Related papers: On the Lie complexity of Sturmian words
In this book chapter, written in French, we consider the classical family of Sturmian words, defined as the aperiodic infinite words containing only $n+1$ factors of a length $n$, which is the minimal possible value. We will discuss several…
Let $R(n)$ denote the number of rich words of length $n$ over a given finite alphabet. In 2017 it was proved that $\lim_{n\rightarrow\infty} \sqrt[n]{R(n)}=1$; it means the number of rich words has a subexponential growth. However, up to…
We compute the exact maximum state complexity for the language consisting of $m$ words of length $N$, and characterize languages achieving the maximum. We also consider a special case, namely languages $C(w)$ consisting of the conjugates of…
A language $L$ is the orthogonal catenation of languages $L_1$ and $L_2$ if every word of $L$ can be written in a unique way as a catenation of a word in $L_1$ and a word in $L_2$. We establish a tight bound for the state complexity of…
For a word $S$, let $f(S)$ be the largest integer $m$ such that there are two disjoints identical (scattered) subwords of length $m$. Let $f(n, \Sigma) = \min \{f(S): S \text{is of length} n, \text{over alphabet} \Sigma \}$. Here, it is…
We present an impossibility result, called a theorem about facts and words, which pertains to a general communication system. The theorem states that the number of distinct words used in a finite text is roughly greater than the number of…
We present a formalization of basics related to infinite words in the generic proof assistant Isabelle/HOL. Furthermore, we present a formalization of purely morphic and morphic languages. Finally, we present a formalized definition of…
A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume…
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 byproduct, we give a new proof of theorem describing the index of a Sturmian…
In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a…
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of…
A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is attained, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a word…
We define a family of natural decompositions of Sturmian words in Christoffel words, called *reversible Christoffel* (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always)…
Group languages are regular languages recognized by finite groups, or equivalently by finite automata in which each letter induces a permutation on the set of states. We investigate the separation problem for this class of languages: given…
The piecewise complexity $h(u)$ of a word is the minimal length of subwords needed to exactly characterise $u$. Its piecewise minimality index $\rho(u)$ is the smallest length $k$ such that $u$ is minimal among its order-$k$ class $[u]_k$…
The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. The goal of this…
In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic…
String complexity is defined as the cardinality of a set of all distinct words (factors) of a given string. For two strings, we introduce the joint string complexity as the cardinality of a set of words that are common to both strings.…
For a given language $L$, we study the languages $X$ such that for all distinct words $u, v \in L$, there exists a word $x \in X$ that appears a different number of times as a factor in $u$ and in $v$. In particular, we are interested in…
In this paper, we analyze the periodic factors of Sturmian words for the findings to lead to a linear-time algorithm for the computation of runs in this class of words which, to our best knowledge, is an open problem in literature.