Generalized quantifiers and pebble games on finite structures

Abstract

First-order logic is known to have a severely limited expressive power on finite structures. As a result, several different extensions have been investigated, including fragments of second-order logic, fixpoint logic, and the infinitary logic L ∞ ω ω in which every formula has only a finite number of variables. In this paper, we study generalized quantifiers in the realm of finite structures and combine them with the infinitary logic L ∞ ω ω to obtain the logics L ∞ ω ω ( Q), where Q = { Q i : iϵ I} is a family of generalized quantifiers on finite structures. Using the logics L ∞ ω ω ( Q), we can express polynomial-time properties that are not definable in L ∞ ω ω , such as “there is an even number of x” and “there exists at least n 2 x ” ( n is the size of the universe), without going to second-order logic. We show that equivalence of finite structures relative to L ∞ ω ω ( Q) can be characterized in terms of certain pebble games that are a variant of the Ehrenfeucht—Fraïssé games. We combine this game-theoretic characterization with sophisticated combinatorial tools from Ramsey theory, such as van der Waerden's Theorem and Folkman's Theorem, in order to investigate the scope and limits of generalized quantifiers in finite model theory. We obtain sharp lower bounds for expressibility in the logics L ∞ ω ω ( Q) and discover an intrinsic difference between adding finitely many simple unary generalized quantifiers to L ∞ ω ω adding infinitely many. In particular, we show that if Qis a finite sequence of simple unary generalized quantifiers, then the equicardinality, or Härtig, quantifier is not definable in L ∞ ω ω ( Q). We also show that the query “does the equivalence relation E have an even number of equivalence classes” is not definable in the extension L ∞ ω ω ( I, Q) of L ∞ ω ω by the Härtig quantifier I and any finite sequence Q of simple unary generalized quantifiers.