Critical Factorization Theorem Research Articles

Abelian bordered factors and periodicity

A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.

A NEW COMPLEXITY FUNCTION FOR WORDS BASED ON PERIODICITY

Motivated by the extension of the critical factorization theorem to infinite words, we study the (local) periodicity function, i.e. the function that, for any position in a word, gives the size of the shortest square centered in that position. We prove that this function characterizes any binary word up to exchange of letters. We then introduce a new complexity function for words (the periodicity complexity) that, for any position in the word, gives the average value of the periodicity function up to that position. The new complexity function is independent from the other commonly used complexity measures as, for instance, the factor complexity. Indeed, whereas any infinite word with bounded factor complexity is periodic, we will show a recurrent non-periodic word with bounded periodicity complexity. Further, we will prove that the periodicity complexity function grows as Θ( log n) in the case of the Fibonacci infinite word and that it grows as Θ(n) in the case of the Thue–Morse word. Finally, we will show examples of infinite recurrent words with arbitrary high periodicity complexity.

Characteristic Sturmian words are extremal for the Critical Factorization Theorem

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If px(n) denotes the local period of an infinite word x at point n, we prove that x is a characteristic Sturmian word if and only if px(n) is smaller than or equal to n+1 for all n≥1 and it is equal to n+1 for infinitely many integers n.This result is extremal with respect to the CFT since a consequence of the CFT is that, for any infinite recurrent word x, either the function px is bounded, and in such a case x is periodic, or px(n)≥n+1 for infinitely many integers n.As a byproduct of the techniques used in the paper we extend a result of Harju and Nowotka (2002) in [18] stating that any finite Fibonacci word fn,n≥5, has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.

On abelian versions of critical factorization theorem

In the paper we study abelian versions of the critical factorization theorem. We investigate both similarities and differences between the abelian powers and the usual powers. The results we obtained show that the constraints for abelian powers implying periodicity should be quite strong, but still natural analogies exist.

Partial words and the critical factorization theorem revisited

In this paper, we consider one of the most fundamental results on the periodicity of words, namely the critical factorization theorem. Given a word w and nonempty words u , v satisfying w = u v , the minimal local period associated with the factorization ( u , v ) is the length of the shortest square at position | u | − 1 . The critical factorization theorem shows that for any word, there is always a factorization whose minimal local period is equal to the minimal period (or global period) of the word. Crochemore and Perrin presented a linear time algorithm (in the length of the word) that finds a critical factorization from the computation of the maximal suffixes of the word with respect to two total orderings on words: the lexicographic ordering related to a fixed total ordering on the alphabet, and the lexicographic ordering obtained by reversing the order of letters in the alphabet. Here, by refining Crochemore and Perrin’s algorithm, we give a version of the critical factorization theorem for partial words (such sequences may contain “do not know” symbols or “holes”). Our proof provides an efficient algorithm which computes a critical factorization when one exists. Our results extend those of Blanchet-Sadri and Duncan for partial words with one hole. A World Wide Web server interface at http://www.uncg.edu/mat/research/cft2/ has been established for automated use of the program.

On unique factorizations of primitive words

We give a short proof of a result by Weinbaum [Unique sunwords in nonperiodic words, Proc. Amer. Math. Soc. 109(3) (1990) 615–619] stating that each primitive word of length at least 2 has a conjugate w ′ = uv such that both u and v have a unique position in the cyclic word of w . We emphasize the connection of Weinbaum's result to the Critical Factorization Theorem.

A multidimensional critical factorization theorem

The Critical Factorization Theorem is one of the principal results in combinatorics on words. It relates local periodicities of a word to its global periodicity. In this paper we give a multidimensional extension of it. More precisely, we give a new proof of the Critical Factorization Theorem, but in a weak form, where the weakness is due to the fact that we loose the tightness of the local repetition order. In exchange, we gain the possibility of extending our proof to the multidimensional case. Indeed, this new proof makes use of the Theorem of Fine and Wilf, that has several classical generalizations to the multidimensional case.

Partial words and the critical factorization theorem

The study of combinatorics on words, or finite sequences of symbols from a finite alphabet, finds applications in several areas of biology, computer science, mathematics, and physics. Molecular biology, in particular, has stimulated considerable interest in the study of combinatorics on partial words that are sequences that may have a number of “do not know” symbols also called “holes”. This paper is devoted to a fundamental result on periods of words, the critical factorization theorem, which states that the period of a word is always locally detectable in at least one position of the word resulting in a corresponding critical factorization. Here, we describe precisely the class of partial words w with one hole for which the weak period is locally detectable in at least one position of w. Our proof provides an algorithm which computes a critical factorization when one exists. A World Wide Web server interface at http://www.uncg.edu/mat/cft/ has been established for automated use of the program.

Linear-time computation of local periods

We present a linear-time algorithm for computing all local periods of a given word. This subsumes (but is substantially more powerful than) the computation of the (global) period of the word and on the other hand, the computation of a critical factorization, implied by the Critical Factorization Theorem.

Density of Critical Factorizations

We investigate the density of critical factorizations of infinte sequences of words. The density of critical factorizations of a word is the ratio between the number of positions that permit a critical factorization, and the number of all positions of a word. We give a short proof of the Critical Factorization Theorem and show that the maximal number of noncritical positions of a word between two critical ones is less than the period of that word. Therefore, we consider only words of index one, that is words where the shortest period is larger than one half of their total length, in this paper. On one hand, we consider words with the lowest possible number of critical points and show, as an example, that every Fibonacci word longer than five has exactly one critical factorization and every palindrome has at least two critical factorizations. On the other hand, sequences of words with a high density of critical points are considered. We show how to construct an infinite sequence of words in four letters where every point in every word is critical. We construct an infinite sequence of words in three letters with densities of critical points approaching one, using square-free words, and an infinite sequence of words in two letters with densities of critical points approaching one half, using Thue-Morse words. It is shown that these bounds are optimal.

Multiple factorizations of words and defect effect

We prove that if X is a finite prefix set and w is a non-periodic bi-infinite word, possessing 3 disjoint X-factorizations, then the combinatorial rank of X is at most card(X)−2 . This is one of the rare cases when a cumulative defect effect is known to hold. Finally, connections to the critical factorization theorem are discussed.

Periodicity and the golden ratio

We prove a periodicity theorem on words that has strong analogies with the Critical Factorization theorem. The Critical Factorization theorem states, roughly speaking, a connection between local and global periods of a word; the local period at any position in the word is there defined as the shortest repetition (a square) “centered” in that position. We here take into account a different notion of local period by considering, for any position in the word, the shortest repetition “immediately to the left” from that position. In this case a repetition which is a square does not suffices and the golden ratio ϑ (more precisely its square ϑ 2 = 2.618 …) surprisingly appears as a threshold for establishing a connection between local and global periods of the word. We further show that the number ϑ 2 is tight for this result. Two applications are then derived. In the firts we give a characterization of ultimately periodic infinite words. The second application concerns the topological perfectness of some families of infinite words.