Abstract
Although Cai, Fürer and Immerman have shown that fixed-point logic with counting (IFP+C) does not express all polynomial-time properties of finite structures, there have been a number of results demonstrating that the logic does capture P on specific classes of structures. Grohe and Mariño showed that IFP+C captures P on classes of structures of bounded treewidth, and Grohe showed that IFP+C captures P on planar graphs. We show that the first of these results is optimal in two senses. We show that on the class of graphs defined by a non-constant bound on the tree-width of the graph, IFP+C fails to capture P. We also show that on the class of graphs whose local tree-width is bounded by a non-constant function, IFP+C fails to capture P. Both these results are obtained by an analysis of the Cai–Fürer–Immerman (CFI) construction in terms of the treewidth of graphs, and cops and robber games; we present some other implications of this analysis. We then demonstrate the limits of this method by showing that the CFI construction cannot be used to show that IFP+C fails to capture P on proper minor-closed classes.
Research partly carried during the programme ‘Logic and Algorithms’ at the Isaac Newton Institute for Mathematical Sciences, University of Cambridge, 20 Clarkson Road, Cambridge, CB3 0EH, UK.
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References
Atserias, A., Bulatov, A., Dawar, A.: Affine systems of equations and counting infinitary logic. In: Proc. 34th International Colloquium on Automata, Languages and Programming. LNCS, vol. 4596, pp. 558–570. Springer, Heidelberg (to appear, 2007)
Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)
Berwanger, D., Grädel, E.: Entanglement: A measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)
Cai, J.-Y., Fürer, M., Immerman, N.: An optimal lower bound on the number of variables for graph identification. Combinatorica 12(4), 389–410 (1992)
Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwan, J. (ed.) Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 193–242. Elsevier, Amsterdam (1990)
Dawar, A., Grohe, M., Kreutzer, S.: Locally excluding a minor. In: Proc. 22nd IEEE Annual Symposium on Logic in Computer Science, IEEE Computer Society, Los Alamitos (to appear, 2007)
Dawar, A., Grohe, M., Kreutzer, S., Schweikardt, N.: Approximation schemes for first-order definable optimisation problems. In: Proc. 21st IEEE Annual Symposium on Logic in Computer Science, pp. 411–420. IEEE Computer Society, Los Alamitos (2006)
Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, 2nd edn. Springer, Heidelberg (1999)
Eppstein, D.: Diameter and treewidth in minor-closed graph families. Algorithmica 27, 275–291 (2000)
Frick, M., Grohe, M.: Deciding first-order properties of locally tree-decomposable structures. Journal of the ACM 48, 1184–1206 (2001)
Grohe, M.: Fixed-point logics on planar graphs. In: Proc. 13th IEEE Annual Symposium on Logic in Computer Science, pp. 6–15. IEEE Computer Society, Los Alamitos (1998)
Grohe, M.: Isomorphism testing for embeddable graphs through definability. In: Proc. 32nd ACM Symposium on Theory of Computing, pp. 63–72. ACM, New York (2000)
Grohe, M., Mariño, J.: Definability and descriptive complexity on databases of bounded tree-width. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 70–82. Springer, Heidelberg (1998)
Hella, L.: Logical hierarchies in PTIME. Information and Computation 129(1), 1–19 (1996)
Immerman, N., Lander, E.S.: Describing graphs: A first-order approach to graph canonization. In: Selman, A. (ed.) Complexity Theory Retrospective, pp. 59–81. Springer, Heidelberg (1990)
Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory Series B 82, 138–155 (2001)
Nešetřil, J., de Ossona Mendez, P.: The grad of a graph and classes with bounded expansion. In: Raspaud, A., Delmas, O. (eds.) International Colloquium on Graph Theory. Electronic Notes in Discrete Mathematics, vol. 22, pp. 101–106. Elsevier, Amsterdam (2005)
Obdržálek, J.: DAG-width: Connectivity measure for directed graphs. In: Proc. 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 814–821 (2006)
Otto, M.: Bounded Variable Logics and Counting — A Study in Finite Models. Lecture Notes in Logic, vol. 9. Springer, Heidelberg (1997)
Robertson, N., Seymour, P.D.: Graph minors II: Algorithmic aspects of tree-width. Journal of Algorithms 7(3), 307–322 (1986)
Robertson, N., Seymour, P.D.: Graph minors V: Excluding a planar graph. Journal of Combinatorial Theory Series B 41(1), 91–114 (1986)
Robertson, N., Seymour, P.D.: Graph minors XX: Wagner’s conjecture. Journal of Combinatorial Theory Series B 92(2), 325–357 (2004)
Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. Journal of Combinatorial Theory Series B 58, 22–33 (1993)
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Dawar, A., Richerby, D. (2007). The Power of Counting Logics on Restricted Classes of Finite Structures. In: Duparc, J., Henzinger, T.A. (eds) Computer Science Logic. CSL 2007. Lecture Notes in Computer Science, vol 4646. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74915-8_10
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DOI: https://doi.org/10.1007/978-3-540-74915-8_10
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