St Andrews Research Repository

St Andrews University Home
View Item 
  •   St Andrews Research Repository
  • Mathematics & Statistics (School of)
  • Pure Mathematics
  • Pure Mathematics Theses
  • View Item
  •   St Andrews Research Repository
  • Mathematics & Statistics (School of)
  • Pure Mathematics
  • Pure Mathematics Theses
  • View Item
  •   St Andrews Research Repository
  • Mathematics & Statistics (School of)
  • Pure Mathematics
  • Pure Mathematics Theses
  • View Item
  • Login
JavaScript is disabled for your browser. Some features of this site may not work without it.

Counting subwords and other results related to the generalised star-height problem for regular languages

Thumbnail
View/Open
ThomasBournePhDThesis.pdf (702.0Kb)
Date
07/12/2017
Author
Bourne, Thomas
Supervisor
Ruškuc, Nik
Roney-Dougal, Colva Mary
Funder
Engineering and Physical Sciences Research Council (EPSRC)
Keywords
Regular language
Generalised star-height
Semigroup
Monoid
Subwords
Metadata
Show full item record
Altmetrics Handle Statistics
Abstract
The Generalised Star-Height Problem is an open question in the field of formal language theory that concerns a measure of complexity on the class of regular languages; specifically, it asks whether or not there exists an algorithm to determine the generalised star-height of a given regular language. Rather surprisingly, it is not yet known whether there exists a regular language of generalised star-height greater than one. Motivated by a theorem of Thérien, we first take a combinatorial approach to the problem and consider the languages in which every word features a fixed contiguous subword an exact number of times. We show that these languages are all of generalised star-height zero. Similarly, we consider the languages in which every word features a fixed contiguous subword a prescribed number of times modulo a fixed number and show that these languages are all of generalised star-height at most one. Using these combinatorial results, we initiate work on identifying the generalised star-height of the languages that are recognised by finite semigroups. To do this, we establish the generalised star-height of languages recognised by Rees zero-matrix semigroups over nilpotent groups of classes zero and one before considering Rees zero-matrix semigroups over monogenic semigroups. Finally, we explore the generalised star-height of languages recognised by finite groups of a given order. We do this through the use of finite state automata and 'count arrows' to examine semidirect products of the form 𝐴 ⋊ ℤ[sub]𝑟, where 𝐴 is an abelian group and ℤ[sub]𝑟 is the cyclic group of order 𝑟.
Type
Thesis, PhD Doctor of Philosophy
Collections
  • Pure Mathematics Theses
URI
http://hdl.handle.net/10023/12024

Items in the St Andrews Research Repository are protected by copyright, with all rights reserved, unless otherwise indicated.

Advanced Search

Browse

All of RepositoryCommunities & CollectionsBy Issue DateNamesTitlesSubjectsClassificationTypeFunderThis CollectionBy Issue DateNamesTitlesSubjectsClassificationTypeFunder

My Account

Login

Open Access

To find out how you can benefit from open access to research, see our library web pages and Open Access blog. For open access help contact: openaccess@st-andrews.ac.uk.

Accessibility

Read our Accessibility statement.

How to submit research papers

The full text of research papers can be submitted to the repository via Pure, the University's research information system. For help see our guide: How to deposit in Pure.

Electronic thesis deposit

Help with deposit.

Repository help

For repository help contact: Digital-Repository@st-andrews.ac.uk.

Give Feedback

Cookie policy

This site may use cookies. Please see Terms and Conditions.

Usage statistics

COUNTER-compliant statistics on downloads from the repository are available from the IRUS-UK Service. Contact us for information.

© University of St Andrews Library

University of St Andrews is a charity registered in Scotland, No SC013532.

  • Facebook
  • Twitter