Relation between the Usual Order and the Enumeration Orders of Elements of r.e. Sets

TL;DR

This paper explores the relationship between enumeration orders of r.e. sets and Turing machine properties, introducing new concepts of uniformity and a Turing Output Binary Search Tree to analyze these relationships.

Contribution

It introduces novel uniformity concepts for Turing machines and r.e. sets, and proposes a new data structure to facilitate analysis of their relationships.

Findings

Defined uniformity for Turing machines and r.e. sets

Studied relationships between uniformity and reducibility notions

Introduced Turing Output Binary Search Tree

Abstract

In this paper, we have compared r.e. sets based on their enumeration orders with Turing machines. Accordingly, we have defined novel concept uniformity for Turing machines and r.e. sets and have studied some relationships between uniformity and both one-reducibility and Turing reducibility. Furthermore, we have defined type-2 uniformity concept and studied r.e. sets and Turing machines based on this concept. In the end, we have introduced a new structure called Turing Output Binary Search Tree that helps us lighten some ideas.