Multi-Structural Games and Number of Quantifiers

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 t...

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Bibliographic Details
Published in:Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science pp. 1 - 13
Main Authors: Fagin, Ronald, Lenchner, Jonathan, Regan, Kenneth W., Vyas, Nikhil
Format: Conference Proceeding
Language:English
Published: IEEE 29.06.2021
Subjects:
Complexity theory
Computer science
Games
Lattices
Machinery
Size measurement
Online Access:Get full text
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Summary: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.
DOI:10.1109/LICS52264.2021.9470756