Computable Mathematics Measured by Enumeration Degrees

数学逻辑和可计算性产生于需要为数学发展稳定和可信的基础，为什么构成证明提供明确的标准，什么基本公理给予足够的力量来表达数学的其余部分，以及什么构成算法或可计算的函数。场的起源可以追溯到大卫·希尔伯特从1900年的S开始的程序，以及哥德尔、丘奇和图灵的工作，这些工作证明了这个程序的几个方面是不可能的：初等数论中的一些问题没有可计算的解。从那时起，在数学的许多不同子领域都发现了无法计算的问题。最著名的是，一些丢番图方程的解集和一些有限表示群的字问题是不可计算的。可计算性理论提出了研究这类问题的相对算法复杂性的框架。本项目对可计算性理论中的两个相关框架及其引起的复杂结构进行了深入研究。该项目提供了一系列不同的研究途径，适用于所有水平的经验。Soskova和Miller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沉浸在这一领域：一本是初级可计算性理论教科书，另一本是关于本项目所研究的精确主题的更专门的高级教科书。他们还维护一个可视化的在线数据库，以便快速识别未解决的问题。该项目将通过Soskova在CIE妇女参与可计算性焦点小组内组织的导师方案，支持妇女参与实地工作。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，图灵度被用来衡量自然数集合的有效内容。这一度量可以扩展到捕捉数学中其他对象的有效内容，例如实数。在其他情况下，图灵可缩减性是不够的。例如，Miller证明了不可能给单位区间上的每个连续函数分配图灵度。在这种情况和许多其他情况下，图灵约简的扩展，列举约化，被证明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问题，这些问题极大地影响了可计算性理论的发展。其中一个问题是问图灵度和枚举度的偏序是否是刚性结构，即没有非平凡的自同构。Slaman和Woodin用二阶算术证明了图灵度的刚性等价于它的可定义关系的完全刻画和它的双解释性。在此基础上，Soskova证明了枚举度的等价性。米勒和索斯科娃(与合作者)回答了列表中的另一个问题：枚举度中有图灵度的一阶可定义副本。这建立了两个结构的刚性问题之间的联系；如果图灵度是刚性的，那么枚举度也是刚性的。枚举度的刚性虽然仍是开放的，但可能会变得更容易接近。枚举度内的可定义性已经被证明更容易接近。这个项目的目标是扩大我们对枚举度的结构及其与有效数学的非平凡联系的理解，重点是可计算的拓扑学。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学位类别。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.