HowtorefutearandomCSPSarahR.AllenRyanO'DonnellDavidWitmerJuly4,2015 - PDF document

HowtorefutearandomCSPSarahR.AllenRyanO'DonnellDavidWitmerJuly4,2015AbstractLetPbeanontrivialk-arypredicateoveranitealphabet.ConsiderarandomCSP(P)instanceIovernvariableswithmconstraints,eachbeingPappliedtokrandomliterals.WhenmntheinstanceIwillbeunsatisablewithhighprobability,andthenaturalassociatedalgorithmictaskistondarefutationofI|i.e.,acerticateofunsatisability.WhenPisthe3-aryBooleanORpredicate,thisisthewellstudiedproblemofrefutingrandom3-SATformulas;inthiscase,anecientalgorithmisknownonlywhenmn3=2.Understandingthedensityrequiredforaverage-caserefutationofotherpredicatesisofimportanceforvariousareasofcomplexity,includingcryptography,proofcomplexity,andlearningtheory.Themainpreviously-knownresultisthatforageneralBooleank-arypredicateP,havingmndk=2erandomconstraintssucesforecientrefutation.Inthisworkwegiveageneralcriterionforarbitraryk-arypredicates,onethatoftenyieldsecientrefutationalgorithmsatmuchlowerdensities.Specically,ifPfailstosupportat-wiseindependent(uniform)probabilitydistribution(2tk),thenthereisanecientalgorithmthatrefutesrandomCSP(P)instancesIwithhighprobability,providedmnt=2.Indeed,ouralgorithmwill\somewhatstrongly"refuteI,certifyingOpt(I)1� k(1);ift=kthenwefurthermoregetthestrongestpossiblerefutation,certifyingOpt(I)E[P]+o(1).Thislastresultisneweveninthecontextofrandomk-SAT.Regardingtheoptimalityofourmnt=2densityrequirement,priorworkonSDPhierar-chieshasgivensomeevidencethatecientrefutationofrandomCSP(P)maybeimpossiblewhenmnt=2.Thusthereisanindicationouralgorithm'sdependenceonmisoptimalforev-eryP,atleastinthecontextofSDPhierarchies.Alongtheselines,weshowthatourrefutationalgorithmcanbecarriedoutbytheO(1)-roundSOSSDPhierarchy.Finally,asanapplicationofourresult,wefalsifythe\SRCSPassumptions"usedtoshowvarioushardness-of-learningresultsintherecent(STOC2014)workofDaniely,Linial,andShalev{Shwartz. DepartmentofComputerScience,CarnegieMellon.fsrallen,odonnell,dwitmerg@cs.cmu.edu.SupportedbyNSFgrantsCCF-0747250andCCF-1116594.Someofthisworkperformedwhilethesecond-namedauthorwasattheBogaziciUniversityComputerEngineeringDepartment,supportedbyMarieCurieInternationalIncomingFellowshipprojectnumber626373.TherstandthirdnamedauthorswerepartiallysupportedbytheNationalScienceFoundationGraduateResearchFellowshipProgramunderGrantNo.DGE-1252522. 1OnrefutationofrandomCSPsConstraintsatisfactionproblems(CSPs)playamajorroleincomputerscience.Thereisavasttheory[BJK05]ofhowalgebraicpropertiesofaCSPpredicateaectitsworst-casesatisabilitycomplexity;thereisasimilarlyvasttheory[Rag09]ofworst-caseapproximabilityofCSPs.Finally,thereisarichrangeofresearch|fromtheeldsofcomputerscience,mathematics,andphysics|ontheaverage-casecomplexityofrandomCSPs;see[Ach09]forasurveyjustofrandomk-SAT.ThispaperisconcernedwithrandomCSPs,andinparticulartheproblemofecientlyrefutingsatisabilityforrandominstances.Thisisawell-studiedalgorithmictaskwithconnectionsto,e.g.,proofcomplexity[BB02],inapproximability[Fei02],SAT-solvers[SAT],cryptography[ABW10],learningtheory[DLSS14],statisticalphysics[CLP02],andcomplexitytheory[BKS13].Historically,randomCSPsareprobablybeststudiedinthecaseofk-SAT,k3.ThemodelhereinvolveschoosingaCNFformulaIovernvariablesbydrawingmclauses(ORsofkliterals)independentlyanduniformlyatrandom.(Theprecisedetailsoftherandommodelareinessential;seeSection3.1formoreinformation.)Thisisoneofthebestknownecientwaysofgeneratinghard-seeminginstancesofNP-completeandcoNP-completeproblems.Thecomputationalhardnessdependscruciallyonthedensity,=m=n.Foreachkthereis(conjecturally)aconstantcriticaldensityksuchthatIissatisablewithhighprobabilitywhenk,andIisunsatisablewithhighprobabilitywhen&#x-277;k.(Hereandthroughout,\withhighprobability(whp)"meanswithprobability1�o(1)asn!1.)ThisphenomenonoccursforallnontrivialrandomCSPs;inthecaseofk-SATit'sbeenrigorouslyproven[DSS15]forsucientlylargek.Thereisanaturalalgorithmictaskassociatedwiththetworegimes.WhenkonewantstondasatisfyingassignmentforI.When&#x-309;konewantstorefuteI;i.e.,ndacerticateofunsatisability.MostheuristicSAT-solversuseDPLL-basedalgorithms;onunsatisableinstances,theyproducecerticatesthatcanbeviewedasrefutationswithintheResolutionproofsystem.Moregenerally,arefutationalgorithmfordensityisanyalgorithmthat:a)outputs\unsatisable"or\fail";b)neverincorrectlyoutputs\unsatisable";c)outputs\fail"withlowprobability(i.e.,probabilityo(1)).1Empiricalworksuggeststhatasincreasestowardsk,ndingsatisfyingassignmentsbecomesmoredicult;andconversely,asincreasesbeyondk,ndingcerticatesofunsatisabilitygraduallybecomeseasier.AseminalpaperofChvatalandSzemeredi[CS88]showedthatforanysucientlylargeintegerc(dependingonk),arandomk-SATinstancewithm=cnrequiresResolutionrefutationsofsize2 (n)(whp).Ontheotherhand,Fu[Fu96]showedthatpolynomial-sizeResolutionrefutationsexist(whp)oncemO(nk�1);Beameetal.