Computable Mathematics Measured by Enumeration Degrees

用枚举度来衡量的可计算数学

基本信息

批准号：
2053848
负责人：
Mariya Soskova
金额：
$ 42万
依托单位：
University of Wisconsin-Madison
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-06-01 至 2025-05-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2053848&HistoricalAwards=false
关键词：
Computable Mathematics Measured Enumeration Degrees

项目摘要

Mathematical logic and computability arose from the need to develop stable and trustworthy foundations for mathematics, provide clear criteria for what constitutes a proof, what basic axioms give sufficient power to express the rest of mathematics, and what constitutes an algorithm or a computable function. The origins of the fields trace back to David Hilbert's program from the 1900's and to the work by Gödel, Church, and Turing that proved several aspects of this program impossible: there are problems in elementary number theory that do not have computable solutions. Since then, incomputable problems have been identified in many different subfields of mathematics. Most famously, the solution sets to some Diophantine equations and the word problem for some finitely presented groups are incomputable. Computability theory proposes frameworks to study the relative algorithmic complexity of such problems. This project proposes an in-depth investigation of two related frameworks in computability theory, and the complex structures that they give rise to. The project presents a diverse array of research avenues, suitable for all levels of experience. Soskova and Miller intend to continue their collaboration with undergraduate, graduate, and postgraduate students and facilitate the quick immersion of young researchers into this area through two textbooks: a beginner level Computability Theory textbook and a more specialized advanced textbook on the precise topic studied within this project. They also maintain a visual online database that allows the quick identification of open problems. The project will support women's engagement in the field through a mentorship program organized by Soskova within the CiE Women in Computability focus group. Soskova and Miller will continue to support the Logic Community by organizing scientific meetings and through their editorial work.In computability theory, the Turing degrees are used to measure the effective content of sets of natural numbers. This measure can be extended to capture the effective content of other objects in mathematics, such as the real numbers. In other cases, Turing reducibility is not sufficient. For example, Miller proved that it is not possible to assign a Turing degree to every continuous function on the unit interval. In that and many other cases, an extension of Turing reducibility, enumeration reducibility, turns out to provide a better framework for effective mathematics. In 1967, Rogers posed a list of problems that strongly influenced the development of computability theory. One of these problems asks whether the partial orders of the Turing degrees and the enumeration degrees are rigid structures, i.e., have no nontrivial automorphisms. Slaman and Woodin proved that the rigidity of the Turing degrees is equivalent to a complete characterization of its definable relations and to its biinterpretability with second order arithmetic. Building on this, Soskova proved that the same equivalence holds for the enumeration degrees. Miller and Soskova (with collaborators) answered another problem from the list: there is a first order definable copy of the Turing degrees inside the enumeration degrees. This established a link between the rigidity problems for the two structures; if the Turing degrees are rigid then so are the enumeration degrees. The rigidity of the enumeration degrees, while still open, could turn out to be more approachable. Definability within the enumeration degrees has already proved more approachable. The goal of this project is to expand our understanding of the structure of the enumeration degrees and its nontrivial connections to effective mathematics, with a focus on computable topology. We would like to accumulate new methods investigate combinatorial properties of the structure, and isolate special classes of degrees that determine the logical character of the structure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

数学逻辑和可计算性源于需要开发稳定且值得信赖的数学基础，为证明的构成、哪些基本公理提供足够的能力来表达数学的其余部分以及算法或可计算函数的起源提供明确的标准。该领域的一些问题可以追溯到 1900 年代 David Hilbert 的规划，以及哥德尔、丘奇和图灵的工作，这些工作证明了该规划的几个方面是不可能的：初等数论中存在以下问题：从那时起，许多不同的数学子领域都发现了不可计算的问题，最著名的是，一些丢番图方程的解集和一些有限群的字问题是不可计算的。该项目提出了对可计算性理论中的两个相关框架及其产生的复杂结构的深入研究。索斯科娃和米勒打算继续与本科生、研究生和研究生合作，并通过两本教科书促进年轻研究人员快速融入这一领域：一本是初级可计算性理论教科书，另一本是更专业的教科书。他们还维护了一个可视化在线数据库，可以快速识别未解决的问题。该项目将通过 Soskova 在 CiE 女性可计算性重点范围内组织的指导计划来支持女性参与该领域。索斯科娃和米勒将继续通过组织科学会议和编辑工作来支持逻辑社区。在可计算性理论中，图灵度用于测量自然数集的有效内容。这种测量可以扩展到捕获。数学中其他对象的有效内容，例如实数，在其他情况下，图灵可约性是不够的，例如，米勒证明不可能为单位区间上的每个连续函数分配图灵度。和许多其他情况，延伸图灵可归约性、枚举可归约性为有效的数学提供了一个更好的框架。1967 年，罗杰斯提出了一系列对可计算性理论的发展产生重大影响的问题，其中一个问题是图灵度的偏序是否存在。枚举度是刚性结构，即不具有非平凡自同构。斯拉曼和伍丁证明了图灵度的刚性相当于其可定义关系的完整表征。在此基础上，索斯科娃证明了枚举度也具有相同的等价性，米勒和索斯科娃（与合作者）回答了列表中的另一个问题：存在图灵度的一阶可定义副本。这在两个结构的刚性问题之间建立了联系；如果图灵度是刚性的，那么枚举度的刚性也是开放的。事实证明，枚举度的可定义性更容易理解，该项目的目标是扩展我们对枚举度的结构及其与有效数学的重要联系的理解，重点是可计算拓扑。我们希望积累新的方法来研究结构的组合属性，并分离出决定结构逻辑特征的特殊级别。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。