{"title":"BSL volume 28 issue 3 Cover and Back matter","authors":"","doi":"10.1017/bsl.2022.32","DOIUrl":"https://doi.org/10.1017/bsl.2022.32","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86012947","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"BSL volume 28 issue 3 Cover and Front matter","authors":"","doi":"10.1017/bsl.2022.31","DOIUrl":"https://doi.org/10.1017/bsl.2022.31","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88692138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"2022 WINTER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC WITH THE AMS Seattle, Washington Joint Mathematics Meeting January 7–8, 2022","authors":"","doi":"10.1017/bsl.2022.25","DOIUrl":"https://doi.org/10.1017/bsl.2022.25","url":null,"abstract":"","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78218609","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending $mathsf {RCA}_0$ and axiomatizable by a $Pi ^1_{k+2}$ sentence, and for any $ngeq k+1$ , $$begin{align*}T_0+ mathrm{RFN}_{varPi^1_{n+2}}(T) = T_0 + mathrm{TI}_{varPi^1_n}(varepsilon_0), end{align*}$$ $$begin{align*}T_0+ mathrm{RFN}_{varSigma^1_{n+1}}(T) = T_0+ mathrm{TI}_{varPi^1_n}(varepsilon_0)^{-}, end{align*}$$ where T is $T_0$ augmented with full induction, and $mathrm {TI}_{varPi ^1_n}(varepsilon _0)^{-}$ denotes the schema of transfinite induction up to $varepsilon _0$ for $varPi ^1_n$ formulas without set parameters.
{"title":"A NOTE ON FRAGMENTS OF UNIFORM REFLECTION IN SECOND ORDER ARITHMETIC","authors":"Emanuele Frittaion","doi":"10.1017/bsl.2022.23","DOIUrl":"https://doi.org/10.1017/bsl.2022.23","url":null,"abstract":"Abstract We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory \u0000$T_0$\u0000 extending \u0000$mathsf {RCA}_0$\u0000 and axiomatizable by a \u0000$Pi ^1_{k+2}$\u0000 sentence, and for any \u0000$ngeq k+1$\u0000 , \u0000$$begin{align*}T_0+ mathrm{RFN}_{varPi^1_{n+2}}(T) = T_0 + mathrm{TI}_{varPi^1_n}(varepsilon_0), end{align*}$$\u0000 \u0000$$begin{align*}T_0+ mathrm{RFN}_{varSigma^1_{n+1}}(T) = T_0+ mathrm{TI}_{varPi^1_n}(varepsilon_0)^{-}, end{align*}$$\u0000 where T is \u0000$T_0$\u0000 augmented with full induction, and \u0000$mathrm {TI}_{varPi ^1_n}(varepsilon _0)^{-}$\u0000 denotes the schema of transfinite induction up to \u0000$varepsilon _0$\u0000 for \u0000$varPi ^1_n$\u0000 formulas without set parameters.","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78419421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Colloquium, A. Mickiewicz, Szymon Chlebowski, Andrzej Gajda, Marta Gawek, Patrycja Kupś, Paweł Łupkowski, Dawid Ratajczyk, Agata Tomczyk, A. Wasielewska, Joanna Golinska-Pilarek, L. Kolodziejczyk, M. Nasieniewski, J. Pogonowski, Tomasz F. Skura, K. Swirydowicz, M. Soskova, B. Monin, L. Ros
of the invited 31st Annual Gödel Lecture ELISABETH BOUSCAREN, The ubiquity of configurations in model theory. CNRS—Université Paris-Saclay, Gif-sur-Yvette, France. E-mail: elisabeth.bouscaren@universite-paris-saclay.fr. Originally in Classification Theory, then in Geometric Stability, and now, beyond Stability, in Tame Model Theory, one common essential feature is the identification and study of some geometric configurations, of combinatorial and dimensional theoretic nature. They can witness the combinatorial and the model theoretic complexity of a theory or indicate the existence of specific definable algebraic structures. This enables model theory to tackle questions from very diverse subjects. We will attempt to