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Discrete Mathematics & Theoretical Computer Science |
Alessandra Cherubini ; Andrzej Kisielewicz ; Brunetto Piochi - On the length of shortest 2-collapsing words
dmtcs:463 - Discrete Mathematics & Theoretical Computer Science, January 1, 2009, Vol. 11 no. 1 - https://doi.org/10.46298/dmtcs.463- 1 Dipartimento di Matematica "F. Brioschi"
- 2 Mathematical Institute [Wroclaw]
- 3 Dipartimento di Matematica "Ulisse Dini"
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.
