Equations on partial words

Blanchet-Sadri, Francine; Blair, D. Dakota; Lewis, Rebeca V.

doi:10.1051/ita:2007041

Blanchet-Sadri, Francine ; Blair, D. Dakota ; Lewis, Rebeca V.

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 23-39.

Résumé

It is well-known that some of the most basic properties of words, like the commutativity ( $x y = y x$ ) and the conjugacy ( $x z = z y$ ), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation $x^{m} y^{n} = z^{p}$ has only periodic solutions in a free monoid, that is, if $x^{m} y^{n} = z^{p}$ holds with integers $m, n, p \geq 2$ , then there exists a word $w$ such that $x, y, z$ are powers of $w$ . This result, which received a lot of attention, was first proved by Lyndon and Schützenberger for free groups. In this paper, we investigate equations on partial words. Partial words are sequences over a finite alphabet that may contain a number of “do not know” symbols. When we speak about equations on partial words, we replace the notion of equality ( $=$ ) with compatibility ( $↑$ ). Among other equations, we solve $x y ↑ y x$ , $x z ↑ z y$ , and special cases of $x^{m} y^{n} ↑ z^{p}$ for integers $m, n, p \geq 2$ .

MR Zbl | 1 citation dans Numdam

DOI : 10.1051/ita:2007041

Classification : 68R15
Mots-clés : equations on words, equations on partial words, commutativity, conjugacy, free monoid

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     title = {Equations on partial words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {23--39},
     publisher = {EDP-Sciences},
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     mrnumber = {2483443},
     zbl = {1170.68032},
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Blanchet-Sadri, Francine; Blair, D. Dakota; Lewis, Rebeca V. Equations on partial words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 23-39. doi : 10.1051/ita:2007041. https://www.numdam.org/articles/10.1051/ita:2007041/

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