Proof complexity

533 Pages·2019·2.657 MB·English

by  Jan Krajíček

PROOF COMPLEXITY
Proofcomplexityisarichsubjectdrawingonmethodsfromlogic,combinatorics,
algebraandcomputerscience.Thisself-containedbookpresentsthebasicconcepts,
classicalresults,currentstateoftheartandpossiblefuturedirectionsinthefield.It
stressesaviewofproofcomplexityasawholeentityratherthanacollectionof
varioustopicsheldtogetherlooselybyafewnotions,anditfavorsmore
generalizablestatements.
Lowerboundsforlengthsofproofs,oftenregardedasthekeyissueinproof
complexity,areofcoursecoveredindetail.However,upperboundsarenot
neglected:thisbookalsoexplorestherelationsbetweenboundedarithmetictheories
andproofsystemsandhowtheycanbeusedtoproveupperboundsonlengthsof
proofsandsimulationsamongproofsystems.Itgoesontodiscusstopicsthat
transcendspecificproofsystems,allowingfordeeperunderstandingofthe
fundamentalproblemsofthesubject.
EncyclopediaofMathematicsandItsApplications
Thisseriesisdevotedtosignificanttopicsorthemesthathavewideapplicationin
mathematicsormathematicalscienceandforwhichadetaileddevelopmentofthe
abstracttheoryislessimportantthanathoroughandconcreteexplorationofthe
implicationsandapplications.
BooksintheEncyclopediaofMathematicsandItsApplicationscovertheir
subjectscomprehensively.Lessimportantresultsmaybesummarizedasexercises
attheendsofchapters.Fortechnicalities,readerscanbereferredtothe
bibliography,whichisexpectedtobecomprehensive.Asaresult,volumesare
encyclopedicreferencesormanageableguidestomajorsubjects.
ENCYCLOPEDIAOFMATHEMATICSAND ITSAPPLICATIONS
AllthetitleslistedbelowcanbeobtainedfromgoodbooksellersorfromCambridge
UniversityPress.Foracompleteserieslistingvisit
www.cambridge.org/mathematics.
122 S.KhrushchevOrthogonalPolynomialsandContinuedFractions
123 H.NagamochiandT.IbarakiAlgorithmicAspectsofGraphConnectivity
124 F.W.KingHilbertTransformsI
125 F.W.KingHilbertTransformsII
126 O.CalinandD.-C.ChangSub-RiemannianGeometry
127 M.Grabischetal.AggregationFunctions
128 L.W.BeinekeandR.J.Wilson(eds.)withJ.L.GrossandT.W.TuckerTopicsinTopological
GraphTheory
129 J.Berstel,D.PerrinandC.ReutenauerCodesandAutomata
130 T.G.FaticoniModulesoverEndomorphismRings
131 H.MorimotoStochasticControlandMathematicalModeling
132 G.SchmidtRelationalMathematics
133 P.KornerupandD.W.MatulaFinitePrecisionNumberSystemsandArithmetic
134 Y.CramaandP.L.Hammer(eds.)BooleanModelsandMethodsinMathematics,ComputerScience,
andEngineering
135 V.Berthe´andM.Rigo(eds.)Combinatorics,AutomataandNumberTheory
136 A.Krista´ly,V.D.Ra˘dulescuandC.VargaVariationalPrinciplesinMathematicalPhysics,Geometry,
andEconomics
137 J.BerstelandC.ReutenauerNoncommutativeRationalSerieswithApplications
138 B.CourcelleandJ.EngelfrietGraphStructureandMonadicSecond-OrderLogic
139 M.FiedlerMatricesandGraphsinGeometry
140 N.VakilRealAnalysisthroughModernInfinitesimals
141 R.B.ParisHadamardExpansionsandHyperasymptoticEvaluation
142 Y.CramaandP.L.HammerBooleanFunctions
143 A.Arapostathis,V.S.BorkarandM.K.GhoshErgodicControlofDiffusionProcesses
144 N.Caspard,B.LeclercandB.MonjardetFiniteOrderedSets
