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On the length of shortest 2-collapsing words

Abstract : Given a word w over a finite alphabet Sigma and a finite deterministic automaton A = < Q,Sigma,delta >, the inequality vertical bar delta(Q,w)vertical bar <= vertical bar Q vertical bar - k means that under the natural action of the word w the image of the state set Q is reduced by at least k states. The word w is k-collapsing (k-synchronizing) if this inequality holds for any deterministic finite automaton ( with k + 1 states) that satisfies such an inequality for at least one word. We prove that for each alphabet Sigma there is a 2-collapsing word whose length is vertical bar Sigma vertical bar(3)+6 vertical bar Sigma vertical bar(2)+5 vertical bar Sigma vertical bar/2. Then we produce shorter 2-collapsing and 2-synchronizing words over alphabets of 4 and 5 letters.
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Alessandra Cherubini, Andrzej Kisielewicz, Brunetto Piochi. On the length of shortest 2-collapsing words. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2009, 11 (1), pp.33--44. ⟨hal-00988185⟩

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