PhD-Computability and Constrained Equations in Free Semigroups
PhD @Loughborough University posted 21 hours agoJob Description
Project details
Given a finite set A, and an associative binary operation o, a semigroup is a set which contains A and is closed under o. If |A| = n, the free semigroup A+ is a particular example of a semigroup. Often in computer science, A is a set of symbols called letters, o is the concatenation operation, and A+ contains all nonempty strings (also called words) comprising symbols from A. The first order theory of a free semigroup is algorithmically undecidable in general, however satisfiability for first order quantifier-free formulas, including single equations, is well-known to be decidable at least in theory. In practice, developing sufficiently efficient algorithms remains a challenge.
The aim of this project is to study equations or quantifier free first-order formulas in free semigroups which are augmented by additional constraints, for example by restricting the values variables may take to rational subsets of A+ or by placing arithmetic constraints on their lengths, and as a result to better understand the limits of algorithmic approaches in solving the resulting satisfiability problems. Such problems have implications in a variety of areas of mathematics and computer science, including formal methods, combinatorial group theory, arithmetic and number theory, formal language theory, theory of computation and combinatorics on words.
The project will consist primarily of producing novel definitions, theorems and proofs, and may include some practical (programming) elements depending on the strengths and interests of a successful applicant. In all cases, a strong background in theoretical computer science and discrete mathematics is required, particularly in topics such as formal language theory, logic and algorithms and complexity.
A successful candidate should have a strong internal motivation and should enjoy problem solving and abstract thinking. They should work well individually, although there will also be opportunities for collaboration and networking both locally and internationally. Good communication skills, both verbal and written, and experience in reading and writing formal mathematics (including proofs) are also desirable. Applicants from diverse backgrounds are strongly encouraged.
The candidate will be part of the theoretical computer science research theme and will work closely with the primary supervisor, supported by regular supervision meetings. They will likely share an office with other PhD students and postdocs working in a mixture of theoretical and applied areas of computer science.
94% of Loughborough’s research impact is rated world-leading or internationally excellent. REF 2021
Supervisors
Primary supervisor:Â Joel Day
Entry requirements
Our entry requirements are listed using standard UK undergraduate degree classifications i.e. first-class honours, upper second-class honours and lower second-class honours.
Entry requirements for United Kingdom
- 2:1 undergraduate degree in Computer Science, Mathematics, or closely related subject.
- Experience with formal mathematical proofs.
English language requirements
Applicants must meet the minimum English language requirements. Further details are available on the International website.
Applicants must meet the minimum English language requirements. Further details are available on the International website.
Fees and funding
Tuition fees for 2025-26 entry
UK fee
£5,006 Full-time degree per annum
International fee
£28,600 Full-time degree per annum
Fees for the 2025-26 academic year apply to projects starting in October 2025, January 2026, April 2026 and July 2026.
Tuition fees cover the cost of your teaching, assessment and operating University facilities such as the library, IT equipment and other support services. University fees and charges can be paid in advance and there are several methods of payment, including online payments and payment by instalment. Fees are reviewed annually and are likely to increase to take into account inflationary pressures.
How to apply
All applications should be made online. Under programme name, select Computer Science. Please quote the advertised reference number: CO/JD-SF2/2025 in your application.
To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.
The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and self-funded projects.
Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project.
