计算理论导论 英文版【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

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0 Introduction1

0.1 Automata,Computability,and Complexity1

Complexity theory2

Computability theory2

Automata theory3

0.2 Mathematical Notions and Terminology3

Sets3

Sequences and tuples6

Functions and relations7

Graphs10

Strings and languages13

Boolean logic14

Summary of mathematical terms16

0.3 Definitions,Theorems,and Proofs17

Finding proofs17

0.4 Types of Proof21

Proof by construction21

Proof by contradiction21

Proof by induction23

Exercises and Problems25

Part One:Automata and Languages29

1 Regular Languages31

1.1 Finite Automata31

Formal definition of a finite automaton35

Examples of finite automata37

Formal definition of computation40

Designing finite automata41

The regular operations44

1.2 Nondeterminism47

Formal definition of a nondeterministic finite automaton53

Equivalence of NFAs and DFAs54

Closure under the regular operations58

1.3 Regular Expressions63

Formal definition ofa regular expression64

Equivalence with finite automata66

1.4 Nonregular Languages77

Thepumping lemma for regular languages77

Exercises and Problems83

2 Context-Free Languages91

2.1 Context-free Grammars92

Formal definition of a context-free grammar94

Examples of context-free grammars95

Designing context-free grammars96

Ambiguity97

Chomsky normal form98

2.2 Pushdown Automata101

Formal definition of a pushdown automaton103

Examples of pushdown automata104

Equivalence with context-free grammars106

The pumping lemma for context-free languages115

2.3 Non-context-free Languages115

Exercises and Problems119

Part Two：Computability Theory123

3 The Church-Turing Thesis125

3.1 Turing Machines125

Formal definition of a Turing machine127

Examples of Turing machines130

3.2 Variants of Turing Machines136

Multitape Turing machines136

Nondeterministic Turing machines138

Enumerators140

Equivalence with other models141

3.3 The Definition of Algorithm142

Hilbert's problems142

Terminology for describing Turing machines144

Exercises and Problems147

4 Decidability151

4.1 Decidable Languages152

Decidable problems concerning regular languages152

Decidable problems concerning context-free languages156

4.2 The Halting Problem159

The diagonalization method160

The halting problem is undecidable165

ATuring-unrecognizable language167

Exercises and Problems168

5 Reducibility171

5.1 Undecidable Problems from Language Theory172

Reductions via computation histories176

5.2 A Simple Undecidable Problem183

5.3 Mapping Reducibility189

Computable functions190

Formal definition ofmapping reducibility191

Exercises and Problems195

6 Advanced Topics in Computability Theory197

6.1 The Recursion Theorem197

Self-reference198

Terminology for the recursion theorem201

Applications202

6.2 Decidability of logical theories204

A decidable theory206

An undecidable theory209

6.3 Turing Reducibility211

6.4 A Definition of Information213

Minimal length descriptions214

Optimality of the definition217

Incompressible strings and randomness217

Exercises and Problems220

Part Three:Complexity Theory223

7 Time Complexity225

7.1 Measuring Complexity225

Big-O and small-o notation226

Analyzing algorithms229

Complexity relationships among models231

7.2 The Class P234

Polynomial time234

Examples of problems in P236

7.3 The Class NP241

Examples of problemsin NP245

The P versus NP question247

7.4 NP-completeness248

Polynomial time reducibility249

Definition of NP-completeness253

The Cook-Levin Theorem254

7.5 Additional NP-complete Problems260

The vertex cover problem261

The Hamiltonian path problem262

The subset sum problem268

Exercises and Problems271

8 Space Complexity277

8.1 Savitch's Theorem279

8.2 The Class PSPACE281

The TQBF problem283

8.3 PSPACE-completeness283

Winning strategies for games287

Generalized geography289

8.4 The Classes L and NL294

8.5 NL-completeness297

Searching in graphs298

8.6 NL equals coNL300

Exercises and Problems302

9 Intractability305

9.1 Hierarchy Theorems306

Exponential space completeness313

9.2 Relativization318

Limits of the diagonalization method319

9.3 Circuit Complexity321

Exercises and Problems330

10 Advanced topics in complexity theory333

10.1 Approximation Algorithms333

10.2 Probabilistic Algorithms335

The class BPP336

Primality339

Read-once branching programs343

10.3 Alternation348

Alternating time and space349

The Polynomial time hierarchy353

10.4 Interactive Proof Systems354

Graph nonisomorphism355

Definition of the model355

IP=PSPACE357

10.5 Parallel Computation366

Uniform Boolean circuits367

The class NC369

P-completeness371

10.6 Cryptography372

Secret keys372

Public-key cryptosystems374

One-way functions374

Trapdoor functions376

Exercises and Problems378

Selected Bibliography381

Index387