Abstract
We characterize several complexity measures for the resolution of Tseitin formulas in terms of a two person cop-robber game. Our game is a slight variation of the one Seymour and Thomas used in order to characterize the tree-width parameter. For any undirected graph, by counting the number of cops needed in our game in order to catch a robber in it, we are able to exactly characterize the width, variable space and depth measures for the resolution of the Tseitin formula corresponding to that graph. We also give an exact game characterization of resolution variable space for any formula.
We show that our game can be played in a monotone way. This implies that the corresponding resolution measures on Tseitin formulas correspond exactly to those under the restriction of regular resolution.
Using our characterizations we improve the existing complexity bounds for Tseitin formulas showing that resolution width, depth and variable space coincide up to a logarithmic factor, and that variable space is bounded by the clause space times a logarithmic factor.
N. Talebanfard—Supported by ERC grant FEALORA 339691.
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
Notes
- 1.
The resolution depth is well know to coincide with the regular resolution depth for any formula.
- 2.
In the original paper [4] it is stated that \({{\mathrm{\mathsf {W}}}}(F\vdash )= {{\mathrm{\mathsf {sd}}}}(F)-1\), by inspecting the proof it can be seen that the formulation involving the width of F is the correct one.
References
Adler, I.: Marshals, monotone marshals, and hypertree width. J. Gr. Theory 47, 275–296 (2004)
Alekhnovich, M., Razborov, A.A.: Satisfiability, branch-width and Tseitin tautologies. Comput. Complex. 20(4), 649–678 (2011)
Alekhnovich, M., Ben-Sasson, E., Razborov, A.A., Wigderson, A.: Space complexity in propositional calculus SIAM. J. Comput. 31(4), 1184–1211 (2002)
Atserias, A., Dalmau, V.: A combinatorial characterization of resolution width. In: 18th IEEE Conference on Computational Complexity, pp. 239–247 (2003)
Beame, P., Beck, C., Impagliazzo, R.: Time-space trade-offs in resolution: superpolynomial lower bounds for superlinear space. SIAM J. Comput. 49(4), 1612–1645 (2016)
Beck, C., Nordström, J., Tang, B.: Some trade-off results for polynomial calculus: extended abstract. In: Proceedings of the 45th ACM Symposium on the Theory of Computing, pp. 813–822 (2013)
Ben-Sasson, E., Wigderson, A.: Short proofs are narrow - resolution made simple. J. ACM 48(2), 149–169 (2001)
Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Inf. Computat. 113(1), 50–79 (1994)
Esteban, J.L., Torán, J.: Space bounds for resolution. Inf. Comput. 171(1), 84–97 (2001)
Fomin, F.V., Thilikos, D.: An annotated bibliography on guaranteed graph searching. Theor. Comput. Sci. 399, 236–245 (2008)
Gottlob, G., Leone, N., Scarello, F.: Robbers, marshals and guards: game theoretic and logical characterizations of hypertree width. J. Comput. Syst. Sci. 66, 775–808 (2003)
Kirousis, L.M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47(3), 205–218 (1986)
LaPaugh, A.S.: Recontamination does not help to search a graph. Technical report Electrical Engineering and Computer Science Department. Princeton University (1883)
Razborov, A.: On space and depth in resolution. Comput. Complex., 1–49 (2017). https://doi.org/10.1007/s00037-017-0163-1
Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem of tree-width. J. Comb. Theory Ser. B 58, 22–35 (1993)
Torán, J.: Space and width in propositional resolution. Comput. Complex. Column Bull. EATCS 83, 86–104 (2004)
Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic, Part 2, pp. 115–125. Consultants Bureau (1968)
Urquhart, A.: Hard examples for resolution. J. ACM 34, 209–219 (1987)
Urquhart, A.: The depth of resolution proofs. Stud. Logica 99, 349–364 (2011)
Acknowledgments
The authors would like to thank Osamu Watanabe and the ELC project were this research was started. We are also grateful to Dimitrios Thilikos and to the anonymous referees for helpful comments.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Galesi, N., Talebanfard, N., Torán, J. (2018). Cops-Robber Games and the Resolution of Tseitin Formulas. In: Beyersdorff, O., Wintersteiger, C. (eds) Theory and Applications of Satisfiability Testing – SAT 2018. SAT 2018. Lecture Notes in Computer Science(), vol 10929. Springer, Cham. https://doi.org/10.1007/978-3-319-94144-8_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-94144-8_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94143-1
Online ISBN: 978-3-319-94144-8
eBook Packages: Computer ScienceComputer Science (R0)