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PART Ⅰ:INTRODUCTION 1

1 Why Study the Theory of Computation? 2

1.1 The Shelf Life of Programming Tools 2

1.2 Applications of the Theory Are Everywhere 5

2 Languages and Strings 8

2.1 Strings 8

2.2 Languages 10

Exercises 19

3 The Big Picture:A Language Hierarchy 21

3.1 Defining the Task:Language Recognition 21

3.2 The Power of Encoding 22

3.3 A Machine-Based Hierarchy of Language Classes 28

3.4 A Tractability Hierarchy of Language Classes 34

Exercises 34

4 Computation 36

4.1 Decision Procedures 36

4.2 Determinism and Nondeterminism 41

4.3 Functions on Languages and Programs 48

Exercises 52

PART Ⅱ:FINITE STATE MACHINES AND REGULAR LANGUAGES 53

5 Finite State Machines 54

5.1 Deterministic Finite State Machines 56

5.2 The Regular Languages 60

5.3 Designing Deterministic Finite State Machines 63

5.4 Nondeterministic FSMs 66

5.5 From FSMs to Operational Systems 79

5.6 Simulators for FSMs 80

5.7 Minimizing FSMs 82

5.8 A Canonical Form for Regular Languages 94

5.9 Finite State Transducers 96

5.10 Bidirectional Transducers 98

5.11 Stochastic Finite Automata:Markov Models and HMMs 101

5.12 Finite Automata,Infinite Strings:Büchi Automata 115

Exercises 121

6 Regular Expressions 127

6.1 What is a Regular Expression? 128

6.2 Kleene's Theorem 133

6.3 Applications of Regular Expressions 147

6.4 Manipulating and Simplifying Regular Expressions 149

Exercises 151

7 Regular Grammars 155

7.1 Definition of a Regular Grammar 155

7.2 Regular Grammars and Regular Languages 157

Exercises 161

8 Regular and Nonregular Languages 162

8.1 How Many Regular Languages Are There? 162

8.2 Showing That a Language Is Regular 163

8.3 Some Important Closure Properties of Regular Languages 165

8.4 Showing That a Language is Not Regular 169

8.5 Exploiting Problem-Specific Knowledge 178

8.6 Functions on Regular Languages 179

Exercises 182

9 Algorithms and Decision Procedures for Regular Languages 187

9.1 Fundamental Decision Procedures 187

9.2 Summary of Algorithms and Decision Procedures for Regular Languages 194

Exercises 196

10 Summary and References 198

References 199

PART Ⅲ:CONTEXT-FREE LANGUAGES AND PUSHDOWN AUTOMATA 201

11 Context-Free Grammars 203

11.1 Introduction to Rewrite Systems and Grammars 203

11.2 Context-Free Grammars and Languages 207

11.3 Designing Context-Free Grammars 212

11.4 Simplifying Context-Free Grammars 212

11.5 Proving That a Grammar is Correct 215

11.6 Derivations and Parse Trees 218

11.7 Ambiguity 220

11.8 Normal Forms 232

11.9 Island Grammars 241

11.10 Stochastic Context-Free Grammars 243

Exercises 245

12 Pushdown Automata 249

12.1 Definition of a(Nondeterministic)PDA 249

12.2 Deterministic and Nondeterministic PDAs 254

12.3 Equivalence of Context-Free Grammars and PDAs 260

12.4 Nondeterminism and Halting 274

12.5 Alternative Equivalent Definitions of a PDA 275

12.6 Alternatives that are Not Equivalent to the PDA 277

Exercises 277

13 Context-Free and Noncontext-Free Languages 279

13.1 Where Do the Context-Free Languages Fit in the Big Picture? 279

13.2 Showing That a Language is Context-Free 280

13.3 The Pumping Theorem for Context-Free Languages 281

13.4 Some Important Closure Properties of Context-Free Languages 288

13.5 Deterministic Context-Free Languages 295

13.6 Ogden's Lemma 303

13.7 Parikh's Theorem 306

13.8 Functions on Context-Free Languages 308

Exercises 310

14 Algorithms and Decision Procedures for Context-Free Languages 314

14.1 The Decidable Questions 314

14.2 The Undecidable Questions 320

14.3 Summary of Algorithms and Decision Procedures for Context-Free Languages 320

Exercises 322

15 Context-Free Parsing 323

15.1 Lexical Analysis 325

15.2 Top-Down Parsing 327

15.3 Bottom-Up Parsing 340

15.4 Parsing Natural Languages 350

Exercises 358

16 Summary and References 360

References 360

