On the boundary sequence of an automatic sequence

Guo, Ying-Jun; Lü, Xiaotao; Zhou, Wen

doi:10.1016/j.disc.2021.112632

Cited by 2 publications

(5 citation statements)

References 10 publications

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“…With Theorem 16, we prove that for a large class of numeration systems S, if x is an S-automatic word, then the boundary sequence ∂ x is again S-automatic. Our approach generalizes the arguments provided by [24]. Considering exotic numeration systems allows a better understanding of underlying mechanisms, which do not arise in the ordinary integer base systems.…”

Section: Our Contributionsmentioning

confidence: 77%

“…Moreover, if ∂ x is automatic, then the abelian complexity of the image of x under a so-called Parikh-constant morphism is automatic [12]. Guo, Lü, and Wen combine this result with theirs in [24] to establish a large family of infinite words with automatic abelian complexity.…”

Section: Motivation and Related Workmentioning

confidence: 97%

“…As already mentioned in [24], the boundary sequence of a k-automatic word x may be defined by means of a first-order formula and therefore automaticity readily follows. This extends to addable systems S: let x ∈ B N be S-automatic for an addable system S. The above theorem implies that, for all b ∈ B, we have a formula ϕ b (n) which is true if and only if…”

Section: Link With First-order Logicmentioning

confidence: 99%

“…The notion of a (1-)boundary sequence was introduced by Chen and Wen in [12] and was further studied in [24], where it is shown that the boundary sequence of a k-automatic word (in the sense of Allouche and Shallit [2]: see Definition 9) is k-automatic. It is well known that a k-automatic word x is morphic, i.e., there exist morphisms f : A → A * and g : A → B and a letter a ∈ A such that x = g(f ω (a)), where f ω (a) = lim n→∞ f n (a).…”

Section: Motivation and Related Workmentioning

confidence: 99%

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On Extended Boundary Sequences of Morphic and Sturmian Words

Rigo,

Stipulanti,

Whiteland

2024

Electron. J. Combin.

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Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for all pairs of factors of length $\ell$ separated by $n-\ell$ symbols. For the large class of addable abstract numeration systems $S$, we show that if an infinite word is $S$-automatic, then the same holds for its $\ell$-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and $S$-automatic words for which the boundary sequence is nevertheless $S$-automatic and conversely, $S$-automatic words with a boundary sequence that is not $S$-automatic. In the second part of the paper, we study the $\ell$-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope. We also show that it is the image under a morphism of some other characteristic Sturmian word.

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Section: Our Contributionsmentioning

confidence: 77%

Section: Motivation and Related Workmentioning

confidence: 97%

Section: Link With First-order Logicmentioning

confidence: 99%

Section: Motivation and Related Workmentioning

confidence: 99%

See 2 more Smart Citations

On Extended Boundary Sequences of Morphic and Sturmian Words

Rigo,

Stipulanti,

Whiteland

2024

Electron. J. Combin.

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“…We can assume that |f (0)| ̸ = |f (1)|. Because otherwise f is Parikhconstant and Guo et al[17, Thm. 3] showed that the Parikh-constant image of a k-automatic sequence has k-automatic abelian complexity.…”

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confidence: 99%

Automaticity and Parikh-Collinear Morphisms

Rigo

Stipulanti

Whiteland

2023

Lecture Notes in Computer Science

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Parikh-collinear morphisms have recently received a lot of attention. They are defined by the property that the Parikh vectors of the images of letters are collinear. We first show that any fixed point of such a morphism is automatic. Consequently, we get under some mild technical assumption that the abelian complexity of a binary fixed point of a Parikh-collinear morphism is also automatic, and we discuss a generalization to arbitrary alphabets. Then, we consider the abelian complexity function of the fixed point of the Parikh-collinear morphism 0 → 010011, 1 → 1001. This 5-automatic sequence is shown to be aperiodic, answering a question of Salo and Sportiello.

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On the boundary sequence of an automatic sequence

Cited by 2 publications

References 10 publications

On Extended Boundary Sequences of Morphic and Sturmian Words

On Extended Boundary Sequences of Morphic and Sturmian Words

Automaticity and Parikh-Collinear Morphisms

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