illustrate the importance of these configurations through some examples. Abstract of invited tutorialsof invited tutorials KRZYSZTOF KRUPIŃSKI, Topological dynamics in model theory. University of Wrocław, Wrocław, Poland. E-mail: kkrup@math.uni.wroc.pl. Some fundamental notions and methods of topological dynamics were introduced to model theory by Newelski in the mid-2000s. In the first part of my tutorial, I will recall some basic notions of topological dynamics, discuss the flows which appear naturally in model theory (as various spaces of types), and give applications of basic topological dynamics to some group covering results of Newelski such as: if an א0-saturated group is covered by countably many 0-type-definable sets Xn , n ∈ , then for some finite A ⊆ G and n ∈ , G = AXnX –1 n . In the second part, I will define the Ellis semigroup and Ellis group of a flow, and focus on connections between the Ellis groups of natural flows in model theory and certain invariants of definable groups (quotients by model-theoretic connected components) or first order theories (Galois groups of first order theories as well as spaces of strong types). In particular, I will discuss the results of Pillay, Rzepecki, and myself which present certain invariants of this kind as quotients of compact (Hausdorff) groups (which are canonical Hausdorff quotients of Ellis groups). This has various consequences obtained by Pillay, Rzepecki, and myself, e.g., it leads to a general result that model-theoretic type-definability of a bounded invariant equivalence relation defined on a single complete type over ∅ is equivalent to descriptive set theoretic smoothness of this relation. 270 LOGIC COLLOQUIUM ’21 In the last part, I will discuss a definable variant of Kechris–Pestov–Todorčević (KPT) theory, developed by Lee, Moconja, and myself. KPT theory studies relationships between dynamical properties of the groups of automorphisms of Fraïssé structures and Ramseytheoretic (so combinatorial) properties of the underlying Fraïssé classes. In our research, the idea is to find interactions between dynamical properties of first order theories (i.e., properties related to the actions of the automorphism group of a sufficiently saturated model on various types spaces ove
受邀参加第31届Gödel年度讲座ELISABETH BOUSCAREN，构型在模型理论中的普遍性。法国巴黎萨克莱cnrs大学。电子邮件:elisabeth.bouscaren@universite-paris-saclay.fr。最初在分类理论，然后在几何稳定性，现在，超越稳定性，在Tame模型理论，一个共同的基本特征是识别和研究一些几何构型，组合和量纲理论的性质。它们可以证明一个理论的组合复杂性和模型论复杂性，或表明特定可定义代数结构的存在。这使得模型理论能够解决各种各样的问题。我们将尝试通过一些示例来说明这些配置的重要性。特邀教程摘要KRZYSZTOF KRUPIŃSKI，拓扑动力学中的模型理论。波兰Wrocław, Wrocław大学。电子邮件:kkrup@math.uni.wroc.pl。Newelski在2000年代中期将拓扑动力学的一些基本概念和方法引入到模型理论中。在我的教程的第一部分,我将记得拓扑动力学的一些基本概念,讨论出现自然的流动模型理论(如各种空间的类型),和给应用程序的基本拓扑动态等集团覆盖Newelski的结果:如果一个א0-saturated Xn集团是由许多0-type-definable可数集,n∈,然后对一些有限⊆G和n∈,G = AXnX 1 n。在第二部分中，我将定义流的Ellis半群和Ellis群，并重点讨论模型论中自然流的Ellis群与可定义群(模型论连通分量商)或一阶理论(一阶理论的伽罗瓦群以及强类型空间)的某些不变量之间的联系。特别地，我将讨论Pillay, Rzepecki和我自己的结果，这些结果将这种不变量作为紧(Hausdorff)群的商(它们是Ellis群的正则Hausdorff商)。这就有了Pillay, Rzepecki和我自己得到的各种结果，例如，它得出了一个一般的结果，即定义在单个完备类型上的有界不变等价关系的模型论类型可定义性等价于该关系的描述集论平滑性。在最后一部分中，我将讨论由Lee、Moconja和我本人提出的kechris - pestov - todor<e:1> eviki (KPT)理论的一个可定义变体。KPT理论研究Fraïssé结构的自同构群的动力学性质与底层Fraïssé类的ramseytheory(即组合)性质之间的关系。在我们的研究中，我们的想法是找到一阶理论的动力学性质(即与该模型上各种类型空间上充分饱和模型的自同构群的作用有关的性质)与该理论的ramsey理论性质的可定义版本之间的相互作用。这导致了类似于KPT理论的各种结果(即，理论的可定义的极端适应性的组合表征)，但也导致了一些相当新颖的定理，例如，产生一阶理论的Ellis群的收益性准则。本文由波兰国家科学中心资助，项目编号2015/19/B/ST1/ 01151、2016/22/E/ST1/00450、2018/31/B/ST1/00357。ANDREW MARKS，描述Borel复杂性和可分解性的应用。加州大学洛杉矶分校，美国加州洛杉矶。电子邮件:marks@math.ucla.edu。我们给出了Borel层次结构中集合Σn难的一个新的表征。利用Antonio Montalban在可计算性理论中进行优先级论证的真阶段方法证明了这一特征。我们用它来证明可分解性猜想，假设射影确定性。可分解性猜想描述了哪些Borel函数可分解为具有Πn域的部分连续函数的可数并。这是和亚当·戴的合作。ARTEM CHERNIKOV，模型理论中的措施。美国加州大学洛杉矶分校数学系，加州洛杉矶900951555电子邮件:chernikov@math.ucla.edu。URL地址:http://www.math.ucla.edu/~chernikov/。在模型理论中，类型是结构中可定义集合的布尔代数上的一个超过滤器，它与有限加性{0,1}值测度是一样的。这是一种特殊的Keisler测度，它是可定义集合的布尔代数上的有限加性实值概率测度。Keisler在80年代末提出，Keisler测量在过去十年中成为研究的中心对象。这是由几条相互交织的研究路线推动的。 其中之一(也许是最古老的一个)是概率和连续逻辑的发展。另一种是研究o-minimal中的可定义群，更普遍的是在NIP理论中，导致与拓扑动力学的有趣联系。进一步的动机来自于加法和极值组合的应用，将上述方向结合起来。我将概述这一学科的一些最新发展。[10] A. Chernikov，模型理论，Keisler测度和类群，《中国科学》，vol. 24 (2018)， no. 1。3，第336-339页。[10] A. Chernikov和K. Gannon，可定义卷积和幂等Keisler测度。以色列数学学报，2021,arXiv:2004.10378。[10]刘建军，刘建军，刘建军，张建军，张建军，张建军，张建军，张建军，张建军，张建军，张建军，张建军。[b] A. Chernikov和P. Simon，明确可服从的NIP组。《美国数学学会学报》，2018年第31卷，第2期。3，第609-641页。[10] A. Chernikov和S. Starchenko，远端结构的正则引理。《欧洲数学学会学报》，2018年第20卷，第2期。10，第2437-2466页。[10]张建军，张建军。超图正则性与vc维的关系，中国科学:自然科学版，2016,37(4):726 - 726。维拉费希尔，实数的组合集。维也纳大学，奥地利维也纳。电子邮件:vera.fischer@univie.ac.at。实数的无限组合集，如几乎不相交族、共有限群、独立族和塔，在实数线的集合论性质的研究中占有中心地位。特别感兴趣的是这样的实数的极值集，即，在包含下最大的组合集，关于期望的性质，它们的可能的基数，可定义性，以及ZFC依赖的存在或不存在。对这种实数组合集的研究与各种强迫技术的发展密切相关。在这次演讲中，我们将看到这个主题的一些最新进展，并指出一些有趣的悬而未决的问题。相对随机序列共有的信息。惠灵顿维多利亚大学，新西兰惠灵顿。电子邮件:noam.greenberg@vuw.ac.nz。如果X和Y是相对随机的，那么X和Y有什么共同的信息?我们使用算法随机性和可计算性理论来解释这个问题。答案涉及一些意想不到的成分，如勒贝格密度定理和线性规划，并揭示了K平凡度中丰富的图灵度层次。BENOÎT莫宁，米利肯树定理的计算内容。克兰斯泰伊大学，克兰斯泰伊，法国。电子邮件:benoit.monin@computability.fr。密立肯树定理是拉姆齐定理在树上的推广。例如，它意味着如果我们对两个长度相同的字符串的所有集合赋值，其中一个在k种颜色中，存在一个无限二叉树其中每一对高度相同的字符串都具有相同的颜色。我们将从可计算性理论和逆向数学的角度给出关于密立肯树定理的一些结果。全共性的广义描述集理论及其应用。都灵大学，意大利都灵。电子邮件:luca.mottoros@unito.it。广义描述集合论是当今一个非常活跃的研究领域。这个想法是发展一个经典描述性集合理论的高级模拟，其中系统地用不可数基数κ代替。除了少数例外，这一领域的论文往往集中在常规枢机的情况下
{"title":"2021 EUROPEAN SUMMER MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC LOGIC COLLOQUIUM ’21 Adam Mickiewicz University Poznań, Poland July 19–24, 2021","authors":"Colloquium, A. Mickiewicz, Szymon Chlebowski, Andrzej Gajda, Marta Gawek, Patrycja Kupś, Paweł Łupkowski, Dawid Ratajczyk, Agata Tomczyk, A. Wasielewska, Joanna Golinska-Pilarek, L. Kolodziejczyk, M. Nasieniewski, J. Pogonowski, Tomasz F. Skura, K. Swirydowicz, M. Soskova, B. Monin, L. Ros","doi":"10.1017/bsl.2022.17","DOIUrl":"https://doi.org/10.1017/bsl.2022.17","url":null,"abstract":"of the invited 31st Annual Gödel Lecture ELISABETH BOUSCAREN, The ubiquity of configurations in model theory. CNRS—Université Paris-Saclay, Gif-sur-Yvette, France. E-mail: elisabeth.bouscaren@universite-paris-saclay.fr. Originally in Classification Theory, then in Geometric Stability, and now, beyond Stability, in Tame Model Theory, one common essential feature is the identification and study of some geometric configurations, of combinatorial and dimensional theoretic nature. They can witness the combinatorial and the model theoretic complexity of a theory or indicate the existence of specific definable algebraic structures. This enables model theory to tackle questions from very diverse subjects. We will attempt to illustrate the importance of these configurations through some examples. Abstract of invited tutorialsof invited tutorials KRZYSZTOF KRUPIŃSKI, Topological dynamics in model theory. University of Wrocław, Wrocław, Poland. E-mail: kkrup@math.uni.wroc.pl. Some fundamental notions and methods of topological dynamics were introduced to model theory by Newelski in the mid-2000s. In the first part of my tutorial, I will recall some basic notions of