Automata Theory and Formal Languages: Fundamental Notions, Theorems, and Techniques

287 Pages·2022·3.158 MB·

by  Alberto Pettorossi

Undergraduate Topics in Computer Science
Alberto Pettorossi
Automata Theory
and Formal
Languages
Fundamental Notions, Theorems,
and Techniques
Undergraduate Topics in Computer
Science
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Science, Technical University of Denmark, Kongens Lyngby, Denmark
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Alberto Pettorossi
Automata Theory
and Formal Languages
Fundamental Notions, Theorems,
and Techniques
123
AlbertoPettorossi
University of RomeTor Vergata
Rome, Italy
IASI-CNR
Rome, Italy
ISSN 1863-7310 ISSN 2197-1781 (electronic)
Undergraduate Topics inComputer Science
ISBN978-3-031-11964-4 ISBN978-3-031-11965-1 (eBook)
https://doi.org/10.1007/978-3-031-11965-1
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Preface
In this book we present some basic notions and results on Automata Theory,
Formal Language Theory, Computability Theory, and Parsing Theory.
In particular, we consider the class of regular languages which are related to
the class of finite automata, and the class of the context-free languages which are
related to the class of pushdown automata. For the finite automata we also study
the problem of their minimalization and the characterization of their behaviour
using regular expressions.
Forcontext-freelanguagesweillustratehowtoderive theirgrammarsinChom-
sky and Greibach normal form. We study the relationship between deterministic
and nondeterministic pushdown automata and the context-free languages they ac-
cept. We present also some fundamental techniques for parsing both regular and
context-free languages.
Then we consider more powerful automata and we illustrate the relationship
between linear bounded automata and context-sensitive languages, and between
Turing Machines and type 0 languages. Chapter 6 of the book is dedicated to
the analysis of various decidability and undecidability problems in context-free
languages.
In the Supplementary Topics chapter we deal with other classes of machines
and languages, such as the counter machines, the stack automata, and the ab-
stract families of languages. We also present some additional properties of finite
automata, regular grammars, and context-free grammars, and we present a suf-
ficient condition for the existence of a bijection between sets and we prove the
existence of functions that are not computable.
This book was written for a course on ‘Automata, Languages, and Translators’
taught at the University of Roma Tor Vergata. A theorem with number k.m.n is
in Chapter k, Section m, and within that section it is identified by the number n.
Analogous numbering system is used for algorithms, corollaries, definitions, exam-
ples, exercises, figures, and remarks. I use ‘iff’ as an abbreviation for ‘if and only
if’.
Many thanks to my colleagues of the Department of Informatics, Systems,
and Production and the Department of Civil Engineering and Informatics of the
University of Roma Tor Vergata, and the IASI Institute of the National Research
Council of Italy. I amalso grateful to allmy students and co-workers for their help
and encouragement, and in particular to Alessandro Cacciotti, Lorenzo Clemente,
v
vi PREFACE
Emanuele De Angelis, Corrado Di Pietro, Fabio Fioravanti, Fulvio Forni, Fabio
Lecca, Maurizio Proietti, Marco Scarlino, and Valerio Senni.
Thanks also to Ms. Michela Castrica, Mr. Ralf Gerstner, Mr. Ronan Nugent,
and Mr. Wayne Wheeler of Springer for their most appreciated cooperation and
help.
Previous editions of this book were published by the Aracne Publishing Com-
pany, Ariccia (RM), Italy.
Roma, June 2022 Alberto Pettorossi
Contents
Preface ..................................................................... v
1. Formal Grammars and Languages....................................... 1
1.1. Free Monoids....................................................... 1
1.2. Formal Grammars .................................................. 2
1.3. The Chomsky Hierarchy ............................................ 5
1.4. Chomsky Normal Form and Greibach Normal Form................. 13
1.5. Epsilon Productions................................................. 14
1.6. Derivations in Context-Free Grammars.............................. 19
1.7. Substitutions and Homomorphisms.................................. 22
2. Finite Automata and Regular Grammars ............................... 25
2.1. Deterministic and Nondeterministic Finite Automata................ 25
2.2. NondeterministicFiniteAutomataandS-extendedType3Grammars . 29
2.3. Finite Automata and Transition Graphs............................. 31
2.4. Left Linear and Right Linear Regular Grammars.................... 36
2.5. Finite Automata and Regular Expressions .......................... 41
2.6. Arden Rule......................................................... 53
2.7. Axiomatization of Equations Between Regular Expressions.......... 55
2.8. Minimization of Finite Automata ................................... 57
2.9. Pumping Lemma for Regular Languages ............................ 70
2.10. A Parser for Regular Languages.................................... 73
2.10.1. A Java Program for Parsing Regular Languages.................. 83
2.11. Generalizations of Finite Automata................................ 91
2.11.1. Moore Machines ................................................. 92
2.11.2. Mealy Machines.................................................. 93
2.11.3. Generalized Sequential Machines................................. 93
2.12. Closure Properties of Regular Languages........................... 95
2.13. Decidability Properties of Regular Languages...................... 98
3. Pushdown Automata and Context-Free Grammars ...................... 101
3.1. Pushdown Automata and Context-Free Languages .................. 101
3.2. From PDA’s to Context-Free Grammars and Back: Some Examples. 115
3.3. Deterministic PDA’s and Deterministic Context-Free Languages.....121
