In 1978 Sakoda and Sipser raised the question of the cost, in terms of size of representations, of the transformation of two-way and one-way nondeterministic automata into equivalent two-way deterministic automata. Despite all the attempts, the question has been answered only for particular cases (e.g., restrictions of the class of simulated automata or of the class of simulating automata). However the problem remains open in the general case, the best upper bound currently known being exponential. We present a new approach in which unrestricted nondeterministic nite automata are simulated by deterministic models extending two-way deterministic nite automata, paying only a polynomial increase of size. Indeed, we study the costs of the conversions of nondeterministic nite automata into some variants of one-This work contains, in an extended form, some material and results which were previously presented in a preliminary form in conference papers [1] and [2].