Understanding Automata and Computability provides a clear and comprehensive exploration of essential concepts in automata theory and computability. This book covers a wide range of topics, including notation, Turing machines, algorithms, regular expressions, and grammars. It offers insights into algorithm analysis and problem complexity, making it an ideal resource for students and enthusiasts interested in the theoretical foundations of computation. Whether you're studying computer science, mathematics, or simply curious about the nature of computation, this book equips you with the knowledge and tools needed to grasp the fundamentals of automata and computability theory.
Author(s): Educohack Press
Publisher: Educohack Press
Year: 2023
Language: English
Pages: 924
Preface vii
Lectures 1
Introduction
Course Roadmap and Historical Perspective . 3
Strings and Sets . . . . . . . . . . . . . . . . 7
Finite Automata and Regular Sets
3
Finite Automata and Regular Sets
14
4
More on Regular Sets . . . . . . .
19
5
Nondeterministic Finite Automata
25
6
The Subset Construction . . . . . .
32
7
Pattern Matching . . . . . . . . . .
40
8
Pattern Matching and Regular Expressions
44
9
Regular Expressions and Finite Automata .
49
A
Kleene Algebra and Regular Expressions .
55
10
Homomorphisms . . . . . . . . .
61
11
Limitations of Finite Automata .
67
12
Using the Pumping Lemma
72
13
DFA State Minimization ..
77
14
A Minimization Algorithm.
84
15
Myhill-Nerode Relations ..
89
16
The Myhill-Nerode Theorem
95
xii Contents
Collapsing Nondeterministic Automata. . . . . . . 100
Automata on Terms . . . . . . . . . . . . . . . . . 108
The Myhill-Nerode Theorem for Term Automata . 114
Two-Way Finite Automata 119
2DFAs and Regular Sets . . . . . . . . . . . . . . . 124
Pushdown Automata and Context-Free Languages
Context-Free Grammars and Languages 129
Balanced Parentheses . . . . . . 135
Normal Forms. . . . . . . . . . . 140
The Pumping Lemma for CFLs . 148
Pushdown Automata. . . . . . . 157
Final State Versus Empty Stack . 164
PDAs and CFGs . . . . . . . . . 167
Simulating NPDAs by CFGs . . 172
Deterministic Pushdown Automata . 176
Parsing . . . . . . . . . . . . . . . . 181
The Cocke-Kasami-Younger Algorithm 191
The Chomsky-Schiitzenberger Theorem 198
Parikh's Theorem. . . . . . . . . . . 201
Turing Machines and Effective Computability
Turing Machines and Effective Computability 206
More on Turing Machines . . . . . . . . 215
Equivalent Models . . . . . . . . . . . . 221
Universal Machines and Diagonalization 228
Decidable and Undecidable Problems . 235
Reduction . . . . . . . . . . . . . . . 239
Rice's Theorem . . . . . . . . . . . . 245
Undecidable Problems About CFLs . 249
Other Formalisms 256
The .X-Calculus . . . . . 262
While Programs . . . . 269
Beyond Undecidability . 274
Godel's Incompleteness Theorem 282
Proof of the Incompleteness Theorem 287
Godel's Proof . . . . . . . . . . . . . . 292
Exercises
299
Homework Sets
Homework 1
301
Homework 2
302
Contents xiii
Miscellaneous Exercises
Finite Automata and Regular Sets ......... . Pushdown Automata and Context-Free Languages . Turing Machines and Effective Computability
Hints and Solutions
Hints for Selected Miscellaneous Exercises . Solutions to Selected Miscellaneous Exercises .
References
Notation and Abbreviations
