Definability and Automorphisms in Computability Theory

可计算性理论中的可定义性和自同构

• 批准号: 0245167
• 负责人: Peter Cholak
• 金额: $ 36.29万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0245167&HistoricalAwards=false
• 关键词: Definability; Automorphisms; Computability; Theory

AbstractAward: DMS-0245167Principal Investigator: Peter A. CholakThe principal investigator plans to study the relationshipbetween definability and automorphisms in various structuresarising in computability theory. Primarily, but not totally, hewill focus on the collection of all computably enumerable sets.The principal investigator will also consider the collection ofall Pi01 classes and the computably enumerable degrees. A longrange goal is to provide a complete understanding of thesestructures and their automorphisms and definable orbits. Somerelated projects in computable structure theory and models ofsecond order arithmetic are also planned.The main focus of these projects is on definability andcomputability. These notions are both important in measuring thecomplexity of an answer to a mathematical problem. Problems suchas "Is there a computer program which can solve all questions ofthis type?" One develops an intertwined hierarchy ofdefinability and computability. Only answers which lay on thelowest level are computable and even then they are not alwaysfeasibly computable given today's computers. Answers of highercomplexity provide useful mathematical information, allow one totest the limits of mathematical techniques, reveal whether thewrong techniques are being used, and, in some very rare cases,can be useful for encoding/decoding information.

Abstractaward：DMS-0245167原理研究者：Peter A. Cholakthe主要研究者计划研究各种结构性的可确定性与可计算理论中的自动形态之间的关系。 首先，但不是完全，将专注于所有可计算的枚举集的集合。 一个漫长的目标是提供对这些词性及其自动形态和可定义轨道的完整理解。 还计划了可计算结构理论和秒算术模型中的突出相关项目。这些项目的主要重点是可确定性和可兼容性。 这些概念在衡量数学问题的答案方面都很重要。 类似问题“是否有一个计算机程序可以解决这种类型的所有问题？” 一个人开发了一个交织在一起的层次结构，可定义性和可计算性。 只有在最高级别上放置的答案是可以计算的，即使如此，考虑到当今计算机，它们并不总是可计算的。 HigherComplexity的答案提供了有用的数学信息，允许一个totsest的数学技术限制，揭示是否正在使用wrong技术，并且在某些非常罕见的情况下，对于编码/解码信息可能很有用。