Multi-Structural Games and Number of Quantifiers

TL;DR

This paper introduces multi-structural games that extend Ehrenfeucht-Fra"{i}ssé games to characterize the exact number of quantifiers needed to distinguish between structures, with applications to linear orders and beyond.

Contribution

It develops a new class of games capturing quantifier counts, providing a complete characterization for linear orders and tools for analyzing more complex structures.

Findings

Characterization of quantifier counts for linear orders.

Development of machinery for analyzing structures beyond linear orders.

Complete game-theoretic framework for quantifier analysis.

Abstract

We study multi-structural games, played on two sets $\mathcal{A}$ and $\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fra\"{i}ss\'{e} games. Whereas Ehrenfeucht-Fra\"{i}ss\'{e} games capture the quantifier rank of a first-order sentence, multi-structural games capture the number of quantifiers, in the sense that Spoiler wins the $r$-round game if and only if there is a first-order sentence $\phi$ with at most $r$ quantifiers, where every structure in $\mathcal{A}$ satisfies $\phi$ and no structure in $\mathcal{B}$ satisfies $\phi$. We use these games to give a complete characterization of the number of quantifiers required to distinguish linear orders of different sizes, and develop machinery for analyzing structures beyond linear orders.