# Computation Theory

**Principal lecturer:** Prof Andrew Pitts

**Taken by:** Part IB CST

**Term:** Lent

**Hours:** 12

**Format:** In-person lectures

**Suggested hours of supervisions:** 3

**Prerequisites:** Discrete Mathematics

**This course is a prerequisite for:** Complexity Theory, Quantum Computing, Types

**Exam:** Paper 6 Question 5, 6

**Past exam questions, timetable**

## Aims

The aim of this course is to introduce several apparently different formalisations of the informal notion of algorithm; to show that they are equivalent; and to use them to demonstrate that there are uncomputable functions and algorithmically undecidable problems.

## Lectures

**Introduction: algorithmically undecidable problems.**Decision problems. The informal notion of algorithm, or effective procedure. Examples of algorithmically undecidable problems. [1 lecture]**Register machines.**Definition and examples; graphical notation. Register machine computable functions. Doing arithmetic with register machines. [1 lecture]**Universal register machine.**Natural number encoding of pairs and lists. Coding register machine programs as numbers. Specification and implementation of a universal register machine. [2 lectures]**Undecidability of the halting problem.**Statement and proof. Example of an uncomputable partial function. Decidable sets of numbers; examples of undecidable sets of numbers. [1 lecture]**Turing machines.**Informal description. Definition and examples. Turing computable functions. Equivalence of register machine computability and Turing computability. The Church-Turing Thesis. [2 lectures]**Primitive and partial recursive functions.**Definition and examples. Existence of a recursive, but not primitive recursive function. A partial function is partial recursive if and only if it is computable. [2 lectures]**Lambda-Calculus.**Alpha and beta conversion. Normalization. Encoding data. Writing recursive functions in the lambda-calculus. The relationship between computable functions and lambda-definable functions. [3 lectures]

## Objectives

At the end of the course students should

- be familiar with the register machine, Turing machine and lambda-calculus models of computability;
- understand the notion of coding programs as data, and of a universal machine;
- be able to use diagonalisation to prove the undecidability of the Halting Problem;
- understand the mathematical notion of partial recursive function and its relationship to computability.

## Recommended reading

* Hopcroft, J.E., Motwani, R. and Ullman, J.D. (2001).
*Introduction to automata theory, languages, and
computation*. Addison-Wesley (2nd ed.).

* Hindley, J.R. and Seldin, J.P. (2008). *Lambda-calculus and
combinators, an introduction*. Cambridge University Press
(2nd ed.).

Cutland, N.J. (1980). *Computability: an introduction to
recursive function theory*. Cambridge University Press.

Davis, M.D., Sigal, R. and Weyuker, E.J. (1994).
*Computability, complexity and languages*. Academic Press
(2nd ed.).

Sudkamp, T.A. (2005). *Languages and machines*.
Addison-Wesley (3rd ed.).