Understanding Computation: Pillars, Paradigms, Principles

577 Pages·2022·8.787 MB·English

by  Arnold L. Rosenberg, Lenwood S. Heath

Texts in Computer Science
Arnold L. Rosenberg
Lenwood S. Heath
Understanding
Computation
Pillars, Paradigms, Principles
Texts in Computer Science
Series Editors
David Gries, Department of Computer Science, Cornell University, Ithaca, NY,
USA
OritHazzan ,FacultyofEducationinTechnologyandScience,Technion—Israel
Institute of Technology, Haifa, Israel
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Arnold L. Rosenberg Lenwood S. Heath
(cid:129)
Understanding Computation
Pillars, Paradigms, Principles
123
ArnoldL. Rosenberg LenwoodS. Heath
University of Massachusetts Virginia Tech
Amherst, MA, USA Blacksburg, VA, USA
ISSN 1868-0941 ISSN 1868-095X (electronic)
Textsin Computer Science
ISBN978-3-031-10054-3 ISBN978-3-031-10055-0 (eBook)
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Contents
PREFACE......................................................... xv
I: INTRODUCTION............................................... 1
1 IntroducingComputationTheory................................ 3
1.1 TheAutobiographical(ALR)SeedsofOurFramework........... 4
1.1.1 Computationby“Shapeless”AgentsandDevices ......... 4
1.1.2 ComputationTheoryasaStudyofComputation .......... 6
1.2 TheHighlightsofOurFramework:HowWeTelltheStory........ 7
1.3 WhyIsaNewComputationTheoryTextNeeded? ............... 10
2 IntroducingtheBook........................................... 15
2.1 ComputationTheoryasaBranchofDiscreteMathematics ........ 17
2.1.1 DynamismWithinTraditionalMathematics .............. 17
2.1.2 DiscreteMathematicswithComputationalObjects ........ 17
2.2 TheFourPillarsofComputationTheory....................... 20
2.2.1 PillarS:STATE ..................................... 21
2.2.2 PillarE:ENCODING ................................. 22
2.2.3 PillarN:NONDETERMINISM.......................... 26
2.2.4 PillarP:PRESENTATION/SPECIFICATION............... 29
2.2.5 SummingUp........................................ 30
2.3 AMapoftheBookbyChapter ............................... 32
2.4 WaysofUsingthisBook .................................... 36
2.4.1 AsaTextfora“Classical”TheoryCourse ............... 36
2.4.1.1 AutomataandFormalLanguages............... 37
2.4.1.2 ComputabilityTheory ........................ 37
2.4.1.3 ComplexityTheory .......................... 38
2.4.2 AsaPrimaryText:“BigIdeasinComputation”........... 39
2.4.3 AsaSupplementalText:“TheoreticalAspectsof–”........ 42
2.4.3.1 PervasiveTopics:EnhanceManyCourses........ 42
2.4.3.2 FocusedTopics:EnhanceSpecializedCourses.... 44
vii
viii Contents
2.5 ToolsforUsingtheBook .................................... 46
II: PillarS:STATE ................................................ 47
3 PureState-BasedComputationalModels ......................... 51
3.1 OnlineAutomataandTheirLanguages ........................ 51
3.1.1 BasicsoftheOAModel............................... 51
3.1.2 PreparingtoUnderstandtheNotionState ................ 57
3.1.3 AMyhill-Nerode-likeTheoremforOAs ................. 62
3.2 FiniteAutomataandRegularLanguages ....................... 64
3.2.1 OverviewandHistory ................................ 64
3.2.2 PerspectivesonFiniteAutomata........................ 67
3.2.2.1 TheFAasanAbstractModel .................. 68
3.2.2.2 OnDesigningFAsforSpecificTasks............ 70
