Improved upper bounds on synchronizing nondeterministic automata

We show that i-directable nondeterministic automata can be i-directed with a word of length O(2(n)) for i = 1, 2, where n stands for the number of states. Since for i = 1, 2 there exist i-directable automata having i-directing words of length Omega(2(n)), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length O(n(2) . (3)root 4(n)), improving the previously known upper bound O(2(n)). Here the best known lower bound is Omega((3)root 3(n)). (C) 2009 Elsevier B.V. All rights reserved.