Introduction to the Theory of Computation PDF
Preview Introduction to the Theory of Computation
Intro duction to the Theory of C N o m p u t a t i o t E hird ditioN m i C h a E l s i p s E r Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Introduction to the Theory of © 2013 Cengage Learning Computation, Third International CENGAGE and CENGAGE LEARNING are registered trademarks of Edition Cengage Learning, Inc., within the United States and certain other Michael Sipser jurisdictions. Editor-in-Chief: Marie Lee ALL RIGHTS RESERVED. 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C O N T E N T S PrefacetotheFirstEdition xi Tothestudent . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Totheeducator . . . . . . . . . . . . . . . . . . . . . . . . . . xii Thefirstedition . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Feedbacktotheauthor . . . . . . . . . . . . . . . . . . . . . . xiii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . xiv PrefacetotheSecondEdition xvii PrefacetotheThirdEdition xxi 0 Introduction 1 0.1 Automata,Computability,andComplexity . . . . . . . . . . . . . 1 Complexitytheory . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computabilitytheory . . . . . . . . . . . . . . . . . . . . . . . 3 Automatatheory . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 MathematicalNotionsandTerminology . . . . . . . . . . . . . . 3 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Sequencesandtuples . . . . . . . . . . . . . . . . . . . . . . . 6 Functionsandrelations . . . . . . . . . . . . . . . . . . . . . . 7 Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Stringsandlanguages . . . . . . . . . . . . . . . . . . . . . . . 13 Booleanlogic. . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Summaryofmathematicalterms . . . . . . . . . . . . . . . . . 16 0.3 Definitions,Theorems,andProofs . . . . . . . . . . . . . . . . . 17 Findingproofs . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0.4 TypesofProof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Proofbyconstruction . . . . . . . . . . . . . . . . . . . . . . . 21 Proofbycontradiction. . . . . . . . . . . . . . . . . . . . . . . 21 Proofbyinduction . . . . . . . . . . . . . . . . . . . . . . . . . 22 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 25 v Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. vi CONTENTS Part One: Automata and Languages 29 1 RegularLanguages 31 1.1 FiniteAutomata . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Formaldefinitionofafiniteautomaton . . . . . . . . . . . . . 35 Examplesoffiniteautomata . . . . . . . . . . . . . . . . . . . . 37 Formaldefinitionofcomputation . . . . . . . . . . . . . . . . 40 Designingfiniteautomata . . . . . . . . . . . . . . . . . . . . . 41 Theregularoperations . . . . . . . . . . . . . . . . . . . . . . 44 1.2 Nondeterminism . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Formaldefinitionofanondeterministicfiniteautomaton. . . . 53 EquivalenceofNFAsandDFAs . . . . . . . . . . . . . . . . . 54 Closureundertheregularoperations. . . . . . . . . . . . . . . 58 1.3 RegularExpressions . . . . . . . . . . . . . . . . . . . . . . . . . 63 Formaldefinitionofaregularexpression . . . . . . . . . . . . 64 Equivalencewithfiniteautomata . . . . . . . . . . . . . . . . . 66 1.4 NonregularLanguages . . . . . . . . . . . . . . . . . . . . . . . . 77 Thepumpinglemmaforregularlanguages . . . . . . . . . . . 77 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 82 2 Context-FreeLanguages 101 2.1 Context-FreeGrammars . . . . . . . . . . . . . . . . . . . . . . . 102 Formaldefinitionofacontext-freegrammar . . . . . . . . . . 104 Examplesofcontext-freegrammars . . . . . . . . . . . . . . . 105 Designingcontext-freegrammars . . . . . . . . . . . . . . . . 106 Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Chomskynormalform . . . . . . . . . . . . . . . . . . . . . . 108 2.2 PushdownAutomata . . . . . . . . . . . . . . . . . . . . . . . . . 111 Formaldefinitionofapushdownautomaton. . . . . . . . . . . 