Finite-State Mutual Dimension
Adam Case, Jack H. Lutz · 2022
In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finite-state dimension (a finite-state version of algorithmic dimension) of a sequence$S$and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences$S$and$T$. 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$S$and$T$. Intuitively, the finite-state dimension of a sequence$S$represents the density of finite-state information contained within$S$, while the finite-state mutual dimension between two sequences$S$and$T$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$R_{1}$and$R_{2}$to achieve$mdim_{FS}(R_{1}:R_{2})=Mdim_{FS}(R_{1}:R_{2})=0$.