Enumeration Order Equivalency

TL;DR

This paper explores the relationships between enumeration orders of r.e. sets via a new reducibility and equivalence, introducing concepts like enumeration order reducibility and type-2 equivalency, and analyzing their properties.

Contribution

It introduces the novel concepts of enumeration order reducibility and equivalency, and studies their properties within computability theory.

Findings

Defined enumeration order reducibility for r.e. sets

Established properties of enumeration order equivalency

Introduced and analyzed type-2 enumeration order equivalency

Abstract

In this paper we have investigated enumeration orders of elements of r.e. sets enumerated by means of Turing machines. We have defined a reducibility based on enumeration orders named "Enumeration Order Reducibility" on computable functions and also r.e. sets and studied their properties. Based on this reducibility we introduce an equivalence relation "Enumeration Order Equivalency". We have reached some properties of it. In subsequent, we have introduced another concept named "type-2 Enumeration Order Equivalency" and studied its properties too.