Quillen Model Categories-Based Notions of Locality of Logics over Finite Structures

Hendrick Maia
{"title":"Quillen Model Categories-Based Notions of Locality of Logics over Finite Structures","authors":"Hendrick Maia","doi":"10.1017/bsl.2021.37","DOIUrl":null,"url":null,"abstract":"Abstract Locality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures \n$\\mathfrak {A}$\n and \n$\\mathfrak {B}$\n realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence \n$\\Phi $\n . Here d depends only on \n$\\Phi $\n ; (ii’) if the d-neighborhoods of two tuples \n$\\vec {a}_1$\n and \n$\\vec {a}_2$\n in a structure \n$\\mathfrak {A}$\n are isomorphic, then \n$\\mathfrak {A} \\models \\Phi (\\vec {a}_1) \\Leftrightarrow \\Phi (\\vec {a}_2)$\n . Again, d depends on \n$\\Phi $\n , and not on \n$\\mathfrak {A}$\n . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring \n$N_d(\\vec {a}) \\cong N_d(\\vec {b})$\n , you should only require \n$N_d(\\vec {a}) \\equiv _k N_d(\\vec {b})$\n , for some \n$k \\geq 0$\n . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic \n$\\mathcal {L}$\n is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and \n$\\mathcal {L}'$\n is a sub-logic of \n$\\mathcal {L}$\n , then \n$\\mathcal {L}'$\n is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k. Abstract prepared by Hendrick Maia. E-mail: hendrickmaia@gmail.com URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334956","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":"39 1","pages":"529 - 530"},"PeriodicalIF":0.0000,"publicationDate":"2022-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Bulletin of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/bsl.2021.37","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Abstract Locality is a property of logics, whose origins lie in the works of Hanf and Gaifman, having their utility in the context of finite model theory. Such a property is quite useful in proofs of inexpressibility, but it is also useful in establishing normal forms for logical formulas. There are generally two forms of locality: (i’) if two structures $\mathfrak {A}$ and $\mathfrak {B}$ realize the same multiset of types of neighborhoods of radius d, then they agree on a given sentence $\Phi $ . Here d depends only on $\Phi $ ; (ii’) if the d-neighborhoods of two tuples $\vec {a}_1$ and $\vec {a}_2$ in a structure $\mathfrak {A}$ are isomorphic, then $\mathfrak {A} \models \Phi (\vec {a}_1) \Leftrightarrow \Phi (\vec {a}_2)$ . Again, d depends on $\Phi $ , and not on $\mathfrak {A}$ . Form (i’) originated from Hanf’s works. Form (ii’) came from Gaifman’s theorem. There is no doubt about the usefulness of the notion of locality, which as seen applies to a huge number of situations. However, there is a deficiency in such a notion: all versions of the notion of locality refer to isomorphism of neighborhoods, which is a fairly strong property. For example, where structures simply do not have sufficient isomorphic neighborhoods, versions of the notion of locality obviously cannot be applied. So the question that immediately arises is: would it be possible to weaken such a condition and maintain Hanf/Gaifman-localities? Arenas, Barceló, and Libkin establish a new condition for the notions of locality, weakening the requirement that