[BKPS99]subsequentlyshowedthatsuchproofscouldbefoundeciently.2Abreakthroughcamein2001,whenGoerdtandKrivelevich[GK01]abandonedcombinatorialrefutationsforspectralones,showingthatrandomk-SATinstancescanbeecientlyrefutedwhenmeO(ndk=2e).Soonthereafter,FriedmanandGoerdt[FG01](seealso[FGK05])showedthatfor3-SAT,ecientspectralrefutationsexistoncemn3=2+(forany&#x-309;0).Thesedensitiesfork-SAT|aroundn3=2for3-SATandndk=2eingeneral|havenotbeenfundamentallyimproveduponinthelast14years.3(SeeTable1foramoredetailedhistoryofresultsinthis 1Wecautionthereaderthatinthispaperwedonotconsidertherelated,butdistinct,scenarioofdistinguishingplantedrandominstancesfromtrulyrandomones.2Inthispaperweusethefollowingnot-fully-standardterminology:Astatementoftheform\Iff(n)O(g(n))thenX"meansthatthereexistsacertainfunctionh(n),withh(n)beingO(g(n)),suchthatthestatement\Iff(n)h(n)thenX"istrue.WealsouseeO(f(n))todenoteO(f(n)
polylog(f(n)),andOk(f(n))todenotethatthehiddenconstanthasadependenceonk(mostoftenoftheform2O(k)).3Actually,itisclaimedin[GJ02]thatonecanobtainnk=2+foroddk\alongthelinesof[FG01]".Ononehand,thisistrue,aswe'llseeinthispaper.Ontheotherhand,noproofwasprovidedin[GJ02],andwehavenotfoundtheclaimrepeatedinanypapersubsequentto2003.1 area.)Improvingthen3=2boundfor3-SATiswidelyregardedasamajoropenproblem[ABW10],withconjecturesregardingitspossibilitygoingbothways[BB02,DLSS13].SeealsotheintriguingworkofFeige,Kim,andOfek[FKO06]showingthatpolynomial-size3-SATrefutationsexistwhponcemO(n1:4).Strongrefutation.Inanotablepaperfrom2002,Feige[Fei02]madeafruitfulconnectionbe-tweenthehardnessofrefutingrandom3-SATinstancesandtheinapproximabilityofcertainop-timizationproblemsthatarechallengingtoanalyzebyothermeans.ThisreferstocertifyingnotonlythatarandominstanceIisunsatisable,butfurthermorethatOpt(I)1�forsomecon-stant�0.HereOpt(I)denotesthemaximumfractionofsimultaneouslysatisableconstraintsinI.Feigespecicallyintroducedthefollowing\R3SATHypothesis":Forallsmall�0andlargec2N,thereisnopolynomial-time-refutationalgorithmforrandom3-SATwithm=cn.Tostress-testFeige'sR3SATHypothesis,onemayaskiftheaforementionedrefutationalgorithmsfork-SATcanbeimprovedto-refutationalgorithms.Coja-Oghlanetal.[COGL04]showedthattheycanbeinthecaseofk=3;4.Indeed,theygavealgorithmsforwhatiscalledstrongrefutationinthesecases.Herestronglyrefutingk-SATreferstocertifyingthatOpt(I)1�2�k+o(1)(notethatOpt(I)1�2�kwhpassumingmO(n)).Beyondk-SAT.AsinthealgebraicandapproximationtheoriesofCSP,it'sofsignicantinteresttoconsiderrandominstancesoftheCSP(P)problemforgeneralpredicatesP:Zkq!f0;1g,besidesjustBooleanOR.(ThoughBooleanpredicatesaremorefamiliar,largerdomainsareofinterest,e.g.,forq-colorabilityofk-uniformhypergraphs.)Specically,weareinterestedinthequestionofhowpropertiesofPaectthenumberofconstraintsneededforecientrefutationofrandomCSP(P)instances.ThisprecisequestionisveryrelevantforworkincryptographybasedonthecandidateOWFsandPRGsofGoldreich[Gol00];seealso[ABW10]andthesurveyofApplebaum[App13].IthasalsoprovenessentialfortherecentexcitingapproachtohardnessoflearningduetoDaniely,Linial,andShalev-Shwartz[DLSS13,DLSS14,DSS14].WediscussthislearningconnectionandourresultsonitinmoredetailinSection5.Thespecialcaseofrandom3-XORhasprovedparticularlyimportant:itisrelatedto3-SATrefutationthroughFeige's\3XORPrinciple"(see[Fei02,FO05,FKO06]);it'sthebasisforcrypto-graphicschemesduetoAlekhnovich[Ale03](andisrelatedtothe\LearningParitieswithNoise"problem);it'susedinthebestknownlowerboundsfortheSOSSDPhierarchy[Gri01,Sch08],whichwediscussfurtherinSection6;and,BarakandMoitra[BM15]haveshownittobeequivalenttoacertain\tensorpredictionproblem"inlearningtheory.Priortothiswork,verylittlewasknownabouthowthepredicatePaectsthecomplexityofrefutingrandomCSP(P)instances.Themainknownresult,followingfromtheworkCoja-Oghlan,Cooper,andFrieze[COCF10],wasthefollowing:ForanyBooleank-arypredicateP,onecanecientlystronglyrefuterandomCSP(P)instancesI(i.e.,certifyOpt(I)E[P]+o(1))providedthenumberofconstraintsmsatisesmeO(ndk=2e).Inthecaseofk-XOR,theveryrecentworkofBarakandMoitra[BM15]showedhowtoimprovethisboundtomeO(nk=2).4 4Thepresentauthorsalsoobtainedthisresultaroundthesametime,butwecredittheresultto[BM15]astheypublishedearlier.Withtheirpermissionwerepeattheproofherein,partlybecauseweneedtoproveaslightlymoregeneralvariant.2 CSP Poly-sizerefutationswhponcem


