Algorithmically independent sequences

TL;DR

This paper introduces two notions of independence for infinite binary sequences within algorithmic information theory, explores their properties, and demonstrates limitations on effectively constructing independent sequences.

Contribution

It proposes and analyzes two new definitions of independence for sequences, and establishes impossibility results for their effective construction.

Findings

Set of sequences independent of a given sequence has measure one.

No effective method exists to produce independent sequences with high complexity from arbitrary sequences.

Identifies limitations in constructing independent sequences with super-logarithmic complexity.

Abstract

Two objects are independent if they do not affect each other. Independence is well-understood in classical information theory, but less in algorithmic information theory. Working in the framework of algorithmic information theory, the paper proposes two types of independence for arbitrary infinite binary sequences and studies their properties. Our two proposed notions of independence have some of the intuitive properties that one naturally expects. For example, for every sequence $x$, the set of sequences that are independent (in the weaker of the two senses) with $x$ has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be effectively constructed via Turing reductions (from one or more input sequences). In this respect, we prove several impossibility results. For example, it is shown that there is no effective way of producing from an arbitrary sequence with positive constructive Hausdorff dimension two sequences that are independent (even in the weaker type of independence) and have super-logarithmic complexity. Finally, a few conjectures and open questions are discussed.