Computability and Constrained Equations in Free Semigroups (Ref: CO/JD-SF1/2026)

About the Project

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.

Name of primary supervisor/CDT lead:
Joel Day J.Day@lboro.ac.uk
https://www.lboro.ac.uk/departments/compsci/staff/joel-day/

Entry requirements:
Undergraduate degree in Computer Science or Mathematics or closely related subjects. Experience writing formal proofs.

English language requirements:
Applicants must meet the minimum English language requirements. Further details are available on the International website (http://www.lboro.ac.uk/international/applicants/english/).

Bench fees required: No

Closing date of advert: 1st January 2027

Start date: 2026, October 2026, February 2027, July 2027

Full-time/part-time availability: Full-time 3 years, Part-time 6 years

Fee band: 2025/26 Band RB (UK £5,006, International £28,600)

How to apply:
All applications should be made online. Under programme name, select Computer Science. Please quote the advertised reference number: CO/JD-SF1/2026 in your application. To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.

Project search terms:
Computer Science, Mathematics, Formal Language Theory, String Solving, Theory of Computation, Computational Complexity, String Algorithms, Combinatorics on Words, Discrete Mathematics, Discrete non-commutative algebra, Monoids, Groups, Software Verification, Formal Logic

Email Address Sci:
sci-pgr@lboro.ac.uk