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 associated resolution measures on Tseitin formulas correspond exactly to those under the restriction of Davis-Putnam resolution, implying that this kind of resolution is optimal on Tseitin formulas for all the considered measures.
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.
- I. Adler. 2004. Marshals, monotone marshals, and hypertree width. Journal of Graph Theory 47 (2004), 275--296.Google ScholarDigital Library
- M. Alekhnovich and A. A. Razborov. 2011. Satisfiability, branch-width and Tseitin tautologies. Computational Complexity 20, 4 (2011), 649--678.Google ScholarDigital Library
- M. Alekhnovich, E. Ben-Sasson, A. A. Razborov, and A. Wigderson. 2002. Space complexity in propositional calculus. SIAM J. Comput. 31, 4 (2002), 1184--1211.Google ScholarDigital Library
- A. Atserias. 2008. On digraph coloring problems and trewidth duality. European Journal of Combinatorics 29 (2008), 796--820.Google ScholarDigital Library
- A. Atserias and V. Dalmau. 2003. A combinatorial characterization of resolution width. In Proceedings of the 18th IEEE Conference on Computational Complexity. 239--247.Google Scholar
- P. Beame, C. Beck, and R. Impagliazzo. 2016. Time-space trade-offs in resolution: Superpolynomial lower bounds for superlinear space. SIAM J. Comput. 49, 4 (2016), 1612--1645.Google ScholarCross Ref
- C. Beck, J. Nordström, and B. Tang. 2013. Some trade-off results for polynomial calculus: Extended abstract. Proceedings of the 45th ACM Symposium on the Theory of Computing (2013), 813--822.Google Scholar
- E. Ben-Sasson and A. Wigderson. 2001. Short proofs are narrow—Resolution made simple. Journal of the ACM 48, 2 (2001), 149--169.Google ScholarDigital Library
- J. A. Ellis, I. H. Sudborough, and J. S. Turner. 1994. The vertex separation and search number of a graph. Information and Computation 113, 1 (1994), 50--79.Google ScholarDigital Library
- J. L. Esteban and J. Torán. 2001. Space bounds for resolution. Information and Computation 171, 1 (2001), 84--97.Google ScholarDigital Library
- F. V. Fomin and D. Thilikos. 2008. An annotated bibliography on guaranteed graph searching. Theoretical Computer Science 399 (2008), 236--245.Google ScholarDigital Library
- G. Gottlob, N. Leone, and F. Scarello. 2003. Robbers, marshals and guards: Game theoretic and logical characterizations of hypertree width. Journal of Computer and System Sciences 66 (2003), 775--808.Google ScholarDigital Library
- P. Hertel. 2008. Applications of games to propositional proof complexity. Ph.D. Thesis. University of Toronto, 2008.Google Scholar
- L. M. Kirousis and C. H. Papadimitriou. 1986. Searching and pebbling. Theoretical Computer Science 47, 3 (1986), 205--218.Google ScholarCross Ref
- A. S. LaPaugh. 1883. Recontamination does not help to search a graph. Tech. Report Electrical Engineering and Comp. Science Dept. Princeton University, 1883.Google Scholar
- A. Razborov. 2018. On space and depth in resolution. Computational Complexity 27, 3 (2018), 511--559.Google ScholarDigital Library
- P. D. Seymour and R. Thomas. 1993. Graph searching and a min-max theorem of tree-width. Journal of Combinatorial Theory Series B 58 (1993), 22--35.Google ScholarDigital Library
- J. Torán. 2004. Space and width in propositional resolution. Computational Complexity Column, Bulletin of EATCS 83 (2004), 86--104.Google Scholar
- G. S. Tseitin. 1968. On the complexity of derivation in propositional calculus. In Studies in Constructive Mathematics and Mathematical Logic, Part 2, pages 115--125. Consultants Bureau, 1968.Google Scholar
- A. Urquhart. 1987. Hard examples for resolution. Journal of the ACM 34 (1987), 209--219.Google ScholarDigital Library
- A. Urquhart. 2011. The depth of resolution proofs. Studia Logica 99 (2011), 349--364.Google ScholarDigital Library
- A. Urquhart. 2012. Width and size of regular resolution proofs. Logical Methods in Computer Science 8 (2012), 1--15.Google ScholarCross Ref
Index Terms
- Cops-Robber Games and the Resolution of Tseitin Formulas
Recommendations
Automating Resolution is NP-Hard
We show that the problem of finding a Resolution refutation that is at most polynomially longer than a shortest one is NP-hard. In the parlance of proof complexity, Resolution is not automatable unless P = NP. Indeed, we show that it is NP-hard to ...
Cops-Robber Games and the Resolution of Tseitin Formulas
Theory and Applications of Satisfiability Testing – SAT 2018AbstractWe 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 ...
On Tseitin Formulas, Read-Once Branching Programs and Treewidth
Computer Science – Theory and ApplicationsAbstractWe show that any nondeterministic read-once branching program that decides a satisfiable Tseitin formula based on an grid graph has size at least . Then using the Excluded Grid Theorem by Robertson and Seymour we show that for arbitrary ...
Comments