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Introduction to the Theory of Computation,9781133187790
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Introduction to the Theory of Computation



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Cengage Learning
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  • Introduction to the Theory of Computation
    Introduction to the Theory of Computation
  • Introduction To The Theory Of Computation
    Introduction To The Theory Of Computation


Gain a clear understanding of even the most complex, highly theoretical computational theory topics in the approachable presentation found only in the market-leading INTRODUCTION TO THE THEORY OF COMPUTATION, 3E. The number one choice for today's computational theory course, this revision continues the book's well-know, approachable style with timely revisions, additional practice, and more memorable examples in key areas. A new first-of-its-kind theoretical treatment of deterministic context-free languages is ideal for a better understanding of parsing and LR(k) grammars. You gain a solid understanding of the fundamental mathematical properties of computer hardware, software, and applications with a blend of practical and philosophical coverage and mathematical treatments, including advanced theorems and proofs. INTRODUCTION TO THE THEORY OF COMPUTATION, 3E's comprehensive coverage makes this a valuable reference for your continued studies in theoretical computing.

Author Biography

Michael Sipser has taught theoretical computer science and other mathematical subjects at the Massachusetts Institute of Technology for the past 32 years. He is the Head of the Mathematics Department and a member of the Computer Science and Artificial Intelligence Laboratory (CSAIL). He enjoys teaching and pondering the many mysteries of complexity theory.

