Understanding Automata and Computability

✍ Scribed by Educohack Press

• Publisher: Educohack Press
• Year: 2023
• Tongue: English
• Leaves: 924
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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