On the State Complexity of the Shuffle of Regular Languages

Abstract : 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 $$2\leqslant m\leqslant 5$$ and $$n\geqslant 2$$, 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 $$m,n \geqslant 2$$. The question of reachability, and hence also of the tightness of the bound f(m, n) in general, remains open.