Algorithmic tests and randomness with respect to a class of measures

TL;DR

This paper explores various notions of algorithmic randomness across different measure classes, establishing relationships between them and extending the framework to non-computable and metric space contexts.

Contribution

It introduces a unified framework for algorithmic randomness tests across computable, uniform, and arbitrary measures, including classes like Bernoulli measures.

Findings

Hyppocratic randomness equals uniform randomness in certain measure classes

Unified framework for randomness tests in metric spaces

Connections between different randomness notions and measure classes

Abstract

The paper considers quantitative versions of different randomness notions: algorithmic test measures the amount of non-randomness (and is infinite for non-random sequences). We start with computable measures on Cantor space (and Martin-Lof randomness), then consider uniform randomness (test is a function of a sequence and a measure, not necessarily computable) and arbitrary constructive metric spaces. We also consider tests for classes of measures, in particular Bernoulli measures on Cantor space, and show how they are related to uniform tests and original Martin-Lof definition. We show that Hyppocratic (blind, oracle-free) randomness is equivalent to uniform randomness for measures in an effectively orthogonal effectively compact class. We also consider the notions of sparse set and on-line randomness and show how they can be expressed in our framework.