On the boundary of regular languages

Authors:
Jozef Jirásek;Galina Jirásková
Affiliations:
Institute of Computer Science, Faculty of Science, P.J. Šafárik University, Košice, Slovakia;Mathematical Institute, Slovak Academy of Sciences, Košice, Slovakia
Venue:
CIAA'13 Proceedings of the 18th international conference on Implementation and Application of Automata
Year:
2013

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Abstract

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.