# Prolog, constraint programming and algebra

### The University of ManchesterDepartment of Mathematics

Applications accepted all year round  Competition Funded PhD Project (Students Worldwide)

My collaborators and I have written two recent papers which are intended to demonstrate that logic and constraint programming are tools which need to be better known and more widely deployed in pure mathematics. In both cases we used Sicstus Prolog and its CLP(FD) library.

The first paper classified a certain type of simple Lie algebra over the field GF(2) up to dimension 31. At an elementary level, this question boils down to filling out a 31x31 matrix with zeros and ones, subject to a series of constraints which enforce the Lie algebra structure and also the more complicated matter of insisting that it should be simple. It also collapses the Lie algebras into isomorphism classes by deploying a fairly sophisticated technique called toral switching.

It is worth saying that attempting to solve this problem by brute force is completely absurd. Prima facie, the number of 31x31 matrices is 2^{961}; this is many orders of magnitude bigger than the number of atoms in the observable universe. In other words, the success of this approach caused us to be extremely surprised by the power of Prolog's constraint handling.

We wanted to see how Prolog would handle a problem in another field of mathematics. Happily, we came across an unsolved problem in combinatorics which had a direct application to the UK national lottery, which resulted in \cite{CS23}. The problem is to find the minimum number of tickets you need to buy before a particular lottery draw so as to guarantee that no matter what numbers come up, at least one of your tickets will match at least two balls from the draw. In case there are 59 balls---as in the UK national lottery---the answer turns out to be that you need 27 tickets. (Of course you have to choose them rather carefully.)

Our lottery paper generated an enormous amount of media interest: national and international newspapers, radio interviews and so on---perhaps the highlight was seeing David Cushing describing our work on the US news network NBC Now. The prominent public science communicator Matt Parker made a great video about our work that you can watch here: https://www.youtube.com/watch?v=zYkmIxS4ksA.

Prolog is thought of by some as a 50 year-old AI predecessor, though it has typically been at the fringe of academic and industrial interest. That is currently changing and Prolog is being applied to problems as diverse as train scheduling, language processing, government grant assessment, and now also pure mathematics. Prolog is an extremely natural environment in which to approach classification questions in pure mathematics.

This project is about finding ways to better focus Prolog towards pure mathematics and to demonstrate the efficacy of the innovations through applications to problems therein.

Principally, I believe that what is needed is the integration of Prolog's rewriting capacities (for example through definite clause grammars) with the constraint programming library. Prolog needs to be capable of rewriting constraints dynamically through substitution and other algebraic manipulations. So there can be a serious coding component to the problem, which needs an understanding and sensitivity to algebraic methods and algorithms.

The coding side of the project can be balanced against finding solutions to mathematics problems using constraint programming. The creativity here would be to find ways to express mathematical problems through constraints. At least one major task in this direction would be to implement finite fields in Prolog. I also suspect that certain calculations with permutation groups may hugely benefit from interfacing with constraints; in particular, the calculation of (co)homology groups seems ideal for contraint solving.

Eligibility

Applicants should have, or expect to achieve, at least a 2.1 honours degree or a master’s (or international equivalent) in a relevant science or engineering related discipline.

Funding

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## Funding Notes

At Manchester we offer a range of scholarships, studentships and awards at university, faculty and department level, to support both UK and overseas postgraduate researchers applying for competition and self-funded projects. Please see the project description for further details.

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