Non-algorithmic theory of randomness

TL;DR

This paper introduces a new, more precise language for expressing results in the algorithmic theory of randomness, aiming to improve practical applicability, especially in defining Bernoulli sequences.

Contribution

It proposes an alternative language that eliminates unspecified constants, enhancing the precision and practical relevance of randomness results.

Findings

A new language for randomness theory is developed.

Application to defining Bernoulli sequences is demonstrated.

Improved precision over traditional methods is achieved.

Abstract

This paper proposes an alternative language for expressing results of the algorithmic theory of randomness. The language is more precise in that it does not involve unspecified additive or multiplicative constants, making mathematical results, in principle, applicable in practice. Our main testing ground for the proposed language is the problem of defining Bernoulli sequences, which was of great interest to Andrei Kolmogorov and his students.