neighborhoods should be isomorphic, establishing only the condition that they must be indistinguishable in a given logic. That is, instead of requiring $N_d(\vec {a}) \cong N_d(\vec {b})$ , you should only require $N_d(\vec {a}) \equiv _k N_d(\vec {b})$ , for some $k \geq 0$ . Using the fact that logical equivalence is often captured by Ehrenfeucht–Fraïssé games, the authors formulate a game-based framework in which logical equivalence-based locality can be defined. Thus, the notion defined by the authors is that of game-based locality. Although quite promising as well as easy to apply, the game-based framework (used to define locality under logical equivalence) has the following problem: if a logic $\mathcal {L}$ is local (Hanf-, or Gaifman-, or weakly) under isomorphisms, and $\mathcal {L}'$ is a sub-logic of $\mathcal {L}$ , then $\mathcal {L}'$ is local as well. The same, however, is not true for game-based locality: properties of games guaranteeing locality need not be preserved if one passes to weaker games. The question that immediately arises is: is it possible to define the notion of locality under logical equivalence without resorting to game-based frameworks? In this thesis, I present a homotopic variation for locality under logical equivalence, namely a Quillen model category-based framework for locality under k-logical equivalence, for every primitive-positive sentence of quantifier-rank k. Abstract prepared by Hendrick Maia. E-mail: hendrickmaia@gmail.com URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334956
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
基于Quillen模型范畴的有限结构上逻辑局部性概念
局部性是逻辑的一个性质,它起源于Hanf和Gaifman的著作,在有限模型理论的背景下有其用途。这种性质在证明不可表达性时非常有用,但在建立逻辑公式的标准形式时也很有用。局部性一般有两种形式:(i ')如果两个结构$\mathfrak {A}$和$\mathfrak {B}$实现相同的半径d的邻域类型的多集,则它们在给定的句子$\Phi $上一致。这里d只依赖于$\Phi $;(ii)如果结构$\mathfrak {A}$中的两个元组$\vec {a}_1$和$\vec {a}_2$的d邻域是同构的,则$\mathfrak {A} \models \Phi (\vec {a}_1) \Leftrightarrow \Phi (\vec {a}_2)$。同样,d取决于$\Phi $,而不是$\mathfrak {A}$。形式(i’)源于汉夫的作品。形式(ii ')来自Gaifman定理。毫无疑问,局部性概念是有用的,正如我们所看到的,它适用于大量的情况。然而,这种概念有一个缺陷:所有版本的局部性概念都是指邻域的同构性,这是一个相当强的性质。例如,当结构根本没有足够的同构邻域时,局部性概念的版本显然不能应用。因此,立即出现的问题是:是否有可能削弱这种条件并维持汉夫/盖夫曼-地方?Arenas, Barceló和Libkin为局部性概念建立了一个新的条件,削弱了邻里应该同构的要求,只建立了它们必须在给定逻辑中不可区分的条件。也就是说,对于某些$k \geq 0$,不需要$N_d(\vec {a}) \cong N_d(\vec {b})$,而应该只需要$N_d(\vec {a}) \equiv _k N_d(\vec {b})$。利用Ehrenfeucht-Fraïssé游戏经常捕获逻辑等价的事实,作者制定了一个基于游戏的框架,其中可以定义基于逻辑等价的局部性。因此,作者定义的概念是基于游戏的局部性。虽然很有前途也很容易应用,但基于游戏的框架(用于定义逻辑等价下的局部性)存在以下问题:如果逻辑$\mathcal {L}$在同构下是局部性的(Hanf-或Gaifman-或弱),并且$\mathcal {L}'$是$\mathcal {L}$的子逻辑,那么$\mathcal {L}'$也是局部性的。然而,基于游戏的局部性却并非如此:如果玩家转向较弱的游戏,保证局部性的游戏属性便不需要被保留。马上出现的问题是:是否有可能在逻辑等价下定义局部性的概念而不诉诸于基于游戏的框架?在本文中,我提出了一个逻辑等价下的局部性的同伦变异,即对于每一个量词秩为k的基本肯定句,在k-逻辑等价下的局部性的Quillen模型基于范畴的框架。电子邮件:hendrickmaia@gmail.com URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/334956
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
自引率
0.00%
发文量
0
期刊最新文献
POUR-EL’S LANDSCAPE CATEGORICAL QUANTIFICATION POINCARÉ-WEYL’S PREDICATIVITY: GOING BEYOND A TOPOLOGICAL APPROACH TO UNDEFINABILITY IN ALGEBRAIC EXTENSIONS OF John MacFarlane, Philosophical Logic: A Contemporary Introduction, Routledge Contemporary Introductions to Philosophy, Routledge, New York, and London, 2021, xx + 238 pp.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1