9 research outputs found
Open and Closed Prefixes of Sturmian Words
A word is closed if it contains a proper factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We deal with the sequence of open and closed prefixes of Sturmian words and
prove that this sequence characterizes every finite or infinite Sturmian word
up to isomorphisms of the alphabet. We then characterize the combinatorial
structure of the sequence of open and closed prefixes of standard Sturmian
words. We prove that every standard Sturmian word, after swapping its first
letter, can be written as an infinite product of squares of reversed standard
words.Comment: To appear in WORDS 2013 proceeding
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
Factorizations of the Fibonacci Infinite Word
The aim of this note is to survey the factorizations of the Fibonacci
infinite word that make use of the Fibonacci words and other related words, and
to show that all these factorizations can be easily derived in sequence
starting from elementary properties of the Fibonacci numbers
On the Number of Closed Factors in a Word
A closed word (a.k.a. periodic-like word or complete first return) is a word
whose longest border does not have internal occurrences, or, equivalently,
whose longest repeated prefix is not right special. We investigate the
structure of closed factors of words. We show that a word of length
contains at least distinct closed factors, and characterize those words
having exactly closed factors. Furthermore, we show that a word of length
can contain many distinct closed factors.Comment: Accepted to LATA 201
23 11 Article 15
Abstract The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers
23 11 Article 15
Abstract The aim of this note is to survey the factorizations of the Fibonacci infinite word that make use of the Fibonacci words and other related words, and to show that all these factorizations can be easily derived in sequence starting from elementary properties of the Fibonacci numbers
On the Properties and Structure of Bordered Words and Generalizations
Combinatorics on words is a field of mathematics and theoretical computer science that
is concerned with sequences of symbols called words, or strings. One class of words that
are ubiquitous in combinatorics on words, and theoretical computer science more broadly,
are the bordered words. The word w has a border u if u is a non-empty proper prefix and
suffix of w. The word w is said to be bordered if it has a border. Otherwise w is said to
be unbordered.
This thesis is primarily concerned with variations and generalizations of bordered and
unbordered words.
In Chapter 1 we introduce the field of combinatorics on words and give a brief overview
of the literature on borders relevant to this thesis.
In Chapter 2 we give necessary definitions, and we present a more in-depth literature
review on results on borders relevant to this thesis.
In Chapter 3 we complete the characterization due to Harju and Nowotka of binary
words with the maximum number of unbordered conjugates. We also show that for every
number, up to this maximum, there exists a binary word with that number of unbordered
conjugates.
In Chapter 4 we give results on pairs of words that almost commute and anti-commute.
Two words x and y almost commute if xy and yx differ in exactly two places, and they
anti-commute if xy and yx differ in all places. We characterize and count the number of
pairs of words that almost and anti-commute. We also characterize and count variations
of almost-commuting words. Finally we conclude with some asymptotic results related to
the number of almost-commuting pairs of words.
In Chapter 5 we count the number of length-n bordered words with a unique border.
We also show that the probability that a length-n word has a unique border tends to a
constant.
In Chapter 6 we present results on factorizations of words related to borders, called
block palindromes. A block palindrome is a factorization of a word into blocks that turns
into a palindrome if each identical block is replaced by a distinct character. Each block is a
border of a central block. We call the number of blocks in a block palindrome the width of
the block palindrome. The largest block palindrome of a word is the block palindrome of the
word with the maximum width. We count all length-n words that have a width-t largest
block palindrome. We also show that the expected width of a largest block palindrome
tends to a constant. Finally we conclude with some results on another extremal variation
of block palindromes, the smallest block palindrome.
In Chapter 7 we present the main results of the thesis. Roughly speaking, a word is
said to be closed if it contains a non-empty proper border that occurs exactly twice in the
word. A word is said to be privileged if it is of length ≤ 1 or if it contains a non-empty
proper privileged border that occurs exactly twice in the word. We give new and improved
bounds on the number of length-n closed and privileged words over a k-letter alphabet.
In Chapter 8 we work with a generalization of bordered words to pairs of words. The
main result of this chapter is a characterization and enumeration result for this generalization
of bordered words to multiple dimensions.
In Chapter 9 we conclude by summarizing the results of this thesis and presenting
avenues for future research
Privileged Words and Sturmian Words
This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions.
The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words.
The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.Siirretty Doriast