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.

中文翻译：

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多结构游戏和数量词

我们研究了在两套$ \ mathcal {A} $和$ \ mathcal {B} $结构上玩的多结构游戏。这些游戏概括了Ehrenfeucht-Fra \“ {i} ss \'{e}游戏。而Ehrenfeucht-Fra \” {i} ss \'{e}游戏则捕获一阶句子的量词排名，即多结构游戏捕获量词的数量，即Spoiler在且仅当具有最多$ r $量词的一阶句子$ \ phi $，其中$ \ mathcal { A} $满足$ \ phi $，而$ \ mathcal {B} $中没有结构满足$ \ phi $。我们使用这些游戏来完全表征区分不同大小的线性阶所需的数量词，并开发出分析线性阶以外的结构的机制。