Multi-Structural Games and Number of Quantifiers

Abstract

We study multi-structural games, played on two sets $\mathcal{A}$ and $\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fraïssé games. Whereas Ehrenfeucht-Fraïssé 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 $\ddot{I}•$ with at most $r$ quantifiers, where every structure in $\mathcal{A}$ satisfies $\ddot{I}•$ and no structure in $\mathcal{B}$ satisfies $\ddot{I}•$. 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.