On the Descriptive Complexity of Linear Algebra
WoLLIC '08 Proceedings of the 15th international workshop on Logic, Language, Information and Computation
CFI Construction and Balanced Graphs
FAW '09 Proceedings of the 3d International Workshop on Frontiers in Algorithmics
Database theory: query languages
Algorithms and theory of computation handbook
Finite model theory on tame classes of structures
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Hi-index | 0.00 |
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.