Table Of ContentTeubner Studienbücher Informatik
J. Berste!
Transductions and Context-Free Languages
LeiHäden der angewandten
Mathematik und Mechanik LAMM
Unter Mitwirkung von
Prof. Dr. E. Becker, Darmstadt
Prof. Dr. G. Hotz, Saarbrücken
Prof. Dr. P. Kali, Zürich
Prof. Dr. K. Magnus, München
Prof. Dr. E. Meister, Darmstadt
Prof. Dr. Dr. h. c. F. K. G. Odqvist, Stockholm
Prof. Dr. Dr. h. c. Dr. h. c. Dr. h. c. E. Stiefelt
herausgegeben von
Prof. Dr. Dr. h. c. H. Görtler, Freiburg
Band 38
Die Lehrbücher dieser Reihe sind einerseits allen mathematischen Theo
rien und Methoden von grundsätzlicher Bedeutung für die Anwendung
der Mathematik gewidmet; andererseits werden auch die Anwendungs
gebiete selbst behandelt. Die Bände der Reihe sollen dem Ingenieur und
Naturwissenschaftler die Kenntnis der mathematischen Methoden, dem
Mathematiker die Kenntnisse der Anwendungsgebiete seiner Wissen
schaft zugänglich machen. Die Werke sind für die angehenden Industrie
und Wirtschaftsmathematiker, Ingenieure und Naturwissenschaftler be
stimmt, darüber hinaus aber sollen sie den im praktischen Beruf Tätigen
zur Fortbildung im Zuge der fortschreitenden Wissenschaft dienen.
Transductions and
Context-Free Languages
By Dr. Jean Berste!
Professor at the Universite P. et M. Curfe, Paris
With 32 figuras, 158 exercises
and numerous examples
Ef3
Springer Fachmedien Wiesbaden GmbH 1979
Prof. Jean Berste)
Born in 1941 at Nimes (France). From 1961 to 1966 studies in
mathematics at the University of Paris. 1967 These de
troisieme cycle, 1972 These d'Etat at the University of Paris.
From 1970 to 1972 Charge d'Enseignement at the University
of Strasbourg. Since 1972, Maitre de Conference, then Profes
sor at the Institut de Programmation, University of Paris VI
Durlog the Summer Semester 1976 on leave at the University
of Saarbrücken. Activity: Mathematicol and Theoretical Compu
ter Science.
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Rentel, Jean:
Transductions and context-free languages I by
Jean Berste!.
(Leitfäden der angewandten Mathematik und
Mechanik ; Bd. 38) (Teubner Studienbücher :
Informatik)
ISBN 978-3-519-02340-1 ISBN 978-3-663-09367-1 (eBook)
DOI 10.1007/978-3-663-09367-1
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© Springer Fachmedien Wiesbaden 1979
Ursprünglich erschienen bei B.G. Teubner, Stuttgart 1979
Softcover reprint of the bardeover 1st edition 1979
Setting: The Universities Press (Belfast) Ltd., Belfast
Cover design: W. Koch, Sindelfingen
Preface
This book presents a theory of formal languages with main emphasis on
rational transductions and their use for the classification of context-free lan
guages. The Ievel of presentation corresponds to that of beginning graduate or
advanced undergraduate work. Prerequisites for this book are covered by a
"standard" first-semester coursein formallanguages and automata theory: e.g.
a knowledge of Chapters 1-3 of Ginsburg [1966], or Chapters 3-4 of
Hopcroft and Ullman [1971], or Chapter 2 of Salomaa [1973], or Chap
ters 2 and 4 of Becker and Walter [1977] would suffice. The book is
self-contained in the sense that complete proofs are given for all theorems
stated, except for some basic results explicitly summarized at the beginning of
the text. Chapter IV and Chapters V-VIII are independent from each other.
The subject matter is divided into two preliminary and six main chapters. The
initial two chapters contain a general survey of the "classical" theory of regular
and context-free languages with a detailed description of several special
languages. Chapter III deals with the general theory of rational transductions,
treated in an algebraic fashion along the lines of Eilenberg, and which will be
used systematically in subsequent chapters. Chapter N is concerned with the
important special case of rational functions, and gives a full treatment of the
latest developments, including subsequential transductions, unambiguous trans
ducers and decision problems.
The study of families of languages (in the sense of Ginsburg) begins with
Chapter V. There, the structures of "rational cone" and "full AFL" are
introduced, and some general results are established. The chapter ends with the
treatment of several examples of cones of linear languages. Chapter VI
contains a theory of operators on families of languages, setting up an algebraic
framework for transformation and comparison of families of languages. Other
general results on cones and full AFLs are easily derived from a series of
inequalities involving only Operators. Chapter VII is concerned with the study
of principal cones and full AFLs, that is families generated by one language
only. Main interest is in subcones of the context-free languages. First, several
languages are proved to generate the entire cone of context-free languages.
Then S. Greibach's "Syntactic Lemma" is proved and used to exhibit nonprin
cipal cones. A detailed study of two important families follows, namely the
family of one counter and the family of quasi-rational (nonexpansive,
derivation-bounded · · ·) languages. Chapter VIII presents a general method
(due to Boassan and Beauquier) for proving strict containment or incompara-
6 Preface
bility of cones of context-free languages, using iterative pairs and systems of
iterative pairs. This method yields easy proofs of the existence of several
wellknown and of new hierarchies of cones of context-free languages, showing
thus the intirnate connection between iterative pairs and the structure of
context-free languages.
Main interest is in languages rather than in grammars or in acceptors. Indeed, a
language (even a context-free one) exists independently from the grammars
generating it, and a great number of context-free languages can be described
by a combinatorial or an algebraic property, without any reference to a
grammar. Moreover, grammatical characterizations of classes of languages
usually require just the existence of one grammar of some special type. Thus to
prove that a given language is not in the family, one must show that all
grammars generating it violate some property. This is usually a very delicate
proof. Finally there are results which are proved by the method of iterative
pairs and which cannot be proved-up to now-by considering only grammars.
Rational cones are treated in greater detail than full AFLs. They are indeed in
a natural relationship with rational transductions; further full AFLs are a
secondary structure in the sense that any full AFL is the rational closure of
some cone. Since we are mainly concerned with context-free languages, a
description of AFLs and "trios" (in Opposition to full AFLs and rational cones)
seemed unnecessary, all the more as some fundamental results such as the
Syntactic Lemma are still lacking for these families.
The notes from which this book derives were used in courses at the University
of Paris and at the University of Saarbrücken. I want to thank Professor G.
Hotz for the opportunity he gave me to stay with the Institut für angewandte
Mathematik and Informatik, and for his encouragements to write this book. I
am grateful to the following people for useful discussions or comments con
cerning various parts of the text: J. M. Autebert, J. Beauquier, Ch. Choffrut,
G. Cousineau, K. Estenfeld, R. Linder, M. Nivat, D. Perrin, J. F. Perrot, J.
Sakarovitch, M. Soria, M. Stadel, H. Walter. I am deeply indebted to M. P.
Schützenherger for his constant interest in this book and for many fruitful
discussions. Special thanks are due to L. Boasson whose comments have been
of an invaluable help in the preparation of many sections of this book. I want
to thank also J. Messerschmidt for his careful reading of the manuscript and
for many pertinent comments, and Ch. Reutenauer for checking the galley
proofs. I owe a special debt to my wife for her active contribution at each step
of the preparation of the book, and to Bruno and Clara for their indulgence.
Paris, Spring 1978 J. Berste!
Contents
I Preliminaries
1.1 Some Notations . 9
1.2 Monoids, Free Monoids . 9
1.3 Morphisms, Congruences. 12
1.4 Finite Automata, Regular Languages 15
II Context-Free Languages
11.1 Grammars, Languages, Equations 22
11.2 Closure Properties, Iteration 31
11.3 Dyck Languages . . . 35
11.4 Two Special Languages . 47
III Rational Transductions
111.1 Recognizable Sets 51
111.2 Rational Sets . . 55
111.3 Rational Relations 61
111.4 Rational Transductions. 65
111.5 Examples . . . . . 72
111.6 Transducers. . . . . 77
111.7 Matrix Representations. 80
111.8 Decision Problems 87
IV Rational Functions
IV.l Rational Functions 92
IV.2 Sequential Transductions . . 96
IV.3 The Cross-Section Theorem . 111
IV.4 Unambiguous Transducers 114
IV.5 Bimachines. . . . 123
IV.6 A Decidable Property 128
V FarnDies of Languages
V.1 Definition. . . . 134
V.2 Rational Equivalence, Rational Cones. 135
V.3 Rationally Closed Families. 138
V.4 Full AFLs . . . . . . . . . . 141
8 Contents
V.5 Substitution . 145
V.6 Example: The Cone of Linear Languages 150
V.7 Examples of Incomparable Languages. 156
VI Operators
VI.l Operators . . . . . 162
Vl.2 Examples of Operators 164
Vl.3 Closure Operators . . 166
VI.4 Subcommutative Relations 171
Vl.5 Marked Substitution 177
VII Generators
VII.1 Generators of the Context-Free Languages 185
VII.2 The Syntactic Lemma. . . 194
VI1.3 Substitution Oosure . . . 197
VII.4 One Counter Languages . · 201
VII.5 Quasi-Rational Languages 208
VID Iterative Pairs
VIII.1 Types of Iterative Pairs . 219
VIII.2 Grammatical Pairs 223
VIII.3 Transfer of Iterative Pairs . 227
VIII.4 Systems of Iterative Pairs 232
VII1.5 Grammatical Systems 237
VIII.6 Transfer of Systems . . 246
VIII. 7 Applications. . . . . 251
IX Open Problems, Further Developments 265
Bibliography . 268
List of Symbols 274
Index . ... 275
I Preliminaries
This chapter is a short review of some basic concepts used in the sequel. Its aim
is to agree on notations and terminology. We first consider monoids, especially
free monoids, and morphisms. Then a collection of definitions and results is
given, dealing with finite automata and regular languages.
1.1 Some Notations
N ={0, 1, 2, ....} is the set of nonnegative integers. Ll' ={· · · -2, -1, 0, 1, ...} is
the set of integers. Let E be a set. Then Card(E) is the number of its elements.
The empty set is denoted by fZ). If A, B are subsets of E, then we write A c B
iff x E A :::} x E B, and A ~ B iff A c B and A ~ B. Further
I
A \B ={xEE xEA and x$B}.
A singleton is a subset of E consisting of just one element. If no confusion
can arise, we shall not distinguish elements of E from singletons. The set of all
subsets of E, i.e. the powerset of E, is denoted by ~(E) or 2E. With the
preceding convention, E c ~(E).
The d o m a in dom( a) of a partial function a : E--+ F is the set of elements x in
E for which a(x) is defined. a can be viewed as a (total) function from E into
~(F), and with the convention Fe \lHF), as a total function from E into
FU{fZ)}. Then dom(a)={xEEia(x)~fZ)}.
1.2 Monoids, Free Monoids
A semigroup consists of a set M and a binary operation on M, usually
denoted by multiplication, and which is postulated tobe associative: For any
m" m2, m3EM, m1(m2m3)=(m1m2)m3• A neutral element or a unit is an
element 1M E M (also noted 1 for short) such that 1Mm = m 1M= m for all
m E M. A semi-group which has a neutral element is a mono i d. The neutral
element of a monoid is unique. lndeed, if 1' is another neutral element then
1'=11'=1.
Given two subsets A, B of a monoid M, the product AB is defined by
AB ={cEM l3a EA, 3bEB :c =ab}. (2.1)
