Optimal Simulations between Unary Automata

We consider the problem of computing the costs---{ in terms of states---of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by $O({{\rm e}^{\sqrt{{n}\ln{n}}}})$-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n 2 )-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape. Further results on simulating unary one-way alternating finite automata are also discussed.