A bounded jump for the bounded Turing degrees

TL;DR

This paper introduces the bounded jump operation for bounded Turing degrees, explores its properties, and relates it to the Ershov hierarchy, extending classical computability results.

Contribution

It defines the bounded jump for bounded Turing degrees, proves its key properties, and establishes its connection to the Ershov hierarchy and Shoenfield inversion.

Findings

Bounded jump is strictly increasing and order-preserving.

Characterization of bounded jump in relation to the Ershov hierarchy.

Extension of Shoenfield inversion to bounded Turing degrees.

Abstract

We define the bounded jump of A by A^b = {x | Exists i <= x [phi_i (x) converges and Phi_x^[A|phi_i(x)](x) converges} and let A^[nb] denote the n-th bounded jump. We demonstrate several properties of the bounded jump, including that it is strictly increasing and order preserving on the bounded Turing (bT) degrees (also known as the weak truth-table degrees). We show that the bounded jump is related to the Ershov hierarchy. Indeed, for n > 1 we have X <=_[bT] 0^[nb] iff X is omega^n-c.e. iff X <=_1 0^[nb], extending the classical result that X <=_[bT] 0' iff X is omega-c.e. Finally, we prove that the analogue of Shoenfield inversion holds for the bounded jump on the bounded Turing degrees. That is, for every X such that 0^b <=_[bT] X <=_[bT] 0^[2b], there is a Y <=_[bT] 0^b such that Y^b =_[bT] X.