33 research outputs found
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
Longest Unbordered Factor in Quasilinear Time
A border u of a word w is a proper factor of w occurring both as a prefix and as a suffix. The maximal unbordered factor of w is the longest factor of w which does not have a border. Here an O(n log n)-time with high probability (or O(n log n log^2 log n)-time deterministic) algorithm to compute the Longest Unbordered Factor Array of w for general alphabets is presented, where n is the length of w. This array specifies the length of the maximal unbordered factor starting at each position of w. This is a major improvement on the running time of the currently best worst-case algorithm working in O(n^{1.5}) time for integer alphabets [Gawrychowski et al., 2015]
Counting, Adding, and Regular Languages
In this thesis we consider two mostly disjoint topics in formal language theory that both involve the study and use of regular languages.
The first topic lies in the intersection of automata theory and additive number theory. We introduce a method of producing results in additive number theory, relying on theorem-proving software and an approximation technique. As an example of the method, we prove that every natural number greater than 25 can be written as the sum of at most 3 natural numbers whose canonical base-2 representations have an equal number of 0's and 1's. We prove analogous results about similarly defined sets using the automata theory approach, but also give proofs using more "traditional" approaches.
The second topic is the study languages defined by criteria involving the number of occurrences of a particular pair of words within other words. That is, we consider languages of words z defined with respect to words x, y where z has the same number of occurrences (resp., fewer occurrences), (resp., fewer occurrences or the same number of occurrences) of x as a subword of z and y as a subword of z. We give a necessary and sufficient condition on when such languages are regular, and show how to check this condition efficiently.
We conclude by briefly considering ideas tying the two topics together
Properties of Two-Dimensional Words
Combinatorics on words in one dimension is a well-studied subfield of theoretical computer science with its origins in the early 20th century. However, the closely-related study of two-dimensional words is not as popular, even though many results seem naturally extendable from the one-dimensional case. This thesis investigates various properties of these two-dimensional words.
In the early 1960s, Roger Lyndon and Marcel-Paul Schutzenberger developed two famous results on conditions where nontrivial prefixes and suffixes of a one-dimensional word are identical and on conditions where two one-dimensional words commute. Here, the theorems of Lyndon and Schutzenberger are extended in the one-dimensional case to include a number of additional equivalent conditions. One such condition is shown to be equivalent to the defect theorem from formal languages and coding theory. The same theorems of Lyndon and Schutzenberger are then generalized to the two-dimensional case.
The study of two-dimensional words continues by considering primitivity and periodicity in two dimensions, where a method is developed to enumerate two-dimensional primitive words. An efficient computer algorithm is presented to assist with checking the property of primitivity in two dimensions. Finally, borders in both one and two dimensions are considered, with some results being proved and others being offered as suggestions for future work. Another efficient algorithm is presented to assist with checking whether a two-dimensional word is bordered.
The thesis concludes with a selection of open problems and an appendix containing extensive data related to one such open problem
Periods and Borders of Random Words
We investigate the behavior of the periods and border lengths of random words over a fixed alphabet. We show that the asymptotic probability that a random word has a given maximal border length k is a constant, depending only on k and the alphabet size l. We give a recurrence that allows us to determine these constants with any required precision. This also allows us to evaluate the expected period of a random word. For the binary case, the expected period is asymptotically about n-1.641. We also give explicit formulas for the probability that a random word is unbordered or has maximum border length one