113 Examplesofpushdownautomata . . . . . . . . . . . . . . . . . 114 Equivalencewithcontext-freegrammars . . . . . . . . . . . . . 117 2.3 Non-Context-FreeLanguages. . . . . . . . . . . . . . . . . . . . 125 Thepumpinglemmaforcontext-freelanguages. . . . . . . . . 125 2.4 DeterministicContext-FreeLanguages. . . . . . . . . . . . . . . 130 PropertiesofDCFLs . . . . . . . . . . . . . . . . . . . . . . . 133 Deterministiccontext-freegrammars . . . . . . . . . . . . . . 135 RelationshipofDPDAsandDCFGs . . . . . . . . . . . . . . . 146 ParsingandLR(k)Grammars . . . . . . . . . . . . . . . . . . . 151 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 154 Part Two: Computability Theory 163 3 TheChurch–TuringThesis 165 3.1 TuringMachines . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 FormaldefinitionofaTuringmachine . . . . . . . . . . . . . . 167 Copyright 2013 Cengage Learning. 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CONTENTS vii ExamplesofTuringmachines . . . . . . . . . . . . . . . . . . . 170 3.2 VariantsofTuringMachines . . . . . . . . . . . . . . . . . . . . . 176 MultitapeTuringmachines . . . . . . . . . . . . . . . . . . . . 176 NondeterministicTuringmachines . . . . . . . . . . . . . . . . 178 Enumerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Equivalencewithothermodels . . . . . . . . . . . . . . . . . . 181 3.3 TheDefinitionofAlgorithm . . . . . . . . . . . . . . . . . . . . 182 Hilbert’sproblems . . . . . . . . . . . . . . . . . . . . . . . . . 182 TerminologyfordescribingTuringmachines . . . . . . . . . . 184 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 187 4 Decidability 193 4.1 DecidableLanguages. . . . . . . . . . . . . . . . . . . . . . . . . 194 Decidableproblemsconcerningregularlanguages . . . . . . . 194 Decidableproblemsconcerningcontext-freelanguages. . . . . 198 4.2 Undecidability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 Thediagonalizationmethod . . . . . . . . . . . . . . . . . . . 202 Anundecidablelanguage . . . . . . . . . . . . . . . . . . . . . 207 ATuring-unrecognizablelanguage . . . . . . . . . . . . . . . . 209 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 210 5 Reducibility 215 5.1 UndecidableProblemsfromLanguageTheory . . . . . . . . . . 216 Reductionsviacomputationhistories. . . . . . . . . . . . . . . 220 5.2 ASimpleUndecidableProblem . . . . . . . . . . . . . . . . . . . 227 5.3 MappingReducibility . . . . . . . . . . . . . . . . . . . . . . . . 234 Computablefunctions . . . . . . . . . . . . . . . . . . . . . . . 234 Formaldefinitionofmappingreducibility . . . . . . . . . . . . 235 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 239 6 AdvancedTopicsinComputabilityTheory 245 6.1 TheRecursionTheorem. . . . . . . . . . . . . . . . . . . . . . . 245 Self-reference . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 Terminologyfortherecursiontheorem . . . . . . . . . . . . . 249 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 6.2 Decidabilityoflogicaltheories . . . . . . . . . . . . . . . . . . . 252 Adecidabletheory . . . . . . . . . . . . . . . . . . . . . . . . . 255 Anundecidabletheory. . . . . . . . . . . . . . . . . . . . . . . 257 6.3 TuringReducibility. . . . . . . . . . . . . . . . . . . . . . . . . . 260 6.4 ADefinitionofInformation . . . . . . . . . . . . . . . . . . . . . 261 Minimallengthdescriptions . . . . . . . . . . . . . . . . . . . 262 Optimalityofthedefinition . . . . . . . . . . . . . . . . . . . . 266 Incompressiblestringsandrandomness . . . . . . . . . . . . . 267 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 270 Copyright 2013 Cengage Learning. 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Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. viii CONTENTS Part Three: Complexity Theory 273 7 TimeComplexity 275 7.1 MeasuringComplexity . . . . . . . . . . . . . . . . . . . . . . . . 275 Big-Oandsmall-onotation . . . . . . . . . . . . . . . . . . . . 276 Analyzingalgorithms . . . . . . . . . . . . . . . . . . . . . . . 