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(e(root 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 at the endmarkers only. Further results on simulating unary alternating finite automata are pointed out. Our results give answers to some questions left open in the literature.