Finite-State Mutual Dimension
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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 ∈Σ^∞ and, in 2018, Case and Lutz defined and investigated the mutual (algorithmic) dimension between two sequences S ∈Σ^∞ and T ∈Σ^∞. 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 ∈Σ^∞ and T ∈Σ^∞ over an alphabet Σ. Intuitively, the finite-state dimension of a sequence S ∈Σ^∞ represents the density of finite-state information contained within S, while the finite-state mutual dimension between two sequences S ∈Σ^∞ and T ∈Σ^∞ 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.
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