Computable Stability Theory

可计算稳定性理论

• 批准号: 1201338
• 负责人: Uri Andrews
• 金额: $ 10.5万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2015-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201338&HistoricalAwards=false
• 关键词: Computable; Stability; Theory

Andrews proposes a program of research spanning classical computability theory, classical model theory, and especially the interplay between computability theory and model theory. More specifically, the proposed research will develop the emerging field of Computable Stability Theory, where the methods of stability theory plays a key role is the study of effective properties of first order theories. Stability theory is the part of model theory which focuses on the underlying geometrical or structural nature of the mathematical objects. Generally speaking, the more underlying structure the mathematical object respects, the easier it is to compute information about the object. This forms a back-and-forth relationship between stability and computability, and this interplay is the focus of Computable Stability Theory. Questions arising from Computable Stability Theory lead to new questions in classical model theory as well as computability, which sheds light on both subjects. Among these questions is when quantifier elimination to some level can be derived from geometrical properties.Computable Stability Theory endeavors to explore the basic statement that "structurally simple objects ought to be easier to compute." Model theory, a branch of mathematical logic, offers tools to analyze structural simplicity by studying objects in the context of their first order language. The understanding of the language associated to an object or class of objects can often be translated to a deeper understanding of the objects themselves. Computability theory studies questions of computation and relative computation with regard to mathematical objects. By combining tools from both fields, Andrews will examine the question of when structural simplicity does and when it does not translate into computational simplicity.

安德鲁斯（Andrews）提出了一项研究计划，该计划涵盖了经典的计算性理论，经典模型理论，尤其是计算理论与模型理论之间的相互作用。更具体地说，拟议的研究将开发可计算稳定理论的新兴领域，其中稳定理论的方法起着关键作用，是研究一阶理论的有效特性。稳定性理论是模型理论的一部分，该理论侧重于数学对象的基本几何或结构性。一般而言，数学对象尊重的较低的结构越基本，就越容易计算有关对象的信息。这形成了稳定性和可计算性之间的来回关系，并且这种相互作用是可计算稳定性理论的重点。可计算稳定性理论引起的问题导致了经典模型理论以及可计算性的新问题，该问题阐明了这两个主题。在这些问题中，何时可以从几何属性中得出量化级别的量化范围。模型理论是数学逻辑的一个分支，它通过在其一阶语言的背景下研究对象来提供分析结构简单性的工具。对与对象或对象类别相关的语言的理解通常可以转化为对对象本身的更深入的理解。计算理论研究计算问题和相对计算的问题。通过结合两个字段的工具，安德鲁斯将检查结构简单何时以及何时无法转化为计算简单性的问题。