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:Rm→Rn 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∈Rm and y∈Rt. 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.Comment: This article is 29 pages and has been submitted to ACM Transactions
on Computation Theory. A preliminary version of part of this material was
reported at the 2013 Symposium on Theoretical Aspects of Computer Science in
Kiel, German