On factorially balanced sets of words

A set of words is factorially balanced if the set of all the factors of its words is balanced. We prove that if all words of a factorially balanced set have a finite index, then this set is a subset of the set of factors of a Sturmian word. Moreover, characterizing the set of factors of a given length n of a Sturmian word by the left special factor of length n−1 of this Sturmian word, we provide an enumeration formula for the number of sets of words that correspond to some set of factors of length n of a Sturmian word.     

Periodic and Sturmian languages

Counting the number of distinct factors in the words of a language gives a measure of complexity for that language similar to the factor-complexity of infinite words. Similarly as for infinite words, we prove that this complexity function f(n) is either bounded or f(n)?n+1. We call languages with bounded complexity periodic and languages with complexity f(n)=n+1Sturmian. We describe the structure of periodic languages and characterize the Sturmian languages as the sets of factors of (one- or two-way) infinite Sturmian words.     

On the structure of bispecial Sturmian words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that palindromic bispecial Sturmian words are precisely the maximal internal factors of primitive Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of all Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.     

A combinatorial property of the factor poset of a word

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.     

Periods in Extensions of Words

Let π(w) denote the minimum period of the word w,let w be a primitive word with period π(w) < |w|, and let z be a prefix of w. It is shown that if π(wz) = π(w), then |z| < π(w) − gcd (|w|, |z|). Detailed improvements of this result are also proven. Finally, we show that each primitive word w has a conjugate w′ = vu, where w = uv, such that π(w′) = |w′| and |u| < π(w). As a corollary we give a short proof of the fact that if u,v,w are words such that u ² is a prefix of v ², and v ² is a prefix of w ², and v is primitive, then |w| > 2|u|.     

Words, univalent factors, and boxes

A factor u of a word w is (right) univalent if there exists a unique letter a such that ua is still a factor of w. A univalent factor is minimal if none of its proper suffixes is univalent. The starting block of w is the shortest prefix of w such that all proper prefixes of w of length are univalent. We study univalent factors of a word and their relationship with the well known notions of boxes, superboxes, and minimal forbidden factors. Moreover, we prove some new uniqueness conditions for words based on univalent factors. In particular, we show that a word is uniquely determined by its starting block, the set of the extensions of its minimal univalent factors, and its length or its terminal box. Finally, we show how the results and techniques presented can be used to solve the problem of sequence assembly for DNA molecules, under reasonable assumptions on the repetitive structure of the considered molecule and on the set of known fragments. Received: 4 November 2000 / 23 November 2001     

Border Correlations of Partial Words

Partial words are finite sequences over a finite alphabet A that may contain a number of “do not know” symbols denoted by ?’s. Setting $A_{\diamond}=A\cup\lbrace{\diamond}\rbracePartial words are finite sequences over a finite alphabet A that may contain a number of “do not know” symbols denoted by ⋄’s. Setting A_à=Aè{à}A_{\diamond}=A\cup\lbrace{\diamond}\rbrace , A _⋄ ^* denotes the set of all partial words over A. In this paper, we investigate the border correlation function b:A_à^*?{a,b}^*\beta:A_{\diamond}^{*}\rightarrow\lbrace a,b\rbrace^{*} that specifies which conjugates (cyclic shifts) of a given partial word w of length n are bordered, that is, β(w)=c ₀ c ₁…c _n−1 where c _i =a or c _i =b according to whether the ith cyclic shift σ ⁱ (w) of w is unbordered or bordered. A partial word w is bordered if a proper prefix x ₁ of w is compatible with a proper suffix x ₂ of w, in which case any partial word x containing both x ₁ and x ₂ is called a border of w. In addition to β, we investigate an extension β′:A _⋄ ^*→ℕ^* that maps a partial word w of length n to m ₀ m ₁…m _n−1 where m _i is the length of a shortest border of σ ⁱ (w). Our results extend those of Harju and Nowotka.     

On the arithmetical complexity of Sturmian words

Using the geometric dual technique by Berstel and Pocchiola, we give a uniform O(n³)$O\left(n^3\right)$ upper bound for the arithmetical complexity of a Sturmian word. We also give explicit expressions for the arithmetical complexity of Sturmian words of slope between 1/3 and 2/3 (in particular, of the Fibonacci word). In this case, the difference between the genuine arithmetical complexity function and our upper bound is bounded, and ultimately 2-periodic. In fact, our formula is valid not only for Sturmian words but for rotation words from a wider class.     

Periodic-like words, periodicity, and boxes

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 , where is the least non-negative integer such that any prefix of w of length $\geq R'_{w}$ is not right special. We derive that if a word w has two periods , then also the greatest common divisor of p andq is a period ofw. 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. Received: 10 July 2000 / Accepted: 24 January 2001     

Some Aspects of Synchronization of DFA

A word w is called synchronizing （recurrent, reset, directable） word of deterministic finite automata （DFA） if w brings all states of the automaton to a unique state. According to the famous conjecture of Cerny from 1964, every n-state synchronizing automaton possesses a synchronizing word of length at most （n - 1）2. The problem is still open. It will be proved that the Cerny conjecture holds good for synchronizing DFA with transition monoid having no involutions and for every n-state （n 〉 2） synchronizing DFA with transition monoid having only trivial subgroups the minimal length of synchronizing word is not greater than （n - 1）2/2. The last important class of DFA involved and studied by Schutzenberger is called aperiodic; its automata accept precisely star-free languages. Some properties of an arbitrary synchronizing DFA were established.     