3.4. Deterministic PDA’s and Grammars in Greibach Normal Form...... 127
3.5. Simplifications of Context-Free Grammars...........................129
3.5.1. Elimination of Nonterminal Symbols That Do Not Generate Words 129
3.5.2. Elimination of Symbols Unreachable from the Start Symbol........130
3.5.3. Elimination of Epsilon Productions................................ 132
vii
viii CONTENTS
3.5.4. Elimination of Unit Productions................................... 133
3.5.5. Elimination of Left Recursion......................................136
3.6. Construction of the Chomsky Normal Form ........................ 137
3.7. Construction of the Greibach Normal Form ......................... 140
3.8. Theory of Language Equations...................................... 148
3.9. Summary on the Transformations of Context-Free Grammars........154
3.10. Self-Embedding Property of Context-Free Grammars .............. 155
3.11. Pumping Lemma for Context-Free Languages ..................... 158
3.12. Ambiguity and Inherent Ambiguity ................................ 164
3.13. Closure Properties of Context-Free Languages......................165
3.14. Basic Decidable Properties of Context-Free Languages ............. 167
3.15. Parsers for Context-Free Languages................................ 168
3.15.1. The Cocke-Younger-Kasami Parser............................... 168
3.15.2. The Earley Parser................................................171
3.16. Parsing Classes of Deterministic Context-Free Languages...........176
3.17. Closure Properties of Deterministic Context-Free Languages........179
3.18. Decidable Properties of Deterministic Context-Free Languages..... 179
4. Linear Bounded Automata and Context-Sensitive Grammars ........... 181
4.1. Recursiveness of Context-Sensitive Languages....................... 190
5. Turing Machines and Type 0 Grammars ................................ 195
5.1. Turing Machines ................................................... 195
5.2. Equivalence Between Turing Machines and Type 0 Languages ...... 203
6. Decidability and Undecidability in Context-Free Languages ............. 209
6.1. Preliminary Definitions and Theorems...............................209
6.2. Basic Decidability and Undecidability Results ...................... 215
6.2.1. Basic Undecidable Properties of Context-Free Languages.......... 217
6.3. Decidability in Deterministic Context-Free Languages............... 221
6.4. Undecidability in Deterministic Context-Free Languages.............222
6.5. Undecidable Properties of Linear Context-Free Languages........... 223
7. Supplementary Topics...................................................225
7.1. Iterated Counter Machines and Counter Machines...................225
7.2. Stack Automata .................................................... 236
7.3. Relationships Among Various Classes of Automata.................. 238
7.4. Decidable Properties of Classes of Languages........................243
7.5. Algebraic and Closure Properties of Classes of Languages ........... 245
7.6. Abstract Families of Languages ..................................... 246
7.7. Finite Automata to/from S-extended Regular Grammars............252
7.8. Context-Free Grammars over Singleton Terminal Alphabets......... 255
7.9. The Bernstein Theorem............................................. 258
7.10. Existence of Functions That Are Not Computable..................261
List of Algorithms and Programs............................................269
Index....................................................................... 271
Bibliography................................................................ 279
CHAPTER 1
Formal Grammars and Languages
In this chapter we introduce some basic notions and some notations we will use
in the book. In particular, we introduce the notions of a free monoid, a formal
grammar and its generated language, the Chomsky hierarchy, the Kuroda normal
form, the Chomsky normal form, and the Greibach normal form. We also examine
the effects of the presence of the epsilon production in formal grammars. Finally,
westudythederivationsincontext-freelanguagesandthenotionsofasubstitution
and a homomorphism.
1.1. Free Monoids
The set of natural numbers {0,1,2,...} is denoted by N.
Given a set A, |A| denotes the cardinality of A, and 2A denotes the power-
set of A, that is, the set of all subsets of A. Instead of 2A, we will also write
Powerset(A). The set of all finite subsets of A is denoted by P (A).
fin
We say that a set S is countable iff either S is finite or there exists a bijection
between S and the set N of natural numbers.
Let us consider a countable set V, also called an alphabet. The elements of V
are called symbols. The free monoid generated by the set V is the set, denoted V∗,
consisting of all finite sequences of symbols in V, that is,
V∗ = {v ...v |n ≥ 0 and for i = 0,...,n, v ∈ V}.
1 n i
The unary operation ∗ (pronounced ‘star’) is called the Kleene star, or the Kleene
closure, or the ∗closure (pronounced ‘the star closure’). Sequences of symbols are
also called words or strings. The length of a sequence v ...v is n. The sequence
1 n
of length 0 is called the empty sequence or empty word and it is denoted by ε. The
length of a sequence w is denoted by |w|. For all w∈V∗, for all a∈V, the number
of occurrences of the symbol a in the word w is denoted by |w| .
a
Given two sequences w and w in V∗, their concatenation, denoted w (cid:1) w or
1 2 1 2
simply w w , is the sequence in V∗ defined by recursion on the length of w as
1 2 1
follows:
(cid:1)
w w = w if w = ε
1 2 2 1
(cid:1)
= v ((v ...v ) w ) if w = v v ...v with n>0.
1 2 n 2 1 1 2 n
We have that |w (cid:1) w | = |w |+|w |. The concatenation operation (cid:1) is associative
1 2 1 2
and its neutral element is the empty sequence ε.
Any set of sequences which is a subset of V∗ is called a language (or a formal
language) over the alphabet V.
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 1
A. Pettorossi, Automata Theory and Formal Languages,
Undergraduate Topics in Computer Science, https://doi.org/10.1007/978-3-031-11965-1_1