We investigate the shuffle operation on regular languages represented by
complete deterministic finite automata. We prove that $f(m,n)=2^{mn-1} +
2^{(m-1)(n-1)}(2^{m-1}-1)(2^{n-1}-1)$ is an upper bound on the state complexity
of the shuffle of two regular languages having state complexities $m$ and $n$,
respectively. We also state partial results about the tightness of this bound.
We show that there exist witness languages meeting the bound if $2\le m\le 5$
and $n\ge2$, and also if $m=n=6$. Moreover, we prove that in the subset
automaton of the NFA accepting the shuffle, all $2^{mn}$ states can be
distinguishable, and an alphabet of size three suffices for that. It follows
that the bound can be met if all $f(m,n)$ states are reachable. We know that an
alphabet of size at least $mn$ is required provided that $m,n \ge 2$. The
question of reachability, and hence also of the tightness of the bound $f(m,n)$
in general, remains open.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-41114-9_
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