On different generalizations of episturmian words

Bucci, Michelangelo; Luca, Alessandro De; Zamboni, Luca Q.

doi:10.1016/j.tcs.2007.10.043

Cited by 22 publications

(34 citation statements)

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“…Theorem 3.13) of all injective ϑ-characteristic morphisms, to whose proof Section 5 is dedicated. This result, which solves a problem posed in [4], is very useful to construct nontrivial examples of ϑ-characteristic morphisms and then of standard ϑ-episturmian words. Moreover, one has a quite simple procedure to decide whether a given injective morphism is ϑ-characteristic.…”

Section: Introductionmentioning

confidence: 82%

“…The following two propositions, proved in [4], give methods for constructing standard ϑ-episturmian words.…”

Section: Standard ϑ-Episturmian Wordsmentioning

confidence: 96%

“…In [4] standard ϑ-episturmian words were naturally defined by substituting, in the definition of standard episturmian words, the closure under reversal with the closure under ϑ. Thus an infinite word s is standard ϑ-episturmian if it satisfies the following two conditions:…”

Section: Standard ϑ-Episturmian Wordsmentioning

confidence: 99%

“…For example, in [10] a generalization was obtained by making suitable hypotheses on the lengths of palindromic prefixes of an infinite word; in [8,5,4,6] different extensions were introduced, all based on the replacement of the reversal operator R by an arbitrary involutory antimorphism ϑ of the free monoid A * . In particular, the so called ϑ-standard and standard ϑ-episturmian words were studied.…”

Section: Introductionmentioning

confidence: 99%

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Characteristic morphisms of generalized episturmian words

Bucci

Luca

2009

Theoretical Computer Science

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a b s t r a c tIn a recent paper with L.Q. Zamboni, the authors introduced the class of ϑ-episturmian words. An infinite word over A is standard ϑ-episturmian, where ϑ is an involutory antimorphism of A * , if its set of factors is closed under ϑ and its left special factors are prefixes. When ϑ is the reversal operator, one obtains the usual standard episturmian words. In this paper, we introduce and study ϑ-characteristic morphisms, that is, morphisms which map standard episturmian words into standard ϑ-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective. In order to prove this result, we also introduce and study a class of biprefix codes which are overlap-free, i.e., any two code words do not overlap properly, and normal, i.e., no proper suffix (prefix) of any code-word is left (right) special in the code. A further result is that any standard ϑ-episturmian word is a morphic image, by an injective ϑ-characteristic morphism, of a standard episturmian word. Proposition 4.2. Let Z be a suffix, left normal, and overlap-free code over A, and let a, b ∈Then we can assume that (20) holds for suitable h ≥ 0, δ ∈ A * , and ξ ∈ A + . We have n > 1, for otherwise the statement is trivial, and δ = ε since va / ∈ Z * . As δ = a, if |δλ| ≤ |z| we obtain aλ ∈ Fact Z and we are done. Therefore assume |δλ| > |z|. In this case ξ is a proper prefix of λ and a proper suffix of z. If λ ∈ Pref Z * we reach a contradiction, since ξ ∈ Suff Z ∩ Pref Z * and this contradicts the hypothesis that Z is a suffix and overlap-free code. Thus λ / ∈ Pref Z * ; this implies, by the previous lemma, that aλ ∈ Fact Z . Proposition 4.3. Let Z be a biprefix, overlap-free, and right normal code over A. If λ ∈ Pref Z * \ {ε}, then there exists a unique word u = z 1 · · · z k with k ≥ 1 and z i ∈ Z , i = 1, . . . , k, such that u = z 1 · · · z k = λζ , z 1 · · · z k−1 δ = λ,where δ ∈ A + and ζ ∈ A * .Proof. Let us suppose that there exist h ≥ 1 and words z 1 , . . . , z h ∈ Z such that z 1 · · · z h = λζ , z 1 · · · z h−1 δ = λ (22) with ζ ∈ A * and δ ∈ A + . From (21) and (22) one obtains u = z 1 · · · z k = z 1 · · · z h−1 δ ζ and z 1 · · · z h = z 1 · · · z k−1 δζ , with z k = δζ and z h = δ ζ . Since Z is a biprefix code, we derive h = k and consequently z i = z i for i = 1, . . . , k − 1. Indeed, if h = k, we would derive by cancellation that δ ζ = ε or δζ = ε, which is absurd as δ, δ ∈ A + .Hence we obtain z k = δ ζ = δζ , whence δ = δ . Thus δ is a common nonempty prefix of z k and z k . Since Z is right normal, by Proposition 1.1 we obtain that z k is a prefix of z k and vice versa, i.e., z k = z k .Proposition 4.4. Let Z be a biprefix, overlap-free, and normal code over A. If u ∈ Z * \ {ε} is a proper factor of z ∈ Z , then there exist p, q ∈ Z * , h, h ∈ A + such that h / ∈ Suff Z , (h ) f / ∈ Pref Z , and z = hpuqh .Proof. Since u is a proper factor of z ∈ Z , there exist ξ , ξ ∈ A * such that z = ξ uξ ; moreover, ξ...

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Section: Introductionmentioning

confidence: 82%

“…The following two propositions, proved in [4], give methods for constructing standard ϑ-episturmian words.…”

Section: Standard ϑ-Episturmian Wordsmentioning

confidence: 96%

Section: Standard ϑ-Episturmian Wordsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characteristic morphisms of generalized episturmian words

Bucci

Luca

2009

Theoretical Computer Science

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“…More precisely, an infinite word over A is said to be θ-episturmian if it has at most one left special factor of each length and its set of factors is closed under an involutory antimorphism θ of the free monoid A * . Generalizing even further, θ-episturmian words with seed are obtained by requiring the condition on special factors only for sufficiently large lengths (see [28]). …”

Section: Discussionmentioning

confidence: 99%

Episturmian words: a survey

Glen

Justin

2009

RAIRO-Theor. Inf. Appl.

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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 properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

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On a Generalization of Standard Episturmian Morphisms

Bucci

Luca

2008

Developments in Language Theory

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On different generalizations of episturmian words

Cited by 22 publications

References 5 publications

Characteristic morphisms of generalized episturmian words

Characteristic morphisms of generalized episturmian words

Episturmian words: a survey

On a Generalization of Standard Episturmian Morphisms

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