Mutual Dimension

Abstract

We define the lower and upper mutual dimensions mdim ( x : y ) and Mdim ( x : y ) between any two points x and y in Euclidean space. Intuitively, these are the lower and upper densities of the algorithmic information shared by x and y . We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that if f : R m → R n is computable and Lipschitz, then the inequalities mdim ( f ( x ): y ) ≤ mdim ( x : y ) and Mdim ( f ( x ): y ) ≤ Mdim ( x : y ) hold for all x ∈ R m and y ∈ R t . We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.