Theory of automata

270 Pages·1969·13.492 MB·English

by  Arto Salomaa

THEORY OF
AUTOMATA
ARTO SALOMAA
Professor of Mathematics,
University of Turku, Finland
PERGAMON PRESS
OXFORD · LONDON · EDINBURGH · NEW YORK
TORONTO · SYDNEY · PARIS · BRAUNSCHWEIG
Pergamon Press Ltd., Headington Hill Hall, Oxford
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(c) 1969 A. Salomaa
AU Rights Reserved. No part of this publication may be
reproduced, stored in a retrieval system, or transmitted, in
any form or by any means, electronic, mechanical, photo­
copying, recording or otherwise, without the prior permission
of Pergamon Press Ltd.
First edition 1969
Library of Congress Catalog Card No. 71-76796
Printed in Hungary
08 013376 2
To my wife
and children Kai and Kirsti
PREFACE
THE past ten years have witnessed the vigorous growth of mathe­
matical disciplines concerning models for information-processing. In
addition to the classical theory of mechanically executable processes
(or Turing machines, algorithms or recursive functions), attention has
been focused on more restricted models such as finite deterministic
automata which were investigated originally in connection with sequen­
tial switching circuits. The results are so numerous and diverse that,
although one would limit the choice to material independent of tech­
nological developments, one could still write several books entitled
"Theory of Automata" with no or very little common material. Even
for the same material, the approach can vary from an engineering
oriented one to a highly abstracted mathematical treatment.
In selecting material for this book, I have had two principles in mind.
In the first place, I have considered the finite deterministic automaton
as the basic model. All other models, such as finite non-deterministic
and probabilistic automata as well as pushdown and linear bounded
automata, are treated as generalizations of this basic model. Secondly,
the formalism chosen to describe finite deterministic automata is that
of regular expressions. To give a detailed exposition regarding this
formalism, I have included a separate chapter on the algebra of regular
expressions.
This book deals with mathematical aspects of automata theory,
rather than applications. However, I have tried to avoid unnecessary
abstractions because no advanced mathematical background is assumed
on the part of the reader. The reader is supposed to be familiar with the
very basic notions in set theory, algebra and probability theory. One
of the reasons for the choice of the formalism of regular expressions
(rather than predicate calculus or more algebraic formalisms) was that
it can be developed very briefly ab ovo.
ix
X PREFACE
This book is self-contained. All results stated as theorems are proven
(except for minor details and proofs analogous to earlier proofs which
are sometimes left as exercises). Some results are mentioned without-
proof s in Sections 3, 4 and 7 of Chapter IV. The level of presentation
corresponds to that of beginning graduate or advanced undergraduate
work. An attempt has been made to cover also the most recent develop­
ments.
Exercises form an important part of the book. They are theoretical
rather than "numerical". Many of them contain material which could
equally well be treated in the text. However, the text is independent of
the exercises (except for some minor points in the proofs). Because
some exercises are very difficult, the reader is encouraged to consult
the given reference works. In addition to the exercises, there are also
"problems" at the end of some sections. The problems are topics
suggested for research.
Both of the Chapters III and IV can be studied also independently,
provided they are supplemented with a few definitions. A reader who
wants only the basic facts concerning various types of automata may
read only Sections 1, 2 and 4 of Chapter I, 1 and 2 of Chapter II and
1-8 of Chapter IV. These sections form a self-contained "short course".
Thereby, some of the more difficult proofs in Chapter IV may be
omitted.
ACKNOWLEDGEMENTS
THIS book has been written during my stay with the Computer
Science Department of the University of Western Ontario. I want to
thank the Head of the Department, Dr. J. F. Hart, for providing me
the opportunity to work within such a stimulating atmosphere. Special
thanks are due to Dr. Neil Jones whose sophistication in formal lan­
guages has been of an invaluable help in the preparation of many
sections in the book. Of the other members of the Department, I want
to thank especially Mr. Andrew Szilard for many fruitful discussions
and critical suggestions. Of my colleagues at the University of Turku,
I express my gratitude to Professor K. Inkeri for much encouragement
and help during many years. I want to thank also Professor Lauri
Pimiä for planning and drawing the figures and Messrs. Magnus Steinby
and Paavo Turakainen for reading the manuscript and making many
useful comments.
ARTO SALOMAA
London, Ontario
NOTE TO THE READER
REFERENCES to theorems, equations, etc., without a roman numeral
mean the item in the chapter where the reference is made. References
to other chapters are indicated by roman numerals. Thus, (1.2.1) means
the equation (2.1) in Chapter I and Problem III.6.1 means Problem
6.1 in Chapter III.
We use customary set theoretic notations. In particular, AXB de­
notes the Cartesian product of the sets A and B and