Strength Ecient/Existential Reference k-SAT O(nk�1) Refutation Existential [Fu96] k-SAT O(nk�1=logk�2(n)) Refutation Ecient [BKPS99,BKPS02] k-SAT eO(ndk=2e) Refutation Ecient [GK01,FGK05] 3-SAT O(n3=2+) Refutation Ecient [FG01,FGK05] k-SAT O(nk=2+) Refutation Ecient Claimedpossiblein[GJ02,GJ03] Exactly-k1-out-of-k-SAT eO(n) Refutation Ecient [BB02] 2-out-of-5-SAT O(n3=2+) Refutation Ecient [GJ02,GJ03] NAE-3-SAT O(n) (1)-Refutation Ecient [KLP96,GJ02,GJ03] k-SAT O(ndk=2e) Refutation Ecient [COGLS04,FO05] 3-SAT eO(n3=2) Refutation Ecient [GL03] 3-SAT eO(n3=2) Strong Ecient [COGL04,COGL07] 4-SAT O(n2) Strong Ecient [COGL04,COGL07] 3-SAT O(n3=2) Refutation Ecient [FO04] 3-SAT O(n1:4) Refutation Existential [FKO06] 3-XOR O(n3=2) 1=n (1)-refutation Ecient Implicitin[FKO06] 3-SAT (n3=2) Refutation Ecient Claimedin[FKO06] Booleank-CSP O(ndk=2e) Strong Ecient [COCF10] k-XOR eO(nk=2) Strong Ecient [BM15](alsoherein) Anyk-CSP eO(nk=2) Quasirandom(=)strong) Ecient Thispaper Anyk-CSPnotsupportingt-wiseindep. eO(nt=2) k(1)-refutation Ecient Thispaper Table1:Uptologarithmicfactorsonm,ourworksubsumesallpreviouslyknownresults.3 2OurresultsandtechniquesHerewedescribeourmainresultsandtechniquesatahighlevel.PrecisetheoremstatementsappearlaterintheworkandthedenitionsoftheterminologyweuseisgiveninSection3.WealsomentionthatinSectionBwewillgeneralizeallofourresultstothecaseoflargeralphabets;butwe'lljustdiscussBooleanpredicatesP:f0;1gk!f0;1gforsimplicity.Ourmainresultgivesa(possiblysharp)boundonthenumberofconstraintsneededtorefuterandomCSP(P)instances.Beforegettingtoit,werstdescribesomemoreconcreteresultsthatgointothemainproof.Allofourresultsrelyonastrongrefutationalgorithmfork-XOR(actually,aslightgeneralizationthereof).FormeO(ndk=2e),sucharesultfollowsfrom[COCF10];however,theexponentdk=2ecanbeimprovedtok=2.Wegiveademonstrationofthisfactherein;asmentionedearlier,itwaspublishedveryrecentlybyBarakandMoitra[BM15,Corollary5.5andExtensions].Theorem2.1.Thereisanecientalgorithmthat(whp)stronglyrefutesrandomk-XORinstanceswithatleasteO(nk=2)constraints;i.e.,itcertiesOpt(I)1 2+o(1).TheproofofTheorem2.1followsideasfrom[COGL07]andearlierworkson\discrepancy"ofrandomk-SATinstances.Thecaseofevenkisnotablyeasier,andwepresenttwo\folklore"argumentsforit.Thecaseofoddkistrickier.Roughlyspeakingweviewtheinstanceasahomogeneousdegree-kmultilinearpolynomial,whichwewanttocertifytakesononlysmallvaluesoninputsinf�1;1gn.Consideringseparatelythecontributionsbasedonthe\last"ofthekvariablesineachconstraint,andthenusingCauchy{Schwarz,itsucestoboundthenormofacarefullydesignedquadraticformofdimensionnk�1,indexedbytuplesofk�1variables.Thisisdoneusingthetracemethod[Wig55,FK81].Similartechniques,includingtheuseofthetracemethod,datebacktothe2001Friedgman{Goerdtwork[FG01]refutingrandom3-SATgivenm=n3=2+constraints.WithTheorem2.1inhand,thenextstepiscertifyingquasirandomnessofrandomk-aryCSPinstanceshavingmeO(nk=2)constraints.RoughlyspeakingwesaythataCSPinstanceisquasirandomif,foreveryassignmentx2f0;1gn,theminducedk-tuplesofliteralvaluesareclosetobeinguniformlydistributedoverf0;1gk.(Notethatthisisonlyapropertyoftheinstances'constraintscopes/negations,andhasnothingtodowithP.)Sincethe\VaziraniXORLemma"impliesthatadistributiononf�1;1gkisuniformifandonlyifallits2kXORsarehavebias1 2,weareabletoleverageTheorem2.1toprove:Theorem2.2.Thereisanecientalgorithmthat(whp)certiesthatarandominstanceofCSP(P)isquasirandom,providedthenumberofconstraintsisatleasteO(nk=2).Ifaninstanceisquasirandom,thennosolutioncanbemuchbetterthanarandomlychosenone.ThusbycertifyingquasirandomnessweareabletostronglyrefuterandominstancesofanyCSP(P):Theorem2.3.Foranyk-arypredicateP,thereisanecientalgorithmthat(whp)stronglyrefutesrandomCSP(P)instanceswhenthenumberofconstraintsisatleasteO(nk=2).Inparticular,thistheoremimprovesupon[COCF10]byafactorofp nwheneverkisodd;thissavingsisneweveninthewell-studiedcaseofk-SAT.TheaboveresultdoesnotmakeuseofanypropertiesofthepredicatePotherthanitsarity,k.Wenowcometoourmainresult,whichshowsthatformanyinterestingP,randomCSP(P)instancescanberefutedwithmanyfewerconstraintsthannk=2.Inthefollowingtheorem,the4 phrase\t-wiseuniform"(oftenimpreciselycalled\t-wiseindependent")referstoadistributiononf�1;1gkinwhichallmarginaldistributionsontoutofkcoordinatesareuniform.Theorem2.4.