The Power of Counting Logics on Restricted Classes of Finite Structures

Abstract. Although Cai, F¨urer and Immerman have shown that fixedpoint logic with counting (IFP + C) does not express all polynomialtime 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 Mari˜no 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–F¨urer–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 demon...

• Type: Conference
• Year: 2007
• Where: CSL
• Authors: Anuj Dawar, David Richerby
• Keywords: CSL 2007, Logic Does Capture, Polynomialtime Properties, finite Structures