The power of counting logics on restricted classes of finite structures

  • Authors:

  • Anuj Dawar;David Richerby

  • Affiliations:

  • University of Cambridge Computer Laboratory, Cambridge, UK;Department of Mathematics, University of Athens, Athens, Greece

  • Venue:
  • CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
  • Year:
  • 2007

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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.