Cognition and Intractability: A Guide to Classical and Parameterized Complexity Analysis

376 Pages·2019·6.5 MB·English

by  Iris van Rooij, Mark Blokpoel, Johan Kwisthout, Todd Wareham

CognitionandIntractability
Intractability is a growing concern across the cognitive sciences: while many
models of cognition can describe and predict human behavior in the lab, it
remains unclear how these models can scale to situations of real-world com-
plexity. Cognition and Intractability is the first book to provide an accessi-
ble introduction to computational complexity analysis and its application to
questions of intractability in cognitive science. Covering both classical and
parameterized complexity analysis, it introduces the mathematical concepts
andprooftechniquesthatcanbeusedtotestone’sintuitionof(in)tractability.
Italsodescribeshowthesetoolscanbeappliedtocognitivemodelingtodeal
with intractability – and its ramifications – in a systematic way. Aimed at
students and researchers in philosophy, cognitive neuroscience, psychology,
artificialintelligence,andlinguisticswhowanttobuildafirmunderstandingof
intractabilityanditsimplicationsintheirmodelingwork,itisanidealresource
forteachingorself-study.
IrisvanRooijisapsychologistandcognitivescientistattheDondersInstitute
forBrain,CognitionandBehaviourandtheSchoolforPsychologyandArtifi-
cialIntelligenceatRadboudUniversity,theNetherlands.
Mark Blokpoel is a computational cognitive scientist at the Donders Institute
forBrain,CognitionandBehaviouratRadboudUniversity,theNetherlands.
JohanKwisthoutisacomputerscientistattheDondersInstituteforBrain,Cog-
nitionandBehaviourandtheSchoolforPsychologyandArtificialIntelligence
atRadboudUniversity,theNetherlands.
ToddWarehamisacomputerscientistattheDepartmentofComputerScience
atMemorialUniversityofNewfoundland,Canada.

