Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Computer Science

Supervisor

Dr. Lila Kari

Abstract

Periodicity, primitivity and borderedness are some of the fundamental notions in combinatorics on words. Motivated by the Watson-Crick complementarity of DNA strands wherein a word (strand) over the DNA alphabet \{A, G, C, T\} and its Watson-Crick complement are informationally ``identical", these notions have been extended to consider pseudo-periodicity and pseudo-borderedness obtained by replacing the ``identity" function with ``pseudo-identity" functions (antimorphic involution in case of Watson-Crick complementarity). For a given alphabet $\Sigma$, an antimorphic involution $\theta$ is an antimorphism, i.e., $\theta(uv)=\theta(v) \theta(u)$ for all $u,v \in \Sigma^{*}$ and an involution, i.e., $\theta(\theta(u))=u$ for all $u \in \Sigma^{*}$. In this thesis, we continue the study of pseudo-periodic and pseudo-bordered words for pseudo-identity functions including involutions.

To start with, we propose a binary word operation, $\theta$-catenation, that generates $\theta$-powers (pseudo-powers) of a word for any morphic or antimorphic involution $\theta$. We investigate various properties of this operation including closure properties of various classes of languages under it, and its connection with the previously defined notion of $\theta$-primitive words.

A non-empty word $u$ is said to be $\theta$-bordered if there exists a non-empty word $v$ which is a prefix of $u$ while $\theta(v)$ is a suffix of $u$. We investigate the properties of $\theta$-bordered (pseudo-bordered) and $\theta$-unbordered (pseudo-unbordered) words for pseudo-identity functions $\theta$ with the property that $\theta$ is either a morphism or an antimorphism with $\theta^{n}=I$, for a given $n \geq 2$, or $\theta$ is a literal morphism or an antimorphism.

Lastly, we initiate a new line of study by exploring the disjunctivity properties of sets of pseudo-bordered and pseudo-unbordered words and some other related languages for various pseudo-identity functions. In particular, we consider such properties for morphic involutions $\theta$ and prove that, for any $i \geq 2$, the set of all words with exactly $i$ $\theta$-borders is disjunctive (under certain conditions).


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