On the boundary of regular languages

We prove that the tight bound on the state complexity of the boundary of regular languages, defined as bd$(L)=L^* \cap ( \, \overline{L} \, )^*$, is 22n−2+22n−3+2n−2+2−2·3n−2−n. Our witness languages are described over a five-letter alphabet. For a four-letter alphabet, the lower bound is smaller by just one, and we conjecture that the upper bound cannot be met in the quaternary case.