145 D.Z.ArovandH.DymBitangentialDirectandInverseProblemsforSystemsofIntegraland
DifferentialEquations
146 G.DassiosEllipsoidalHarmonics
147 L.W.BeinekeandR.J.Wilson(eds.)withO.R.OellermannTopicsinStructuralGraphTheory
148 L.Berlyand,A.G.KolpakovandA.NovikovIntroductiontotheNetworkApproximationMethodfor
MaterialsModeling
149 M.BaakeandU.GrimmAperiodicOrderI:AMathematicalInvitation
150 J.Borweinetal.LatticeSumsThenandNow
151 R.SchneiderConvexBodies:TheBrunnMinkowskiTheory(SecondEdition)
152 G.DaPratoandJ.ZabczykStochasticEquationsinInfiniteDimensions(SecondEdition)
153 D.Hofmann,G.J.SealandW.Tholen(eds.)MonoidalTopology
154 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasI:TheVidav–Palmer
andGelfand–NaimarkTheorems
155 C.F.DunklandY.XuOrthogonalPolynomialsofSeveralVariables(SecondEdition)
156 L.W.BeinekeandR.J.Wilson(eds.)withB.ToftTopicsinChromaticGraphTheory
157 T.MoraSolvingPolynomialEquationSystemsIII:AlgebraicSolving
158 T.MoraSolvingPolynomialEquationSystemsIV:BuchbergerTheoryandBeyond
159 V.Berthe´andM.Rigo(eds.)Combinatorics,WordsandSymbolicDynamics
160 B.RubinIntroductiontoRadonTransforms:WithElementsofFractionalCalculusandHarmonicAnalysis
161 M.GherguandS.D.TaliaferroIsolatedSingularitiesinPartialDifferentialInequalities
162 G.MolicaBisci,V.D.RadulescuandR.ServadeiVariationalMethodsforNonlocalFractionalProblems
163 S.WagonTheBanach–TarskiParadox(SecondEdition)
164 K.BroughanEquivalentsoftheRiemannHypothesisI:ArithmeticEquivalents
165 K.BroughanEquivalentsoftheRiemannHypothesisII:AnalyticEquivalents
166 M.BaakeandU.Grimm(eds.)AperiodicOrderII:CrystallographyandAlmostPeriodicity
167 M.CabreraGarc´ıaandA´.Rodr´ıguezPalaciosNon-AssociativeNormedAlgebrasII:Representation
TheoryandtheZel’manovApproach
168 A.Yu.Khrennikov,S.V.KozyrevandW.A.Zu´n˜iga-GalindoUltrametricPseudodifferentialEquations
andApplications
169 S.R.FinchMathematicalConstantsII
170 J.Kraj´ıcˇekProofComplexity
171 D.Bulacu,S.Caenepeel,F.PanaiteandF.VanOystaeyenQuasi-HopfAlgebras
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS
Proof Complexity
JAN KRAJ´ICˇEK
CharlesUnivesity,Prague
UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom
OneLibertyPlaza,20thFloor,NewYork,NY10006,USA
477WilliamstownRoad,PortMelbourne,VIC3207,Australia
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www.cambridge.org
Informationonthistitle:www.cambridge.org/9781108416849
DOI:10.1017/9781108242066
©JanKraj´ıcˇek2019
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
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Firstpublished2019
PrintedandboundinGreatBritainbyClaysLtd.,ElcografS.P.A.
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary.
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Names:Kraj´ıcˇek,Jan,author.
Title:Proofcomplexity/JanKraj´ıcˇek.
Description:Cambridge;NewYork,NY:CambridgeUniversityPress,2019.|
Series:Encyclopediaofmathematicsanditsapplications;170|Includes
bibliographicalreferencesandindex.
Identifiers:LCCN2018042527|ISBN9781108416849(hardback:alk.paper)
Subjects:LCSH:Prooftheory.|Computationalcomplexity.
