpolation theorems, lower bounds for proof systems and independence results for bounded arithmetic. The Journal of Symbolic Logic, 62 (2): 457--486, 1997.

[35]

Jan Krajíček and Pavel Pudlák. Propositional proof systems, the consistency of first order theories and the complexity of computations. The Journal of Symbolic Logic, 54 (3): 1063--1079, 1989.

[36]

Jan Krajíček and Pavel Pudlák. Quantified propositional calculi and fragments of bounded arithmetic. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 36: 29--46, 1990.

[37]

Jan Krajíček and Pavel Pudlák. Some consequences of cryptographical conjectures for S^1/2 and EF. Information and Computation, 140 (1): 82--94, 1998.

Digital Library

[38]

Jan Krajíček, Pavel Pudlák, and Alan Woods. Exponential lower bounds to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7 (1): 15--39, 1995.

Digital Library

[39]

David E. Muller and Franco P. Preparata. Bounds to complexities of networks for sorting and for switching. J. ACM, 22 (2): 195--201, 1975.

Digital Library

[40]

Toniann Pitassi, Paul Beame, and Russell Impagliazzo. Exponential lower bounds for the pigeonhole principle. Computational Complexity, 3: 97--140, 1993.

Digital Library

[41]

Pavel Pudlák. Lower bounds for resolution and cutting planes proofs and monotone computations. The Journal of Symbolic Logic, 62 (3): 981--998, 1997.

[42]

Alexander A. Razborov. Lower bounds for the size of circuits of bounded depth with basis {&, ⊕}. Math. Notes Acad. Sci. USSR, 41 (4): 333--338, 1987.

[43]

Jussi Rintanen. Asymptotically optimal encodings of conformant planning in QBF. In AAAI, pages 1045--1050. AAAI Press, 2007.

Digital Library

[44]

Ronald L. Rivest. Learning decision lists. Machine Learning, 2 (3): 229--246, 1987.

Digital Library

[45]

John Alan Robinson. A machine-oriented logic based on the resolution principle. J. ACM, 12 (1): 23--41, 1965.

Digital Library

[46]

Nathan Segerlind. The complexity of propositional proofs. Bulletin of Symbolic Logic, 13 (4): 417--481, 2007.

[47]

Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proc. of 19th ACM STOC, pages 77--82, 1987.

Digital Library

[48]

Allen Van Gelder. Contributions to the theory of practical quantified Boolean formula solving. In CP, pages 647--663, 2012.

Digital Library

Cited By

Beyersdorff OBlinkhorn JMahajan MPeitl TSood G(2024)Hard QBFs for Merge ResolutionACM Transactions on Computation Theory10.1145/363826316:2(1-24)Online publication date: 14-Mar-2024
https://dl.acm.org/doi/10.1145/3638263
Bulatov AŽivný S(2020)Approximate Counting CSP Seen from the Other SideACM Transactions on Computation Theory10.1145/338939012:2(1-19)Online publication date: 13-May-2020
https://dl.acm.org/doi/10.1145/3389390
Beyersdorff OBonacina IChew LPich J(2020)Frege Systems for Quantified Boolean LogicJournal of the ACM10.1145/338188167:2(1-36)Online publication date: 5-Apr-2020
https://dl.acm.org/doi/10.1145/3381881
Show More Cited By

Index Terms

Lower Bounds: From Circuits to QBF Proof Systems
1. Theory of computation
  1. Computational complexity and cryptography
    1. Interactive proof systems
    2. Proof complexity
  2. Logic
    1. Proof theory

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cover image ACM Conferences

ITCS '16: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

January 2016

422 pages

ISBN:9781450340571

DOI:10.1145/2840728

Program Chair:
Madhu Sudan
Harvard University

Copyright © 2016 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 14 January 2016

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ITCS'16: Innovations in Theoretical Computer Science

January 14 - 17, 2016

Massachusetts, Cambridge, USA

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ITCS '16 Paper Acceptance Rate 40 of 145 submissions, 28%;

Overall Acceptance Rate 172 of 513 submissions, 34%

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Cited By

Beyersdorff OBlinkhorn JMahajan MPeitl TSood G(2024)Hard QBFs for Merge ResolutionACM Transactions on Computation Theory10.1145/363826316:2(1-24)Online publication date: 14-Mar-2024
https://dl.acm.org/doi/10.1145/3638263
Bulatov AŽivný S(2020)Approximate Counting CSP Seen from the Other SideACM Transactions on Computation Theory10.1145/338939012:2(1-19)Online publication date: 13-May-2020
https://dl.acm.org/doi/10.1145/3389390
Beyersdorff OBonacina IChew LPich J(2020)Frege Systems for Quantified Boolean LogicJournal of the ACM10.1145/338188167:2(1-36)Online publication date: 5-Apr-2020
https://dl.acm.org/doi/10.1145/3381881
Beyersdorff OHinde LPich J(2020)Reasons for Hardness in QBF Proof SystemsACM Transactions on Computation Theory10.1145/337866512:2(1-27)Online publication date: 10-Feb-2020
https://dl.acm.org/doi/10.1145/3378665
Beyersdorff OBlinkhorn JMahajan MHermanns HZhang LKobayashi NMiller D(2020)Hardness Characterisations and Size-Width Lower Bounds for QBF ResolutionProceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science10.1145/3373718.3394793(209-223)Online publication date: 8-Jul-2020
https://dl.acm.org/doi/10.1145/3373718.3394793
Beyersdorff OBlinkhorn JMahajan M(2020)Building Strategies into QBF ProofsJournal of Automated Reasoning10.1007/s10817-020-09560-1Online publication date: 22-May-2020
https://doi.org/10.1007/s10817-020-09560-1
Beyersdorff OBlinkhorn JPeitl T(2020)Strong (D)QBF Dependency Schemes via Tautology-Free Resolution PathsTheory and Applications of Satisfiability Testing – SAT 202010.1007/978-3-030-51825-7_28(394-411)Online publication date: 26-Jun-2020
https://doi.org/10.1007/978-3-030-51825-7_28
Chew LClymo J(2020)How QBF Expansion Makes Strategy Extraction HardAutomated Reasoning10.1007/978-3-030-51074-9_5(66-82)Online publication date: 24-Jun-2020
https://doi.org/10.1007/978-3-030-51074-9_5
Beyersdorff OChew LJanota M(2019)New Resolution-Based QBF Calculi and Their Proof ComplexityACM Transactions on Computation Theory10.1145/335215511:4(1-42)Online publication date: 12-Sep-2019
https://dl.acm.org/doi/10.1145/3352155
Reif SSchmidt AHönig THerfet TSchröder-Preikschat W(2019)ΔeltaACM SIGBED Review10.1145/3314206.331421116:1(33-38)Online publication date: 20-Feb-2019
https://dl.acm.org/doi/10.1145/3314206.3314211
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