Algorithmically Independent Sequences

Abstract

Two objects are independent if they do not a ect each other. Independence is wellunderstood 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 in nite 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 with x has measure one. For both notions of independence we investigate to what extent pairs of independent sequences, can be e ectively 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 e ective way of producing from an arbitrary sequence with positive constructive Hausdor 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.