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## Computation Theory

Lecturer: Professor A.M. Pitts

No. of lectures: 12

Prerequisite course: Discrete Mathematics

This course is a prerequisite for Complexity Theory (Part IB), Quantum Computing (Part II).

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 recursive functions. Definition and examples. Primitive recursive partial function are computable and total. [1 lecture]

• Partial recursive functions. Definition. Existence of a recursive, but not primitive recursive function. Ackermann's function. A partial function is partial recursive if and only if it is computable. [2 lectures]

• Recursive and recursively enumerable sets. Decidability and recursive sets. Generability and recursive enumeration. Example of a set that is not recursively enumerable. Example of a recursively enumerable set that is not recursive. Alternative characterisations of recursively enumerable sets as the images and the domains of definition of partial recursive functions. [2 lectures]

Objectives

At the end of the course students should

• be familiar with the register machine and Turing machine 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

• be able to develop simple mathematical arguments to show that particular sets are not recursively enumerable    