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Aspects algorithmiques du retournement de mot

Abstract : First part: Word reversing is a rewriting operation associated to a presentation (of a semigroup, here). In good cases, reversing yields a solution to the word problem. Otherwise, there is a way to complete a presentation. Now, Gröbner bases also provide a mean to complete a presentation and solve the word problem. We show that these two methods are not the same and exhibit a classification of the different behaviours. We then introduce an extension of word reversing to circumvent the defect of completeness of certain pre- sentations and illustrate its effectiveness on Heisenberg presentation—which is incomplete. Second part: We restrict the study to Artin-Tits presentations of braid monoids. We show that the maximal combinatorial distance between two equivalent braid words is at least quartic in their width. We show simple criteria for a van Kampen diagram (or a reversing diagram) to realize the combinatorial distance between two words. We then compute bounds for two numbers associated to word reversing, and, in particular, for braid words whose width is arbitrarily large: starting from a word, the first one is the maximal length of a sequence of reversings and the second one is the maximal length of the last word (which exists and is unique) of such a sequence. We compute a lower bound (quartic in the length of the initial word) for the first number. For the second number, we establish a cubic (in the length of the initial word) upper bound.
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https://tel.archives-ouvertes.fr/tel-00439023
Contributor : Marc Autord <>
Submitted on : Saturday, December 5, 2009 - 7:02:18 PM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM
Long-term archiving on: : Thursday, June 17, 2010 - 8:26:20 PM

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  • HAL Id : tel-00439023, version 1

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Marc Autord. Aspects algorithmiques du retournement de mot. Mathématiques [math]. Université de Caen, 2009. Français. ⟨tel-00439023⟩

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