  Two languages are "finitely different" if their symmetric difference is
finite. We consider the DFAs of finitely different regular languages and find
major structural similarities. We proceed to consider the smallest DFAs that
recognize a language finitely different from some given DFA. Such "f-minimal"
DFAs are not unique, and this non-uniqueness is characterized. Finally, we
offer a solution to the minimization problem of finding such f-minimal DFAs.