(Main.)LetPbeak-arypredicatesuchthatthereisnot-wiseuniformdistributionsupportedonitssatisfyingassignments,t2.Thenthereisanecientalgorithmthat(whp) k(1)-refutesrandominstancesofCSP(P)withatleasteO(nt=2)constraints.WeremarkthatpropertyofapredicatePsupportingapairwiseuniformdistributionhasplayedanimportantroleinapproximabilitytheoryforCSPs,eversinceAustrinandMossel[AM09]showedthatsuchpredicatesarehereditarilyapproximation-resistantundertheUGC.Also,notethatthelargesttforwhichapredicatePsupportsat-wiseuniformdistributiondeterminestheminimumnumberofconstraintsrequiredbyouralgorithm.ThisvalueiscloselyrelatedtothenotionofdistributioncomplexitystudiedbyFeldman,Perkins,andVempala[FPV14,FPV15]inthecontextofplantedrandomCSPs.Informally,thedistributioncomplexityofaplantedCSPisthelargesttforwhichthedistributionoverconstraintinputsf�1;1gkinducedbytheplantedassignmentist-wiseuniform.Despitethissimilarity,thealgorithmictechniquesusedbyFeldman,Perkins,andVempalaintheplantedcase[FPV14]donotseemtodirectlyapplytorefutation.TheideabehindtheproofofTheorem2.4isthatwitheO(nt=2)constraintswecanusethealgorithmofTheorem2.2tocertifyquasirandomness(closenesstouniformity)forallsubsetsoftoutofkcoordinates.Thusforeveryassignmentx2f0;1gn,theinduceddistributiononconstraintk-tuplesis(o(1)-closeto)t-wiseuniform.SincePdoesnotsupportat-wiseuniformdistribution,thisessentiallyshowsthatnoxcaninduceafully-satisfyingdistributiononconstraintinputs.Tohandletheo(1)-closenesscaveat,weshowthatifPdoesnotsupportat-wiseuniformdistribution,thenitis-farfromsupportingsuchadistribution,for= k(1).Thealgorithmcantheninfact(�o(1))-refuterandomCSP(P)instances.Example2.5.Tobrie yillustratetheresult,considertheExactly-k-out-of-2k-SATCSP,studiedpreviouslyin[BB02,GJ03].Theassociatedpredicatesupportsa1-wiseuniformdistribution,namelytheuniformdistributionoverstringsinf0;1g2kofHammingweightk.However,itisnothardtoshowthatitdoesnotsupportanypairwiseuniformdistribution.Asaconsequence,randominstancesofthisCSPcanberefutedwithonlyeO(n)constraints,independentofk.2.1AnapplicationfromlearningtheoryRecently,anexcitingapproachtoprovinghardness-of-learningresultshasbeendevelopedbyDaniely,Linial,andShalev-Shwartz[DLSS13,DLSS14,DSS14,Dan15].Themostgeneralresultsappearin[DLSS14].Inthiswork,Danielyetal.provecomputationalhardnessofseveralcen-trallearningtheoryproblems,basedontwoassumptionsconcerningthehardnessofrandomCSPrefutation.Whiletheassumptionsmadein[DSS14,Dan15]appeartobeplausible,ourworkun-fortunatelyshowsthatthemoregeneralassumptionsmadein[DLSS14]arefalse.Belowwestatethe(admittedlystrong)assumptionsfrom[DLSS14](uptosomeveryminortechnicaldetailswhicharediscussedandtreatedinSection5).Wewillneedonepieceoftermi-nology:the0-variabilityVAR0(P)ofapredicateP:f�1;1gk!f0;1gistheleastcsuchthatthereexistsarestrictiontosomecinputcoordinatesforcingPtobe0.Essentially,theassumptionsstatethatonecanobtainhardness-of-refutationwithanarbitrarilylargepolynomialnumberofcon-straintsbyusingafamilyofpredicates(Pk)that:a)haveunbounded0-variability;b)don'tsupportpairwiseuniformity.However,ourworkshowsthatavoidingt-wiseuniformityforunboundedtisalsonecessary.5 SRCSPAssumption1.([DLSS14].)Foralld2NthereisalargeenoughCsuchthatthefollowingholds:IfP:f�1;1gk!f0;1ghasVAR0(P)Candishereditarilyapproximationresistantonsatisableinstances,thenthereisnopolynomial-timealgorithmrefuting(whp)randominstancesofCSP(P)withm=ndconstraints.SRCSPAssumption2.([DLSS14],generalizingthe\RCSPAssumption"of[BKS13]tosuper-linearlymanyconstraints.)Foralld2NthereisalargeenoughCsuchthatthefollowingholds:IfP:f�1;1gk!f0;1ghasVAR0(P)Candis-farfromsupportingapairwiseuniformdistri-bution,thenforall�0thereisnopolynomial-timealgorithmthat(+)-refutes(whp)randominstancesofCSP(P)withm=ndconstraints.In[DLSS14]itisshownhowtoobtainthreeverynotablehardness-of-learningresultsfromtheseassumptions.Howeverasstated,ourworkfalsiestheSRCSPAssumptions.Indeed,theassumptionsarefalseeveninthethreespeciccasesusedby[DLSS14]toobtainhardness-of-learningresults.Wenowdescribethesecases.Case1.TheHuangpredicates(H)arearity-(3)predicatesintroducedin[Hua13];theyarehereditarilyapproximationresistantonsatisableinstancesandhave0-variability ().In[DLSS14]theyareusedinSRCSPAssumption1todeducehardnessofPAC-learningDNFswith!