Abstract
We present and study a framework in which one can present alternation-based lower bounds on proof length in proof systems for quantified Boolean formulas. A key notion in this framework is that of proof system ensemble, which is (essentially) a sequence of proof systems where, for each, proof checking can be performed in the polynomial hierarchy. We introduce a proof system ensemble called relaxing QU-res that is based on the established proof system QU-resolution. Our main results include an exponential separation of the treelike and general versions of relaxing QU-res and an exponential lower bound for relaxing QU-res; these are analogs of classical results in propositional proof complexity.
- Albert Atserias, Johannes Klaus Fichte, and Marc Thurley. 2011. Clause-learning algorithms with many restarts and bounded-width resolution. J. Artif. Intell. Res. 40 (2011), 353--373. Google Scholar
Cross Ref
- Valeriy Balabanov and Jie-Hong R. Jiang. 2011. Resolution proofs and skolem functions in QBF evaluation and applications. In Proceedings of the 23rd International Conference on Computer Aided Verification (CAV’11). 149--164. Google Scholar
Digital Library
- Valeriy Balabanov, Magdalena Widl, and Jie-Hong R. Jiang. 2014. QBF resolution systems and their proof complexities. In Proceedings of the 17th International Conference on Theory and Applications of Satisfiability Testing (SAT’14). 154--169.Google Scholar
- 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 Scholar
Cross Ref
- Paul Beame and Toniann Pitassi. 1998. Propositional proof complexity: Past, present and future. Bull. EATCS 65 (1998), 66--89.Google Scholar
- Eli Ben-Sasson, Russell Impagliazzo, and Avi Wigderson. 2004. Near optimal separation of tree-like and general resolution. Combinatorica 24, 4 (2004), 585--603. Google Scholar
Digital Library
- Eli Ben-Sasson and Avi Wigderson. 2001. Short proofs are narrow - Resolution made simple. J. ACM 48, 2 (2001), 149--169. Google Scholar
Digital Library
- Marco Benedetti. 2005. sKizzo: A suite to evaluate and certify QBFs. In Proceedings of the 20th International Conference on Automated Deduction (CADE-20). 369--376. Google Scholar
Digital Library
- Olaf Beyersdorff, Leroy Chew, and Mikolas Janota. 2014. On unification of QBF resolution-based calculi. In Proceedings of the 39th International Symposium on Mathematical Foundations of Computer Science 2014 (MFCS’14). 81--93.Google Scholar
Cross Ref
- Olaf Beyersdorff, Leroy Chew, and Mikolás Janota. 2015. Proof complexity of resolution-based QBF calculi. In Proceedings of the 32nd International Symposium on Theoretical Aspects of Computer Science (STACS’15). 76--89.Google Scholar
- Olaf Beyersdorff, Leroy Chew, Meena Mahajan, and Anil Shukla. Understanding cutting planes for QBFs. In Proceedings of the 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS’16).Google Scholar
- Olaf Beyersdorff, Leroy Chew, and Karteek Sreenivasaiah. 2014. A game characterisation of tree-like Q-resolution size. Electronic Colloquium on Computational Complexity (ECCC) (2014).Google Scholar
- 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 Scholar
Digital Library
- 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 Scholar
Digital Library
- Hans Kleine Büning, Marek Karpinski, and Andreas Flögel. 1995. Resolution for quantified boolean formulas. Inf. Comput. 117, 1 (1995), 12--18. Google Scholar
Digital Library
- Hubie Chen. 2014. Beyond Q-resolution and prenex form: A proof system for quantified constraint satisfaction. Logical Methods Comput. Sci. 10, 4 (2014).Google Scholar
- Hubie Chen. 2014. Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness. CoRR abs/1410.5369 (2014).Google Scholar
- Hubie Chen. 2016. Proof complexity modulo the polynomial hierarchy: Understanding alternation as a source of hardness. In Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP’16). 94:1--94:14.Google Scholar
- Stephen A. Cook and Robert A. Reckhow. 1974. On the lengths of proofs in the propositional calculus (preliminary version). In Proceedings of the 6th Annual ACM Symposium on Theory of Computing. 135--148. Google Scholar
Digital Library
- Uwe Egly, Florian Lonsing, and Magdalena Widl. 2013. Long-distance resolution: Proof generation and strategy extraction in search-based QBF solving. In Proceedings of the 19th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR-19). 291--308.Google Scholar
Cross Ref
- Allen Van Gelder. 2012. Contributions to the theory of practical quantified boolean formula solving. In Proceedings of the 18th International Conference on Principles and Practice of Constraint Programming (CP’12). 647--663.Google Scholar
Cross Ref
- 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 Proceedings of the 22nd International Joint Conference on Artificial Intelligence (IJCAI’11). 546--553. Google Scholar
Digital Library
- Armin Haken. 1985. The intractability of resolution. Theor. Comput. Sci. 39 (1985), 297--308.Google Scholar
Cross Ref
- Marijn Heule, Martina Seidl, and Armin Biere. 2014. A unified proof system for QBF preprocessing. In Proceedings of the 7th International Joint Conference Automated Reasoning (IJCAR’14). 91--106.Google Scholar
Cross Ref
- Mikolás Janota, Radu Grigore, and João Marques-Silva. 2013. On QBF proofs and preprocessing. In Proceedings of the 19th International Conference Logic for Programming, Artificial Intelligence, and Reasoning (LPAR-19). 473--489.Google Scholar
Cross Ref
- Mikolás Janota and Joao Marques-Silva. 2013. On propositional QBF expansions and Q-resolution. In Proceedings of the International Conference on Theory and Applications of Satisfiability Testing (SAT’13). 67--82. Google Scholar
Digital Library
- Pavel Pudlák and Russell Impagliazzo. 2000. A lower bound for DLL algorithms for k-SAT (preliminary version). In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms. 128--136. Google Scholar
Digital Library
- Horst Samulowitz and Fahiem Bacchus. 2005. Using SAT in QBF. In Proceedings of the 11th International Conferenceon Principles and Practice of Constraint Programming (CP’05). 578--592. Google Scholar
Digital Library
- N. Segerlind. 2007. The complexity of propositional proofs. Bull. Symbol. Logic 13 (2007), 417--626. Issue 4.Google Scholar
Cross Ref
- Yinlei Yu and Sharad Malik. 2005. Validating the result of a quantified boolean formula (QBF) solver: Theory and practice. In Proceedings of the 2005 Conference on Asia South Pacific Design Automation (ASP-DAC’05). 1047--1051. Google Scholar
Digital Library
Index Terms
Proof Complexity Modulo the Polynomial Hierarchy: Understanding Alternation as a Source of Hardness
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