Strong Medvedev reducibilities and the KL-randomness problem

TL;DR

This paper investigates the relationships between different notions of randomness and Medvedev reducibilities, demonstrating limitations of stronger reducibilities in reducing Martin-Löf randomness to Kjos-Hanssen and Webb's class.

Contribution

It establishes that Degtev's stronger reducibilities are insufficient for certain reductions between randomness classes, clarifying the boundaries of these computational reductions.

Findings

Stronger reducibilities do not suffice for reducing MLR to Either(MLR)

MLR is truth-table Medvedev reducible to KLR

Limits of Degtev's reducibilities in randomness reductions

Abstract

While it is not known whether each real that is Kolmogorov-Loveland random is Martin-L\"of random, i.e., whether $\mathrm{KLR}\subseteq\mathrm{MLR}$, Kjos-Hanssen and Webb (2021) showed that $\mathrm{MLR}$ is truth-table Medvedev reducible ($\le_{s,tt}$) to $\mathrm{KLR}$. They did this by studying a natural class Either(MLR) and showing that $\mathrm{MLR}\le_{s,tt}\mathrm{Either(MLR)}\supseteq\mathrm{KLR}$. We show that Degtev's stronger reducibilities (positive and linear) do not suffice for the reduction of MLR to Either(MLR), and some related results.