The probability distribution as a computational resource for randomness testing

TL;DR

The paper explores the concept of randomness testing under the constraint of not accessing distribution parameters, showing that for Bernoulli measures, Hippocratic randomness aligns with ordinary randomness, but no universal test exists.

Contribution

It introduces the notion of Hippocratic randomness tests that do not access distribution parameters and proves their equivalence to ordinary randomness for Bernoulli measures, highlighting the absence of a universal test.

Findings

Hippocratic randomness coincides with ordinary randomness for Bernoulli measures.

No single Hippocratic randomness test can compute the distribution parameter.

There is no universal Hippocratic randomness test.

Abstract

When testing a set of data for randomness according to a probability distribution that depends on a parameter, access to this parameter can be considered as a computational resource. We call a randomness test Hippocratic if it is not permitted to access this resource. In these terms, we show that for Bernoulli measures $\mu_p$, $0\le p\le 1$ and the Martin-L\"of randomness model, Hippocratic randomness of a set of data is the same as ordinary randomness. The main idea of the proof is to first show that from Hippocrates-random data one can Turing compute the parameter $p$. However, we show that there is no single Hippocratic randomness test such that passing the test implies computing $p$, and in particular there is no universal Hippocratic randomness test.