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Composing short 3-compressing words on a 2-letter alphabet

Abstract : A finite deterministic (semi)automaton A = (Q, Σ, δ) is k-compressible if there is some word w ∈ Σ + such that the image of its state set Q under the natural action of w is reduced by at least k states. Such word w, if it exists, is called a k-compressing word for A and A is said to be k-compressed by w. A word is k-collapsing if it is k-compressing for each k-compressible automaton, and it is k-synchronizing if it is k-compressing for all k-compressible automata with k+1 states. We compute a set W of short words such that each 3-compressible automaton on a two-letter alphabet is 3-compressed at least by a word in W. Then we construct a shortest common superstring of the words in W and, with a further refinement, we obtain a 3-collapsing word of length 53. Moreover, as previously announced, we show that the shortest 3-synchronizing word is not 3-collapsing, illustrating the new bounds 34 ≤ c(2, 3) ≤ 53 for the length c(2, 3) of the shortest 3-collapsing word on a two-letter alphabet.
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https://hal.archives-ouvertes.fr/hal-01184460
Contributor : Achille Frigeri <>
Submitted on : Thursday, June 1, 2017 - 4:23:44 PM
Last modification on : Tuesday, August 13, 2019 - 11:10:04 AM
Long-term archiving on: : Wednesday, December 13, 2017 - 2:44:37 AM

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Alessandra Cherubini, Achille Frigeri, Zuhua Liu. Composing short 3-compressing words on a 2-letter alphabet. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2017, Vol 19 no. 1. ⟨hal-01184460v3⟩

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