New upper bounds to the limitedness of distance automata

A distance automaton is a finite nondeterministic automaton with a distance function which assigns zero or one to each atomic transition and assigns a nonnegative integer to each accepted word by the plus-min principle. In this paper, we prove that the distances of all accepted words of a distance automaton is bounded by some constant if and only if they are bounded by 2 4m 3 +m log (m+2)+m , where m is the number of states of the automaton.