Pseudojump inversion in special r. b. $\Pi^0_1$ classes

TL;DR

This paper explores the possibility of refining classical theorems in computability theory by finding solutions within special a0^0_1 classes, revealing limitations and possibilities of such refinements.

Contribution

It demonstrates that certain theorems can be refined within some special a0^0_1 classes but not in others, highlighting nuanced limitations.

Findings

Refinements are possible in some a0^0_1 classes.

Refinements are impossible in certain other a0^0_1 classes.

The results delineate the boundaries of such refinements.

Abstract

The Jump Inversion Theorem says that for every real $A \ge_T 0'$ there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0'$. A known refinement of this theorem says that we can choose $B$ to be a member of any special $\Pi^0_1$ subclass of $\{0,1\}^\omega$. We now consider the possibility of analogous refinements of two other well-known theorems: the Join Theorem -- for all reals $A$ and $Z$ such that $A \ge_T Z \oplus 0'$ and $Z >_T 0$, there is a real $B$ such that $A \equiv_T B' \equiv_T B \oplus 0' \equiv_T B \oplus Z$ -- and the Pseudojump Inversion Theorem -- for all reals $A \ge_T 0'$ and every $e \in \omega$, there is a real $B$ such that $A \equiv_T B \oplus W_e^B \equiv_T B \oplus 0'$. We show that in these theorems, $B$ can be found in some special $\Pi^0_1$ subclasses of $\{0,1\}^\omega$ but not in others.