The "paradox" of computability and a recursive relative version of the Busy Beaver function

TL;DR

This paper explores the relativity of uncomputability within subrecursive classes, introducing a recursive relative Busy Beaver function and demonstrating a paradoxical situation where a function is both computable and uncomputable depending on the perspective.

Contribution

It defines Turing submachines and a recursive relative Busy Beaver Plus function, revealing a computability paradox and proposing a hierarchy of negative Turing degrees.

Findings

The Busy Beaver Plus function is uncomputable on any Turing submachine.

Uncomputability is shown to be a relative property within subrecursive classes.

A paradox of computability is demonstrated, where a function appears both computable and uncomputable.

Abstract

In this article, we will show that uncomputability is a relative property not only of oracle Turing machines, but also of subrecursive classes. We will define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function which we will call Busy Beaver Plus function. Therefore, we will prove that the computable Busy Beaver Plus function defined on any Turing submachine is not computable by any program running on this submachine. We will thereby demonstrate the existence of a "paradox" of computability a la Skolem: a function is computable when "seen from the outside" the subsystem, but uncomputable when "seen from within" the same subsystem. Finally, we will raise the possibility of defining universal submachines, and a hierarchy of negative Turing degrees.