Theory of Computing

Learning Outcomes

Subject Specific Intellectual and Research Skills

Having successfully completed this module you will be able to:

• Use polynomial-time reduction to reason about the complexity class of a problem
• Ascertain and prove whether or not a given language is regular
• Use the reduction technique to show that a problem is undecidable
• Analyse the complexity of a given algorithm or problem
• Ascertain and prove whether or not a given language is context-free

Knowledge and Understanding

Having successfully completed this module, you will be able to demonstrate knowledge and understanding of:

• The complexity of algorithms and problems, and key complexity classes
• The relationship between the regular, context-free and recursively enumerable classes of languages, and the state-machines that accept them
• The nature and examples of undecidable problems
• The diagonalisation proof technique

Automata theory

• Finite state automata, regular expressions and regular languages
• The pumping lemma for regular languages
• Closure properties of regular languages
• Context-free grammars and pushdown automata
• Closure properties of context-free languages
• The pumping lemma for context-free languages

Computability theory

• Turing machines, recursively enumerable and recursive languages
• Church-Turing thesis
• Limitations of algorithms: universality, the halting problem and undecidability

Computational complexity theory

• Complexity of algorithms and of problems
• Complexity classes
• Polynomial-time reduction
• NP-Completeness and Cook's theorem

Teaching and learning methods

The content of this module is delivered through lectures, the module website, directed reading, pre-recorded materials and tutorials.

Students work on their understanding through a combination of independent study, preparation for timetabled activities, tutorials, and problem classes, along with formative assessments in the form of problem sheets.

Resources & Reading list

Textbooks

J. Hein (2002). Discrete Structures, Logic and Computability. Jones and Bartlett.

A.J.G. Hey (1996). Feynman Lectures on Computation. Addison Wesley.

D.C. Kozen (1999). Automata and Computability. Springer.

D. Cohen (1996). Introduction to Computer Theory. Wiley.

M. Sipser (1997). Introduction to the Theory of Computation. PWS.

J. Barwise and J. Etchemendy (1993). Turing's World. Stanford.

D. Harel (1992). Algorithmics: The Spirit of Computing. Addison Wesley.

A.K. Dewdney (2001). The (new) Turing Omnibus. Henry Holt.

N.D. Jones (1999). Computability and Complexity. MIT Press.

J. Gruska (1996). Foundations of Computing. Thomson.

Assessment strategy

This module is assessed by a combination of problem sheets and a final assessment in the form of a written examination.

Summative

This is how we’ll formally assess what you have learned in this module.