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arXiv · 2104.14709

Multi-Structural Games and Number of Quantifiers

Ronald Fagin·Jonathan Lenchner·Kenneth W. Regan·Nikhil Vyas
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Abstract

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.

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Ronald Fagin, Jonathan Lenchner, Kenneth W. Regan, Nikhil Vyas. 2021-04-30. Multi-Structural Games and Number of Quantifiers. https://doi.org/10.46298/lmcs-21(1:10)2025

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