Table Of ContentAutomata
Macmillan Computer Science SeriJs
Consulting Editor: Professor J. S. Rohl, UMIST
A. Learner and A. J. Powell: An Introduction to Algol 68 through Problems
A. M. Lister: Fundamentals o[Operating Systems
Automata
David Hopkin
Barbara Moss
Department ofM athematics,
University of Salford
M
© David Hopkin and Barbara Moss 1976
All rights reserved. No part of this publication may be reproduced or
transmitted, in any form or by any means, without permission
First published 1976 by
THE MACMILLAN PRESS LTD
London and Basingstoke
Associated companies in New York Dublin
Melbourne Johannesburg and Madras
ISBN 978-0-333-21217-2 ISBN 978-1-349-15691-7 (eBook)
DOI 10.1007/978-1-349-15691-7
This book is sold subject to the standard conditions of the Net Book Agreement.
The paperback edition of this book is sold subject to the condition that it shall
not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circul
ated without the publisher's prior consent in any form of binding or cover other
than that in which it is published and without a similar condition including this
condition being imposed on the subsequent purchaser.
Contents
Preface vi
Introduction 1.
1. Finite Automata 5
1.1 Finite Transducers 6
1.2 Finite Acceptors 12
1.3 Languages Accepted by Finite Automata 18
1.4 Regular Grammars and Non-deterministic Automata 24
2. Intermediate Automata 36
2.1 Pushdown Automata 36
2.2 Context-free Grammars 45
2.3 The Equivalence Theorem 52
2.4 Context-sensitive Languages 55
2.5 Unear Bounded Automata 59
2.6 Context-sensitive Languages and Unear Bounded Automata 63
3. Turing Machines and Computable Functions 67
3.1 Turing Machines 67
3.2 Computable Functions 76
3.3 Primitive Recursive Functions 88
3.4 Partial Recursive and Recursive Functions 99
3.5 Arithmetisation of the Theory of Turing Machines 102
4. Computability 110
4.1 Recursively Enumerable Sets 110
4.2 Computability and Proof: Godel's Theorem 116
4.3 Definitions of Computability 120
Appendix A: Logic 130
A.l From Bits to Truth Values 130
A.2 Rules of Deduction 134
A.3 Sets 141
A.4 Relations and Functions 145
A.5 Numbers 153
Appendix B: Biographical Notes 159
References 165
Further Reading 167
Index 168
Preface
This book has grown out of a course of lectures given to undergraduates reading
mathematics with computer science; however, no specific mathematical background
is assumed, and we hope that the result will be of interest not only to mathe
maticians and computer scientists, but also to those, such as linguists and brain
theorists, whose subject has been influenced by the development of automata
theory.
Our main object is to clarify what is meant by the claim that a function may
be computed, or that a question may be decided, by an effective procedure.
Answers to questions like these may be phrased either in terms of logic or in
terms of some particular machine that carries out the procedure. In order to
present a self-contained treatment, we include as appendix A a brief survey of
the logic underlying the theory of computability.
We should like to express our thanks to J. S. Rohl, Jane Hughes and Derek
Coleman, for their many useful comments and suggestions, and to Susan Heywood,
whose help has been invaluable in translating our ideas into typescript.
Salford, 1976 DAVID HOPKIN
BARBARA MOSS
Introduction
A computer is a large machine capable of solving a wide range of problems. The
problems have certain common features, whether they relate to calculation, data
processing, or other fields; they must be precisely formulated both to the human
solver and to the computer, and it must be possible to find an algorithm, or
program, of simple, precise instructions that leads to the solution of the problem.
In general, the machine receives an input of information related to the problem
(program, data, etc.), which affects the state of the machine and produces an
output related to the solution. Both input and output may be represented by
finite strings of symbols. A common, though not universal, feature of the programs
used is iteration, exemplified by such instructions as 'repeat until all the data
has been used' or 'replace x by y and repeat the last 10 lines until I x - y 1< 1' ;
it will be shown that this feature may occur even in very simple machines.
The object of this book is to study the logic of computability in terms of
simple machines, or automata. These machines are models for certain aspects
of computing; like the digital computer, they receive input, may take on one of a
number of different states determined wholly or partly by this input, and emit
an output determined by the input and the state of the machine. Some of the
machines resemble parts of a computer (for example, adders and pushdown
stores), but their physical realisation will not concern us here. Others may be
imaginary; the Turing machine, whose tape may grow beyond any given
bound, can only be imitated in a world with finite supplies of raw materials.
