Universality, optimality, and randomness deficiency

TL;DR

This paper investigates the computational distinctions between universal and optimal Martin-Löf tests, examining their impact on layerwise computability and the Weihrauch degree of the randomness deficiency function, with robustness and reducibility results.

Contribution

It provides a detailed analysis of the differences between universal and optimal ML-tests and their influence on computational properties like layerwise computability and Weihrauch degrees.

Findings

Layerwise computability is more restrictive than Weihrauch reducibility to LAY.

Robustness and idempotence results for the Weihrauch degree of LAY.

Analysis of the principle RD as a variant of LAY with exact randomness deficiency.

Abstract

A Martin-L\"of test $\mathcal U$ is universal if it captures all non-Martin-L\"of random sequences, and it is optimal if for every ML-test $\mathcal V$ there is a $c \in \omega$ such that $\forall n(\mathcal{V}_{n+c} \subseteq \mathcal{U}_n)$. We study the computational differences between universal and optimal ML-tests as well as the effects that these differences have on both the notion of layerwise computability and the Weihrauch degree of LAY, the function that produces a bound for a given Martin-L\"of random sequence's randomness deficiency. We prove several robustness and idempotence results concerning the Weihrauch degree of LAY, and we show that layerwise computability is more restrictive than Weihrauch reducibility to LAY. Along similar lines we also study the principle RD, a variant of LAY outputting the precise randomness deficiency of sequences instead of only an upper bound as LAY.