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Pré-Publication, Document De Travail Année : 2015

Composing short 3-compressing words on a 2 letter alphabet

Résumé

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, if it exists, is called a k-compressing word for A. A word is k-collapsing if it is k-compressing for each k-compressible automaton. We compute a set W of short words such that each 3-compressible automata 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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Dates et versions

hal-01184460 , version 1 (17-08-2015)
hal-01184460 , version 2 (11-03-2017)
hal-01184460 , version 3 (01-06-2017)

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Paternité - Pas d'utilisation commerciale

Identifiants

  • HAL Id : hal-01184460 , version 1

Citer

Alessandra Cherubini, Achille Frigeri, Zuhua Liu. Composing short 3-compressing words on a 2 letter alphabet. 2015. ⟨hal-01184460v1⟩
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