Computer Laboratory – Course material 2008

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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

Recommended reading

* Hopcroft, J.E., Motwani, R. & Ullman, J.D. (2001). Introduction to automata theory, languages, and computation. Addison-Wesley (2nd ed.).
Cutland, N.J. (1980). Computability. An introduction to recursive function theory. Cambridge University Press.
Davis, M.D., Sigal, R. & Weyuker E.J. (1994). Computability, complexity and languages. Academic Press (2nd ed.).
Sudkamp, T.A. (2005). Languages and machines. Addison-Wesley (3rd ed.).

Next: Computer Design Up: Michaelmas Term 2008: Part Previous: Algorithms II Contents