Multi-Structural Games and Number of Quantifiers
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866915136152797184 |
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| author | Fagin, Ronald Lenchner, Jonathan Regan, Kenneth W. Vyas, Nikhil |
| author_facet | Fagin, Ronald Lenchner, Jonathan Regan, Kenneth W. Vyas, Nikhil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_14709 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Multi-Structural Games and Number of Quantifiers Fagin, Ronald Lenchner, Jonathan Regan, Kenneth W. Vyas, Nikhil Logic in Computer Science 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. |
| title | Multi-Structural Games and Number of Quantifiers |
| topic | Logic in Computer Science |
| url | https://arxiv.org/abs/2104.14709 |