Lower bounds for invariant queries in logics with counting

  • Authors:

  • Leonid Libkin;Limsoon Wong

  • Affiliations:

  • Department of Computer Science, University of Toronto, Toronto, Ont., Canada M5S 3H5, Canada;Kent Ridge Digital Labs, 21 Heng Mui Keng Terrace, Singapore 119613, Singapore

  • Venue:
  • Theoretical Computer Science - Complexity and logic
  • Year:
  • 2002

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Abstract

We study the expressive power of counting logics in the presence of auxiliary relations such as orders and preorders. The simplest such logic is the first-order logic with counting. This logic captures the complexity class TC0 over ordered structures. We also consider first-order logic with arbitrary unary quantifiers and with infinitary extensions.We start by giving a simple direct proof that first-order logic with counting, in the presence of pre-orders that are almost-everywhere linear orders, cannot express the transitive closure of a binary relation. The proof is based on locality of formulae. We then show that the technique cannot be extended to linear orders. We further show that this result does not say anything about the power of invariant queries in first-order logic with counting vs. the class TC0, in the presence of these preorders.In the second part of the paper, we prove a separation result showing that, for all the counting logics above, a linear order is more powerful than a preorder that is a linear order almost everywhere. In fact, we prove that the expressive power of invariant queries in the presence of such preorders can be characterized by a property normally associated with first-order definability over unordered structures. We do this by using locality techniques from finite-model theory. However, as some standard notions of locality fail in this setting, we have to modify them to prove the main result.