Abelian bordered factors and periodicity

A finite word u is said to be bordered if u has a proper prefix which is also a suffix of u, and unbordered otherwise. Ehrenfeucht and Silberger proved that an infinite word is purely periodic if and only if it contains only finitely many unbordered factors. We are interested in abelian and weak abelian analogues of this result; namely, we investigate the following question(s): Let w be an infinite word such that all sufficiently long factors are (weakly) abelian bordered; is w (weakly) abelian periodic? In the process we answer a question of Avgustinovich et al. concerning the abelian critical factorization theorem.Comment: 14 page

On the Number of Unbordered Factors

We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) = 4 and that f(n) = n infinitely often. We also give examples of automatic sequences having exactly 2 unbordered factors of every length

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

On Generating Binary Words Palindromically

We regard a finite word $u=u_1u_2\cdots u_n$ up to word isomorphism as an equivalence relation on $\{1,2,\ldots, n\}$ where $i$ is equivalent to $j$ if and only if $x_i=x_j.$ Some finite words (in particular all binary words) are generated by "{\it palindromic}" relations of the form $k\sim j+i-k$ for some choice of $1\leq i\leq j\leq n$ and $k\in \{i,i+1,\ldots,j\}.$ That is to say, some finite words $u$ are uniquely determined up to word isomorphism by the position and length of some of its palindromic factors. In this paper we study the function $\mu(u)$ defined as the least number of palindromic relations required to generate $u.$ We show that every aperiodic infinite word must contain a factor $u$ with $\mu(u)\geq 3,$ and that some infinite words $x$ have the property that $\mu(u)\leq 3$ for each factor $u$ of $x.$ We obtain a complete classification of such words on a binary alphabet (which includes the well known class of Sturmian words). In contrast for the Thue-Morse word, we show that the function $\mu$ is unbounded

Biinfinite words with maximal recurrent unbordered factors

A finite non-empty word z is said to be a border of a finite non-empty word w if w=uz=zv for some non-empty words u and v. A finite non-empty word is said to be bordered if it admits a border, and it is said to be unbordered otherwise. In this paper, we give two characterizations of the biinfinite words of the form ...uuuvuuu..., where u and v are finite words, in terms of its unbordered factors. The main result of the paper states that the words of the form ...uuuvuuu... are precisely the biinfinite words w=...a_{-2}a_{-1}a_0a_1a_2... for which there exists a pair (l_0,r_0) of integers with l_0<r_0 such that, for every integers l\leq l_0 and r\geq r_0, the factor a_l...a_{l_0}...a_{r_0}... a_r is a bordered word. The words of the form ...uuuvuuu... are also characterized as being those biinfinite words w that admit a left recurrent unbordered factor (i.e., an unbordered factor of w that has an infinite number of occurrences "to the left'' in w) of maximal length that is also a right recurrent unbordered factor of maximal length. This last result is a biinfinite analogue of a result known for infinite words.Fundação para a Ciência e a Tecnologia (FCT) – Programa Operacional “Ciência, Tecnologia, Inovação” (POCTI), União Europeia (UE). Fundo Europeu de Desenvolvimento Regional (FEDER) - POCTI/32817/MAT/2000

Unbordered partial words

An unbordered word is a string over a finite alphabet such that none of its proper prefixes is one of its suffixes. In this paper, we extend the results on unbordered words to unbordered partial words. Partial words are strings that may have a number of ?do not know? symbols. We extend a result of Ehrenfeucht and Silberger which states that if a word u can be written as a concatenation of nonempty prefixes of a word v, then u can be written as a unique concatenation of nonempty unbordered prefixes of v. We study the properties of the longest unbordered prefix of a partial word, investigate the relationship between the minimal weak period of a partial word and the maximal length of its unbordered factors, and also investigate some of the properties of an unbordered partial word and how they relate to its critical factorizations (if any)

Representations of Circular Words

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The latter form can also be used as a graphical representation of periodic properties of finite (in some cases, infinite) words. We also define iterative representations which can be seen as an encoding utilizing the flexible properties of circular words. Every word over the two letter alphabet can be constructed starting from ab by applying the fractional power and the cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527