Abstract

In this talk, I will discuss recent work with Jack H. Lutz on a notion of finite-state mutual dimension. Intuitively, the finite-state dimension of a sequence S represents the density of finite-state information contained within S, while the finite-state mutual dimension between two sequences S and T 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 that will be discussed 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 finite-state mutual dimension zero.