Table Of Content

WHATCANBECOMPUTED? (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) WHAT CAN BE COMPUTED? (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) A PRACTICAL GUIDE TO THE THEORY OF COMPUTATION (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) J M C OHN AC ORMICK PRINCETON UNIVERSITY PRESS • PRINCETON AND OXFORD (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) Copyright©2018byPrincetonUniversityPress PublishedbyPrincetonUniversityPress, 41WilliamStreet,Princeton,NewJersey08540 IntheUnitedKingdom:PrincetonUniversityPress, 6OxfordStreet,Woodstock,OxfordshireOX201TR press.princeton.edu Curved crease sculpture by Erik Demaine and Martin Demaine. Cover photo courtesy of the artists. AllRightsReserved ISBN978-0-691-17066-4 LibraryofCongressControlNumber2018935138 BritishLibraryCataloging-in-PublicationDataisavailable ThisbookhasbeencomposedinSabonandAvenir Printedonacid-freepaper∞ TypesetbyNovaTechsetPvtLtd,Bangalore,India PrintedintheUnitedStatesofAmerica 10987654321 CONTENTS (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) Acknowledgments xiii Prefaceforinstructors xv TheinspirationofGEB xv Which“theory”coursearewetalkingabout? xv Thefeaturesthatmightmakethisbookappealing xvi What’sinandwhat’sout xvii Possiblecoursesbasedonthisbook xvii Computerscienceasaliberalart xviii OVERVIEW 1 1 INTRODUCTION:WHATCANANDCANNOTBECOMPUTED? 3 1.1 Tractableproblems 4 1.2 Intractableproblems 5 1.3 Uncomputableproblems 6 1.4 Amoredetailedoverviewofthebook 6 OverviewofpartI:Computabilitytheory 6 OverviewofpartII:Complexitytheory 7 OverviewofpartIII:Originsandapplications 8 1.5 Prerequisitesforunderstandingthisbook 8 1.6 Thegoalsofthebook 9 Thefundamentalgoal:Whatcanbecomputed? 9 Secondarygoal1:Apracticalapproach 9 Secondarygoal2:Somehistoricalinsight 10 1.7 Whystudythetheoryofcomputation? 10 Reason1:Thetheoryofcomputationisuseful 10 Reason2:Thetheoryofcomputationisbeautifulandimportant 11 Exercises 11 PartI: COMPUTABILITYTHEORY 13 2 WHATISACOMPUTERPROGRAM? 15 2.1 SomePythonprogrambasics 15 EditingandrerunningaPythonprogram 17 RunningaPythonprogramoninputfromafile 17 RunningmorecomplexexperimentsonPythonprograms 18 2.2 SISOPythonprograms 18 Programsthatcallotherfunctionsandprograms 20 2.3 ASCIIcharactersandmultilinestrings 21 2.4 Someproblematicprograms 22 vi (cid:129) Contents 2.5 FormaldefinitionofPythonprogram 23 2.6 Decisionprogramsandequivalentprograms 25 2.7 Real-worldprogramsversusSISOPythonprograms 26 Exercises 27 3 SOMEIMPOSSIBLEPYTHONPROGRAMS 30 3.1 Proofbycontradiction 30 3.2 Programsthatanalyzeotherprograms 31 Programsthatanalyzethemselves 33 3.3 TheprogramyesOnString.py 33 3.4 TheprogramyesOnSelf.py 34 3.5 TheprogramnotYesOnSelf.py 36 3.6 yesOnString.pycan’texisteither 37 AcompactproofthatyesOnString.pycan’texist 37 3.7 Perfectbug-findingprogramsareimpossible 39 3.8 Wecanstillfindbugs,butwecan’tdoitperfectly 41 Exercises 42 4 WHATISACOMPUTATIONALPROBLEM? 