scholarly journals The Overlap Gap Between Left-Infinite and Right-Infinite Words

2021 ◽  
pp. 1-9
Author(s):  
José Carlos Costa ◽  
Conceição Nogueira ◽  
Maria Lurdes Teixeira
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
Positive Integers

We study ultimate periodicity properties related to overlaps between the suffixes of a left-infinite word [Formula: see text] and the prefixes of a right-infinite word [Formula: see text]. The main theorem states that the set of minimum lengths of words [Formula: see text] and [Formula: see text] such that [Formula: see text] or [Formula: see text] is finite, where [Formula: see text] runs over positive integers and [Formula: see text] and [Formula: see text] are respectively the suffix of [Formula: see text] and the prefix of [Formula: see text] of length [Formula: see text], if and only if [Formula: see text] and [Formula: see text] are ultimately periodic words of the form [Formula: see text] and [Formula: see text] for some finite words [Formula: see text], [Formula: see text] and [Formula: see text].

scholarly journals Defect Effect of Bi-infinite Words in the Two-element Case

2001 ◽  
Vol Vol. 4 no. 2 ◽  
Author(s):  
Ján Maňuch
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finite Alphabet ◽  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
International Audience ◽  
Necessary And Sufficient ◽  
Primitive Roots ◽  
Element Set

International audience Let X be a two-element set of words over a finite alphabet. If a bi-infinite word possesses two X-factorizations which are not shiftequivalent, then the primitive roots of the words in X are conjugates. Note, that this is a strict sharpening of a defect theorem for bi-infinite words stated in \emphKMP. Moreover, we prove that there is at most one bi-infinite word possessing two different X-factorizations and give a necessary and sufficient conditions on X for the existence of such a word. Finally, we prove that the family of sets X for which such a word exists is parameterizable.

scholarly journals Lyndon factorization of generalized words of Thue

2002 ◽  
Vol Vol. 5 ◽  
Author(s):  
Anton Černý
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
International Audience

International audience The i-th symbol of the well-known infinite word of Thue on the alphabet \ 0,1\ can be characterized as the parity of the number of occurrences of the digit 1 in the binary notation of i. Generalized words of Thue are based on counting the parity of occurrences of an arbitrary word w∈\ 0,1\^+-0^* in the binary notation of i. We provide here the standard Lyndon factorization of some subclasses of this class of infinite words.

The Markoff Property

2018 ◽  
pp. 29-34
Author(s):  
Christophe Reutenauer
Keyword(s):  
Combinatorial Property ◽  
Periodic Pattern ◽  
Infinite Words ◽  
Infinite Word ◽  
Markoff Property

The Markoff property is a combinatorial property of infinite words on the alphabet {a,b}, and of bi-infinite words. Such a word has this property if whenever there is a factor xy in the word,with x,y equal to the letters a,b (in some order), then itmay be extended into a factor of the formym’xymx, wherem’ is the reversal ofm, and where the length ofmis bounded (the bound depends only on the infinite word). As discussed in this chapter, the main theorem, due toMarkoff, is that this property implies periodicity, with a periodic pattern which must be a Christoffel word. It is one of the crucial results inMarkoff’s theory.

scholarly journals Entropy ratio for infinite sequences with positive entropy

2018 ◽  
Vol 40 (3) ◽  
pp. 751-762 ◽  
Author(s):  
CHRISTIAN MAUDUIT ◽  
CARLOS GUSTAVO MOREIRA
Keyword(s):  
Exponential Growth ◽  
Fractal Dimensions ◽  
Combinatorial Structure ◽  
Finite Alphabet ◽  
Exponential Growth Rate ◽  
Positive Entropy ◽  
Infinite Words ◽  
Complexity Function ◽  
Infinite Word ◽  
Infinite Sequences

The complexity function of an infinite word $w$ on a finite alphabet $A$ is the sequence counting, for each non-negative $n$, the number of words of length $n$ on the alphabet $A$ that are factors of the infinite word $w$. For any given function $f$ with exponential growth, we introduced in [Complexity and fractal dimensions for infinite sequences with positive entropy. Commun. Contemp. Math. to appear] the notion of word entropy$E_{W}(f)$ associated to $f$ and we described the combinatorial structure of sets of infinite words with a complexity function bounded by $f$. The goal of this work is to give estimates on the word entropy $E_{W}(f)$ in terms of the limiting lower exponential growth rate of $f$.

scholarly journals Characterization of Infinite LSP Words and Endomorphisms Preserving the LSP Property

2019 ◽  
Vol 30 (01) ◽  
pp. 171-196 ◽  
Author(s):  
Gwenaël Richomme
Keyword(s):  
Infinite Words ◽  
Infinite Word ◽  
The Family ◽  
Finite Set ◽  
Special Factors

Answering a question of G. Fici, we give an [Formula: see text]-adic characterization of the family of infinite LSP words, that is, the family of infinite words having all their left special factors as prefixes. More precisely we provide a finite set of morphisms [Formula: see text] and an automaton [Formula: see text] such that an infinite word is LSP if and only if it is [Formula: see text]-adic and one of its directive words is recognizable by [Formula: see text]. Then we characterize the endomorphisms that preserve the property of being LSP for infinite words. This allows us to prove that there exists no set [Formula: see text] of endomorphisms for which the set of infinite LSP words corresponds to the set of [Formula: see text]-adic words. This implies that an automaton is required no matter which set of morphisms is used.

