Solovay reducibility implies S2a-reducibility

TL;DR

This paper proves that Solovay reducibility implies S2a-reducibility for computably approximable reals, establishing a hierarchical relationship between these two measures of relative randomness.

Contribution

It demonstrates that Solovay reducibility implies S2a-reducibility on c.a. reals with the same constant, clarifying their relationship.

Findings

Solovay reducibility implies S2a-reducibility on c.a. reals

The implication holds even with the same constant

The reverse implication does not hold

Abstract

The original notion of Solovay reducibility was introduced by Robert M. Solovay (unpublished notes) in 1975 as a measure of relative randomness. The S2a-reducibility introduced by Xizhong Zheng and Robert Rettinger (DOI:10.1007/978-3-540-27798-9_39) in 2004 is a modification of Solovay reducibility suitable for computably approximable (c.a.) reals. We demonstrate that Solovay reducibility implies S2a-reducibility on the set of c.a. reals, even with the same constant, but not vice versa.