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.

@article{paperbot3451,
  title = {Multi-Structural Games and Number of Quantifiers},
  author = {Ronald Fagin and Jonathan Lenchner and Kenneth W. Regan and Nikhil Vyas},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 21, Issue 1},
  year = {2025},
  doi = {10.46298/lmcs-21(1:10)2025}
}