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.

安德鲁斯提出了一个跨越经典可计算性理论，经典模型理论，特别是可计算性理论和模型理论之间的相互作用的研究计划。更具体地说，拟议的研究将发展新兴领域的可计算稳定性理论，其中稳定性理论的方法起着关键作用是一阶理论的有效性质的研究。稳定性理论是模型论的一部分，它侧重于数学对象的基本几何或结构性质。一般来说，数学对象越尊重底层结构，就越容易计算关于该对象的信息。这形成了稳定性和可计算性之间的来回关系，这种相互作用是可计算稳定性理论的焦点。从可计算稳定性理论中产生的问题导致了经典模型理论和可计算性中的新问题，这对这两个主题都有启发。其中一个问题是什么时候可以从几何性质中推导出某种程度上的量词消去。可计算稳定性理论致力于探索“结构简单的对象应该更容易计算”的基本命题。模型论是数理逻辑的一个分支，它通过在一阶语言的上下文中研究对象，提供了分析结构简单性的工具。对与对象或对象类相关联的语言的理解通常可以转化为对对象本身的更深入理解。可计算性理论研究数学对象的计算及相关计算问题。通过结合这两个领域的工具，安德鲁斯将研究什么时候结构简单，什么时候不转化为计算简单的问题。