计算理论基础：可计算性、复杂性和语言

Contents

I Preliminaries 1

1. Sets and n-tuples 1

2. Functions 3

3. Alphabets and Strings 4

4. Predicates 5

5. Quantifiers 6

6. Proof by Contradiction 8

7. Mathematical Induction 9

Part 1 Computability 15

2 Programs and Computable Functions 17

1. A Programming Language 17

2. Some Examples of Programs 18

3. Syntax 25

4. Computable Functions 28

5. More about Macros 32

3 Primitive Recursive Functions 39

1. Composition 39

2. Recursion 40

3. PRC Classes 42

4. Some Primitive Recursive Functions 44

5. Primitive Recursive Predicates 49

6. Iterated Operations and Bounded Quantifiers 52

7. Minimalization 55

8. Pairing Functions and G6del Numbers 59

4 A Universal Program 65

1. Coding Programs by Numbers 65

2. The Halting Problem 68

3. Universality 70

4. Recursively Enumerable Sets 78

5. The Parameter Theorem 85

6. Diagonalization and Reducibility 88

7. Rice s Theorem 95

*8. The Recursion Theorem 97

*9. A Computable Function That Is Not Primitive Recursive 105

5 Calculations on Strings 113

1. Numerical Representation of Strings 113

2. A Programming Language for String Computations 121

3. The Languages and n 126

4. Post-Turing Programs 129

5. Simulation of n in 135

6. Simulation of in 140

6 Turing Machines 145

1. Internal States 145

2. A Universal Turing Machine 152

3. The Languages Accepted by Turing Machines 153

4. The Halting Problem for Turing Machines 157

5. Nondeterministic Turing Machines 159

6. Variations on the Turing Machine Theme 162

7 Processes and Grammars 169

1. Semi-Thue Processes 169

2. Simulation of Nondeterministic Turing Machines by Semi-Thue Processes 171

3. Unsolvable Word Problems 176

4. Posts Correspondence Problem 181

5. Grammars 186

6. Some Unsolvable Problems Concerning Grammars 191

7. Normal Processes 192

8 Classifying Unsolvable Problems 197

1. Using Oracles 197

2. Relativization of Universality 201

3. Reducibility 207

4. Sets r.e. Relative to an Oracle 211

5. The Arithmetic Hierarchy 215

6. Posts Theorem 217

7. Classifying Some Unsolvable Problems 224

8. Rices Theorem Revisited 230

9. Recursive Permutations 231

Part 2 Grammars and Automata 235

9 Regular Languages 237

1. Finite Automata 237

2. Nondeterministic Finite Automata 242

3. Additional Examples 247

4. Closure Properties 249

5. Kleenes Theorem 253

6. The Pumping Lemma and Its Applications 260

7. The Myhill-Nerode Theorem 263

10 Context-Free Languages 269

1. Context-Free Grammars and Their Derivation Trees 269

2. Regular Grammars 280

3. Chomsky Normal Form 285

4. Bar-Hillels Pumping Lemma 287

5. Closure Properties 291

*6. Solvable and Unsolvable Problems 297

7. Bracket Languages 301

8. Pushdown Automata 308

9. Compilers and Formal Languages 323

11 Context-Sensitive Languages 327

1. The Chomsky Hierarchy 327

2. Linear Bounded Automata 330

3. Closure Properties 337

Part 3 Logic 345

12 Propositional Calculus 347

1. Formulas and Assignments 347

2. Tautological Inference 352

3. Normal Forms 353

4. The Davis-Putnam Rules 360

5. Minimal Unsatisfiability and Subsumption 366

6. Resolution 367

7. The Compactness Theorem 370

13 Quantification Theory 375

1. The Language of Predicate Logic 375

2. Semantics 377

3. Logical Consequence 382

4. Herbrands Theorem 388

5. Unification 399

6. Compactness and Countability 404

*7. G6dels Incompleteness Theorem 407

*8. Unsolvability of the Satisfiability Problem in Predicate Logic 410

Part 4 Complexity 417

14 Abstract Complexity 419

1. The Blum Axioms 419

2. The Gap Theorem 425

3. Preliminary Form of the Speedup Theorem 428

4. The Speedup Theorem Concluded 435

15 Polynomial-Time Computability 439

1. Rates of Growth 439

2. P versus NP 443

3. Cooks Theorem 451

4. Other NP-Complete Problems 457

Part 5 Semantics 465

16 Approximation Orderings 467

1. Programming Language Semantics 467

2. Partial Orders 472

3. Complete Partial Orders 475

4. Continuous Functions 486

5. Fixed Points 494

17 Denotational Semantics of Recursion Equations 505

1. Syntax 505

2. Semantics of Terms 511

3. Solutions to W-Programs 520

4. Denotational Semantics of W-Programs 530

5. Simple Data Structure Systems 539

6. Infinitary Data Structure Systems 544

18 Operational Semantics of Recursion Equations 557

1. Operational Semantics for Simple Data Structure Systems 557

2. Computable Functions 575

3. Operational Semantics for lnfinitary Data Structure Systems 584

Suggestions for Further Reading 593

Notation Index 595

Index 599

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