A defect theorem for bi-infinite words

Theoretical Computer Science, 2002

A word of length n over a ÿnite alphabet A is a map from {0; : : : ; n − 1} into A. A partial word of length n over A is a partial map from {0; : : : ; n − 1} into A. In the latter case, elements of {0; : : : ; n − 1} without image are called holes (a word is just a partial word without holes). In this paper, we extend a fundamental periodicity result on words due to Fine and Wilf to partial words with two or three holes. This study was initiated by Berstel and Boasson for partial words with one hole. Partial words are motivated by molecular biology.

Information and Computation, 2008

The concept of periodicity has played over the years a centra1 role in the development of combinatorics on words and has been a highly valuable too1 for the design and analysis of algorithms. Fine and Wilf's famous periodicity result, which is one of the most used and known results on words, has extensions to partia1 words, or sequences that may have a number of "do not know" symbols. These extensions fal1 into two categories: the ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we obtain consequences by generalizing, in particular, the combinatoria1 property that "for any word u over {a, b}, ua or ub is primitive," which proves in some sense that there exist very many primitive partia1 words.

Theoretical Computer Science, 2003

We prove that, given a double sequence w over the alphabet A (i.e. a mapping from Z 2 to A), if there exists a pair (n0; m0) ∈ Z 2 such that pw(n0; m0) ¡ 1 100 n0m0, then w has a periodicity vector, where pw is the complexity function in rectangles of w.

Theoretical Computer Science, 2009

We investigate a special variant of the shuffle decomposition problem for regular languages; namely, when the given regular language is the shuffle of finite languages. The shuffle decomposition into finite languages is, in general, not unique. That is, there are

The concept of periodicity has played over the years a central role in the development of combinatorics on words and has been a highly valuable tool for the design and analysis of algorithms. There are many fundamental periodicity results on words. Among them is the famous result of Fine and Wilf which intuitively determines how far two periodic events have to match in order to guarantee a common period. This result states that for positive integers p and q, if the word u has periods p and q and the length of u is not less than p + q − gcd(p, q), then u has also period gcd(p, q). Fine and Wilf's result, which is one of the most used and known results on words, has extensions to partial words, or sequences that may have a number of "do not know" symbols. These extensions fall into two categories: The ones that relate to strong periodicity and the ones that relate to weak periodicity. In this paper, we study some consequences of these results.

Acta Informatica, 2001

We introduce the notion of periodic-like word. It is a word whose longest repeated prefix is not right special. Some different characterizations of this concept are given. In particular, we show that a word w is periodic-like if and only if it has a period not larger than |w| − R w , where R w is the least non-negative integer such that any prefix of w of length ≥ R w is not right special. We derive that if a word w has two periods p, q ≤ |w| − R w , then also the greatest common divisor of p and q is a period of w. This result is, in fact, an improvement of the theorem of Fine and Wilf. We also prove that the minimal period of a word w is equal to the sum of the minimal periods of its components in a suitable canonical decomposition in periodic-like subwords. Moreover, we characterize periodic-like words having the same set of proper boxes, in terms of the important notion of root-conjugacy. Finally, some new uniqueness conditions for words, related to the maximal box theorem are given.

Information Processing Letters, 2002

We prove the following interesting combinatorial property of the poset of the factors of a word. Let w be a word and n = G w + 2, where G w is the maximal length of a repeated factor of w. If v is any word such that the posets of the factors of v and of w up to length n are isomorphic, then v can be obtained by renaming the letters of w or of the reversal of w. 

Information and Computation/information and Control, 2011

One of the particularities of information encoded as DNA strands is that a string u contains basically the same information as its Watson-Crick complement, denoted here as θ(u). Thus, any expression consisting of repetitions of u and θ(u) can be considered in some sense periodic. In this paper, we give a generalization of Lyndon and Schützenberger's classical result about equations of the form u l = v n w m , to cases where both sides involve repetitions of words as well as their complements. Our main results show that, for such extended equations, if l 5, n, m 3, then all three words involved can be expressed in terms of a common word t and its complement θ(t). Moreover, if l 5, then n = m = 3 is an optimal bound. These results are established based on a complete characterization of all possible overlaps between two expressions that involve only some word u and its complement θ(u), which is also obtained in this paper. Crown

Theoretical Computer Science, 2008

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their directive words. Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular normalized directive words, which we introduced in an earlier paper. Also of importance is the use of return words to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.

Theoretical Computer Science, 2011

Initially stated in terms of Beatty sequences, the Fraenkel conjecture can be reformulated as follows: for a k-letter alphabet A, with a fixed k ≥ 3, there exists a unique balanced infinite word, up to letter permutations and shifts, that has mutually distinct letter frequencies. Motivated by the Fraenkel conjecture, we study in this paper whether two Christoffel words can be superimposed. Following from previous works on this conjecture using Beatty sequences, we give a necessary and sufficient condition for the superimposition of two Christoffel words having same length, and more generally, of two arbitrary Christoffel words. Moreover, for any two superimposable Christoffel words, we give the number of different possible superimpositions and we prove that there exists a superimposition that works for any two superimposable Christoffel words. Finally, some new properties of Christoffel words are obtained as well as a geometric proof of a classic result concerning the money problem, using Christoffel words.