Powers in a class of -strict standard episturmian words

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz [D. Damanik, D. Lenz, Powers in Sturmian sequences, European J. Combin. 24 (2003) 377–390, doi:10.1016/S0195-6698(03)00026-X], who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the k-bonacci word, a generalization of the Fibonacci word to a k-letter alphabet (k≥2).     

Quasiperiodic and Lyndon episturmian words

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.     

Relation between powers of factors and the recurrence function characterizing Sturmian words

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a by-product, we give a new proof of the theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved in [A. Carpi, A. de Luca, Special factors, periodicity, and an application to Sturmian words, Acta Inform. 36 (2000) 983–1006] and [D. Damanik, D. Lenz, The index of Sturmian sequences, European J. Combin. 23 (2002) 23–29].     

Sturmian words and a criterium by Michaux–Villemaire

Michaux and Villemaire's proof of Cobham's theorem relies on the characterization of ultimately periodic words by means of the behaviour of certain repetitions in the word. Namely, they consider the length of the smallest shift between repetitions of a given length and the first position at which that smallest shift is observed. In this paper we study those properties for characteristic Sturmian words. In particular we answer a question posed by Michaux and Villemaire in that context.     

On Word Equations in One Variable

For a word equation E of length n in one variable x occurring # _x times in E a resolution algorithm of O(n+# _x log n) time complexity is presented here. This is the best result known and for the equations that feature #_x < \fracnlogn\#_{x}<\frac{n}{\log n} it yields time complexity of O(n) which is optimal. Additionally it is proven here that the set of solutions of any one-variable word equation is either of the form F or of the form F∪(uv)⁺ u where F is a set of O(log n) words and u, v are some words such that uv is a primitive word.     

On a word avoiding near repeats

In this paper we construct an infinite binary word w with the following property: the minimal distance among two occurrences of a same factor of length n cannot be polynomially upperbounded. In particular, for all positive ε the number of distinct factors of w with exponent larger than 1+ε is finite.     

Sturmian Trees

We consider Sturmian trees as a natural generalization of Sturmian words. A Sturmian tree is a tree having n+1 distinct subtrees of height n for each n. As for the case of words, Sturmian trees are irrational trees of minimal complexity.     

Deciding word neighborhood with universal neighborhood automata

Given some form of distance between words, a fundamental operation is to decide whether the distance between two given words w and v is within a given bound. In earlier work, we introduced the concept of a universal Levenshtein automaton for a given distance bound n. This deterministic automaton takes as input a sequence χ of bitvectors computed from w and v. The sequence χ is accepted iff the Levenshtein distance between w and v does not exceed n. The automaton is called universal since the same automaton can be used for arbitrary input words w and v, regardless of the underlying input alphabet. Here, we extend this picture. After introducing a large abstract family of generalized word distances, we exactly characterize those members where word neighborhood can be decided using universal neighborhood automata similar to universal Levenshtein automata. Our theoretical results establish several bridges to the theory of synchronized finite-state transducers and dynamic programming. For small neighborhood bounds, universal neighborhood automata can be held in main memory. This leads to very efficient algorithms for the above decision problem. Evaluation results show that these algorithms are much faster than those based on dynamic programming.     

The Structure of Infinite Solutions of Marked and Binary Post Correspondence Problems

In the infinite Post Correspondence Problem an instance (h,g) consists of two morphisms h and g, and the problem is to determine whether or not there exists an infinite word ω such that h(ω) = g(ω). This problem is undecidable in general, but it is known to be decidable for binary and marked instances. A morphism is binary if the domain alphabet is of size 2, and marked if each image of a letter begins with a different letter. We prove that the solutions of a marked instance form a set E^ω ⋃ E^* (P ⋃ F), where P is a finite set of ultimately periodic words, E is a finite set of solutions of the PCP, and F is a finite set of morphic images of fixed points of D0L systems. We also establish the structure of infinite solutions of the binary PCP.     

Words derivated from Sturmian words

A return word of a factor of a Sturmian word starts at an occurrence of that factor and ends exactly before its next occurrence. Derivated words encode the unique decomposition of a word in terms of return words. Vuillon has proved that each factor of a Sturmian word has exactly two return words. We determine these two return words, as well as their first occurrence, for the prefixes of characteristic Sturmian words. We then characterize words derivated from a characteristic Sturmian word and give their precise form. Finally, we apply our results to obtain a new proof of the characterization of characteristic Sturmian words which are fixed points of morphisms.