Table of Contents

Preface to the First Editionp. xi
To the studentp. xi
To the educatorp. xii
The first editionp. xiii
Feedback to the authorp. xiii
Acknowledgmentsp. xiv
Preface to the Second Editionp. xvii
Preface to the Third Editionp. xxi
Introductionp. 1
Automata, Computability, and Complexityp. 1
Complexity theoryp. 2
Computability theoryp. 3
Automata theoryp. 3
Mathematical Notions and Terminologyp. 3
Setsp. 3
Sequences and tuplesp. 6
Functions and relationsp. 7
Graphsp. 10
Strings and languagesp. 13
Boolean logicp. 14
Summary of mathematical termsp. 16
Definitions, Theorems, and Proofsp. 17
Finding proofsp. 17
Types of Proofp. 21
Proof by constructionp. 21
Proof by contradictionp. 21
Proof by inductionp. 22
Exercises, Problems, and Solutionsp. 25
Automata and Languagesp. 29
Regular Languagesp. 31
Finite Automatap. 31
Formal definition of a finite automatonp. 35
Examples of finite automatap. 37
Formal definition of computationp. 40
Designing finite automatap. 41
The regular operationsp. 44
Nondeterminismp. 47
Formal definition of a nondeterministic finite automatonp. 53
Equivalence of NFAs and DFAsp. 54
Closure under the regular operationsp. 58
Regular Expressionsp. 63
Formal definition of a regular expressionp. 64
Equivalence with finite automatap. 66
Nonregular Languagesp. 77
The pumping lemma for regular languagesp. 77
Exercises, Problems, and Solutionsp. 82
Context-Free Languagesp. 101
Context-Free Grammarsp. 102
Formal definition of a context-free grammarp. 104
Examples of context-free grammarsp. 105
Designing context-free grammarsp. 106
Ambiguityp. 107
Chomsky normal formp. 108
Pushdown Automatap. 111
Formal definition of a pushdown automatonp. 113
Examples of pushdown automatap. 114
Equivalence with context-free grammarsp. 117
Non-Context-Free Languagesp. 125
The pumping lemma for context-free languagesp. 125
Deterministic Context-Free Languagesp. 130
Properties of DCFLsp. 133
Deterministic context-free grammarsp. 135
Relationship of DPDAs and DCFGsp. 146
Parsing and LR(k) Grammarsp. 151
Exercises, Problems, and Solutionsp. 154
Computability Theoryp. 163
The Church-Turing Thesisp. 165
Turing Machinesp. 165
Formal definition of a Turing machinep. 167
Examples of Turing machinesp. 170
Variants of Turing Machinesp. 176
Multitape Turing machinesp. 176
Nondeterministic Turing machinesp. 178
Enumeratorsp. 180
Equivalence with other modelsp. 181
The Definition of Algorithmp. 182
Hilbert's problemsp. 182
Terminology for describing Turing machinesp. 184
Exercises, Problems, and Solutionsp. 187
Decidabilityp. 193
Decidable Languagesp. 194
Decidable problems concerning regular languagesp. 194
Decidable problems concerning context-free languagesp. 198
Undecidabilityp. 201
The diagonalization methodp. 202
An undecidable languagep. 207
A Turing-unrecognizable languagep. 209
Exercises, Problems, and Solutionsp. 210
Reducibilityp. 215
Undecidable Problems from Language Theoryp. 216
Reductions via computation historiesp. 220
A Simple Undecidable Problemp. 227
Mapping Reducibilityp. 234
Computable functionsp. 234
Formal definition of mapping reducibilityp. 235
Exercises, Problems, and Solutionsp. 239
Advanced Topics in Computability Theoryp. 245
The Recursion Theoremp. 245
Self-referencep. 246
Terminology for the recursion theoremp. 249
Applicationsp. 250
Decidability of logical theoriesp. 252
A decidable theoryp. 255
An undecidable theoryp. 257
Turing Reducibilityp. 260
A Definition of Informationp. 261
Minimal length descriptionsp. 262
Optimality of the definitionp. 266
Incompressible strings and randomnessp. 267
Exercises, Problems, and Solutionsp. 270
Complexity Theoryp. 273
Time Complexityp. 275
Measuring Complexityp. 275
Big-O and small-o notationp. 276
Analyzing algorithmsp. 279
Complexity relationships among modelsp. 282
The Class Pp. 284
Polynomial timep. 284
Examples of problems in Pp. 286
The Class NPp. 292
Examples of problems in NPp. 295
The P versus NP questionp. 297
NP-completenessp. 299
Polynomial time reducibilityp. 300
Definition of NP-completenessp. 304
The Cook-Levin Theoremp. 304
Additional NP-complete Problemsp. 311
The vertex cover problemp. 312
The Hamiltonian path problemp. 314
The subset sum problemp. 319
Exercises, Problems, arid Solutionsp. 322
Space Complexityp. 331
SavitchÆs Theoremp. 333
The Class PSPACEp. 336
PSPACE-completenessp. 337
The TQBF problemp. 338
Winning strategies for gamesp. 341
Generalized geographyp. 343
The Classes L and NLp. 348
NL-completenessp. 351
Searching in graphsp. 353
NL equals coNLp. 354
Exercises, Problems, and Solutionsp. 356
Intractabilityp. 363
Hierarchy Theoremsp. 364
Exponential space completenessp. 371
Relativizationp. 376
Limits of the diagonalization methodp. 377
Circuit Complexityp. 379
Exercises, Problems, and Solutionsp. 388
Advanced Topics in Complexity Theoryp. 393
Approximation Algorithmsp. 393
Probabilistic Algorithmsp. 395
The class BPPp. 395
Primalityp. 399
Read-once branching programsp. 404
Alternationp. 408
Alternating time and spacep. 410
The Polynomial time hierarchyp. 414
Interactive Proof Systemsp. 415
Graph nonisomorphismp. 415
Definition of the modelp. 415
IP = PSPACEp. 418
Parallel Computationp. 427
Uniform Boolean circuitsp. 428
The class NCp. 430
P-completenessp. 432
Cryptographyp. 433
Secret keysp. 433
Public-key cryptosystemsp. 435
One-way functionsp. 435
Trapdoor functionsp. 437
Exercises, Problems, and Solutionsp. 439
Selected Bibliographyp. 443
Indexp. 448
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