A length bound for binary equality words

Jana Hadravová

Displaying similar documents to “A length bound for binary equality words”

Borders and roots of a word

Silberger, D.M. (1971)

Portugaliae mathematica

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Periodicity problem of substitutions over ternary alphabets

Bo Tan, Zhi-Ying Wen (2008)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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In this paper, we characterize the substitutions over a three-letter alphabet which generate a ultimately periodic sequence.

On low-complexity bi-infinite words and their factors

Alex Heinis (2001)

Journal de théorie des nombres de Bordeaux

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In this paper we study bi-infinite words on two letters. We say that such a word has stiffness $k$ if the number of different subwords of length $n$ equals $n + k$ for all $n$ sufficiently large. The word is called $k$ -balanced if the numbers of occurrences of the symbol a in any two subwords of the same length differ by at most $k$ . In the present paper we give a complete description of the class of bi-infinite words of stiffness $k$ and show that the number of subwords of length $n$ from this class has...

Binary equality words with two $b$ ’s

Štěpán Holub, Jiří Sýkora (2018)

Commentationes Mathematicae Universitatis Carolinae

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Deciding whether a given word is an equality word of two nonperiodic morphisms is also known as the dual Post correspondence problem. Although the problem is decidable, there is no practical decision algorithm. Already in the binary case, the classification is a large project dating back to 1980s. In this paper we give a full classification of binary equality words in which one of the letters has two occurrences.

Complexity of infinite words associated with beta-expansions

Christiane Frougny, Zuzana Masáková, Edita Pelantová (2004)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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We study the complexity of the infinite word $u_{β}$ associated with the Rényi expansion of $1$ in an irrational base $β > 1$ . When $β$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ (n) = n + 1$ . For $β$ such that $d_{β} (1) = t_{1} t_{2} \dots t_{m}$ is finite we provide a simple description of the structure of special factors of the word $u_{β}$ . When $t_{m} = 1$ we show that $ℂ (n) = (m - 1) n + 1$ . In the cases when $t_{1} = t_{2} = \dots = t_{m - 1}$ or $t_{1} > max {t_{2}, \dots, t_{m - 1}}$ we show that the first difference of the complexity function $ℂ (n + 1) - ℂ (n)$ takes value in ${m - 1, m}$ for every $n$ , and consequently...

Equations on partial words

Francine Blanchet-Sadri, D. Dakota Blair, Rebeca V. Lewis (2009)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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It is well-known that some of the most basic properties of words, like the commutativity ( $x y = y x$ ) and the conjugacy ( $x z = z y$ ), can be expressed as solutions of word equations. An important problem is to decide whether or not a given equation on words has a solution. For instance, the equation $x^{m} y^{n} = z^{p}$ has only periodic solutions in a free monoid, that is, if $x^{m} y^{n} = z^{p}$ holds with integers $m, n, p \geq 2$ , then there exists a word $w$ such that $x, y, z$ are powers of $w$ . This result, which received a lot of attention, was first proved...

*-sturmian words and complexity

Izumi Nakashima, Jun-Ichi Tamura, Shin-Ichi Yasutomi (2003)

Journal de théorie des nombres de Bordeaux

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We give analogs of the complexity $p (n)$ and of Sturmian words which are called respectively the $*$ -complexity $p_{*} (n)$ and $*$ -Sturmian words. We show that the class of $*$ -Sturmian words coincides with the class of words satisfying $p_{*} (n) \leq n + 1$ , and we determine the structure of $*$ -Sturmian words. For a class of words satisfying $p_{*} (n) = n + 1$ , we give a general formula and an upper bound for $p (n)$ . Using this general formula, we give explicit formulae for $p (n)$ for some words belonging to this class. In general, $p (n)$ can take large...

A test-set for $k$ -power-free binary morphisms

F. Wlazinski (2001)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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A morphism $f$ is $k$ -power-free if and only if $f (w)$ is $k$ -power-free whenever $w$ is a $k$ -power-free word. A morphism $f$ is $k$ -power-free up to $m$ if and only if $f (w)$ is $k$ -power-free whenever $w$ is a $k$ -power-free word of length at most $m$ . Given an integer $k \geq 2$ , we prove that a binary morphism is $k$ -power-free if and only if it is $k$ -power-free up to $k^{2}$ . This bound becomes linear for primitive morphisms: a binary primitive morphism is $k$ -power-free if and only if it is $k$ -power-free up to $2 k + 1$ ...