  This paper addresses the problem of determining the distance between two
regular languages. It will show how to expand Jaccard distance, which works on
finite sets, to potentially-infinite regular languages. The entropy of a
regular language plays a large role in the extension. Much of the paper is
spent investigating the entropy of a regular language. This includes addressing
issues that have required previous authors to rely on the upper limit of
Shannon's traditional formulation of entropy, because its limit does not always
exist. The paper also includes proposing a new limit based formulation for the
entropy of a regular language and proves that formulation to both exist and be
equivalent to Shannon's original formulation (when it exists). Additionally,
the proposed formulation is shown to equal an analogous but formally quite
different notion of topological entropy from Symbolic Dynamics -- consequently
also showing Shannon's original formulation to be equivalent to topological
entropy. Surprisingly, the natural Jaccard-like entropy distance is trivial in
most cases. Instead, the {\it entropy sum} distance metric is suggested, and
shown to be granular in certain situations.
