# MMath&Phys Mathematics and Physics

Year of entry: 2023

## Course unit details:Computation and Complexity

Unit code MATH43011 15 Level 4 Semester 1 Department of Mathematics 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 MATH63011 Anti-requisite Compulsory

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 unit students should

• 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 prove some of the main theorems of computation and complexity theory;
• classify and compare the computability and complexity of decision problems in simple cases.

### Syllabus

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

1. COMPUTABILITY (approx 9 hours): 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 (approx 7 hours): 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 (approx 8 hours): 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 (approx 3 hours): Savitch's Theorem; the Immerman-Szelepcsenyi Theorem.

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework: two take home tests weighting 10% each.
• End of course examination: weighting 80%.

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

The course notes are self-contained 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
• Sipser, Introduction to the Theory of Computation (second edition), 2006.

### Study hours

Scheduled activity hours
Lectures 28
Tutorials 5
Independent study hours
Independent study 117

### Teaching staff

Staff member Role
Mark Kambites Unit coordinator