(1)terms.However:Theorem2.6.Forall9,thepredicateHdoesnotsupporta4-wiseuniformdistribution.ThusbyTheorem2.4wecanecientlyrefuterandominstancesofCSP(H)withjusteO(n2)constraints,independentof.ThiscontradictsSRCSPAssumption1.Case2.ThemajoritypredicateMajkhas0-variabilitydk=2eandisshownin[DLSS14]tobe1 k+1-farfromsupportingapairwiseuniformdistribution.In[DLSS14]thesepredicatesareusedinSRCSPAssumption2todeducehardnessofagnsoticallylearningBooleanhalfspacestowithinanyconstantfactor.However:Theorem2.7.Foroddk25,thepredicateMajkdoesnotsupporta4-wiseuniformdistribution;infact,itis:1-farfromsupportinga4-wiseuniformdistribution.Theorem2.4thenimplieswecaneciently-refuterandominstancesofCSP(Majk)witheO(n2)constraints,where=:11 k+1.ThiscontradictsSRCSPAssumption2.Case3.Finally,wealsoprovethatSRCSPAssumption1isfalseforanotherfamilyofpredi-cates(Tk)usedby[DLSS14]toshowhardnessofPAC-learningintersectionsof4Booleanhalfspaces.Ourresultsdescribedinthesethreecasesalluselinearprogrammingduality.Specically,inLemma3.16weshowthatPis-farfromsupportingat-wiseuniformdistributionifandonlyifthereexistsak-variablemultilinearpolynomialQthatsatisescertainpropertiesinvolvingPand.WethenexplicitlyconstructthesedualpolynomialsfortheHuang,Majority,andTkpred-icates.WeconcludethissectionbyemphasizingtheimportanceoftheDaniely{Linial{Shalev-Shwartzhardness-of-learningprogram,despitetheaboveresults.Indeed,subsequentlyto[DLSS14],DanielyandShalev-Shwartz[DSS14]showedhardnessofimproperlylearningDNFformulaswith!(logn)termsunderamuchweakerassumptionthanSRCSPAssumption1.Specically,theirworkonly6 assumesthatforalldthereisalargeenoughksuchthatrefutingrandomk-SATinstancesishardwhentherearem=ndconstraints.Thisassumptionlooksquiteplausibletous,andmayevenbetruewithknotmuchlargerthan2d.Mostrecently,DanielyshowedhardnessofapproximatelyagnosticallylearninghalfspacesusingtheXORpredicateratherthanmajority[Dan15].Thisresultshowsthatthereisnoecientalgorithmthatagnosticallylearnshalfspacestowithinaconstantapproximationratioundertheassumptionthatrefutingrandomk-XORinstancesishardwhenm=ncp klogkforsomec�0.Healsoshowshardnessoflearninghalfspacestowithinanapproximationfactorof2log1�nforany�0assumingthatthereexistssomeconstantc�0suchthatforalls,refutingrandomk-XORinstanceswithk=logsnishardwhenm=nck.2.2EvidencefortheoptimalityofourresultsIt'snaturaltoaskwhetherthent=2dependenceinourmainTheorem2.4canbeimproved.Aswedon'texpecttoproveunconditionalhardnessresults,weinsteadmerelyseekgoodevidencethatrefutingt-wisenon-supportingpredicatesPishardwhenmnt=2.Anaturalformofevidencewouldbeshowingthatvariousstrongclassesofpolynomial-timerefutationalgorithmsfailwhenmnt=2.Tomakesenseofthisweneedtotalkabouttheformofsuchalgorithms;i.e.,propositionalproofsystems.Recently,therehasbeensignicantstudyofthe\SOS"(Sum-Of-Squares)proofsystem,intro-ducedbyGrigorievandVorobjov[GV01];see,e.g.,[OZ13,BS14]fordiscussion.Ithasthefollowingvirtues:a)itisverypowerful,beingabletoecientlysimulateotherproofsystems(e.g.,Reso-lution,Lovasz{Schrijver);b)itisautomatizable[Las00,Par00],meaningthatn-variable\degree-dSOSproofs"canbefoundinnO(d)timewhenevertheyexist;c)wedoknowsomeexamplesoflowerboundsfordegree-dSOSproofs.InSection6ofthispaperweshowthefollowing:Theorem2.8.Allofourrefutationalgorithmsfork-arypredicatescanbeextendedtoproducedegree-2kSOSproofs.Wenowreturntothequestionofevidencefortheoptimalityofconstraintdensityusedinourresults.DatingbacktoFranco{Paull[FP83]andChvatal{Szemeredi[CS88],therehasbeenalonglineofworkinproofcomplexityshowinglowerboundsforrefutingrandom3-SATinstances,especiallyintheResolutionproofsystem.ThisculminatedintheworkofBen-SassonandWigder-son[BSW99],whichshowedthatforrandom3-SAT(and3-XOR)withm=O(n3=2�),Resolutionrefutationsrequiresize2n ()(whp).Morerecently,Schoenbeck[Sch08]showed(usingtheexpan-sionanalysisof[BSW99])thatrandomk-XORandk-SATinstanceswithmnk=2�requireSOSproofsofdegreen (),andhencetake2n ()timetorefutebythe\SOSMethod".See[Tul09,Cha13]forrelatedlarger-alphabetfollowups.TheseresultshowthattheBarak{MoitraeO(nk=2)boundforrefutingrandomk-XOR(whichalsoworksinO(k)-degreeSOS)andourboundforrandomk-SATaretight(uptosubpolynomialfactors)withintheSOSframework.GiventhepoweroftheSOSframework,thisarguablyconstitutessomereasonableevidencethatnopolynomial-timealgorithmcanrefuterandomk-SATinstanceswithmnk=2.Wenowdiscussourmaintheorem'snt=2boundforpredicatesPnotsupportingt-wiseuniformdistributions.InthecontextofinapproximabilityandSDP-hierarchyintegralitygaps,thisconditiononPhasbeensignicantlystudiedinthecaseoft=2.ForPnotsupportingpairwiseuniformity,itisknown[BGMT12,TW13]thattheSherali{AdamsandLovasz{Schrijver+SDPhierarchiesrequiredegree (n)torefuterandomCSP(P)instances(whp)whenm=O(n).ThisresultwasalsorecentlyprovenforthestrongerSOSproofsystembyBarak,Chan,andKothari[BCK15],exceptthattheCSP(P)instancesarenotquiteuniformlyrandom;theyare\slightlypruned"randominstances.7 NowsupposePisapredicatethatdoesnotsupportat-wiseuniformdistribution,wheret�2.Thesecondandthirdauthorsrecentlyessentiallyshowed[OW14]thatfortheSherali{Adams+SDPhierarchy,degreen ()is(whp)necessarytorefuterandomCSP(P)instanceswhenmnt=2�.Asacaveat,againtheinstancesareslightlyprunedrandominstances,ratherthanbeingpurelyuniformlyrandom.