ON THE REPETITIVITY INDEX OF INFINITE WORDS

2009 ◽  
Vol 19 (02) ◽  
pp. 145-158 ◽  
Author(s):  
ARTURO CARPI ◽  
VALERIO D'ALONZO
Keyword(s):  
Negative Integer ◽  
Minimal Distance ◽  
Infinite Words ◽  
Infinite Word

With any infinite word w we associate a function, called the repetitivity index, giving, for any non-negative integer n, the minimal distance between any two occurrences of identical factors of length n in w. Some properties of the repetitivity index are studied concerning, in particular, periodic words, synchronized sequences, smooth words.

On maximal pattern complexity of some automatic words

2010 ◽  
Vol 31 (5) ◽  
pp. 1463-1470 ◽  
Author(s):  
TETURO KAMAE ◽  
PAVEL V. SALIMOV
Keyword(s):  
Fixed Point ◽  
Linear Function ◽  
Finite Subset ◽  
Constant Length ◽  
Sequence Entropy ◽  
Pattern Complexity ◽  
Infinite Words ◽  
Maximum Value ◽  
Infinite Word

AbstractThe pattern complexity of a word for a given pattern S, where S is a finite subset of {0,1,2,…}, is the number of distinct restrictions of the word to S+n (with n=0,1,2,…). The maximal pattern complexity of the word, introduced in the paper of T. Kamae and L. Zamboni [Sequence entropy and the maximal pattern complexity of infinite words. Ergod. Th. & Dynam. Sys.22(4) (2002), 1191–1199], is the maximum value of the pattern complexity of S with #S=k as a function of k=1,2,…. A substitution of constant length on an alphabet is a mapping from the alphabet to finite words on it of constant length not less than two. An infinite word is called a fixed point of the substitution if it stays the same after the substitution is applied. In this paper, we prove that the maximal pattern complexity of a fixed point of a substitution of constant length on {0,1} (as a function of k=1,2,…) is either bounded, a linear function of k, or 2k.

scholarly journals WORD COMPLEXITY AND REPETITIONS IN WORDS

2004 ◽  
Vol 15 (01) ◽  
pp. 41-55 ◽  
Author(s):  
LUCIAN ILIE ◽  
SHENG YU ◽  
KAIZHONG ZHANG
Keyword(s):  
Data Compression ◽  
Fast Algorithms ◽  
Complexity Measure ◽  
Combinatorics On Words ◽  
Complexity Measures ◽  
Infinite Words ◽  
Infinite Word ◽  
De Bruijn ◽  
Subword Complexity

With ideas from data compression and combinatorics on words, we introduce a complexity measure for words, called repetition complexity, which quantifies the amount of repetition in a word. The repetition complexity of w, R (w), is defined as the smallest amount of space needed to store w when reduced by repeatedly applying the following procedure: n consecutive occurrences uu…u of the same subword u of w are stored as (u,n). The repetition complexity has interesting relations with well-known complexity measures, such as subword complexity, SUB , and Lempel-Ziv complexity, LZ . We have always R (w)≥ LZ (w) and could even be that the former is linear while the latter is only logarithmic; e.g., this happens for prefixes of certain infinite words obtained by iterated morphisms. An infinite word α being ultimately periodic is equivalent to: (i) [Formula: see text], (ii) [Formula: see text], and (iii) [Formula: see text]. De Bruijn words, well known for their high subword complexity, are shown to have almost highest repetition complexity; the precise complexity remains open. R (w) can be computed in time [Formula: see text] and it is open, and probably very difficult, to find fast algorithms.

ITERATIVE DEVICES GENERATING INFINITE WORDS

1994 ◽  
Vol 05 (01) ◽  
pp. 69-97 ◽  
Author(s):  
KAREL CULIK II ◽  
JUHANI KARHUMÄKI
Keyword(s):  
Real Time ◽  
Infinite Words ◽  
Automatic Sequences ◽  
Infinite Word

We consider various TAG-like devices that generate one-way infinite words in real time. The simplest types of these devices are equivalent to iterative morphisms (also called substitutions), automatic sequences and iterative DGSM’s. We consider also a few new types. Mainly we study the comparative power of these mechanisms and develop some techniques for proving that certain devices cannot produce a particular infinite word.

scholarly journals Approximations of Additive Squares in Infinite Words

Integers ◽  
2012 ◽  
Vol 12 (5) ◽  
Author(s):  
Tom Brown
Keyword(s):  
Infinite Words ◽  
Infinite Word

Abstract.We show that every infinite word

Sign in / Sign up

Export Citation Format

Share Document