9
{x\Pi,...,P }
k
denotes the set of all elements x which possess each of the properties
Pi, ..., P .
k
The symbol D is used to mark the end of the proof of a theorem.
Throughout the book, the expression "if and only if" is abbreviated as
"iff".
Xli
CHAPTER I
FINITE DETERMINISTIC AUTOMATA
AUTOMATA are mathematical models of devices which process informa­
tion by giving responses to inputs. Models for discrete deterministic
computing devices possessing a finite memory will be our concern in
this chapter. Their behavior at a particular time instant depends not
only on the latest input but on the entire sequence of past inputs.
They are capable of only a finite number of inputs, internal states and
outputs and are, furthermore, deterministic in the sense that an input
sequence uniquely determines the output behavior.
§ 1. Regular languages
By an alphabet we mean a non-empty set. The elements of an alpha­
bet / are called letters. In most cases, we shall consider only finite
alphabets. Sometimes a subscript is used to indicate the number of
letters. Thus, I is an alphabet with r letters.
r
A word over an alphabet /is a finite string consisting of zero or more
letters of/, whereby the same letter may occur several times. The string
consisting of zero letters is called the empty word, written λ. Thus
Λ, X\ #I#2^2J Χ±Χ±Χ±Χ±Χχ X^K^K\X
9 9 2
are words over the alphabet / = {x x }. The set of all words over
2 l9 2
an alphabet /is denoted by W(J). Clearly, W(I) is infinite.
If P and Q are words over an alphabet /, then their catenation PQ is
also a word over /. Clearly, catenation is an associative operation and
the empty word λ is an identity with respect to catenation: Ρλ = λΡ=Ρ,
for any word P. For a word P and a natural number /, the notation P'
means the word obtained by catenating i copies of the word P. P° de­
notes the empty word λ.
i
2 THEORY OF AUTOMATA
By the length of a word P, in symbols lg (P), is meant the number of
letters in P when each letter is counted as many times as it occurs.
By definition, lg (X) = 0. The length function possesses some of the
formal properties of logarithm:
lg(P0 = lg(P)+lg(0),
lg(/*) = /lg(P), for/SET 0.
A word P is a subword of a word Q iff there are words Pi and P,
2
possibly empty, such that Q = PiPP. If Pi = A (P = A), then P is
2 2
termed an initial (a ywa/) subword of Q. If P is a subword of g, then
clearly lg(P)^lg(ß).
Subsets of W(I) are referred to as languages over the alphabet /.
For instance, the following sets are languages over the alphabet
h = {xi X2} ·
9
L\ = |^iX2^2> #i#2-£l5 Xi#2^1^2j»
Z/2 = {A, -*lj *2j ^l^lj X\X2i ·*2*1> <*2*2/>
£3 = {xi \p prime},
£4 = {x%y\i natural number},
L5 = {χ[χ 1 i natural number}.
2
The languages Lx and L% are finite, i.e. contain only a finite number of
words, whereas the languages L3-L5 are infinite. The language L con­
2
sists of all words over the alphabet {x x ] with length less than or
l9 2
equal to 2.
We consider also the empty language Ζ,, i.e. the language containing
φ
no words. Note that Ζ, is not the same as the language L or {A}
φ x
consisting of the empty word A. Sometimes we identify an element and
its unit set to simplify notation: we may denote simply by P the lan­
guage {P} consisting of the word P. In such cases, the meaning of the
notation will be clear from the context.
We shall now introduce some operations for the family of languages
over a fixed alphabet /. Because languages are sets, we may consider
the Boolean operations. The sum or union of two languages Lx and L*
is denoted by L1+L2, their intersection by LxC\L and the complement
2i
of a language L with respect to the set W(I) by ~L. We use also the
notation
L1—L2 = Iifi (^£2).
§ 1. REGULAR LANGUAGES 3
The catenation (or product) of two languages L\ and L2, in symbols
L1L2, is defined to be the language
UU = {PQ\P £L Q<l L }.
l9 2
Catenation of languages is associative because catenation of words
is associative. Thus, the notation L\ i ^ 1, is meaningful for a language
L. Furthermore, we define L° to be the language L consisting of the
x
empty word. Note that the languages L^ and L are zero and identity
x
elements with respect to catenation of languages: for any language
L, ΖΧψ = Ζ,φ-L = Ι,ψ and LL = L L = L.
X A
The catenation closure (or iteration) of a language L, in symbols L*
9
is defined to be the sum of all powers of L:
00
v = Σ L'.
Later on, we shall define some further operations for the family of
languages.
We shall now introduce the notion of a regular expression which
will be one of the basic notions in our subsequent discussions. Consider
the auxiliary alphabet
/' = {+,*,Φ, e»
and any alphabet / such that / and /' are disjoint. A regular expression
over the alphabet / is any word over the union of the alphabets / and /'
which satisfies the condition formulated in the following
DEFINITION, (i) Each letter belonging to the alphabet / is a regular
expression over /, and so is the letter φ.
(ii) If a and β are regular expressions over /, then so are also (a+/?),
(aß) and a*.
(iii) Nothing is a regular expression over /, unless its being so follows
from a finite number of applications of (i) and (ii).
Thus, for instance,
φ, φ*, xi, (X1+X2), ((xi+xi)(xi+X2)y (1.1)
are regular expressions over the alphabet h = {xi, ^2}.
DEFINITION. Each regular expression y over an alphabet / denotes a
language j γ \ over / according to the following conventions :