Périodes locales et propagation de périodes dans un mot

• Published in: Theoretical computer science Vol. 204; no. 1; pp. 87 - 98
• Main Author: Duval, Jean-Pierre
• Format: Journal Article
• Language: French
• Published: Elsevier B.V 06.09.1998
• Subjects: Combinatory, Local period; Periods; Repetition; Words; Combinatory, Local period; Periods; Repetition; Words
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/S0304-3975(98)00033-4

A word of length n over an alphabet A is a sequence a 1 … a n of letters of A. It is convenient to consider a “long enough” word over A as an infinite right word, that is an infinite sequence a 1 … a i … of elements of A. An integer λ is a period of the word in the interval [ j … k] if we have a i = a i+ λ for those indices i and i + λ in the considered interval. The period of a word designates its smallest period over its whole length. A point p of a word is the cut ( a 1 … a p , a p+1 …). A non-negative integer λ is a local period at point p iff λ is a period in the interval [ p − λ + 1 … p + λ]. According to the critical point's theorem [1,2], the period of a “long enough (or not)” word is the maximum of the minimal local periods taken in each point of this word. M.P. Schützenberger, who was at the origin of the research work on the relations between local periods and periods of a word obtained by concatenation of periodical words, and our ability to characterize its period from the observation of the local period at the concatenation points only. This is the formulation of the unpublished answer we offered him that we suggest here. A word of length n over an alphabet A is a sequence a 1 … a n of letters of A. It is convenient to consider a “long enough” word over A as an infinite right word, that is an infinite sequence a 1 … a i … of elements of A. An integer λ is a period of the word in the interval [ j … k] if we have a i = a i+ λ for those indices i and i + λ in the considered interval. The period of a word designates its smallest period over its whole length. A point p of a word is the cut ( a 1 … a p , a p+1 …). A non-negative integer λ is a local period at point p iff λ is a period in the interval [ p − λ + 1 … p + λ]. According to the critical point's theorem [1,2], the period of a “long enough (or not)” word is the maximum of the minimal local periods taken in each point of this word. M.P. Schützenberger, who was at the origin of the research work on the relations between local periods and periods of a word obtained by concatenation of periodical words, and our ability to characterize its period from the observation of the local period at the concatenation points only. This is the formulation of the unpublished answer we offered him that we suggest here.