Randomness Conservation Inequalities; Information and Independence in Mathematical Theories

2026-07-26

AI Summary

This work addresses a central challenge in algorithmic information theory: rigorously defining randomness, mutual information, and independence at the level of individual infinite sequences while satisfying strong invariance properties. By refining Kolmogorov complexity theory, the paper introduces a novel notion of randomness that fulfills conservation inequalities, enabling—for the first time—the formalization of mutual information and independence between mathematical objects on individual infinite sequences. This framework not only provides a more streamlined foundation for probability theory, algorithmic theory, and intuitionistic logic but also substantially simplifies the associated axiomatic systems and derivations.

Abstract

The article develops further Kolmogorov's Algorithmic Complexity Theory. The definition of Randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as Mutual Information in individual infinite sequences. Applications to several areas, like Probability Theory, Theory of Algorithms, Intuitionistic Logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property.

Problem

• Randomness
• Mutual Information
• Algorithmic Complexity
• Independence
• Conservation Inequalities

Innovation

• Randomness Conservation Inequalities
• Algorithmic Complexity
• Mutual Information
• Independence
• Kolmogorov Complexity