Thesis-2016-Day.pdf (860.29 kB)

On restricting the ambiguity in morphic images of words

Download (860.29 kB)
thesis
posted on 16.11.2016 by Joel D. Day
For alphabets Delta_1, Delta_2, a morphism g : Delta_1* to Delta_2* is ambiguous with respect to a word u in Delta_1* if there exists a second morphism h : Delta_1* to Delta_2* such that g(u) = h(u) and g not= h. Otherwise g is unambiguous. Hence unambiguous morphisms are those whose structure is fully preserved in their morphic images. A concept so far considered in the free monoid, the first part of this thesis considers natural extensions of ambiguity of morphisms to free groups. It is shown that, while the most straightforward generalization of ambiguity to a free monoid results in a trivial situation, that all morphisms are (always) ambiguous, there exist meaningful extensions of (un)ambiguity which are non-trivial - most notably the concepts of (un)ambiguity up to inner automorphism and up to automorphism. A characterization is given of words in a free group for which there exists an injective morphism which is unambiguous up to inner automorphism in terms of fixed points of morphisms, replicating an existing result for words in the free monoid. A conjecture is presented, which if correct, is sufficient to show an equivalent characterization for unambiguity up to automorphism. A rather counterintuitive statement is also established, that for some words, the only unambiguous (up to automorphism) morphisms are non-injective (or even periodic). The second part of the thesis addresses words for which all non-periodic morphisms are unambiguous. In the free monoid, these take the form of periodicity forcing words. It is shown using morphisms that there exist ratio-primitive periodicity forcing words over arbitrary alphabets, and furthermore that it is possible to establish large and varied classes in this way. It is observed that the set of periodicity forcing words is spanned by chains of words, where each word is a morphic image of its predecessor. It is shown that the chains terminate in exactly one direction, meaning not all periodicity forcing words may be reached as the (non-trivial) morphic image of another. Such words are called prime periodicity forcing words, and some alternative methods for finding them are given. The free-group equivalent to periodicity forcing words - a special class of C-test words - is also considered, as well as the ambiguity of terminal-preserving morphisms with respect to words containing terminal symbols, or constants. Moreover, some applications to pattern languages and group pattern languages are discussed.

Funding

Loughborough University, School of Science, Dept of Computer Science.

History

School

  • Science

Department

  • Computer Science

Publisher

© Joel D. Day

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2016

Notes

A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.

Language

en

Exports

Logo branding

Keyword(s)

Exports