Computability Theory and its Applications

直到19世纪中叶左右，数学主要是算法。存在的证明通常是通过给出物体的实际结构来完成的。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。第一个绊脚石是关于什么是算法或可计算的定义。当面对一种算法时，人们会同意这是一种算法，但如何证明某些东西不能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个可接受的可计算性概念。自20世纪30年代以来，可计算性理论已经远远超出了仅仅检查哪些是可计算的，哪些是不可计算的。我们有相对可计算性的概念，其中我们说A是从B计算的，如果有一些程序可以计算A，如果它被允许(有限地经常)引用一个无限的磁带编码B。在这个项目中，我建议学习数学的各个领域，并使用可计算性理论来检查证明的复杂性。这可以帮助回答这样的问题：证明一件事比证明另一件事有多难？原始证据是最有效的吗？区分相似的物体实际上是相似的有多难？通过研究这些性质，我们可以发现与正在研究的数学领域有关的有趣现象，通常会导致进一步的洞察。

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 discover interesting phenomena about the area of mathematics being studied, often leading to further insight.