Computable model theory and invariant descriptive computability theory

数理逻辑产生于为数学发展严谨基础的需要。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。可计算性理论通过从计算复杂性的角度考虑数学对象来理解它们。集合论通过数学的基本公理以及这些公理如何暗示对象的存在来理解数学对象。这个项目集中在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集合嵌入到另一个集合中来探索二维集合的复杂性。根据并集和补集的基本集合论运算，这些嵌入是可构造的，但它们可能是不可计算的。该项目将探索一种类似的理论，在这种理论中，人们认为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询问该理论的模型的哪些维度是可计算的。可计算模型理论中的谱问题自70年代以来一直是一个重大的公开问题，它要求哪些集合可能是不可数范畴理论的可计算谱，重点是强极小理论。在这个项目中，目标是将问题从一个完全一般的框架归结为局部模强极小理论，这些理论在几何上是驯服的，并且与群密切相关。在那里，人们希望能够给出具体的答案，说明哪些集合是光谱。另外，这个项目将检查等价关系的可计算约简。一个主要的方向是使用这个复杂性概念来详细地研究代数决策问题。过去，图灵度被用来分析代数决策问题，但它们形成了一个粗略的尺度，因此所有可计算的可枚举度似乎都包含了所有自然的代数决策问题。使用可计算的等价关系缩减，应该会出现一个更有趣的结构。这个奖项反映了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 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.