Counting and Locality over Finite Structures: A Survey

  • Authors:

  • Leonid Libkin;Juha Nurmonen

  • Affiliations:

  • -;-

  • Venue:
  • ESSLLI '97 Revised Lectures from the 9th European Summer School on Logic, Language, and Information: Generalized Quantifiers and Computation
  • Year:
  • 1997

Quantified Score

Hi-index 0.00

Visualization

Abstract

We survey recent results on logics with counting and their local properties. We first consider game-theoretic characterizations of first-order logic and its counting extensions provided by unary generalized quantifiers. We then study Gaifman's and Hanf's locality theorems, their connection with game characterizations, and examples of their usage in proving expressivity bounds for first-order logic and its extensions. We review the abstract notions of Gaifman's and Hanf's locality, and show how they are related. We also consider a closely related bounded degree property, and demonstrate its usefulness in proving expressivity bounds. We discuss two applications. One deals with proving lower bounds for the complexity class TC0. In particular, we use logical characterization of TC0 and locality theorems for first-order with counting quantifiers to provide lower bounds. We then explain how the notions of locality are used in database theory to prove that extensions of relational calculus with aggregate functions and grouping still lack the power to express fixpoint computation.