45 4.1 Graphs,alphabets,strings,andlanguages 46 Graphs 46 Treesandrootedtrees 49 Alphabets 49 Strings 50 Languages 51 4.2 Definingcomputationalproblems 53 Positiveandnegativeinstances 55 Notationforcomputationalproblems 56 4.3 Categoriesofcomputationalproblems 57 Searchproblems 57 Optimizationproblems 57 Thresholdproblems 57 Functionproblems 58 Decisionproblems 58 Convertingbetweengeneralanddecisionproblems 59 Complementofadecisionproblem 60 Computationalproblemswithtwoinputstrings 61 4.4 Theformaldefinitionof“solving”aproblem 62 Computablefunctions 63 4.5 Recognizinganddecidinglanguages 63 Recognizablelanguages 65 Recursiveandrecursivelyenumerablelanguages 66 Exercises 66 5 TURINGMACHINES:THESIMPLESTCOMPUTERS 71 5.1 DefinitionofaTuringmachine 72 Haltingandlooping 76 Acceptersandtransducers 77 Contents (cid:129) vii Abbreviatednotationforstatediagrams 78 CreatingyourownTuringmachines 78 5.2 SomenontrivialTuringmachines 79 ThemoreCsThanGsmachine 80 ThecountCsmachine 81 ImportantlessonsfromthecountCsexample 85 5.3 Fromsingle-tapeTuringmachinestomulti-tapeTuringmachines 86 Two-tape,single-headTuringmachines 87 Two-wayinfinitetapes 88 Multi-tape,single-headTuringmachines 89 Two-tape,two-headTuringmachines 90 5.4 Frommulti-tapeTuringmachinestoPythonprogramsandbeyond 91 Multi-tapeTuringmachine→random-accessTuringmachine 92 Random-accessTuringmachine→realcomputer 92 Moderncomputer→Pythonprogram 95 5.5 Goingbacktheotherway:SimulatingaTuringmachine withPython 95 Aseriouscaveat:Memorylimitationsandothertechnicalities 97 5.6 Classicalcomputerscansimulatequantumcomputers 98 5.7 AllknowncomputersareTuringequivalent 98 Exercises 99 6 UNIVERSALCOMPUTERPROGRAMS:PROGRAMSTHATCANDOANYTHING 103 6.1 UniversalPythonprograms 104 6.2 UniversalTuringmachines 105 6.3 Universalcomputationintherealworld 107 6.4 Programsthatalterotherprograms 110 Ignoringtheinputandperformingafixedcalculationinstead 112 6.5 Problemsthatareundecidablebutrecognizable 113 Exercises 114 7 REDUCTIONS:HOWTOPROVEAPROBLEMISHARD 116 7.1 Areductionforeasiness 116 7.2 Areductionforhardness 118 7.3 FormaldefinitionofTuringreduction 120 Why“Turing”reduction? 120 Oracleprograms 121 Whyis≤TusedtodenoteaTuringreduction? 121 Bewarethetruemeaningof“reduction” 121 7.4 PropertiesofTuringreductions 122 7.5 Anabundanceofuncomputableproblems 123 ThevariantsofYESONSTRING 123 Thehaltingproblemanditsvariants 126 Uncomputableproblemsthataren’tdecisionproblems 128 7.6 Evenmoreuncomputableproblems 130 ThecomputationalproblemCOMPUTESF 132 Rice’stheorem 134 7.7 Uncomputableproblemsthataren’taboutprograms 134 viii (cid:129) Contents 7.8 Noteveryquestionaboutprogramsisuncomputable 135 7.9 Prooftechniquesforuncomputability 136 Technique1:Thereductionrecipe 137 Technique2:ReductionwithexplicitPythonprograms 138 Technique3:ApplyRice’stheorem 139 Exercises 140 8 NONDETERMINISM:MAGICORREALITY? 143 8.1 NondeterministicPythonprograms 144 8.2 Nondeterministicprogramsfornondecisionproblems 148 8.3 Computationtrees 149 8.4 Nondeterminismdoesn’tchangewhatiscomputable 153 8.5 NondeterministicTuringmachines 154 8.6 FormaldefinitionofnondeterministicTuringmachines 156 8.7 Modelsofnondeterminism 158 8.8 Unrecognizableproblems 158 8.9 Whystudynondeterminism? 