(Theinstancesin[OW14]arealsointhe\Goldreich[Gol00]style";i.e.,therearenoliterals,butthe\right-handsides"arerandom.Howeveritisnothardtoshowtheproofsin[OW14]gothroughinthestandardrandommodelofthispaper.)Futurework[MWW15]isdevotedtoremovingthepruningintheseinstances.AlthoughtheSherali{Adams+SDPhierarchyiscertainlyweakerthantheSOShierarchy,theseworksstillconstitutesomeevidencethatourmaintheorem'srequirementofmeO(nt=2)fornon-t-wisesupportingpredicatesmaybeessentiallyoptimal.FurtherevidencefortheoptimalityofourresultsisprovidedbytheworkofFeldman,Perkins,andVempalaonstatisticalalgorithmsforrandomplantedCSPs[FPV15].TheyshowthattheirlowerboundsagainststatisticalalgorithmsforsolvingrandomplantedCSPsalsoimplylowerboundsagainststatisticalalgorithmsforrefutinguniformlyrandomCSPs.Specically,theyprovethatwhenPsupportsat-wiseuniformdistribution,anystatisticalalgorithmusingqueriesthattakeeO(nt=2)possiblevaluescanonlyrefuterandominstancesofCSP(P)withatleaste (nt=2)constraints.Asanapplicationofthisresult,theyalsoshowthatanyconvexprogramrefutingsuchaninstanceofCSP(P)musthavedimensionatleaste (nt=2).2.3Certifyingindependencenumberandchromaticnumberofrandomhyper-graphsCoja-Oghlan,Goerdt,andLanka[COGL07]alsousetheirCSPrefutationtechniquestocertifythatrandom3-and4-uniformhypergraphshavesmallindependencenumberandlargechromaticnumber.Weextendtheseresultstorandomk-uniformhypergraphs.Theorem2.9.Forarandomk-uniformhypergraphH,thereisapolynomialtimealgorithmcer-tifyingthattheindependencenumberofHisatmostwithhighprobabilitywhenHhasatleasteOn5k=2 2khyperedges.Theorem2.10.Forarandomk-uniformhypergraphH,thereisapolynomialtimealgorithmcertifyingthatthechromaticnumberofHisatleastwithhighprobabilitywhenHhasatleasteO�2knk=2
hyperedges.Theproofsofthesetheoremsfollowtheoutlineof[COGL07].WeshowTheorem2.9usingaslightlymoregeneralformofTheorem2.1.Theorem2.10followsalmostdirectlyfromTheorem2.9usingthefactthateverycolorclassofavalidcoloringisanindependentset.DetailsaregiveninAppendixC.3Preliminariesandnotation3.1ConstraintsatisfactionproblemsWereviewsomebasicdenitionsandfactsrelatedtoconstraintsatisfactionproblems(CSPs).InthissectionwediscussonlytheBooleandomain,whichweprefertowriteasf�1;1gratherthanf0;1g.ThestraightforwardextensionstolargerdomainsappearinSectionB.Wewillneedthefollowingnotation:Forx2RnandS[n]wewritexS2RjSjfortherestrictionofxtocoordinatesS;i.e.,(xi)i2S.Wealsousetodenotetheentrywiseproductforvectors.8 Denition3.1.GivenapredicateP:f�1;1gk!f0;1g,aninstanceIoftheCSP(P)problemovervariablesx1;:::;xnisamultisetofP-constraints.EachP-constraintconsistsofapair(c;S),whereS2[n]kisthescopeandc2f�1;1gkisthenegationpattern;thisrepresentstheconstraintP(cxS)=1.Wetypicallywritem=jIj.LetValI(x)bethefractionofconstraintssatisedbyassignmentx2f�1;1gn,i.e.,ValI(x)=1 mP(c;S)2IP(cxS).TheobjectiveistondanassignmentxmaximizingValI(x).TheoptimumofI,denotedbyOpt(I),ismaxx2f�1;1gkValI(x).IfOpt(I)=1,wesaythatIissatisable.Wealsowrite PforthequantityEzf�1;1gk[P(z)];i.e.,thefractionofassignmentsthatsatisfyP.ForanyinstanceIinwhicheachconstraintinvolveskdierentvariables,wehaveOpt(I) P.5WenextdeneastandardrandommodelforCSPs.ForP:f�1;1gk!f0;1g,letFP(n;p)bethedistributionoverCSPinstancesgivenbyincludingeachofthe2knkpossibleconstraintsindependentlywithprobabilityp.Notethatwemayincludeconstraintsondierentpermutationsofthesamesetofvariables,constraintsonthesametupleofvariableswithdierentnegationsc,andconstraintswiththesamevariableoccurringasmorethanoneargument.ItisreasonabletoincludesuchconstraintsinthecasethatthepredicatePisnotasymmetricfunction.Weuse mtodenotetheexpectednumberofconstraints,namely2knkp.AsnotedinFact3.6below,thenumberofconstraintsminadrawfromFP(n;p)isverytightlyconcentratedaround m,andweoftenblurthedistinctionbetweentheseparameters.AppendixDexplicitlydescribesamethodforsimulatinganinstancedrawnfromFP(n;p)whenthenumberofconstraintsisxed.Quasirandomness.Wenowintroduceanimportantnotionforthispaper:thatofaCSPinstancebeingquasirandom.VersionsofthisnotionoriginateintheworksofGoerdtandLanka[GL03](underthename\discrepancy"),Khot[Kho06](\quasi-randomness"),AustrinandHastad[AH13](\adaptiveuselessness"),andChan[Cha13](\lowcorrelation"),amongotherplaces.Todeneit,werstneedtodenetheinduceddistributionofaninstanceandanassignment.Denition3.2.GivenaCSPinstanceIandandanassignmentx2f�1;1gn,theinduceddis-tribution,denotedDI;x,istheprobabilitydistributiononf�1;1gkwheretheprobabilitymasson2f�1;1gkisgivenbyDI;x()=1 jIj
#f(c;S)2IjcxS=g:Inotherwords,itistheempiricaldistributiononinputstoPgeneratedbytheconstraintscopes/negationsonassignmentx.NotethatthepredicatePitselfisirrelevanttothisnotion.WewilldropthesubscriptIwhenitisclearfromthecontext.WedeneDI;x=2k
DI;xtobethedensityfunctionassociatedwithDI;x.Wecannowdenequasirandomness.Denition3.3.ACSPinstanceIis-quasirandomifDI;xis-closetotheuniformdistributionforallx2f�1;1gn;i.e.,ifdTV(DI;x;Uk)forallx2f�1;1gn.HereweusethenotationUkfortheuniformdistributiononf�1;1gkaswellasthefollowing:Denition3.4.IfDandD0areprobabilitydistributionsonthesamenitesetAthendTV(D;D0)denotestheirtotalvariationdistance,1 2P2AjD()�D0()j.IfdTV(D;D0)wesaythatDandD0are-close.IfdTV(D;D0)wesaytheyare-far.(Asneitherinequalityisstrict,thesenotionsarenotquiteopposites.)Animmediateconsequenceofaninstancebeingquasirandomisthatitsoptimumiscloseto P: 5Technically,ourdenitionsallowconstraintswithavariableappearingmorethanonce,soOpt(I) Pdoesn'talwaysholdforus.ButsinceweonlyconsiderrandomI,we'llinfacthaveOpt(I) PwhpoverIanyway.9 Fact3.5.IfIis-quasirandom,thenOpt(I) P+(andinfact,Opt(I)� P).WeconcludethediscussionofCSPsbyrecordingsomefactsthatareproveneasilywiththeChernobound:Fact3.6.LetIFP(n;p).Thenthefollowingstatementsholdwithhighprobability.1.m=jIj2 m
1Oq logn m.2.Opt(I) P
1+Oq 1 P
n m.3.IisOq 2k
n m-quasirandom.3.2AlgorithmsandrefutationsonrandomCSPsDenition3.7.LetPbeaBooleanpredicate.WesaythatAis-refutationalgorithmforrandomCSP(P)with mconstraintsifAhasthefollowingproperties.First,onallinstancesItheoutputofAiseitherthestatement\Opt(I)1�"oris\fail".Second,Aisneverallowedtoerr,whereerringmeansoutputting\Opt(I)1�"onaninstancewhichactuallyhasOpt(I)�1�.Finally,AmustsatisfyPrIFP(n;p)[A(I)=\fail"]o(1)(asn!1);wherepisdenedby m=2knkp.AlthoughAisoftenadeterministicalgorithm,wedoallowittoberandomized,inwhichcasetheaboveprobabilityisalsooverthe\internalrandomcoins"ofA.Werefertothisnotionasweakrefutation,orsimplyrefutation,whenthecerticationstatementisoftheform\Opt(I)1"(equivalently,when=1=jIj).Werefertothenotionasstrongrefutationwhenthestatementisoftheform\Opt(I) P+o(1)"(equivalently,when=1� P+o(1)).Remark3.8.InSection5wewillencountera\two-sidederror"variantofthisdenition.Thisistheslightlyeasieralgorithmictaskinwhichwerelaxtheconditiononerring:itisonlyrequiredthatforeachinstanceIwithOpt(I)�1�,itholdsthatPr[A(I)=\Opt(I)1�"]1=4,wheretheprobabilityisjustovertherandomcoinsofA.Remark3.9.Wewillalsousetheanalogousdenitionforcerticationofrelatedproperties;e.g.,wewilldiscuss-quasirandomnesscerticationalgorithmsinwhichthestatement\Opt(I)1�"isreplacedbythestatement\Iis-quasirandom".3.3t-wiseuniformityAnimportantnotionforthispaperisthatoft-wiseuniformity.Recall:Denition3.10.ProbabilitydistributionDonf�1;1gkissaidtobet-wiseuniform,1tk,ifforallS[k]withjSj=ttherandomvariablexSisuniformonf�1;1gtwhenxD.(Weremarkthatthisconditionissometimesinaccuratelycalled\t-wiseindependence"intheliterature.)Wewillalsoconsiderthemoregeneralnotionof(;t)-wiseuniformity.ThisistypicallydenedusingFouriercoecients:10 Denition3.11.ProbabilitydistributionDonf�1;1gkissaidtobe(;t)-wiseuniformifjbD(S)jforallS[k]with0jSjt,whereD=2k
DistheprobabilitydensityassociatedwithdistributionD.HereweareusingstandardnotationfromFourieranalysisofBooleanfunctions[O'D14].Inparticular,foranyf:f�1;1gk!Rwewritef(x)=PS[k]bf(S)xSforitsexpansionasamultilinearpolynomialoverR,withxSdenotingQi2Sxi(nottobeconfusedwiththeprojectionxS2RjSj).Remark3.12.Itisasimplefact(anditfollowsfromLemma3.13below)that(0;t)-wiseuniformityisequivalenttot-wiseuniformity.Alsoimportantforusisarelatedbutdistinctnotion,thatofbeing-closetoat-wiseuniformdistribution.It'seasytoshowthatifDis-closetoat-wiseuniformdistributionthenDis(2;t)-wiseuniform.Intheotherdirection,wehavethefollowing(seealso[AAK+07]forsomequantitativeimprovement):Lemma3.13.(Alon{Goldreich-Mansour[AGM03,Theorem2.1]).IfDisan(;t)-wiseuniformdistributiononf�1;1gk,thenthereexistsat-wiseuniformdistributionD0onf�1;1gkwithdTV(D;D0)tXi=1�kt

kt
:Inparticularift=kwehavethebound2k
(andthiscanalsobeimproved[Gol11]to2k=2�1