topological dynamics, discuss the flows which appear naturally in model theory (as various spaces of types), and give applications of basic topological dynamics to some group covering results of Newelski such as: if an א0-saturated group is covered by countably many 0-type-definable sets Xn , n ∈ , then for some finite A ⊆ G and n ∈ , G = AXnX –1 n . In the second part, I will define the Ellis semigroup and Ellis group of a flow, and focus on connections between the Ellis groups of natural flows in model theory and certain invariants of definable groups (quotients by model-theoretic connected components) or first order theories (Galois groups of first order theories as well as spaces of strong types). In particular, I will discuss the results of Pillay, Rzepecki, and myself which present certain invariants of this kind as quotients of compact (Hausdorff) groups (which are canonical Hausdorff quotients of Ellis groups). This has various consequences obtained by Pillay, Rzepecki, and myself, e.g., it leads to a general result that model-theoretic type-definability of a bounded invariant equivalence relation defined on a single complete type over ∅ is equivalent to descriptive set theoretic smoothness of this relation. 270 LOGIC COLLOQUIUM ’21 In the last part, I will discuss a definable variant of Kechris–Pestov–Todorčević (KPT) theory, developed by Lee, Moconja, and myself. KPT theory studies relationships between dynamical properties of the groups of automorphisms of Fraïssé structures and Ramseytheoretic (so combinatorial) properties of the underlying Fraïssé classes. In our research, the idea is to find interactions between dynamical properties of first order theories (i.e., properties related to the actions of the automorphism group of a sufficiently saturated model on various types spaces ove","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75972883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as $mathbb Q_{p}langle Xrangle $ , $mathbb Z_{p}langle Xrangle $ , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4
什么时候是环的理想基或素数?通过检验它的产生源，人们可以在许多情况下明确而一致地检验理想的性质。我们试图在p进幂级数环中建立这样的可定义公式，例如$mathbb Q_{p}langle Xrangle $， $mathbb Z_{p}langle Xrangle $，以及在更一般的赋值环及其分数域上幂级数的相关环。我们得到了一个可定义的、一致的根性检验，并在一维情况下得到了素数检验。这建立在Denef和van den Dries对p进整数解析理论的量词消去结果的证明以及Hermann和Seidenberg的线性代数方法所产生的技术之上。摘要由Madeline G. Barnicle制备。电子邮件:barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4
{"title":"Uniform Properties of Ideals in Rings of Restricted Power Series","authors":"Madeline Grace Barnicle","doi":"10.1017/bsl.2020.26","DOIUrl":"https://doi.org/10.1017/bsl.2020.26","url":null,"abstract":"Abstract When is an ideal of a ring radical or prime? By examining its generators, one may in many cases definably and uniformly test the ideal’s properties. We seek to establish such definable formulas in rings of p-adic power series, such as \u0000$mathbb Q_{p}langle Xrangle $\u0000 , \u0000$mathbb Z_{p}langle Xrangle $\u0000 , and related rings of power series over more general valuation rings and their fraction fields. We obtain a definable, uniform test for radicality, and, in the one-dimensional case, for primality. This builds upon the techniques stemming from the proof of the quantifier elimination results for the analytic theory of the p-adic integers by Denef and van den Dries, and the linear algebra methods of Hermann and Seidenberg. Abstract prepared by Madeline G. Barnicle. E-mail: barnicle@math.ucla.edu URL: https://escholarship.org/uc/item/6t02q9s4","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84290436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and (effective) descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy. We first analyze the strength of the open and clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $mathbb {N}$ or is single-valued. These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $mathsf {DS}$ , and the Weihrauch equivalent problem $mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $mathsf {DS}$ and $mathsf {BS}$ by considering $boldsymbol {Gamma }$ -presented orders, where $boldsymbol {Gamma }$ is a Borel pointclass or $boldsymbol {Delta }^1_1$ , $boldsymbol {Sigma }^1_1$ , $boldsymbol {Pi }^1_1$ . We study the obtained $mathsf {DS}$ -hierarchy and $mathsf {BS}$ -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $mathbf {K}([0,1])$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $boldsymbol {Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $mathbf {K}([0,1]^d)$ . We also generalize the results by relaxing the compactness of the ambient space and sho
{"title":"A journey through computability, topology and analysis","authors":"Manlio Valenti","doi":"10.1017/bsl.2022.13","DOIUrl":"https://doi.org/10.1017/bsl.2022.13","url":null,"abstract":"Abstract This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and (effective) descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy. We first analyze the strength of the open and clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions (five corresponding to the open Ramsey theorem and three corresponding to the clopen Ramsey theorem) and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain \u0000$mathbb {N}$\u0000 or is single-valued. These notions proved to be extremely useful when exploring the Weihrauch degree of the problem \u0000$mathsf {DS}$\u0000 of computing descending sequences in ill-founded linear orders. They allow us to show that \u0000$mathsf {DS}$\u0000 , and the Weihrauch equivalent problem \u0000$mathsf {BS}$\u0000 of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize \u0000$mathsf {DS}$\u0000 and \u0000$mathsf {BS}$\u0000 by considering \u0000$boldsymbol {Gamma }$\u0000 -presented orders, where \u0000$boldsymbol {Gamma }$\u0000 is a Borel pointclass or \u0000$boldsymbol {Delta }^1_1$\u0000 , \u0000$boldsymbol {Sigma }^1_1$\u0000 , \u0000$boldsymbol {Pi }^1_1$\u0000 . We study the obtained \u0000$mathsf {DS}$\u0000 -hierarchy and \u0000$mathsf {BS}$\u0000 -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level. Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace \u0000$mathbf {K}([0,1])$\u0000 of compact subsets of \u0000$[0,1]$\u0000 and show that the closed Salem sets form a \u0000$boldsymbol {Pi }^0_3$\u0000 -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in \u0000$mathbf {K}([0,1]^d)$\u0000 . We also generalize the results by relaxing the compactness of the ambient space and sho","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80848571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract From the interaction among areas such as Computer Science, Formal Logic, and Automated Deduction arises an important new subject called Logic Programming. This has been used continuously in the theoretical study and practical applications in various fields of Artificial Intelligence. After the emergence of a wide variety of non-classical logics and the understanding of the limitations presented by first-order classical logic, it became necessary