3.2.3 WhyFAsGetConfused:aConsequenceofFiniteness...... 75
4 TheMyhill-NerodeTheorem:ImplicationsandApplications........ 79
4.1 TheMyhill-NerodeTheoremforFAs.......................... 80
4.1.1 TheTheorem:StatesAreEquivalenceClasses ............ 81
4.1.2 WhatDo≡ -EquivalenceClassesLookLike?............ 83
L
4.1.2.1 Language#1:L = {ai|i≡0mod3}.......... 83
1
4.1.2.2 Language#2:L ={x1111y|x;y∈{0;1}?} ..... 85
2
4.1.2.3 Language#3:L ={anbn|n∈N}.............. 86
3
4.1.2.4 TheLanguageof(Binary)Palindromes.......... 87
4.2 SampleApplicationsoftheMyhill-NerodeTheorem ............. 88
4.2.1 ProvingthatLanguagesAreNotRegular ................ 88
4.2.2 OnMinimizingFiniteAutomata........................ 91
4.2.2.1 TheFAState-MinimizationAlgorithm .......... 92
4.2.3 Two-Way(Offline)FiniteAutomata..................... 96
4.2.4 ⊕FiniteAutomatawithProbabilisticTransitions..........101
4.2.4.1 PFAsandTheirLanguages ....................101
4.2.4.2 PFALanguagesandRegularLanguages .........103
A.PFAlanguageswhicharenotRegular ........103
B.⊕⊕PFAlanguageswhichareRegular ........109
4.2.5 ⊕StateasaMemory-ConstrainingResource .............112
4.2.5.1 ApproximatingOAsbyGrowingFAs:Examples..114
4.2.5.2 ApproximatingOAsbyGrowingFAs:General
Case.......................................115
5 OnlineTuringMachinesandtheImplicationsofOnlineComputing..119
5.1 OnlineTuringMachines:RealizationsofInfiniteOAs ............120
5.1.1 OTMswithAbstractStorageDevices ...................120
5.1.2 OTMswithLinearTapesforStorage....................122
5.2 TheNatureofOnlineComputing .............................129
5.2.1 OnlineTMswithMultipleComplexTapes ...............130
5.2.2 AnInformation-RetrievalProblemasaLanguage .........132
Contents ix
5.2.3 TheImpactofTapeStructureonMemoryLocality ........134
5.2.4 TapeDimensionalityandtheTimeComplexityofL .....135
DB
6 Pumping:ComputationalPigeonholesinFinitarySystems..........139
6.1 IntroductionandSynopsis ...................................139
6.2 PumpinginAlgebraicSettings ...............................140
6.3 PumpinginRegularLanguages...............................142
6.3.1 PumpinginGeneralRegularLanguages .................143
6.3.2 PumpinginRegularTallyLanguages ...................147
6.3.2.1 ACharacterizationoftheRegularTallyLanguages147
6.3.2.2 ⊕APurelyCombinatorialCharacterization ......150
6.4 PumpinginaRoboticSetting:aMobileFAonaMesh ...........150
6.4.1 TheMeshM andItsSubdivisions.....................150
n
6.4.2 TheMobileFiniteAutomaton(MFA)Model .............153
6.4.3 AnInherentLimitationonSoloAutonomousMFAs .......153
7 MobilityinComputing:AnFANavigatesaMesh ..................157
7.1 IntroductionandSynopsis ...................................157
7.2 MFAsThatComputeinRead-OnlyMode ......................160
7.2.1 HowanMFACanExploittheStructureofItsHostMesh...160
7.2.1.1 “BouncingOffWalls”toExploitSymmetries.....160
7.2.1.2 ApproximatingRational-SlopePathsWithinM .161
n
7.2.2 SoloMFAsScalablyRecognizeNon-RegularLanguages...163
7.2.2.1 MatchingForwardandBackwardPatterns .......164
7.2.2.2 MatchingDistinctForwardPatterns.............166
7.3 MFAsasObject-TransportingRobots..........................169
7.3.1 ReversingM’sTop-RowPatterntoItsBottomRow.......170
7.3.2 ⊕RotatingM’sTop-RowPatterntoItsBottomRow......171
7.3.3 AlgorithmsthatCircumnavigateM’s“Walls” ............173
7.3.3.1 Sorting-BasedRearrangements.................176
