Periodicity of Morphic Words

Mitrofanov, Ivan

doi:10.1007/s10958-015-2344-2

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…the image under a coding of the fixed point of a morphism) is ultimately periodic is a generalization of the periodicity problem for b-recognizable sets mentioned in the previous paragraph. The HD0L ultimate periodicity problem was shown to be decidable in its full generality [21,31]. The proofs rely on return words, primitive substitutions or evolution of Rauzy graphs.…”

Section: Introductionmentioning

confidence: 99%

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Charlier

Cisternino

Massuir

2021

Electron. J. Combin.

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The Thue-Morse set $\mathcal{T}$ is the set of those non-negative integers whose binary expansions have an even number of $1$'s. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word $${\tt 0110100110010110\cdots},$$ which is the fixed point starting with ${\tt 0}$ of the word morphism ${\tt 0\mapsto 01}$, ${\tt 1\mapsto 10}$. The numbers in $\mathcal{T}$ are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set $m\mathcal{T}+r$ (i.e. the number of states of its minimal automaton) with respect to any base $b$ which is a power of $2$. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all $2^p$-expansions of the set of integers $m\mathcal{T}+r$ for any positive integers $p$ and $m$ and any remainder $r\in\{0,\ldots,m{-}1\}$. The proposed method is general for any $b$-recognizable set of integers.

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Section: Introductionmentioning

confidence: 99%

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Charlier

Cisternino

Massuir

2021

Electron. J. Combin.

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show abstract

“…the image under a coding of the fixed point of a morphism) is ultimately periodic is a generalization of the periodicity problem for b-recognizable sets mentioned in the previous paragraph. The HD0L ultimate periodicity problem was shown to be decidable in its full generality [21,32]. The proofs rely on return words, primitive substitutions or evolution of Rauzy graphs.…”

Section: Introductionmentioning

confidence: 99%

Minimal automaton for multiplying and translating the Thue-Morse set

Charlier¹,

Cisternino²,

Massuir³

2019

Preprint

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The Thue-Morse set T is the set of those non-negative integers whose binary expansions have an even number of 1. The name of this set comes from the fact that its characteristic sequence is given by the famous Thue-Morse word abbabaabbaababba • • •, which is the fixed point starting with a of the word morphism a → ab, b → ba. The numbers in T are commonly called the evil numbers. We obtain an exact formula for the state complexity of the set mT + r (i.e. the number of states of its minimal automaton) with respect to any base b which is a power of 2. Our proof is constructive and we are able to explicitly provide the minimal automaton of the language of all 2 p -expansions of the set of integers mT + r for any positive integers p and m and any remainder r ∈ {0, . . . , m − 1}. The proposed method is general for any b-recognizable set of integers. As an application, we obtain a decision procedure running in quadratic time for the problem of deciding whether a given 2 p -recognizable set is equal to a set of the form mT + r.

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On almost periodicity of morphic sequences

Mitrofanov

2016

Dokl. Math.

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Periodicity of Morphic Words

Cited by 3 publications

References 6 publications

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Minimal Automaton for Multiplying and Translating the Thue-Morse Set

Minimal automaton for multiplying and translating the Thue-Morse set

On almost periodicity of morphic sequences

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