Cognition and Intractability
A Guide to Classical and Parameterized
Complexity Analysis
IRIS VAN ROOIJ
RadboudUniversityNijmegen
MARK BLOKPOEL
RadboudUniversityNijmegen
JOHAN KWISTHOUT
RadboudUniversityNijmegen
TODD WAREHAM
MemorialUniversityofNewfoundland
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107043992
DOI:10.1017/9781107358331
©IrisvanRooij,MarkBlokpoel,JohanKwisthout,andToddWareham2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2019
PrintedandboundinGreatBritainbyClaysLtd,ElcografS.p.A.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
LibraryofCongressCataloging-in-PublicationData
Names:Rooij,Irisvan,1973-author.|Blokpoel,Mark,1984-author.|
Kwisthout,Johan,1976-author.|Wareham,Todd,1963-author.
Title:Cognitionandintractability:aguidetoclassicalandparameterized
complexityanalysis/IrisvanRooij(RadboudUniversiteitNijmegen),Mark
Blokpoel(RadboudUniversityNijmegen),JohanKwisthout(Universiteit
Leiden),ToddWareham(MemorialUniversityofNewfoundland).
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Includesbibliographicalreferencesandindex.
Identifiers:LCCN2018057971|ISBN9781107043992(hardback:alk.paper)|
ISBN9781108728973(pbk.:alk.paper)
Subjects:LCSH:Computationalcomplexity.|Decisionmaking.|Probabilities.
Classification:LCCQA267.7.C642019|DDC511.3/52–dc23LCrecordavailableat
https://lccn.loc.gov/2018057971
ISBN978-1-107-04399-2Hardback
ISBN978-1-108-72897-3Paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
ofURLsforexternalorthird-partyinternetwebsitesreferredtointhispublication
anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain,
accurateorappropriate.
Contents
ListofFigures pageix
ListofTables xii
Preface xiii
P.1 WhyThisBook? xiii
P.2 ForWhomIsThisBook? xiv
P.3 WhatNottoExpect xiv
P.4 OrganizationoftheBook xv
P.5 GuidefortheReader xvi
P.6 ReuseofMaterials xviii
P.7 Acknowledgments xix
PartI Introduction
1 Introduction 3
1.1 SelectingPizzaToppings 3
1.2 ConceptualFoundations 8
1.3 Exercises 20
1.4 FurtherReading 21
PartII ConceptsandTechniques
2 PolynomialversusExponentialTime 25
2.1 TypesofProblems:Optimization,Search,Decision 25
2.2 SolvingProblemswithAlgorithms 31
2.3 Time-ComplexityofAlgorithms 35
2.4 Time-ComplexityofProblems 45
2.5 Exercises 47
2.6 FurtherReading 51
v
vi Contents
3 Polynomial-TimeReductions 53
3.1 Exponential-TimeLowerBounds 53
3.2 Polynomial-TimeReductions 54
3.3 *AMoreFormalViewonReductions 69
3.4 Exercises 71
3.5 FurtherReading 76
4 ClassicalComplexityClasses 78
4.1 TravelingonaTightBudget 78
4.2 TheClassesPandNP 81
4.3 NP-Hardness 89
4.4 *AdvancedTopics 93
4.5 Exercises 105
4.6 FurtherReading 106
5 Fixed-ParameterTractableTime 108
5.1 Fixed-ParameterTractability 108
5.2 ProvingFixed-ParameterTractability 111
5.3 Helpful Strategies for Proving Fixed-Parameter
Tractability 125
5.4 Exercises 125
5.5 FurtherReading 127
6 ParameterizedReductions 128
6.1 Fixed-ParameterIntractability:InitialThoughts 128
6.2 ParameterizedReductions 129
6.3 Helpful Strategies for Deriving Parameterized
Reductions 140
6.4 *Templates for Parameterized Reduction by
ComponentDesign 140
6.5 Exercises 146
6.6 FurtherReading 147
7 ParameterizedComplexityClasses 148
7.1 Fixed-ParameterIntractability:PuttingItAllTogether 148
7.2 TheW-Hierarchy 150
7.3 ProvingFixed-ParameterIntractability 156
7.4 ParameterizedComplexityAnalysis 158
7.5 Exercises 161
7.6 FurtherReading 162
Contents vii
PartIII ReflectionsandElaborations
8 DealingwithIntractability 167
8.1 RecapandSettingtheStage 167
8.2 How(Not)toDealwithIntractability 171
8.3 IdentifyingSourcesofIntractability 185
8.4 Exercises 187
8.5 FurtherReading 187
9 RepliestoCommonObjections 188
9.1 TheEmpiricistObjection 189
9.2 TheCognition-Is-Not-ComputationObjection 189
9.3 TheSuper-HumanObjection 192
9.4 TheHeuristicsObjection 192
9.5 TheAverage-CaseObjection 194
9.6 TheParallelismObjection 195
9.7 TheNon-DeterminismObjection 196
9.8 TheQuantumComputingObjection 198
9.9 TheSmall-InputsObjection 201
9.10 TheNothing-NewObjection 202
9.11 TheToo-LiberalObjection 203
9.12 TheTuring-MachineObjection 203
9.13 TheApproximationObjection 206
9.14 TheAs-IfObjection 208
9.15 Exercises 212
9.16 FurtherReading 213
PartIV Applications
10 CoherenceasConstraintSatisfaction 217
10.1 Introduction 217
10.2 FormalizingtheTheory 218
10.3 ComplexityAnalyses 221
10.4 Discussion 230
10.5 Exercises 231
10.6 FurtherReading 232
11 AnalogyasStructureMapping 234
11.1 Introduction 234
11.2 FormalizingtheTheory 235
viii Contents
11.3 ComplexityAnalysis 239
11.4 Discussion 242
11.5 Exercises 243
11.6 FurtherReading 245
12 CommunicationasBayesianInference 246
12.1 Introduction 246
12.2 FormalizingtheTheory 248
12.3 ComplexityAnalysis 250
12.4 Discussion 256
12.5 Exercises 257
12.6 FurtherReading 257
AppendixA MathematicalBackground 258
A.1 Turing’sModelofComputation 258
A.2 SetNotation 262
A.3 GraphTheory 264
A.4 PropositionalLogic 269
A.5 Probabilities 272
A.6 MiscellaneousTopics 278
AppendixB ListofComputationalProblems 280
AppendixC CompendiumofComplexityResults 293
C.1 HowtoUseThisCompendium 293
C.2 List of Computational-Level Theories with
ComplexityResults 300
References 336
Index 350