Reversal of binary regular languages

We present binary deterministic finite automata of n states that meet the upper bound 2n on the state complexity of reversal. The automata have a single final state and are one-cycle-free-path automata; thus the witness languages are deterministic union-free languages. This result allows us to describe a binary language such that the nondeterministic state complexity of the language and of its complement is n and n+1, respectively, while the state complexity of the language is 2n. Next, we show that if the nondeterministic state complexity of a language and of its complement is n, then the state complexity of the language cannot be 2n. We also provide lower and upper bounds on the state complexity of such a language.