Computer Science

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 (e.g., to find the shortest round tour of 30 cities requires a computation which would take even the fastest computers longer than the age of the universe). 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).

Pre-requisites: ECM1414, ECM2418

To introduce the Mathematical basis and practical implications of the classical theory of computability 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 manipulate formal languages and create abstract automata to a given specification;
2 explain what is meant by a general model of computation and work with some specific examples of such models;
3 describe the mathematical basis of the theory of computability;
4 appreciate the relationship between the theory of computation and practical computing.
Discipline Specific Skills and Knowledge:
5 appreciate the power of abstraction to support a general understanding of some subject matter;
6 appreciate the role of theoretical understanding in underpinning disciplined and responsible practice.
Personal and Key Transferable / Employment Skills and Knowledge:
7 approach problems analytically at an appropriate level of abstraction.

- regular languages and regular expressions, context-free languages and grammars, pumping lemmas and other methods of proof;
- finite-state automata, elimination of non-determinism, Kleene's theorem, push-down automata;
- Turing machines, the universal Turing machine;
- recursive and recursively enumerable functions;
- recursive insolubility of the halting problem for Turing machines; variants of the halting problem; other recursively insoluble problems;
- the Church-Turing thesis and its variants; other models of computation; the quest for hypercomputation;
- time complexity vs space complexity; overview of the complexity class landscape;
- recapitulation of the P vs NP problem; more rigorous presentation of P vs NP in terms of Turing machine model; proof of Cook’s Theorem.

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.
 

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Reading list for this module: