Computability and Complexity - 2025 entry

It is popularly supposed that there is no limit to the power of computers to perform any task, so long as it is sufficiently well defined, and to do so quickly and efficiently. In fact this is not so, and it can be proved mathematically that there are well-defined computational tasks which cannot, in principle, be performed by computers as we know them; and other tasks which, while they can be performed, cannot be completed in a feasible amount of time. This module will introduce you to the Turing Machine model of computation which underpins the fundamental theories of computability (concerned with what can be computed at all) and complexity (concerned with how efficiently things which can be computed can be computed). These theories will be introduced in a precise and formal way, and the main results and theorems will be stated and proven.

To introduce the Mathematical basis and practical implications of the classical theory of computability and complexity and to consider the extent to which this is still relevant to modern developments in computing.

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1. Explain what is meant by a general model of computation and work with some specific examples of such models;

2. Describe the mathematical basis of the theory of computability and complexity;

3. Appreciate the relationship between the theory of computation and complexity and practical computing.

Discipline Specific Skills and Knowledge:

4. Appreciate the power of abstraction to support a general understanding of some subject matter;

5. Appreciate the role of theoretical understanding in underpinning disciplined and responsible practice.

Personal and Key Transferable / Employment Skills and Knowledge:

6. Approach problems analytically at an appropriate level of abstraction.

• Turing machines, and the universal Turing machine;

• Recursive and recursively enumerable languages;

• Undecidability of the halting problem for Turing machines; variants of the halting problem; other undecidable languages;

• The Church-Turing thesis;

• Properties of languages and Rice’s theorem;

• Time complexity;

• Complexity classes P and NP, and the P vs NP problem;

• NP-complete problems;

• Cook’s theorem;

• Space complexity classes and their relationships with the other complexity classes seen in the module

Not applicable.

Reassessment will be by coursework and/or written exam in the failed or deferred element only. For referred candidates, the module mark will be capped at 40%. For deferred candidates, the module mark will be uncapped.

• ELE

Reading list for this module: