Ronald Fagin ; Jonathan Lenchner ; Kenneth W. Regan ; Nikhil Vyas - Multi-Structural Games and Number of Quantifiers

lmcs:9181 - Logical Methods in Computer Science, January 29, 2025, Volume 21, Issue 1 - https://doi.org/10.46298/lmcs-21(1:10)2025
Multi-Structural Games and Number of QuantifiersArticle

Authors: Ronald Fagin ORCID; Jonathan Lenchner ORCID; Kenneth W. Regan ORCID; Nikhil Vyas ORCID

    We study multi-structural games, played on two sets $\mathcal{A}$ and $\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fra\"{i}ssé games. Whereas Ehrenfeucht-Fra\"{i}ssé 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.


    Volume: Volume 21, Issue 1
    Published on: January 29, 2025
    Accepted on: November 6, 2024
    Submitted on: March 7, 2022
    Keywords: Computer Science - Logic in Computer Science

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