Proofs of conservation inequalities for Levin's notion of mutual information of 1974

TL;DR

This paper proves the conservation inequalities for Levin's mutual information in infinite sequences, filling a gap by providing the first published proof of the probabilistic conservation inequality and reviewing related properties.

Contribution

It provides the first published proof of Levin's probabilistic conservation inequality and reviews key properties of Levin's mutual information in infinite sequences.

Findings

Proof of Levin's probabilistic conservation inequality.

Short proofs of other properties of Levin's mutual information.

Clarification of the foundational aspects of Levin's information measure.

Abstract

In this paper we consider Levin's notion of mutual information in infinite 0-1-sequences, as defined in [Leonid Levin. Laws of Information Conservation (Nongrowth) and Aspects of the Foundation of Probability Theory. Problems of information transmission, vol. 10 (1974), pp. 206--211]. The respective information conservation inequalities were stated in that paper without proofs. Later some proofs appeared in the literature, however no proof of the probabilistic conservation inequality has been published yet. In this paper we prove that inequality and for the sake of completeness we present also short proofs of other properties of the said notion.