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 work 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 sup 4m sup 3 +mlog(m+2)+m, where m is the number of states of the automaton.
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