# Computational logic: super-counting quantifiers

### Primary supervisor

### Additional information

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### Other projects with the same supervisor

### Funding

- Competition Funded Project (Students Worldwide)

This research project is one of a number of projects at this institution. It is in competition for funding with one or more of these projects. Usually the project which receives the best applicant will be awarded the funding. Applications for this project are welcome from suitably qualified candidates worldwide. Funding may only be available to a limited set of nationalities and you should read the full department and project details for further information.

### Project description

Many problems in computational logic reduce to the so-called satisfiability problem: given a formula of some logic, determine whether that formula is satisfiable (i.e., represents a logically possible situation). It is well known that, for first-order logic as a whole, this problem is undecidable; however, for various fragments of first-order logic, it admits of an algorithmic solution. One large such fragment is the two-variable fragment with counting quantifiers, usually denoted C2. In this fragment, only two logical variables, x and y may appear; however, we can use counting quantifiers of the form "There exist M x such that ....", where M is a natural number. The complexity-theoretic properties of this fragment have been understood for some years.

Very recently, it has been shown that counting quantifiers can be generalized without losing decidability. Thus, for example, C2 can be extended with counting modulo k for some fixed k, thus giving us expressions like "There exist an even number of x such that ....". It seems clear that this extension is a special case of a more general kind of quantifier in which certain "local patterns" of relations are specified using various computational models. The aim of this project is to establish the extent to which counting quantifiers can be thus generalized. It is expected that this work will have an impact developements in description logics and related applications. The research may taken in either a purely theoretical direction, or via a mixture of theory and implementation.

The project requires a good understanding of logic, and the willingness to learn the necessary mathematical background in graph theory, model theory and comptational complexity theory.

### Person specification

#### For information

- Candidates must hold a minimum of an upper Second Class UK Honours degree or international equivalent in a relevant science or engineering discipline.
- Candidates must meet the School's minimum English Language requirement.
- Candidates will be expected to comply with the University's policies and practices of equality, diversity and inclusion.

#### Essential

Applicants will be required to evidence the following skills and qualifications.

- This project requires mathematical engagement and ability substantially greater than for a typical Computer Science PhD. Give evidence for appropriate competence, as relevant to the project description.
- You must be capable of performing at a very high level.
- You must have a self-driven interest in uncovering and solving unknown problems and be able to work hard and creatively without constant supervision.

#### Desirable

Applicants will be required to evidence the following skills and qualifications.

- You will have good time management.
- You will possess determination (which is often more important than qualifications) although you'll need a good amount of both.

#### General

Applicants will be required to address the following.

- Comment on your transcript/predicted degree marks, outlining both strong and weak points.
- Discuss your final year Undergraduate project work - and if appropriate your MSc project work.
- How well does your previous study prepare you for undertaking Postgraduate Research?
- Why do you believe you are suitable for doing Postgraduate Research?