PART Ⅳ:TURING MACHINES AND UNDECIDABILITY 363

17 Turing Machines 364

17.1 Definition,Notation and Examples 364

17.2 Computing With Turing Machines 375

17.3 Adding Multiple Tapes and Nondeterminism 382

17.4 Simulating a"Real"Computer 393

17.5 Alternative Turing Machine Definitions 396

17.6 Encoding Turing Machines as Strings 400

17.7 The Universal Turing Machine 404

Exercises 407

18 The Church-Turing Thesis 411

18.1 The Thesis 411

18.2 Examples of Equivalent Formalisms 414

Exercises 424

19 The Unsolvability of the Halting Problem 426

19.1 The Language H is Semidecidable but Not Decidable 428

19.2 Some Implications of the Undecidability of H 431

19.3 Back to Turing,Church,and the Entscheidungsproblem 432

Exercises 433

20 Decidable and Semidecidable Languages 435

20.1 D:The Big Picture 435

20.2 SD:The Big Picture 435

20.3 Subset Relationships between D and SD 437

20.4 The Classes D and SD Under Complement 438

20.5 Enumerating a Language 440

20.6 Summary 444

Exercises 445

21 Decidability and Undecidability Proofs 448

21.1 Reduction 449

21.2 Using Reduction to Show that a Language is Not Decidable 452

21.3 Are All Questions About Turing Machines Undecidable? 466

2 1.4 Rice's Theorem 468

2 1.5 Undecidable Questions About Real Programs 472

2 1.6 Showing That a Language is Not Semidecidable 474

21.7 Summary of D,SD/D and ？SD Languages that Include Turing Machine Descriptions 482

Exercises 483

22 Decidability of Languages That Do Not(Obviously)Ask Questions about Turing Machines 487

22.1 Diophantine Equations and Hilbert's 10th Problem 488

22.2 Post Correspondence Problem 489

22.3 Tiling Problems 492

22.4 Logical Theories 495

22.5 Undecidable Problems about Context-Free Languages 499

Exercises 508

23 Unrestricted Grammars 510

23.1 Definition and Examples 510

23.2 Equivalence of Unrestricted Grammars and Turing Machines 516

23.3 Grammars Compute Functions 518

23.4 Undecidable Problems About Unrestricted Grammars 521

23.5 The Word Problem for Semi-Thue Systems 522

Exercises 524

24 The Chomsky Hierarchy and Beyond 526

24.1 The Context-Sensitive Languages 526

24.2 The Chomsky Hierarchy 539

24.3 Attribute,Feature,and Unification Grammars 540

24.4 Lindenmayer Systems 544

Exercises 553

25 Computable Functions 555

25.1 What is a Computable Function? 555

25.2 Recursive Function Theory 565

25.3 The Recursion Theorem and Its Use 573

Exercises 580

26 Summary and References 581

References 582

PART Ⅴ:COMPLEXITY 585

27 Introduction to the Analysis of Complexity 586

27.1 The Travelina Salesman Problem 586

27.2 The Complexity Zoo 589

27.3 Characterizing Problems 590

27.4 Measuring Time and Space Complexity 593

27.5 Growth Rates of Functions 597

27.6 Asymptotic Dominance 598

27.7 Algorithmic Gaps 604

27.8 Examples 605

Exercises 617

28 Time Complexity Classes 621

28.1 The Language Class P 621

28.2 The Language Class NP 633

28.3 Does P=NP? 642

28.4 Using Reduction in Complexity Proofs 644

28.5 NP-Completeness and the Cook-Levin Theorem 647

28.6 Other NP-Complete Probiems 656

28.7 The Relationship between P and NP-Complete 672

28.8 The Language Class Co-NP 679

28.9 The Time Hierarchy Theorems,EXPTIME,and Beyond 681

28.10 The Problem Classes FP and FNP 689

Exercises 690

29 Space Complexity Classes 695

29.1 Analyzing Space Complexity 695

29.2 PSPACE,NPSPACE,and Savitch's Theorem 699

29.3 PSPACE-Completeness 704

29.4 Sublinear Space Complexity 713

29.5 The Closure of Space Complexity Classes Under Complement 718

29.6 Space Hierarchy Theorems 719

Exercises 720

30 Practical Solutions for Hard Problems 721

30.1 Approaches 721

30.2 Randomized Algorithms and the Language Classes BPP,RP,Co-RP and ZPP 723

30.3 Heuristic Search 731

Exercises 740

31 Summary and References 742

References 742

APPENDICES 745

A Review of Mathematical Background:Logic,Sets,Relations,Functions,and Proof Techniques 745

