The Power of Counting Logics on Restricted Classes of Finite Structures

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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.