Abstract
Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10–15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)
Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Computers 48(5), 506–521 (1999)
Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: engineering an efficient SAT solver. In: Proc. DAC, pp. 530–535. ACM (2001)
Marques-Silva, J.P., Lynce, I., Malik, S.: Conflict-driven clause learning SAT solvers. In: [1], pp. 131–153
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem proving. Communications of the ACM 5(7), 394–397 (1962)
Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of the ACM 7(3), 201–215 (1960)
Buss, S.R., Hoffmann, J., Johannsen, J.: Resolution trees with lemmas: Resolution refinements that characterize DLL-algorithms with clause learning. Logical Methods in Computer Science 4(4:13) (2008)
Pipatsrisawat, K., Darwiche, A.: On the power of clause-learning SAT solvers as resolution engines. Artificial Intelligence 175, 512–525 (2011)
Atserias, A., Fichte, J.K., Thurley, M.: Clause-learning algorithms with many restarts and bounded-width resolution. Journal of Artificial Intelligence Research 40, 353–373 (2011)
Alekhnovich, M., Razborov, A.A.: Resolution is not automatizable unless W[P] is tractable. In: Proc. FOCS, pp. 210–219. IEEE (2001)
Bonet, M.L., Galesi, N.: Optimality of size-width tradeoffs for resolution. Computational Complexity 10(4), 261–276 (2001)
Atserias, A., Dalmau, V.: A combinatorial characterization of resolution width. Journal of Computer and System Sciences 74(3), 323–334 (2008)
Ben-Sasson, E., Nordström, J.: Short proofs may be spacious: An optimal separation of space and length in resolution. In: Proc. FOCS, pp. 709–718. IEEE (2008)
Ansótegui, C., Bonet, M.L., Levy, J., Manyà, F.: Measuring the hardness of SAT instances. In: Proc. AAAI, pp. 222–228. AAAI Press (2008)
Esteban, J.L., Torán, J.: Space bounds for resolution. Inf. Comput. 171(1), 84–97 (2001)
Esteban, J.L., Torán, J.: A combinatorial characterization of treelike resolution space. Information Processing Letters 87(6), 295–300 (2003)
Williams, R., Gomes, C.P., Selman, B.: Backdoors to typical case complexity. In: Proc. IJCAI, pp. 1173–1178. Morgan Kaufmann (2003)
Nordström, J.: Pebble games, proof complexity and time-space trade-offs. Logical Methods in Computer Science (to appear, 2012), http://www.csc.kth.se/~jakobn/research
Haken, A.: The intractability of resolution. Theoret. Comp. Sci. 39(2-3), 297–308 (1985)
Urquhart, A.: Hard examples for resolution. Journal of the ACM 34(1), 209–219 (1987)
Ben-Sasson, E., Wigderson, A.: Short proofs are narrow—resolution made simple. Journal of the ACM 48(2), 149–169 (2001)
Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Space complexity in propositional calculus. SIAM Journal on Computing 31(4), 1184–1211 (2002)
Ben-Sasson, E., Galesi, N.: Space complexity of random formulae in resolution. Random Structures and Algorithms 23(1), 92–109 (2003)
Nordström, J.: New wine into old wineskins: A survey of some pebbling classics with supplemental results. Foundations and Trends in Theoretical Computer Science (to appear, 2012), Draft version available at http://www.csc.kth.se/~jakobn/research/
Ben-Sasson, E., Nordström, J.: Understanding space in proof complexity: Separations and trade-offs via substitutions. In: Proc. ICS, pp. 401–416 (2011)
Nordström, J.: Narrow proofs may be spacious: Separating space and width in resolution. SIAM Journal on Computing 39(1), 59–121 (2009)
Nordström, J., Håstad, J.: Towards an optimal separation of space and length in resolution (Extended abstract). In: Proc. STOC, pp. 701–710 (2008)
Gilbert, J.R., Tarjan, R.E.: Variations of a pebble game on graphs. Technical Report STAN-CS-78-661, Stanford University (1978)
Beame, P., Kautz, H., Sabharwal, A.: Towards understanding and harnessing the potential of clause learning. Journal of Artificial Intelligence Research 22, 319–351 (2004)
Eén, N., Sörensson, N.: An Extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)
Biere, A.: Lingeling, Plingeling, PicoSAT and PrecoSAT at SAT Race 2010. FMV Tech. Report 10/1, Johannes Kepler University (2010)
Dilkina, B., Gomes, C.P., Sabharwal, A.: Backdoors in the Context of Learning. In: Kullmann, O. (ed.) SAT 2009. LNCS, vol. 5584, pp. 73–79. Springer, Heidelberg (2009)
Beame, P., Beck, C., Impagliazzo, R.: Time-space tradeoffs in resolution: Superpolynomial lower bounds for superlinear space. In: Proc. STOC, pp. 213–232. ACM (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Järvisalo, M., Matsliah, A., Nordström, J., Živný, S. (2012). Relating Proof Complexity Measures and Practical Hardness of SAT. In: Milano, M. (eds) Principles and Practice of Constraint Programming. CP 2012. Lecture Notes in Computer Science, vol 7514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33558-7_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-33558-7_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-33557-0
Online ISBN: 978-3-642-33558-7
eBook Packages: Computer ScienceComputer Science (R0)