279 Complexityrelationshipsamongmodels . . . . . . . . . . . . . 282 7.2 TheClassP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Polynomialtime . . . . . . . . . . . . . . . . . . . . . . . . . . 284 ExamplesofproblemsinP . . . . . . . . . . . . . . . . . . . . 286 7.3 TheClassNP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 ExamplesofproblemsinNP . . . . . . . . . . . . . . . . . . . 295 ThePversusNPquestion . . . . . . . . . . . . . . . . . . . . 297 7.4 NP-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 Polynomialtimereducibility . . . . . . . . . . . . . . . . . . . 300 DefinitionofNP-completeness . . . . . . . . . . . . . . . . . . 304 TheCook–LevinTheorem . . . . . . . . . . . . . . . . . . . . 304 7.5 AdditionalNP-completeProblems . . . . . . . . . . . . . . . . . 311 Thevertexcoverproblem . . . . . . . . . . . . . . . . . . . . . 312 TheHamiltonianpathproblem . . . . . . . . . . . . . . . . . 314 Thesubsetsumproblem . . . . . . . . . . . . . . . . . . . . . 319 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 322 8 SpaceComplexity 331 8.1 Savitch’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.2 TheClassPSPACE . . . . . . . . . . . . . . . . . . . . . . . . . 336 8.3 PSPACE-completeness . . . . . . . . . . . . . . . . . . . . . . . 337 TheTQBFproblem . . . . . . . . . . . . . . . . . . . . . . . . 338 Winningstrategiesforgames . . . . . . . . . . . . . . . . . . . 341 Generalizedgeography . . . . . . . . . . . . . . . . . . . . . . 343 8.4 TheClassesLandNL . . . . . . . . . . . . . . . . . . . . . . . . 348 8.5 NL-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Searchingingraphs . . . . . . . . . . . . . . . . . . . . . . . . 353 8.6 NLequalscoNL . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 356 9 Intractability 363 9.1 HierarchyTheorems . . . . . . . . . . . . . . . . . . . . . . . . . 364 Exponentialspacecompleteness . . . . . . . . . . . . . . . . . 371 9.2 Relativization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 Limitsofthediagonalizationmethod . . . . . . . . . . . . . . 377 9.3 CircuitComplexity . . . . . . . . . . . . . . . . . . . . . . . . . . 379 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 388 10 AdvancedTopicsinComplexityTheory 393 10.1 ApproximationAlgorithms . . . . . . . . . . . . . . . . . . . . . 393 Copyright 2013 Cengage Learning. 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CONTENTS ix 10.2 ProbabilisticAlgorithms . . . . . . . . . . . . . . . . . . . . . . . 396 TheclassBPP . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Primality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Read-oncebranchingprograms . . . . . . . . . . . . . . . . . . 404 10.3 Alternation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 Alternatingtimeandspace . . . . . . . . . . . . . . . . . . . . 410 ThePolynomialtimehierarchy . . . . . . . . . . . . . . . . . . 414 10.4 InteractiveProofSystems . . . . . . . . . . . . . . . . . . . . . . 415 Graphnonisomorphism . . . . . . . . . . . . . . . . . . . . . . 415 Definitionofthemodel . . . . . . . . . . . . . . . . . . . . . . 416 IP=PSPACE . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 10.5 ParallelComputation . . . . . . . . . . . . . . . . . . . . . . . . 427 UniformBooleancircuits . . . . . . . . . . . . . . . . . . . . . 428 TheclassNC . . . . . . . . . . . . . . . . . . . . . . . . . . . 430 P-completeness . . . . . . . . . . . . . . . . . . . . . . . . . . 432 10.6 Cryptography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Secretkeys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 Public-keycryptosystems . . . . . . . . . . . . . . . . . . . . . 435 One-wayfunctions. . . . . . . . . . . . . . . . . . . . . . . . . 435 Trapdoorfunctions . . . . . . . . . . . . . . . . . . . . . . . . 437 Exercises,Problems,andSolutions . . . . . . . . . . . . . . . . . . . 439 SelectedBibliography 443 Index 448 Copyright 2013 Cengage Learning. 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