Computability Theory and its Applications

可计算性理论及其应用

基本信息

批准号：
312501-2013
负责人：
Csima, Barbara
金额：
$ 1.09万
依托单位：
University of Waterloo
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2017
资助国家：
加拿大
起止时间：
2017-01-01 至 2018-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635499
关键词：
Computability Theory its Applications

项目摘要

Up to the mid 19th century or so, mathematics was mostly algorithmic. Proofs of existence were usually done by giving an actual construction of the object. As mathematics began to become more abstract, people began to study the notions of algorithms and computability. The first stumbling block was on the definition of what it meant to be algorithmic or computable. When faced with an algorithm, people would agree that it was one, but how would one show that something could not be computed, or solved by an algorithm? By the 1930s, work of Turing and others culminated in an acceptable notion of computability.Since the 1930s, computability theory has developed far beyond just checking what is computable and what is not. We have notions of relative computability, where we say A is computed from B if there is some program that can compute A if it is allowed to refer (finitely often) to an infinite tape coding B.In this project, I propose to study various areas of mathematics, and use computability theory to examine the complexity of the proofs. This can help answer questions such as how difficult is it to prove one thing compared to another? Is the original proof the most efficient? How difficult is it to tell that similar objects are actually similar? By studying these properties, we can find out interesting phenomena about the area of mathematics being studied, often leading to further insight.

直到 19 世纪中叶左右，数学主要是算法。存在性证明通常是通过给出对象的实际构造来完成的。随着数学开始变得更加抽象，人们开始研究算法和可计算性的概念。第一个障碍是算法或可计算的定义。当面对一种算法时，人们会同意它是一种算法，但是如何证明某些东西无法通过算法计算或解决呢？到 20 世纪 30 年代，图灵和其他人的工作最终形成了一种可以接受的可计算性概念。自 20 世纪 30 年代以来，可计算性理论的发展远远超出了仅仅检查什么是可计算的、什么是不可计算的。我们有相对可计算性的概念，我们说 A 是从 B 计算出来的，如果有一些程序可以计算 A，如果它被允许引用（有限经常）无限磁带编码 B。在这个项目中，我建议研究各种数学领域，并使用可计算性理论来检查证明的复杂性。这可以帮助回答一些问题，例如证明一件事与另一件事相比有多困难？原始证明是最有效的吗？判断相似的物体实际上相似有多困难？通过研究这些性质，我们可以发现正在研究的数学领域的有趣现象，通常会带来进一步的见解。