The simplest 'machines' need not be mechanical at all; a code table receives
an input message and produces a coded message as its output, but the energy
involved is human energy to read off and transcribe the output. Such a machine
may be regarded as having one state if a given input letter always produces
the same letter as output.
By considering these simple models, it is possible to build up a picture of how
a machine computes a function or solves a problem. As the structure of a class
of machines becomes more complex, a wider range of problems can be solved; so
we shall develop a notion of what it means for a function to be computable,
or for a question to be decidable, in terms of a given type of machine. The last
example we shall consider - the class of Turing -computable functions - coincides
with so many definitions of computability that arise in mathematics or in
automata theory that we shall then be able to give precise and useful definitions
of computability and decidability.
Of particular interest will be machines whose only function is to answer 'yes'
or 'no' to a question about the input string. These machines are called acceptors;
2 AUTOMATA
with each acceptor we associate a family of strings, or [onnallanguage, containing
those strings for which the answer is 'yes'. The languages defined in this way share
certain properties with natural languages, such as Swahili, and programming
languages, such as Algol. At every level, there is a very close correspondence
between the machine theory and the theory of the language.
The following examples may help to clarify the notion of an automaton and
to introduce some of the terms used in classifying automata.
Example 1 A Slide Rule
We regard the slide rule as a machine for multiplying pairs of numbers. If x and
yare to be multiplied, treat x as the first input. The only important feature of
the initial state of the slide rule is that it has received no input to the current
problem. To read in the input x, set the rule with 1 on the lower scale in line with
x on the upper scale. There is no output, but the new state consists of the
position of the rule, together with the knowledge that one input has been
received. In this state, y is 'read in' by sliding the cursor to cover y on the lower
scale; the output is then the number covered on the upper scale and the rule
returns to a 'passive' state, since no further input is involved· in this problem.
In general, active and passive states may be distinguished by leaving the cursor
in line with 1 on the lower scale after the first input is read in.
Since there are infinitely many points on a line of finite length, the slide rule
has infmitely many states, but in practice the number of states is restricted by
the number of points that can be distinguished by the user.
An automaton is called finite if it has a finite number of states and operates
with finite strings of input and output. Infinite automata have some accessory
device, such as a stack or a tape that may be moved to left or right, which can
grow to meet the demands made on the machine.
Example 2 Quality Control
Quality control has two aims: the removal of substandard products and the
inhibition of their production. A simple mechanism might test items from a
production line for weight, size, conductivity, or some other quality, representing
a satisfactory item by 1, a defective item by O. Not only are defective items
removed; if they occur too frequently, the production line is halted. Suppose the
line is halted if two out of three consecutive items are defective. The control
machine has two possible inputs, 0 and 1; three possible outputs, accept, reject,
reject and stop line; and enough states to reflect the past history of the machine,
which, in this example, is determined by the last two inputs.
State A The last input was 0, but either the preceding input was 1 or there has
been only one input since the line was restarted. (The machine 'remembers' the
pair (1,0>.)
INTRODUCTION 3
State B The last input was 1, but the preceding input was 0 and there have been
at least two inputs since the line was restarted. (Remember (0,1>.)
State C Either no 0 has appeared since the line restarted or the last two inputs
were 1. ,(Remember (1-, D.) (If the last two inputs were (0, 0>, the line would
have been stopped and the next input would be deferred until the line restarts in
state C.)
Consider all possible combinations of states and inputs
Input State New State Output
A B accept
B C accept
1 C C accept
o A C reject and stop
o B C reject and stop
o
C A reject
Thus we have a finite automaton with a small number of states, in which the
input and initial state determine the final state and the output. The machine
carries out the simple procedure of identifying places in a sequence of Os and Is
where two out of three consecutive digits are zero. This is a typical example of
the finite automata which will be discussed in chapter 1.
Example 3 A Non-deterministic Automaton
A non-deterministic automaton sounds like a contradiction in terms. However,
to give a simple illustration of this very useful concept, let us consider a machine
that is assembling coloured bricks into a pattern according to various rules. It is
easy to embody in the type of machine discussed above such rules as
If two consecutive bricks are blue, the next must be red.
On the other hand, the rules
The first brick must be either blue or white.
and
The fourth and fifth bricks may not be the same colour.
suggest that the state of the machine does not specify the output completely;
instead, it dictates that one of several colours should be chosen - perhaps which
ever brick is nearest to hand.
A non-deterministic automaton, then, is one that in at least one configuration
has a choice of possible moves.