Classification:LCCQA9.54.K722019|DDC511.3/6–dc23
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Contents
Preface pagexiii
Acknowledgements xv
Introduction 1
PartI BasicConcepts 9
1 ConceptsandProblems 11
1.1 PropositionalLogic 11
1.2 First-OrderLogicandFiniteStructures 18
1.3 ComplexityClasses 22
1.4 BooleanCircuitsandComputations 27
1.5 ProofSystemsandFundamentalProblems 32
1.6 BibliographicalandOtherRemarks 37
2 FregeSystems 39
2.1 FregeRulesandProofs 39
2.2 Tree-LikeProofs 42
2.3 Reckhow’sTheorem 47
2.4 ExtendedandSubstitutionFregeSystems 52
2.5 OverviewofProofComplexityMeasures 58
2.6 BibliographicalandOtherRemarks 62
3 SequentCalculus 64
3.1 SequentCalculusLK 64
3.2 TheStrengthofLK 66
3.3 InterpolationforCut-FreeProofs 70
3.4 Constant-DepthSubsystemsofLK 73
3.5 BibliographicalandOtherRemarks 79
4 QuantifiedPropositionalCalculus 81
4.1 SequentCalculusG 81
4.2 RelativizedFregesystemsandBooleanPrograms 84
viii Contents
4.3 SkolemizationofG 87
4.4 BibliographicalandOtherRemarks 91
5 Resolution 93
5.1 TheResolutionRuleandItsCompleteness 93
∗
5.2 R andtheDPLLProcedure 95
5.3 RegularResolutionandRead-OnceBranchingPrograms 97
5.4 WidthandSize 98
5.5 HeightandSpace 102
5.6 Interpolation 106
5.7 DNF-Resolution 108
5.8 ExtendedResolution 111
5.9 BibliographicalandOtherRemarks 112
6 AlgebraicandGeometricProofSystems 115
6.1 EquationalCalculus 116
6.2 AlgebraicGeometryandtheNullstellensatz 117
6.3 IntegerLinearProgrammingandtheCuttingPlaneMethod 121
6.4 Semi-AlgebraicGeometryandthePositivstellensatz 124
6.5 BetweentheDiscreteandtheContinuum 128
6.6 BeyondPropositionalLogic 130
6.7 BibliographicalandOtherRemarks 131
7 FurtherProofSystems 134
7.1 CombinedProofSystems 134
7.2 CircuitandWPHPFregeSystems 138
7.3 ImplicitProofSystems 141
7.4 AuxiliaryLogicProofSystems 145
7.5 AuxiliaryAlgebraicProofSystems 149
7.6 OutsidetheBox 154
7.7 BibliographicalandOtherRemarks 157
PartII UpperBounds 163
8 BasicExampleoftheCorrespondencebetweenTheories
andProofSystems 165
8.1 Parikh’sTheoryI(cid:2) 165
0
8.2 TheParis–WilkieTranslation(cid:3)...(cid:4) 167
8.3 ModelsofI(cid:2) (R) 170
0
8.4 SoundnessofAC0-FregeSystems 174
8.5 Ajtai’sArgument 179
8.6 TheCorrespondence–ASummary 180
8.7 BibliographicalandOtherRemarks 182
Contents ix
9 TheTwoWorldsofBoundedArithmetic 185
9.1 TheLanguagesofBoundedArithmetic 186
9.2 OneorTwoSorts 187
9.3 Buss’sTheories 191
9.4 FromTwoSortstoOneandBack 194
9.5 BibliographicalandOtherRemarks 196
10 UptoEFviathe(cid:3)...(cid:4)Translation 197
10.1 AC0[m]-FregeSystems 197
10.2 TC0-FregeSystems 199
10.3 ExtendedFregeSystems 201
10.4 FregeSystems 203
10.5 R-LikeProofSystems 205
10.6 BibliographicalandOtherRemarks 209
11 ExamplesofUpperBoundsandp-Simulations 211
11.1 ModularCountingPrinciplesinAC0[m]-Frege 211
11.2 PHPinTC0-Frege 212
11.3 SimulationofTC0-FregebyFrege 212
11.4 WPHPinAC0-Frege 215
11.5 SimulationsbyEF 221
11.6 SubexponentialSimulationsbyAC0-FregeSystems 223
11.7 SimulationsofAlgebro-GeometricSystemsbyLogicalSystems 225
11.8 BibliographicalandOtherRemarks 230
12 BeyondEFviathe||...||Translation 233
12.1 Cook’sTheoryPVandBuss’sTheoryS1 233
2
12.2 Herbrand’sTheorem 236
12.3 The||...||Translation 241
12.4 PV,S1andER 244
2
∗
12.5 G,G andG 246
i i
12.6 Jeˇra´bek’sTheoriesAPC andAPC 250
1 2
12.7 HigherUp:V1andiER 252
2
12.8 LimitsoftheCorrespondence 254
12.9 BibliographicalandOtherRemarks 257
PartIII LowerBounds 261
13 RandR-LikeProofSystems 263
13.1 AdversaryArguments 263
13.2 Relativization 269
13.3 WidthandExpansion 276
13.4 RandomFormulas 281
13.5 Interpolation 286