On Higher Order Busy Beaver Function

TL;DR

This paper extends the Busy Beaver function to higher orders using Turing oracle machines, exploring its relation to decidability, computability, and proposing new conjectures.

Contribution

It introduces higher order Busy Beaver functions based on oracle machines and analyzes their connections to decidability and computability, along with new conjectures.

Findings

Established relationships between decidability of number theoretical formulas and higher order Busy Beaver functions.

Linked computability of max-min partial recursive functions to higher order Busy Beaver functions.

Presented conjectures on the properties of higher order Busy Beaver functions.

Abstract

In this paper, we extend Busy Beaver function to a class of higher order Busy Beaver functions based on Turing oracle machine. We prove some results about the relation between decidability of number theoretical formula and higher order Busy Beaver functions, and the relation between computability of max-min partial recursive functions and higher order Busy Beaver functions. We also present some conjectures on higher order Busy Beaver functions.