Finite-State Mutual Dimension

Abstract

In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the finitestate 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 F S (S : T ) and M dim F S (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 finitestate 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 F S (S : T ) = M dim F S (S : T ) = 0.

Finite-State Relative Dimension, Dimensions of AP Subsequences and a Finite-State van Lambalgen’s Theorem

“…Case and Lutz [4] introduce finite-state mutual dimension as a finite state analogue of mutual dimension [8]. Intuitively, it represents the density of finite-state information shared between two sequences.…”

“…This demonstrates the mathematical robustness of our definition. Our notion is a finite-state analogue of conditional entropy, in a similar manner as mutual dimension introduced by Case and Lutz [4] is an analogue of mutual information.…”