Mutual Dimension and Random Sequences

Abstract

If S and T are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions mdim(S : T ) and M dim(S : T ) are the upper and lower densities of the algorithmic information that is shared by S and T . In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences R 1 and R 2 with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions mdim(R 1 : R 2 ) and M dim(R 1 : R 2 ), and we show that the condition M dim(R 1 : R 2 ) = 0 is necessary but not sufficient for R 1 and R 2 to be independently random.We also identify conditions under which Billingsley generalizations of the mutual dimensions mdim(S : T ) and M dim(S : T ) can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the "correct" relationships with the Billingsley generalizations of dim(S), Dim(S), dim(T ), and Dim(T ) that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.

Algorithmic Information, Plane Kakeya Sets, and Conditional Dimension

“…Third, and most crucially, this principle implies that, in order to prove a lower bound dim H (E) ≥ α, it suffices to show that, for every A ⊆ N and every ε > 0, there is a point x ∈ E such that dim A (x) ≥ α − ε. 5 For the (≥) direction of this principle, we construct the minimizing oracle A. The oracle encodes, for a carefully chosen sequence of increasingly refined covers for E, the approximate locations and diameters of all cover elements.…”

“…The above-described dimensions dim(x) and Dim(x) of a point x in Euclidean space (or an infinite sequence x over a finite alphabet) are analogous by limit theorems [27,1] to K(u) and hence to H(X). Case and the first author have recently developed and investigated the mutual dimension mdim(x : y) and the dual strong mutual dimension Mdim(x : y), which are densities of the algorithmic information shared by points x and y in Euclidean spaces [4] or sequences x and y over a finite alphabet [5]. These mutual dimensions are analogous to I(u : v) and I(X; Y ).…”

Finite-State Mutual Dimension

“…where I(S ↾ n : T ↾ n) is the algorithmic mutual information between the first n bits of S and T [7]. The algorithmic mutual information I(u : w) between two strings u ∈ Σ * and w ∈ Σ * is…”

“…In the same paper, the authors demonstrate that, if two sequences S ∈ Σ ∞ and T ∈ Σ ∞ are independently random, then M dim(S : T ) = 0. However, they also show that not all pairs of sequences that achieve mutual dimension zero are necessarily independently random [7]. The purpose of this article is to develop a notion of finite-state mutual dimension, which includes defining it using information-lossless finite-state compressors, proving that it can be characterized in terms of block entropy rates, and exploring its relationship with normal sequences.…”

Bounded Turing Reductions and Data Processing Inequalities for Sequences

“…In this paper, we discuss several new data processing inequalities for sequences. We use mutual dimension, a recent development in constructive dimension, as the means for measuring the quantity of shared information between two sequences [3,4]. Lutz defined and explored the constructive dimension of sequences in [11], and Mayordomo showed that constructive dimension can be characterized in terms of Kolmogorov complexity in [12].…”