Inverse star, borders, and palstars

peer reviewedA language L is closed if L = L*. We consider an operation on closed languages, L-*, that is an inverse to Kleene closure. It is known that if L is closed and regular, then L-* is also regular. We show that the analogous result fails to hold for the context-free languages. Along the way we find a new relationship between the unbordered words and the prime palstars of Knuth, Morris, and Pratt. We use this relationship to enumerate the prime palstars, and we prove that neither the language of all unbordered words nor the language of all prime palstars is context-free

On the Combinatorics of Palindromes and Antipalindromes

We prove a number of results on the structure and enumeration of palindromes and antipalindromes. In particular, we study conjugates of palindromes, palindromic pairs, rich words, and the counterparts of these notions for antipalindromes.Comment: 13 pages/ submitted to DLT 201

Factorization in Formal Languages

We consider several novel aspects of unique factorization in formal languages. We reprove the familiar fact that the set uf(L) of words having unique factorization into elements of L is regular if L is regular, and from this deduce an quadratic upper and lower bound on the length of the shortest word not in uf(L). We observe that uf(L) need not be context-free if L is context-free. Next, we consider variations on unique factorization. We define a notion of "semi-unique" factorization, where every factorization has the same number of terms, and show that, if L is regular or even finite, the set of words having such a factorization need not be context-free. Finally, we consider additional variations, such as unique factorization "up to permutation" and "up to subset"

Automaticity of Primitive Words and Irreducible Polynomials

If L is a language, the automaticity function AL(n) (resp. NL(n)) of L counts the number of states of a smallest deterministic (resp. non-deterministic) finite automaton that accepts a language that agrees with L on all inputs of length at most n. We provide bounds for the automaticity of the language of primitive words and the language of unbordered words over a k-letter alphabet. We also give a bound for the automaticity of the language of base-b representations of the irreducible polynomials over a finite field. This latter result is analogous to a result of Shallit concerning the base-k representations of the set of prime numbers.https://hal.archives-ouvertes.fr/hal-00990604v

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

DNA Computing: Modelling in Formal Languages and Combinatorics on Words, and Complexity Estimation

DNA computing, an essential area of unconventional computing research, encodes problems using DNA molecules and solves them using biological processes. This thesis contributes to the theoretical research in DNA computing by modelling biological processes as computations and by studying formal language and combinatorics on words concepts motivated by DNA processes. It also contributes to the experimental research in DNA computing by a scaling comparison between DNA computing and other models of computation. First, for theoretical DNA computing research, we propose a new word operation inspired by a DNA wet lab protocol called cross-pairing polymerase chain reaction (XPCR). We define and study a word operation called word blending that models and generalizes an unexpected outcome of XPCR. The input words are uwx and ywv that share a non-empty overlap w, and the output is the word uwv. Closure properties of the Chomsky families of languages under this operation and its iterated version, the existence of a solution to equations involving this operation, and its state complexity are studied. To follow the XPCR experimental requirement closely, a new word operation called conjugate word blending is defined, where the subwords x and y are required to be identical. Closure properties of the Chomsky families of languages under this operation and the XPCR experiments that motivate and implement it are presented. Second, we generalize the sequence of Fibonacci words inspired by biological concepts on DNA. The sequence of Fibonacci words is an infinite sequence of words obtained from two initial letters f(1) = a and f(2)= b, by the recursive definition f(n+2) = f(n+1)*f(n), for all positive integers n, where * denotes word concatenation. After we propose a unified terminology for different types of Fibonacci words and corresponding results in the extensive literature on the topic, we define and explore involutive Fibonacci words motivated by ideas stemming from theoretical studies of DNA computing. The relationship between different involutive Fibonacci words and their borderedness and primitivity are studied. Third, we analyze the practicability of DNA computing experiments since DNA computing and other unconventional computing methods that solve computationally challenging problems often have the limitation that the space of potential solutions grows exponentially with their sizes. For such problems, DNA computing algorithms may achieve a linear time complexity with an exponential space complexity as a trade-off. Using the subset sum problem as the benchmark problem, we present a scaling comparison of the DNA computing (DNA-C) approach with the network biocomputing (NB-C) and the electronic computing (E-C) approaches, where the volume, computing time, and energy required, relative to the input size, are compared. Our analysis shows that E-C uses a tiny volume compared to that required by DNA-C and NB-C, at the cost of the E-C computing time being outperformed first by DNA-C and then by NB-C. In addition, NB-C appears to be more energy efficient than DNA-C for some input sets, and E-C is always an order of magnitude less energy efficient than DNA-C

Automatic Sequences and Decidable Properties: Implementation and Applications

In 1912 Axel Thue sparked the study of combinatorics on words when he showed that the Thue-Morse sequence contains no overlaps, that is, factors of the form ayaya. Since then many interesting properties of sequences began to be discovered and studied. In this thesis, we consider a class of infinite sequences generated by automata, called the k-automatic sequences. In particular, we present a logical theory in which many properties of k-automatic sequences can be expressed as predicates and we show that such predicates are decidable. Our main contribution is the implementation of a theorem prover capable of practically characterizing many commonly sought-after properties of k-automatic sequences. We showcase a panoply of results achieved using our method. We give new explicit descriptions of the recurrence and appearance functions of a list of well-known k-automatic sequences. We define a related function, called the condensation function, and give explicit descriptions for it as well. We re-affirm known results on the critical exponent of some sequences and determine it for others where it was previously unknown. On the more theoretical side, we show that the subword complexity p(n) of k-automatic sequences is k-synchronized, i.e., the language of pairs (n, p(n)) (expressed in base k) is accepted by an automaton. Furthermore, we prove that the Lyndon factorization of k-automatic sequences is also k-automatic and explicitly compute the factorization for several sequences. Finally, we show that while the number of unbordered factors of length n is not k-synchronized, it is k-regular