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A synergic approach to the minimal uncompletable words problem

Sandrine Julia 1 Arnaud Malapert 2 Julien Provillard 1
1 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe MC3
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
2 Laboratoire d'Informatique, Signaux, et Systèmes de Sophia-Antipolis (I3S) / Equipe CEP
Laboratoire I3S - MDSC - Modèles Discrets pour les Systèmes Complexes
Abstract : A finite language X over an alphabet Σ is complete if any word in Σ* is a factor of a word in X*. A word which is not a factor of X* is said uncompletable. Among them, some are minimal as all their proper factors belong to Fact(X*). The problem is to find bounds on the length of the shortest minimal uncompletable words depending on k, the maximal length of words in X. Though Restivo’s conjecture stating an upper bound in 2k^2 was already contradicted twice, the problem of the existence of a quadratic upper bound is still open. Our approach is original and synergic. We start by characterizing minimal uncompletable words. An efficient algorithm is given to speed up the search of such words. Finally, a genetic algorithm using a SAT-solver allows us to obtain new results for the first values of k.
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https://hal.archives-ouvertes.fr/hal-01342479
Contributor : Sandrine Julia <>
Submitted on : Wednesday, July 6, 2016 - 10:54:12 AM
Last modification on : Thursday, March 5, 2020 - 12:20:25 PM

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  • HAL Id : hal-01342479, version 1

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Sandrine Julia, Arnaud Malapert, Julien Provillard. A synergic approach to the minimal uncompletable words problem. 16th Mons Theoretical Computer Science Days , Sep 2016, Liège, Belgium. ⟨hal-01342479⟩

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