7.3.3.2 ⊕Pattern-Checkingvs.PatternGeneration.......179
8 ThePowerofCooperation:TeamsofMFAsonaMesh .............183
8.1 BasicsofMFATeams.......................................183
8.2 StrictlyCoupledMFAs:ParallelismwithNoAddedPower........186
8.3 SynchronizedComputing:A2-MFARecognizes{akbk}..........188
8.4 QueuedComputing:MFAsProceedThroughaPipeline ..........192
8.4.1 The“Standard”FormofPipelining .....................192
8.4.2 StreamingaPipeline/QueueofAgents ..................194
8.5 UsheredComputing:TwoMFAsSweepaMesh-Quadrant ........196
8.6 Sentry-EnabledComputing:MFAsIdentifyHomeWedges........198
III: PillarE:ENCODING ............................................201
x Contents
9 CountabilityandUncountability:ThePrecursorsofENCODING....203
9.1 EncodingFunctionsandProofsofCountability .................206
9.2 Diagonalization:ProofsofUncountability......................212
9.3 WhereHas(Un)CountabilityLedUs? .........................214
10 ComputabilityTheory ..........................................217
10.1 IntroductionandHistory.....................................217
10.1.1 FormalizingMathematicalReasoning ...................217
10.1.2 AbstractComputationalModels:theChurch-TuringThesis .218
10.1.3 ThinkingAboutThinkingAboutComputation............221
10.2 Preliminaries ..............................................223
10.2.1 ComputationalProblemsasFormalLanguages ...........223
10.2.2 FunctionsandPartialFunctions ........................225
10.2.3 Self-ReferentialPrograms:InterpretersandCompilers .....227
10.3 TheHaltingProblem:The“Oldest”UnsolvableProblem .........228
10.3.1 TheHaltingProblemIsSemisolvablebutNotSolvable.....229
10.3.2 WhyWeCareAbouttheHaltingProblem—AnExample...232
10.4 MappingReducibility.......................................234
10.4.1 BasicPropertiesofm-Reducibility......................236
10.4.2 Thes-m-nTheorem:AnInvaluableSourceofEncodings ...238
10.5 TheRice-Myhill-ShapiroTheorem............................240
10.6 Complete—or“Hardest”—SemidecidableProblems .............245
10.7 SomeImportantLimitationsofComputability ..................248
11 AChurch-TuringZooofComputationalModels...................251
11.1 TheChurch-TuringThesisandUniversalModels................251
11.2 UniversalityinSimplifiedOTMs..............................253
11.2.1 AnOTMwithaOne-EndedWorktape...................253
11.2.2 AnOTMwithTwoStacksInsteadofaWorktape ..........256
11.2.3 AnOTMwithaFIFOQueueInsteadofaWorktape .......260
11.2.4 AnOTMwitha“Paper”Worktape .....................265
11.2.5 ⊕AnOTMwithRegistersInsteadofaWorktape .........269
11.2.6 ⊕AMobileFAonaSemi-InfiniteMesh.................276
11.3 EnhancedOTMsthatAreNoMorePowerfulthanOTMs .........278
11.3.1 AnOTMWhichHasSeveralLinearWorktapes ...........278
11.3.2 AnOTMHavingMultidimensionalWorktapes............283
11.3.3 AnOTMHavinga“Random-Access”Worktape ..........287
11.4 ⊕ModelsOutsidetheClassicalMold .........................291
11.4.1 CellularAutomataandKindredModels .................292
11.4.1.1 TheCellularAutomatonModel ................293
11.4.1.2 CAsAreComputationallyEquivalenttoOTMs ...296
11.4.2 TMsasVolunteers;SimulationbyDovetailing............297
11.4.2.1 AbstractVolunteerComputing(VC) ............297
11.4.2.2 TheAbstractVCModelIsComputable..........298
11.5 LearningfromtheZoo ......................................299