Universal test for Hippocratic randomness

TL;DR

This paper introduces a universal test for Hippocratic randomness, extending the concept of Martin-Lof randomness to cases where probability computability is not assumed, under certain approximation conditions.

Contribution

It establishes the existence of a universal test for Hippocratic randomness using an approximation of probability, generalizing prior results to non-computable probabilities.

Findings

Universal test exists for Hippocratic randomness with approximated probability

Levin-Schnorr theorem extended to non-computable probability scenarios

Provides a foundation for randomness testing without computability assumptions

Abstract

Hippocratic randomness is defined in a similar way to Martin-Lof randomness, however it does not assume computability of the probability and the existence of universal test is not assured. We introduce the notion of approximation of probability and show the existence of the universal test (Levin-Schnorr theorem) for Hippocratic randomness when the logarithm of the probability is approximated within additive constant.