MSc Pure Mathematics

Quite a lot of the mathematics you have studied so far involves using algorithms to solve computational problems. For example, you have probably used Euclid's algorithm to solve the problem of finding the greatest common divisor of two integers. In this course, we abstract a level further, and study the properties of problems and algorithms themselves. The kind of questions we ask are "is there an algorithm to solve EVERY problem?" and "what problems can be solved by an EFFICIENT algorithm?".

Compared with most of mathematics, this area is in its infancy, and many important things remain unknown. The course will take you to the point where you understand the statement of one of the most important open questions in mathematics and computer science: the "P vs NP" problem, for which the Clay Mathematics Foundation is offering a $1,000,000 prize. And who knows, perhaps one day you will be the one to solve it!

Students are not permitted to take, for credit, MATH43012 in an undergraduate programme and then MATH63012 in a postgraduate programme at the University of Manchester, as the courses are identical.

The course unit aims

• to introduce the main model of computation currently being employed in the theory of computation, Turing machines;
• to introduce the key parameters quantifying computational complexity (deterministic, non-deterministic, time, space) and the relationships between them.

On successful completion of the course unit students should

• be familiar with Turing machines and their capabilities and limitations, and be able to construct and analyse examples to solve simple problems;
• have a basic overview of classical complexity theory, including the main parameters quantifying computational complexity classes.
• be able to classify and compare the decidability and complexity of decision problems in simple cases.
• understand the statement of, and have an appreciation of some of the issues and concepts surrounding, the "P vs NP" problem.

0. INTRODUCTION (1 lecture): outline introduction to computability and complexity; course practicalities.

1. COMPUTABILITY (11 lectures): problems and solutions; alphabets and languages; Turing machines; recursiveness and the Church-Turing Thesis; multitape machines; coding machines and non-recursive languages; universal computation; non-determinism.

2. COMPUTATIONAL COMPLEXITY (8 lectures): time and space; linear speed up and space reduction; complexity classes; lower bounds and crossing arguments; space and time hierarchy theorems; tractability and P vs NP; polynomial time reduction.

3. COMPLETENESS (9 lectures): NP-completeness; SAT and the Cook-Levin Theorem; NP-completeness by reduction; further examples of NP-complete languages; NP-intermediacy and Ladner's Theorem; PSpace-completeness; oracles and the Baker-Gill-Solovay Theorem.

4. SPACE COMPLEXITY (3 lectures): Savitch's Theorem; the Immerman-Szelepcsenyi Theorem.

• Mid-semester coursework: two take home tests weighting 20%
• End of course examination: weighting 80%.

Tutorials will provide an opportunity for students’ work to be discussed and provide feedback on their understanding.

Printed notes will be supplied and you should not need to refer to any books. But if you would like an alternative viewpoint, the following texts cover most of the course material:

• Bovet and Crescenzi, Introduction to the Theory of Complexity, 1994,
• Papadimitriou, Computational Complexity, 1994,
• Sipser, Introduction to the Theory of Computation (second edition), 2006.

This course unit detail provides the framework for delivery in 20/21 and may be subject to change due to any additional Covid-19 impact.  

Please see Blackboard / course unit related emails for any further updates.