arXiv subjects

Hayden R. Jananthan

Publications and source records attributed to Hayden R. Jananthan.

2 recordsLinked to original sources

arXiv2024

Turing Degrees of Hyperjumps

The Posner-Robinson Theorem states that for any reals $Z$ and $A$ such that $Z \oplus 0' \leq_\mathrm{T} A$ and $0 <_\mathrm{T} Z$, there exists $B$ such that $A \equiv_\mathrm{T} B' \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus 0'$. Consequently, any nonzero Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a Turing jump relative to some $B$. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals $Z$ and $A$ such that $Z \oplus \mathcal{O} \leq_\mathrm{T} A$ and $0 <_\mathrm{HYP} Z$, there exists $B$ such that $A \equiv_\mathrm{T} \mathcal{O}^B \equiv_\mathrm{T} B \oplus Z \equiv_\mathrm{T} B \oplus \mathcal{O}$. As an analogous consequence, any nonhyperarithmetical Turing degree $\operatorname{deg}_\mathrm{T}(Z)$ is a hyperjump relative to some $B$.

math.LO↗

arXiv2021

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

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 $Π^0_1$ subclass of $\{0,1\}^ω$. 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 $e \in ω$, 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 $Π^0_1$ subclasses of $\{0,1\}^ω$ but not in others.

math.LO↗