A Note on Recursively Enumerable Sets and Computable Functions

• Main Authors: Chao-Wei Chen, 陳昭瑋
• Other Authors: Hsing-Chien Tsai
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/96648458207305269166

碩士 === 國立中正大學 === 哲學所 === 97 === The main goal of this thesis is to survey the theorems and properties related to recursively enumerable function, and rephrase or reorganize these theorems and properties. In Chapter one, I will first explain the concept of ``computable'' in an intuitive way, and then describe two different formal ways to define computability: by (partial) recursive functions or by Turing computable functions. I will look into the recursive function approach and introduce some basic results of it. Two important relevant concepts, recursively enumerable sets and index sets will also be introduced in this chapter. I will then show that several problems are unsolvable. In chapter two, I will focus on recursively enumerable sets and the Recursion Theorem. Two different definitions of recursively enumerable sets and some basic properties of $r.e.$ sets will be presented. Then I will show some properties for recursive sets and finite sets. Finally, I will present the proof of Recursion Theorem as well as the proofs for Recursion Theorem with parameters and Recursion Theorem for partial recursive functions. And then several applications of Recursion Theorem will be given.