A.1 Logic 745

A.2 Sets 753

A.3 Relations 756

A.4 Functions 769

A.5 Closures 776

A.6 Proof Techniques 779

A.7 Reasoning about Programs 792

A.8 A General Definition of Closure 800

Exercises 804

B The Theory:Working with Logical Formulas 808

B.1 Working with Boolean Formulas:Normal Forms,Resolution and OBDDs 808

B.2 Working with First-Order Formulas:Clause Form and Resolution 820

Exercises 833

C The Theory:Finite State Machines and Regular Languages 835

D The Theory:Context-Free Languages and PDAs 839

D.1 Proof of the Greibach Normal Form Theorem 839

D.2 Proof that the Deterministic Context-Free Languages are Closed Under Complement 846

D.3 Proof of Parikh's Theorem 852

E The Theory:Turing Machines and Undecidability 856

E.1 Proof that Nondeterminism Does Not Add Power to Turing Machines 856

E.2 An Analysis of Iterative Deepening 861

E.3 The Power of Reduction 862

E.4 The Undecidability of the Post Correspondence Problem 864

F The Theory:Complexity 869

F.1 Asymptotic Dominance 869

F.2 The Linear Speedup Theorem 875

APPENDICES G-Q:APPLICATIONS: 879

G Applications:Programming Languages and Compilers 880

G.1 Defining the Syntax of Programming Languages 880

G.2 Are Programming Languages Context-Free? 883

G.3 Designing Programming Languages and Their Grammars 885

G.4 Compilers for Programming Languages 887

G.5 Functional Programming and the Lambda Calculus 890

H Applications:Tools for Programming,Databases and Software Engineering 899

H.1 Proving Correctness Properties of Programs and Hardware 899

H.2 Statecharts:A Technique for Specifying Complex Systems 910

H.3 Model-Based Test Case Generation 913

H.4 Reverse Engineering 914

H.5 Normal Forms for Data and for Querying Relational Databases 916

I Applications:Networks 918

1.1 Network Protocols 918

1.2 Modeling Networks as Graphs 927

1.3 Exploiting Knowledge:The Semantic Web 929

J Applications:Security 948

J.1 Physical Security Systems as FSMs 948

J.2 Computer System Safety 949

J.3 Cryptography 955

J.4 Hackers and Viruses 959

K Applications:Computational Biology 962

K.1 A(Very)Short Introduction to Molecular Biology and Genetics 962

K.2 The Sequence Matching Problem 968

K.3 DNA and Protein Sequence Matching Using the Tools of Regular Languages 970

K.4 RNA Sequence Matching and Secondary Structure Prediction Using the Tools of Context-Free Languages 975

K.5 Complexity of the Algorithms Used in Computational Biology 977

L Applications:Natural Language Processing 978

L.1 Morphological Analysis 978

L.2 Part of Speech Tagging 981

L.3 The Grammar of English 983

L.4 Building a Complete NL System 998

L.5 Speech Understanding Systems 999

M Applications:Artificial Intelligence and Computational Reasoning 1004

M.1 The Role of Search 1006

M.2 A Logical Foundation for Artificial Intelligence 1007

M.3 A Rule-Based Foundation for Artificial Intelligence and Cognition 1022

N Applications:Art and Entertainment:Music and Games 1028

N.1 Music 1028

N.2 Classic Games and Puzzles 1034

N.3 Interactive Video Games 1046

O Applications:Using Regular Expressions 1050

P Applications:Using Finite State Machines and Transducers 1054

P.1 Finite State Machines Predate Computers 1054

P.2 The Towers of Hanoi:Regular Doesn't Always Mean Tractable 1058

P.3 The Arithmetic Logic Unit(ALU) 1060

P.4 Controlling a Soccer-Playing Robot 1062

Q Applications:Using G rammars 1065

Q.1 Describing Artificial Languages Designed for Person/Machine Interaction 1066

Q.2 Describing Naturally Occurring Phenomena 1071

References 1073