Multi-Structural Games and Number of Quantifiers

We study multi-structural games, played on two sets A and 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 A satisfies ϕ and no structure in 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 we develop machinery for analyzing structures beyond linear orders. * This paper is an expanded and reorganized version of the LICS (Logic in Computer Science) 2021 paper [FLRV21a], which points to the Computer Science arXiv, version 1, entry [FLRV21b] for full proofs of its theorems.