to consider logic programming based on other types of reasoning in addition to classical reasoning. A type of reasoning that has been well studied is the paraconsistent, that is, the reasoning that tolerates contradictions. However, although there are many paraconsistent logics with different types of semantics, their application to logic programming is more delicate than it first appears, requiring an in-depth study of what can or cannot be transferred directly from classical first-order logic to other types of logic. Based on studies of Tarcisio Rodrigues on the foundations of Paraconsistent Logic Programming (2010) for some Logics of Formal Inconsistency (LFIs), this thesis intends to resume the research of Rodrigues and place it in the specific context of LFIs with three- and four-valued semantics. This kind of logics are interesting from the computational point of view, as presented by Luiz Silvestrini in his Ph.D. thesis entitled “A new approach to the concept of quase-truth” (2011), and by Marcelo Coniglio and Martín Figallo in the article “Hilbert-style presentations of two logics associated to tetravalent modal algebras” [Studia Logica (2012)]. Based on original techniques, this study aims to define well-founded systems of paraconsistent logic programming based on well-known logics, in contrast to the ad hoc approaches to this question found in the literature. Abstract prepared by Kleidson Êglicio Carvalho da Silva Oliveira. E-mail: kecso10@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322632
{"title":"Paraconsistent Logic Programming in Three and Four-Valued Logics","authors":"Kleidson Êglicio Carvalho da Silva Oliveira","doi":"10.1017/bsl.2021.34","DOIUrl":"https://doi.org/10.1017/bsl.2021.34","url":null,"abstract":"Abstract From the interaction among areas such as Computer Science, Formal Logic, and Automated Deduction arises an important new subject called Logic Programming. This has been used continuously in the theoretical study and practical applications in various fields of Artificial Intelligence. After the emergence of a wide variety of non-classical logics and the understanding of the limitations presented by first-order classical logic, it became necessary to consider logic programming based on other types of reasoning in addition to classical reasoning. A type of reasoning that has been well studied is the paraconsistent, that is, the reasoning that tolerates contradictions. However, although there are many paraconsistent logics with different types of semantics, their application to logic programming is more delicate than it first appears, requiring an in-depth study of what can or cannot be transferred directly from classical first-order logic to other types of logic. Based on studies of Tarcisio Rodrigues on the foundations of Paraconsistent Logic Programming (2010) for some Logics of Formal Inconsistency (LFIs), this thesis intends to resume the research of Rodrigues and place it in the specific context of LFIs with three- and four-valued semantics. This kind of logics are interesting from the computational point of view, as presented by Luiz Silvestrini in his Ph.D. thesis entitled “A new approach to the concept of quase-truth” (2011), and by Marcelo Coniglio and Martín Figallo in the article “Hilbert-style presentations of two logics associated to tetravalent modal algebras” [Studia Logica (2012)]. Based on original techniques, this study aims to define well-founded systems of paraconsistent logic programming