Infinite Lyndon words

doi:10.1016/0020-0190(94)90016-7

Information Processing Letters

Volume 50, Issue 2, 22 April 1994, Pages 101-104

https://doi.org/10.1016/0020-0190(94)90016-7 Get rights and content

Abstract

We define an infinite Lyndon word as the limit of an increasing sequence of prefix preserving lyndon words and show that some of the interesting properties of Lyndon words generalize to the infinite case. We construct a queue automaton that recognizes the set of Lyndon words and show that it can be extended to recognize infinite Lyndon words. We discuss certain topological properties of the set of infinite Lyndon words such as homeomorphism with a subspace of the Cantor space.

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Notice that the prefix normal condition is different from the Lyndon condition2: for finite words, there are words which are both Lyndon and prefix normal (e.g. 110010), words which are Lyndon but not prefix normal (11100110110), words which are prefix normal but not Lyndon (110101), and words which are neither (101100). We study infinite prefix normal words and their prefix normal forms in the context of lexicographic orderings, and compare them to infinite Lyndon words [32] and the max- and min-words of [26] (Corollary 5). Finally, we give conditions for periodicity and ultimate periodicity of prefix normal words in terms of their minimum density, a parameter introduced in [14] (Theorem 8).
Prefix normal words are binary words with the property that no factor has more 1s than the prefix of the same length. Finite prefix normal words were introduced in Fici and Lipták (2011) [18]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.¹
Reprint of: Generalized Lyndon factorizations of infinite words
2020, Theoretical Computer Science
Citation Excerpt :
This follows directly from Theorems 9, 13, and 18. Siromoney et al. showed in [19] that the infinite Lyndon words are precisely the limits of prefix-preserving increasing sequences of finite Lyndon words. We show that this result still holds when Lyndon words are replaced with generalized Lyndon words provided that the infinite word is primitive.
A generalized lexicographic order on words is a lexicographic order where the total order of the (possibly infinite) alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We positively answer a question of Dolce, Restivo, and Reutenauer by proving that every infinite word has a unique nonincreasing factorization into finite and infinite generalized Lyndon words. When this factorization has finitely many terms, we characterize the last term of the factorization. Our methods also show that the infinite generalized Lyndon words are precisely the words with infinitely many generalized Lyndon prefixes.
Reprint of: ω-Lyndon words
2020, Theoretical Computer Science
Citation Excerpt :
As with Lemma 2.1, we omit the proof as it is identical to that of Theorem 16 in [3]. The following lemma constitutes a generalisation of a characterisation of infinite Lyndon words given in [5] (see also Proposition 10 in [4]): We now turn to the question of unicity of ω-Lyndon factorisations for infinite words.
Let $A$ be a finite non-empty set and ⪯ a total order on $A^{N}$ verifying the following lexicographic like condition: For each $n \in N$ and $u, v \in A^{n}$ , if $u^{ω} ≺ v^{ω}$ then $u x ≺ v y$ for all $x, y \in A^{N}$ . A word $x \in A^{N}$ is called ω-Lyndon if $x ≺ y$ for each proper suffix y of x. A finite word $w \in A^{+}$ is called ω-Lyndon if $w^{ω} ≺ v^{ω}$ for each proper suffix v of w. In this note we prove that every infinite word may be written uniquely as a non-increasing product of ω-Lyndon words.
ω-Lyndon words
2020, Theoretical Computer Science
Citation Excerpt :
As with Lemma 2.1, we omit the proof as it is identical to that of Theorem 16 in [3]. The following lemma constitutes a generalisation of a characterisation of infinite Lyndon words given in [5] (see also Proposition 10 in [4]): We now turn to the question of unicity of ω-Lyndon factorisations for infinite words.
Let $A$ be a finite non-empty set and ⪯ a total order on $A^{N}$ verifying the following lexicographic like condition: For each $n \in N$ and $u, v \in A^{n}$ , if $u^{ω} ≺ v^{ω}$ then $u x ≺ v y$ for all $x, y \in A^{N}$ . A word $x \in A^{N}$ is called ω-Lyndon if $x ≺ y$ for each proper suffix y of x. A finite word $w \in A^{+}$ is called ω-Lyndon if $w^{ω} ≺ v^{ω}$ for each proper suffix v of w. In this note we prove that every infinite word may be written uniquely as a non-increasing product of ω-Lyndon words.
Generalized Lyndon factorizations of infinite words
2020, Theoretical Computer Science
Citation Excerpt :
This follows directly from Theorems 9, 13, and 18. Siromoney et al. showed in [19] that the infinite Lyndon words are precisely the limits of prefix-preserving increasing sequences of finite Lyndon words. We show that this result still holds when Lyndon words are replaced with generalized Lyndon words provided that the infinite word is primitive.
A generalized lexicographic order on words is a lexicographic order where the total order of the (possibly infinite) alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We positively answer a question of Dolce, Restivo, and Reutenauer by proving that every infinite word has a unique nonincreasing factorization into finite and infinite generalized Lyndon words. When this factorization has finitely many terms, we characterize the last term of the factorization. Our methods also show that the infinite generalized Lyndon words are precisely the words with infinitely many generalized Lyndon prefixes.

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Infinite Lyndon words

Abstract

Theoret. Comput. Sci.

Factorizing words over an ordered alphabet

J. Algorithms