MSc Pure Mathematics and Mathematical Logic / Course details

Year of entry: 2025

Course unit details:
Computation and Complexity

Course unit fact file
Unit code MATH63012
Credit rating 15
Unit level FHEQ level 7 – master's degree or fourth year of an integrated master's degree
Teaching period(s) Semester 2
Available as a free choice unit? No

Overview

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!

Pre/co-requisites

Unit title Unit code Requirement type Description
Computation and Complexity MATH63012 Anti-requisite Compulsory

Pre-Requisites: There are no explicit prerequisites; however, this is a heavily proof-based course and experience with proof-based mathematics is essential – eg MATH21120 or MATH20101 etc.

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.

Aims

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.

Learning outcomes

On successful completion of the course, students will be able to:

  • Define finite state automata and regular languages, explain the relationship between these, construct and analyse simple examples;
  • define the syntactic monoid of a language, discuss its relevance, and calculate it in simple examples; 
  • define Turing machines, discuss their capabilities and limitations, and construct and analyse simple examples;
  • define the key concepts of computation and complexity (including the main computability and complexity classes), discuss the relationships between them, and prove simple seen and unseen facts about them;
  • recall and analyse a range of computational problems and their properties;
  • state, apply and

Syllabus

0. INTRODUCTION: outline introduction to computability and complexity; course practicalities.

 

1. FINITE STATE AUTOMATA: alphabets and languages; finite state automata as models of computation; regular languages; Kleene’s theorem; semigroups and syntactic monoids; Schützenberger’s theorem.

 

2. COMPUTABILITY: problems and solutions; Turing machines; recursiveness and the Church-Turing Thesis; multitape machines; coding machines and non-recursive languages; universal computation; non-determinism.

 

3. COMPUTATIONAL COMPLEXITY: 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.

 

4. COMPLETENESS: NP-completeness; SAT and the Cook-Levin Theorem; NP-completeness by reduction; further examples of NP-complete languages.

Assessment methods

Method Weight
Other 20%
Written exam 80%
  • Mid-semester coursework: two take home tests weighting 20%
  • End of course examination: weighting 80%.

Feedback methods

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

Recommended reading

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.

Study hours

Scheduled activity hours
Lectures 11
Tutorials 11
Independent study hours
Independent study 128

Teaching staff

Staff member Role
Nora Szakacs Unit coordinator

Additional notes

The independent study hours will normally comprise the following. During each week of the taught part of the semester:


· You will normally have approximately 75-120 minutes of video content. Normally you would spend approximately 2.5-4 hrs per week studying this content independently
· You will normally have exercise or problem sheets, on which you might spend approximately 2-2.5hrs per week
· There may be other tasks assigned to you on Blackboard, for example short quizzes, short-answer formative exercises or directed reading
· In some weeks you may be preparing coursework or revising for mid-semester tests


Together with the timetabled classes, you should be spending approximately 9 hours per week on this course unit.


The remaining independent study time comprises revision for and taking the end-of-semester assessment.


The above times are indicative only and may vary depending on the week and the course unit. More information can be found on the course unit’s Blackboard page.

Return to course details