# Solovay reduction and continuity

## TL;DR

This paper explores the connections between different types of reductions and continuity notions in computable analysis, introducing quasi Solovay reduction linked to H"older continuity and examining their implications for randomness and completeness.

## Contribution

It introduces quasi Solovay reduction, characterizes it via H"older continuity, and distinguishes it from existing reductions, advancing the understanding of reducibility and continuity in computable analysis.

## Key findings

- Solovay reduction characterized by Lipschitz continuous functions.
- Introduction of quasi Solovay reduction related to H"older continuity.
- Distinctness of quasi Solovay reduction from Solovay and Turing reductions.

## Abstract

The objective of this study is a better understanding of the relationships between reduction and continuity. Solovay reduction is a variation of Turing reduction based on the distance of two real numbers. We characterize Solovay reduction by the existence of a certain real function that is computable (in the sense of computable analysis) and Lipschitz continuous. We ask whether there exists a reducibility concept that corresponds to H\"older continuity. The answer is affirmative. We introduce quasi Solovay reduction and characterize this new reduction via H\"older continuity. In addition, we separate it from Solovay reduction and Turing reduction and investigate the relationships between complete sets and partial randomness.