159 Exercises 160 9 FINITEAUTOMATA:COMPUTINGWITHLIMITEDRESOURCES 164 9.1 Deterministicfiniteautomata 164 9.2 Nondeterministicfiniteautomata 167 Statediagramsfornfas 168 Formaldefinitionofannfa 169 Howdoesannfaacceptastring? 170 Sometimesnfasmakethingseasier 170 9.3 Equivalenceofnfasanddfas 170 Nondeterminismcanaffectcomputability:Theexampleofpdas 173 Practicalityofconvertednfas 174 Minimizingthesizeofdfas 175 9.4 Regularexpressions 175 Pureregularexpressions 176 Standardregularexpressions 177 Convertingbetweenregexesandfiniteautomata 178 9.5 Somelanguagesaren’tregular 181 ThenonregularlanguageGnTn 181 ThekeydifferencebetweenTuringmachinesandfiniteautomata 183 9.6 Manymorenonregularlanguages 183 Thepumpinglemma 185 9.7 Combiningregularlanguages 187 Exercises 188 PartII: COMPUTATIONALCOMPLEXITYTHEORY 193 10 COMPLEXITYTHEORY:WHENEFFICIENCYDOESMATTER 195 10.1 Complexitytheoryusesasymptoticrunningtimes 195 10.2 Big-Onotation 197 Dominanttermsoffunctions 199 Apracticaldefinitionofbig-Onotation 201 Contents (cid:129) ix Superpolynomialandsubexponential 202 Otherasymptoticnotation 202 Compositionofpolynomialsispolynomial 203 Countingthingswithbig-O 203 10.3 Therunningtimeofaprogram 204 RunningtimeofaTuringmachine 204 RunningtimeofaPythonprogram 206 ThelackofrigorinPythonrunningtimes 209 10.4 Fundamentalsofdeterminingtimecomplexity 210 Acrucialdistinction:Thelengthoftheinputversusthenumerical valueoftheinput 210 Thecomplexityofarithmeticoperations 212 Bewareofconstant-timearithmeticoperations 214 Thecomplexityoffactoring 215 Theimportanceofthehardnessoffactoring 216 Thecomplexityofsorting 217 10.5 Forcomplexity,thecomputationalmodeldoesmatter 217 Simulationcostsforcommoncomputationalmodels 217 Multi-tapesimulationhasquadraticcost 218 Random-accesssimulationhascubiccost 219 Universalsimulationhaslogarithmiccost 219 Realcomputerscostonlyaconstantfactor 220 Pythonprogramscostthesameasrealcomputers 220 Pythonprogramscansimulaterandom-accessTuring machinesefficiently 220 Quantumsimulationmayhaveexponentialcost 220 Allclassicalcomputationalmodelsdifferbyonly polynomialfactors 221 Ourstandardcomputationalmodel:Pythonprograms 221 10.6 Complexityclasses 221 Exercises 224 11 PolyANDExpo:THETWOMOSTFUNDAMENTALCOMPLEXITYCLASSES 228 11.1 DefinitionsofPolyandExpo 228 PolyandExpocomparedtoP,Exp,andFP 229 11.2 PolyisasubsetofExpo 230 11.3 AfirstlookattheboundarybetweenPolyandExpo 231 ALL3SETSandALLSUBSETS 231 Travelingsalespeopleandshortestpaths 232 Multiplyingandfactoring 235 BacktotheboundarybetweenPolyandExpo 235 PrimalitytestingisinPoly 237 11.4 PolyandExpodon’tcareaboutthecomputationalmodel 238 11.5 HALTEX:AdecisionprobleminExpobutnotPoly 238 11.6 OtherproblemsthatareoutsidePoly 243 11.7 Unreasonableencodingsoftheinputaffectcomplexity 244 11.8 WhystudyPoly,really? 245 Exercises 246

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405 Pages·2018·4.874 MB·English
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#Computers #Algorithms and Data Structures
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