On the Number of Quantifiers as a Complexity Measure

TL;DR

This paper advances the understanding of the number of quantifiers as a complexity measure in first-order logic by providing explicit examples that distinguish between quantifier rank and total quantifier count, especially in finite structures.

Contribution

It introduces new results that differentiate minimum quantifier number from quantifier rank and variable count, and constructs properties requiring exponentially more quantifiers than their quantifier rank suggests.

Findings

Explicit properties with quantifier rank k need 2^{Ω(k^2)} quantifiers.

Distinction between minimum quantifier number and other complexity measures.

Extension of game-theoretic analysis to quantify logical complexity.

Abstract

In 1981, Neil Immerman described a two-player game, which he called the "separability game" \cite{Immerman81}, that captures the number of quantifiers needed to describe a property in first-order logic. Immerman's paper laid the groundwork for studying the number of quantifiers needed to express properties in first-order logic, but the game seemed to be too complicated to study, and the arguments of the paper almost exclusively used quantifier rank as a lower bound on the total number of quantifiers. However, last year Fagin, Lenchner, Regan and Vyas rediscovered the games, provided some tools for analyzing them, and showed how to utilize them to characterize the number of quantifiers needed to express linear orders of different sizes. In this paper, we push forward in the study of number of quantifiers as a bona fide complexity measure by establishing several new results. First we carefully distinguish minimum number of quantifiers from the more usual descriptive complexity measures, minimum quantifier rank and minimum number of variables. Then, for each positive integer $k$, we give an explicit example of a property of finite structures (in particular, of finite graphs) that can be expressed with a sentence of quantifier rank $k$, but where the same property needs $2^{\Omega (k^2)}$ quantifiers to be expressed.