从可计算的观点分类

Isomorphism and classification for countable structures

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we u…

There is no classification of the decidably presentable structures

A computable structure $\mathcal{A}$ is decidable if, given a formula $φ(\bar{x})$ of elementary first-order logic, and a tuple $\bar{a} \in \mathcal{A}$, we have a decision procedure to decide whether $φ$ holds of $\bar{a}$. We show that there is no reasonable classification of the decidably presentable structures. Formally, we show that the index set of the computable structures with decidable presentations is $Σ^1_1$-complete. This result holds even if we restrict out attention to groups, graphs, or fields. We also show that the index sets of the computable structures with $n$-decidable pre…

连续函数、transducer函数与正则函数的可计算分类

Computable classifications of continuous, transducer, and regular functions

本文开发了基于索引集的算法框架，统一了全局与局部分类问题，证明了连续正则函数与transducer函数的$Σ^0_2$-完全性，并给出了$C[0,1]$空间的算术分类。

可计算分析中的有效选择与有界性原则

Effective Choice and Boundedness Principles in Computable Analysis

在本文中，我们研究了一种新的方法来分类数学定理根据其计算内容。基本上，我们问的问题是，哪些定理可以连续地或可计算地相互转换？为了这个目的，定理被认为是通过其实现，这是与某些输入和输出数据的操作。Weihrauch约简及其导出的偏序度结构是表达这些运算之间连续或可计算关系的技术工具，我们确定了Weihrauch度中的某些选择原则，并证明了分析中的某些核心定理可以在这种结构中自然地分类。特别地，我们研究了诸如中值定理、Baire范畴定理、Banach逆映射定理等定理。我们还探讨了现有的分类的哈恩-巴拿赫定理和弱K“onig引理适合这幅图片。我们比较我们的分类结果与现有的分类建设性和逆向数学，我们声称，在一定意义上，我们的分类更精细，并揭示了一些新的计算内容的相应定理。我们开发了一些分离技术的基础上，一个新的并行化原则，在一定的不变性性质的Weihrauch约简，低基定理的Jockusch和Soare和Baire类定理。最后，我们提出了一些元定理，允许推导出上界的分类Weihrauch度的许多定理，我们讨论了布劳威尔不动点定理作为一个例子。

有限长可计算阿贝尔p-群的同构问题

The Isomorphism Problem for Computable Abelian p-Groups of Bounded Length

分类理论将具有好的结构定理的类与不可能具有好的结构定理的类区分开来。有些类（例如稠密线性序）一般是不可分类的，但当我们只考虑可数成员时是可分类的。本文通过一系列的例子来探讨可计算结构类的这种概念。我们遵循Goncharov和Knight最近的工作，使用类的同构问题的程度来区分可分类类和不可分类类。本文计算了有界Ulm长的交换$p$ -群同构问题的度。其结果是一个序列的类，其同构问题是共尾的超算术层次。在这个过程中，新的来回关系，这样的群体计算。

Degrees of and lowness for isometric isomorphism

We contribute to the program of extending computable structure theory to the realm of metric structures by investigating lowness for isometric isomorphism of metric structures. We show that lowness for isomorphism coincides with lowness for isometric isomorphism and with lowness for isometry of metric spaces. We also examine certain restricted notions of lowness for isometric isomorphism with respect to fixed computable presentations, and, in this vein, we obtain classifications of the degrees that are low for isometric isomorphism with respect to the standard copies of certain Lebesgue spaces…