Finite-State Mutual Dimension

TL;DR

This paper introduces finite-state mutual dimension, a measure of shared finite-state information between sequences, extending previous notions of finite-state and algorithmic dimensions, with properties akin to mutual information.

Contribution

It defines finite-state mutual dimensions, characterizes them via block mutual information rates, and establishes conditions for sequences with zero mutual dimension.

Findings

Finite-state mutual dimension has properties of mutual information.

Characterization via block mutual information rates.

Conditions for sequences with zero mutual dimension.

Abstract

In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finite-state dimension (a finite-state version of algorithmic dimension) of a sequence $S \in \Sigma^\infty$ and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences $S \in \Sigma^\infty$ and $T \in \Sigma^\infty$. In this paper, we propose a definition for the lower and upper finite-state mutual dimensions $mdim_{FS}(S:T)$ and $Mdim_{FS}(S:T)$ between two sequences $S \in \Sigma^\infty$ and $T \in \Sigma^\infty$ over an alphabet $\Sigma$. Intuitively, the finite-state dimension of a sequence $S \in \Sigma^\infty$ represents the density of finite-state information contained within $S$, while the finite-state mutual dimension between two sequences $S \in \Sigma^\infty$ and $T \in \Sigma^\infty$ represents the density of finite-state information shared by $S$ and $T$. Thus ``finite-state mutual dimension'' can be viewed as a ``finite-state'' version of mutual dimension and as a ``mutual'' version of finite-state dimension. The main results of this investigation are as follows. First, we show that finite-state mutual dimension, defined using information-lossless finite-state compressors, has all of the properties expected of a measure of mutual information. Next, we prove that finite-state mutual dimension may be characterized in terms of block mutual information rates. Finally, we provide necessary and sufficient conditions for two normal sequences to achieve $mdim_{FS}(S:T) = Mdim_{FS}(S:T) = 0$.