based on well-known logics, in contrast to the ad hoc approaches to this question found in the literature. Abstract prepared by Kleidson Êglicio Carvalho da Silva Oliveira. E-mail: kecso10@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322632","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84846068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron’s semantics. In this thesis, we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by the very general techniques introduced by Blok and Pigozzi. We also will obtain representation theorems for some LFIs and we will prove that, within out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbC. Abstract prepared by Ana Claudia de Jesus Golzio. E-mail: anaclaudiagolzio@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322436
我们称多操作为为偶数参数返回一组值而不是单个值的任何操作。通过多重运算，我们可以定义一个至少具有一个多重运算的代数结构。这种结构叫做多重代数。对它们的研究始于1934年，当时马蒂发表了一篇论文。在逻辑领域，Avron和他的合作者以非确定性矩阵(或Nmatrices)的名义考虑了多重代数，并将其用作表征某些不能用单个有限矩阵表征的逻辑的语义工具。卡尼elli和Coniglio为lfi(形式不一致逻辑)引入了交换结构的语义，lfi是布尔代数中定义在三元组上的n矩阵，推广了Avron的语义。在本文中，我们将介绍一种新的基于多代数和交换结构的逻辑代数化方法，它类似于经典的Lindenbaum-Tarski代数化方法，但由于它可以应用于某些算子非同余的系统，因此它的应用范围更广。特别地，这种方法将被应用于非正态模态逻辑和一些不能被Blok和Pigozzi引入的非常一般的技术代数化的lfi。我们也将得到一些lfi的表示定理，并且我们将证明，在我们的方法中，一些公理扩展的交换结构类是逻辑mbC的交换结构类的子类。摘要由Ana Claudia de Jesus Golzio准备。电子邮件:anaclaudiagolzio@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322436
{"title":"Non-Deterministic Matrices: Theory and Applications to Algebraic Semantics","authors":"A. C. Golzio","doi":"10.1017/bsl.2021.35","DOIUrl":"https://doi.org/10.1017/bsl.2021.35","url":null,"abstract":"Abstract We call multioperation any operation that return for even argument a set of values instead of a single value. Through multioperations we can define an algebraic structure equipped with at least one multioperation. This kind of structure is called multialgebra. The study of them began in 1934 with the publication of a paper of Marty. In the realm of Logic, multialgebras were considered by Avron and his collaborators under the name of non-deterministic matrices (or Nmatrices) and used as semantics tool for characterizing some logics which cannot be characterized by a single finite matrix. Carnielli and Coniglio introduced the semantics of swap structures for LFIs (Logics of Formal Inconsistency), which are Nmatrices defined over triples in a Boolean algebra, generalizing Avron’s semantics. In this thesis, we will introduce a new method of algebraization of logics based on multialgebras and swap structures that is similar to classical algebraization method of Lindenbaum-Tarski, but more extensive because it can be applied to systems such that some operators are non-congruential. In particular, this method will be applied to a family of non-normal modal logics and to some LFIs that are not algebraizable by the very general techniques introduced by Blok and Pigozzi. We also will obtain representation theorems for some LFIs and we will prove that, within out approach, the classes of swap structures for some axiomatic extensions of mbC are a subclass of the class of swap structures for the logic mbC. Abstract prepared by Ana Claudia de Jesus Golzio. E-mail: anaclaudiagolzio@yahoo.com.br URL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/322436","PeriodicalId":22265,"journal":{"name":"The Bulletin of Symbolic Logic","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87018721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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