  In 2004, Dai, Lathrop, Lutz, and Mayordomo defined and investigated the
finite-state dimension (a finite-state version of algorithmic dimension) of a
sequence $S \in \Sigma^\infty$ and, in 2018, Case and Lutz defined and
investigated the mutual (algorithmic) dimension between two sequences $S \in
\Sigma^\infty$ and $T \in \Sigma^\infty$. In this paper, we propose a
definition for the lower and upper finite-state mutual dimensions
$mdim_{FS}(S:T)$ and $Mdim_{FS}(S:T)$ between two sequences $S \in
\Sigma^\infty$ and $T \in \Sigma^\infty$ over an alphabet $\Sigma$.
Intuitively, the finite-state dimension of a sequence $S \in \Sigma^\infty$
represents the density of finite-state information contained within $S$, while
the finite-state mutual dimension between two sequences $S \in \Sigma^\infty$
and $T \in \Sigma^\infty$ 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 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
$mdim_{FS}(S:T) = Mdim_{FS}(S:T) = 0$.
