通过可计算性,拓扑和分析的旅程

Manlio Valenti
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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 \n$\\mathbb {N}$\n or is single-valued. These notions proved to be extremely useful when exploring the Weihrauch degree of the problem \n$\\mathsf {DS}$\n of computing descending sequences in ill-founded linear orders. 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引用次数: 2

摘要

本文利用可计算分析和(有效)描述集合论的框架来探讨一些数学问题的复杂性。我们将特别关注Weihrauch可约性,作为比较问题统一计算强度的一种手段。在简要介绍了相关的背景概念之后,我们研究了在逆向数学层次的较高层次上由定理引起的问题的统一计算内容。我们首先分析开拉姆齐定理和闭拉姆齐定理的强度。由于没有一种规范的方式将这些定理表述为多值函数,我们确定了8种不同的多值函数(5种对应于开拉姆齐定理,3种对应于闭拉姆齐定理),并从Weihrauch,强Weihrauch和算术Weihrauch可约性的角度研究了它们的程度。然后讨论了一些新的多值函数上的算子,并研究了它们的代数性质以及它们与已有问题上算子的关系。特别地,我们研究了一个问题f的一阶部分和确定性部分,分别捕获了可约为f和具有上域$\mathbb {N}$或单值的最强多值问题的Weihrauch度。这些概念被证明是非常有用的,当探索问题的Weihrauch度$\mathsf {DS}$计算在不正确的线性顺序下降的序列。它们使我们能够证明$\mathsf {DS}$和Weihrauch等效问题$\mathsf {BS}$(通过非良好准阶查找不良序列)虽然非常“难以”解决,但在统一计算强度方面相当弱。然后,我们通过考虑$\boldsymbol {\Gamma }$表示的顺序来推广$\mathsf {DS}$和$\mathsf {BS}$,其中$\boldsymbol {\Gamma }$是Borel点类或$\boldsymbol {\Delta }^1_1$、$\boldsymbol {\Sigma }^1_1$、$\boldsymbol {\Pi }^1_1$。我们将所得的$\mathsf {DS}$ -层次和$\mathsf {BS}$ -层次与(有效的)Baire层次进行了比较,并证明它们在任何有限的水平上都不会崩溃。最后,我们在几何测量理论的背景下工作,从描述性集合理论的角度,我们关注一些涉及豪斯多夫/傅里叶维数和塞勒姆集合概念的条件的表征。我们首先研究了$[0,1]$的紧子集的超空间$\mathbf {K}([0,1])$,并证明了闭Salem集合构成了一个$\boldsymbol {\Pi }^0_3$ -完全族。这是通过描述具有足够大的豪斯多夫维数或傅里叶维数的集合族的复杂性来实现的。我们还表明,如果我们增加环境空间的维度并在$\mathbf {K}([0,1]^d)$中工作,复杂性不会改变。我们还通过放松环境空间的紧性来推广结果,并证明当我们赋予$\mathbf {F}(\mathbb {R}^d)$ Fell拓扑时,封闭的Salem集仍然是$\boldsymbol {\Pi }^0_3$ -完全的。类似的结果也适用于Vietoris拓扑。最后,我们从有效描述集理论和2型效率理论的角度分析了相同的概念,并表明在lightface情况下复杂性也保持不变。特别地,我们证明了所有闭塞勒姆集合的族是$\Pi ^0_3$ -完全的。进一步刻画了计算闭集的Hausdorff维数和Fourier维数的函数的Weihrauch度。摘要由Manlio Valenti准备。电子邮件:manliovalenti@gmail.com
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A journey through computability, topology and analysis
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 show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}(\mathbb {R}^d)$ with the Fell topology. A similar result holds also for the Vietoris topology. We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets. Abstract prepared by Manlio Valenti. E-mail: manliovalenti@gmail.com
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