Multi-Structural Games and Number of Quantifiers - Fagin, Ronald; Lenchner, Jonathan; Regan, Kenneth W.; Vyas, Nikhil (Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science, 2021)
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 ϕ with at most r quantifiers, where every structure in {\mathcal{A}} satisfies ϕ and no structure in {\mathcal{B}} satisfies ϕ. 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.