Computable model theory and invariant descriptive computability theory

数学逻辑产生于为数学建立严格基础的需要。在数理逻辑中，研究了三个主要的子领域。模型理论通过形式语言的视角来理解数学对象。可计算性理论通过计算复杂性的视角来理解数学对象。集合论通过数学的基本公理来理解数学对象，以及这些公理如何暗示对象的存在。这个项目的重点是可计算性理论和逻辑的其他两个子领域之间的一些联系。关于模型理论，该项目涉及探索两个数学对象在形式语言方面看起来相同，但一个可以计算而另一个不能计算的现象。在集合论中，有一个丰富的理论，通过建设性地将一个二维集合嵌入另一个来探索二维集合的复杂性。这些嵌入可以用基本集合论的并和补运算来构造，但它们可能不是可计算的。该项目将探索一个类似的理论，其中考虑嵌入必须是可计算的。这个项目涉及本科生和研究生的工作。一阶理论的可计算谱问的是该理论模型的哪些维度是可计算的。可计算模型理论中的谱问题是20世纪70年代以来的一个主要开放问题，主要研究不可数范畴理论中哪些集合可以是可计算谱，重点研究强极小理论。在这个项目中，目的是将问题从一个完全一般的框架简化到局部模强最小理论，这些理论在几何上是驯服的，并且与群密切相关。从那里，希望能够给出具体的答案，哪些集合是光谱。另外，本项目将研究等价关系上的可计算约简。一个主要的方向是使用这个复杂性概念来详细检查代数决策问题。在过去，图灵度已经被用来分析代数决策问题，但它们形成了一个粗糙的尺度，因此所有可计算枚举的度似乎包含了所有自然的代数决策问题。在等价关系上使用可计算约简，应该会出现一个更有趣的结构。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Mathematical logic grew out of a need to develop rigorous foundations for mathematics. Within mathematical logic, three major subfields are studied. Model theory understands mathematical objects by considering them through the lens of a formal language. Computability theory understands mathematical objects by considering them through the lens of computational complexity. Set theory understands mathematical objects through the foundational axioms of mathematics and how those axioms imply the object’s existence. This project focuses on some connections between computability theory and the other two subfields of logic. Regarding model theory, the project involves exploring the phenomenon when two mathematical objects look the same in terms of their formal languages, but one can be computed while the other cannot. In set theory, there is a rich theory exploring the complexity of 2-dimensional sets in terms of constructively embedding one into another. These embeddings are constructible in terms of the basic set-theoretic operations of unions and complements, but they may not be computable. The project will explore an analogous theory where one considers embeddings that must be computable. This project involves work with undergraduate and graduate students. The computable spectrum of a first-order theory asks which dimensions of models of that theory are computable. The spectrum problem in computable model theory, which has been a major open problem since the 70s, asks for which sets may be computable spectra of uncountably categorical theories, with a focus on strongly minimal theories. In this project, the aim is to give a reduction of the problem from a fully general framework down to the locally modular strongly minimal theories, which are geometrically tame and are closely related to groups. From there, the hope is to be able to give concrete answers as to which sets are spectra. Separately, this project will examine computable reduction on equivalence relations. One major direction is to use this complexity notion to examine algebraic decision problems in detail. In the past, the Turing degrees have been used to analyze algebraic decision problems, but these form a coarse yardstick, so all computably enumerable degrees seem to contain all natural algebraic decision problems. Using computable reductions on equivalence relations, there should be a much more interesting structure emerging.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.