Francine Blanchet-Sadri

Partial words and a theorem of Fine and Wilf revisited

A word of length n over a finite 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,...,n1} 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 biolog

Periodicity on Partial Words

A partial word of length n over a finite alphabet A is a partial map from {0, … , n - 1} into A. Elements of {0, … , n-1} without image are called holes (a word is just a partial word without holes). A fundamental periodicity result on words due to Fine and Wilf [1] intuitively determines how far two periodic events have to match in order to guarantee a common period. This result was extended to partial words with one hole by Berstel and Boasson [2] and to partial words with two or three holes by Blanchet-Sadri and Hegstrom [3]. In this paper, we give an extension to partial words with an arbitrary number of holes.

Conjugacy on partial words

The study of the combinatorial properties of strings of symbols from a finite alphabet (also referred to as words) is profoundly connected to numerous fields such as biology, computer science, mathematics, and physics. In this paper, we examine to which extent some fundamental combinatorial properties of words, such as conjugacy, remain true for partial words. The motivation behind the notion of a partial word is the comparison of two genes (alignment of two such strings can be viewed as a construction of two partial words that are said to be compatible). This study on partial words was initiated by Berstel and Boasson.

Local periods and binary partial words: an algorithm

The study of the combinatorial properties of strings of symbols from a finite alphabet (also referred to as words) is profoundly connected to numerous fields such as biology, computer science, mathematics, and physics. Research in combinatorics on words goes back roughly a century. There is a renewed interest in combinatorics on words as a result of emerging new application areas such as molecular biology. Partial words were recently introduced in this context. The motivation behind the notion of a partial word is the comparison of genes (or proteins). Alignment of two genes (or two proteins) can be viewed as a construction of partial words that are said to be compatible. While a word can be described by a total function, a partial word can be described by a partial function. More precisely, a partial word of length n over a finite alphabet A is a partial function from {1,...,n} into A. Elements of {1,...,n} without an image are called holes. A word is just a partial word without holes. The notion of period of a word is central in combinatorics on words. In the case of partial words, there arc two notions: one is that of period, the other is that of local period. This paper extends to partial words with one hole the well known result of Guibas and Odlyzko which states that for every word u, there exists a word v of same length as u over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v. Our result states that for every partial word u with one hole, there exists a partial word v of same length as u with at most one hole over the alphabet {0, 1} such that the set of all periods of u coincides with the set of all periods of v and the set of all local periods of u coincides with the set of all local periods of v. To prove our result, we use the technique of Halava, Harju and Ilie which they used to characterize constructively the set of periods of a given word. As a consequence of our constructive proof, we obtain a linear time algorithm which, given a partial word with one hole, computes a partial word with at most one hole over the alphabet {0, 1} with the same length and the same sets of periods and local periods. A World Wide Web server interface at http://www.uncg.edu/mat/AlgBin/ has been established for automated use of the program.

Equations and monoid varieties of dot-depth one and two

Abstract Each level of the Straubings hierarchy of aperiodic monoids can be parametrized in a natural way. This paper studies this parametrization for dot-depth one and two monoids. For level one, it is shown that the m th level is defined by a finite sequence of equations if and only if m = 1, 2 or 3. For level two, and for m ⩾ 1, a sequence of equations is given which is satisfied in the m th level and shown to ultimately define the 1st level.

Games, equations and the dot-depth hierarchy

Abstract This paper studies the fine structure of the Straubing hierarchy of star-free languages. The monoid varieties of some sublevels of level one of the hierarchy are shown to be characterized by certain natural equation systems. Those are then generalized to definitions of equation systems satisfied in monoid varieties of higher sublevels. A version of the Ehrenfeucht-Fraisse game is used to verify equations.

Some Logical Characterizations of the Dot-Depth Hierarchy and Applications

A logical characterization of natural subhierarchies of the dot-depth hierarchy refining a theorem of Thomas and a congruence characterization related to a version of the Ehrenfeucht-Fra??sse game generalizing a theorem of Simon are given. For a sequence m=(m1, ..., mk) of positive integers, subclasses L(m1, ..., mk) of languages of level k are defined. L(m1, ..., mk) are shown to be decidable. Some properties of the characterizing congruences are studied, among them, a condition which insures L(m1, ..., mk) to be included in L(m?1, ..., m?k?). A conjecture of Pin concerning tree hierarchies of monoids (the dot-depth being a particular case) is shown to be false.

Avoidable binary patterns in partial words

The problem of classifying all the avoidable binary patterns in (full) words has been completely solved (see Chap. 3 of M. Lothaire, Algebraic Combinatorics on Words, Cambridge University Press, 2002). In this paper, we classify all the avoidable binary patterns in partial words, or sequences that may have some undefined positions called holes. In particular we show that, if we do not substitute any variable of the pattern by a partial word consisting of only one hole, the avoidability index of the pattern remains the same as in the full word case.

Equations and dot-depth one

This paper studies the fine structure of the Straubing hierarchy of star-free languages. Sequences of equations are defined and are shown to be sufficiently strong to characterize completely the monoid varieties of a natural subhierarchy of level one. In a few cases, it is also shown that those sequences of equations are equivalent to finite ones. Extensions to a natural sublevel of level two are discussed.

An Answer to a Conjecture on Overlaps in Partial Words Using Periodicity Algorithms

We propose an algorithm that given as input a full word w of length n , and positive integers p and d , outputs (if any exists) a maximal p -periodic partial word contained in w with the property that no two holes are within distance d . Our algorithm runs in O (nd ) time and is used for the study of freeness of partial words. Furthermore, we construct an infinite word over a five-letter alphabet that is overlap-free even after the insertion of an arbitrary number of holes, answering affirmatively a conjecture from Blanchet-Sadri, Mercas, and Scott.