).Finally,wemakeacrucialdenition:Denition3.14.ApredicateP:f�1;1gk!f0;1gissaidtobet-wisesupportingifthereisat-wiseuniformdistributionDwhosesupportiscontainedinP�1(1).WesayPis-farfromt-wisesupportingifeveryt-wiseuniformdistributionDis-farfrombeingsupportedonP;i.e.,hasprobabilitymassatleastonP�1(0).3.4AdualcharacterizationoflimiteduniformityItisknownthattheconditionofPsupportingat-wiseuniformdistributionisequivalenttothefeasibilityofacertainlinearprogram;henceonecanshowthatPisnott-wisesupportingbyexhibitingacertaindualobject,namelyapolynomial.Thisappears,e.g.,inworkofAustrinandHastad[AH09,Theorem3.1].Hereinwewillextendthisfactbygivingadualcharacterizationofbeingfarfromt-wisesupporting.Denition3.15.Let01.ForamultilinearpolynomialQ:f�1;1gk!R,wesaythatQ-separatesP:f�1;1gk!f0;1gifthefollowingconditionshold:Q(z)�18z2f�1;1gk;Q(z)8z2P�1(1);bQ(;)=0,i.e.,Qhasnoconstantcoecient.Wenowprovidethequantitativeversionoftheaforementioned[AH09,Theorem3.1]:Lemma3.16.LetP:f�1;1gk!f0;1gandlet01.ThenPis-farfromt-wisesupportingifandonlyifthereisa-separatingpolynomialforPofdegreeatmostt.11 Proof.Theproofisanapplicationoflinearprogrammingduality.ConsiderthefollowingLP,whichhasvariablesD(z)foreachz2f�1;1gk.minimizeXz2f�1;1gk(1�P(z))D(z)(1)s.t.Xz2f�1;1gkD(z)zS=2k
bD(S)=08S[k]0jSjt(2)Xz2f�1;1gkD(z)=1(3)D(z)08z2f�1;1gk Constraint(3)andthenonnegativityconstraintensurethatDisaprobabilitydistributiononf�1;1gk.Constraint(2)expressesthatDist-wiseuniform(seeRemark3.12).Theobjective(1)isminimiz-ingtheprobabilitymassthatDputsonassignmentsinP�1(0).ThustheoptimalvalueoftheLPisequaltothesmallest suchthatPis -closetot-wisesupporting;equivalently,thelargest suchthatPis -farfromt-wisesupporting.ThefollowingisthedualoftheaboveLP.Ithasavariablec(S)foreach0jSjtaswellasavariablecorrespondingtoconstraint(3).maximize(4)s.t.XS[k]0jSjtc(S)zS1�P(z)�8z2f�1;1gk:(5) Observethatafeasiblesolution(fc(S)gS;)ispreciselyequivalenttoamultilinearpolynomialQofdegreeatmostt,namelyQ(z)=�PSc(S)zS,that-separatesP.ThusPis-farfromt-wisesupportingifandonlyiftheLP'sobjective(1)isatleast,ifandonlyifthedual'sobjective(4)isatleast,ifandonlyifthereisa-separatingpolynomialforPofdegreeatmostt. FromthisproofwecanalsoderivethatifPfailstobet-wisesupportingthenitmustinfactbe k(1)-farfrombeingt-wisesupporting:Corollary3.17.SupposeP:f�1;1gk!f0;1gisnott-wisesupporting.Thenitisinfact-farfromt-wisesupportingfor=2�eO(kt)(or=2�eO(2k)whent=k).Proof.LetK=1+Pti=1�kt
bethenumberofvariablesinthedualLPfromLemma3.16,soKkt+1ingeneral,withK2kwhent=k.Byassumption,theobjective(4)ofthedualLP'soptimalsolutionisstrictlypositive.Thisoptimumoccursatavertex,whichisthesolutionofalinearsystemgivenbyaKKmatrixof1entriesanda\right-handside"vectorwith0;1entries.ByCramer'srule,thesolution'sentriesareratiosofdeterminantsofintegermatriceswithentriesinf�1;0;1g.Thusanystrictlypositiveentryisatleast1=N,whereNisthemaximumpossiblesuchdeterminant.ByHadamard'sinequality,N=KK=2andtheclaimedresultfollows. 12 4Quasirandomnessanditsimplicationsforrefutation4.1Strongrefutationofk-XORInthissection,westateourresultonstrongrefutationofrandomk-XORinstanceswithm=~O�nk=2
constraints.(RecallthatessentiallythisresultwasveryrecentlyobtainedbyBarakandMoitra[BM15].)Weactuallyhaveaslightlymoregeneralresult,allowingvariablesandcoecientstotakevaluesin[�1;1]andnotjustinf�1;1g.WewillusethisadditionalfreedomtoproverefutationresultsforCSPsoverlargeralphabetsinAppendixBandrefutationresultsforindependencenumberandchromaticnumberofrandomhypergraphsinAppendixC.Theorem4.1.Fork2andpn�k=2,letfw(T)gT2[n]kbeindependentrandomvariablessuchthatforeachT2[n]k,E[w(T)]=0(6)Pr[w(T)6=0]p(7)jw(T)j1:(8)ThenthereisanecientalgorithmcertifyingthatXT2[n]kw(T)xT2O(k)p pn3k=4log3=2n:forallx2Rnsuchthatkxk11withhighprobability.Inthisform,thetheoremisnotreallyaboutCSPrefutationatall.Itsaysthatthevalueofapolynomialwithrandomcoecientsisclosetoitsexpectationwhenitsinputsarebounded.WegivetheproofinAppendixA.Itfollowstechniquesfrom[COGL07]fairlycloselyandisessentiallythesameastheproofof[BM15].Wewillusethistheoremtoproveourresultsinsubsequentsections.Weobtainstrongrefutationofk-XORasasimplecorollary.Corollary4.2.Fork2,letIFk-XOR(n;p).Then,withhighprobability,thereisadegree-2kSOSproofthatOpt(I)1 2+ when m2O(k)nk=2log3n 2.Proof.Wecanwritethek-XORpredicateask-XOR(z)=1�Qki=1zi 2;soforak-XORinstanceIFk-XOR(n;p),ValI(x)=1 2�1 2mXT2[n]kXc2f1gk1f(T;c)2IgxTYi2[k]ci=1 2+2k�1 mXT2[n]kw(T)xT;wherew(T)=�2�kPc2f1gk1f(T;c)2IgQi2[k]ci.Thew(T)'sarerandomvariablesdependingonthechoiceofI;observethatE[w(T)]=0,Pr[w(T)6=0]2kp,andjw(T)j1forallT2[n]k.ByTheorem4.1,thereisanalgorithmcertifyingthatOpt(I)1 2+2O(k)p pn3k=4log3=2n m:withhighprobabilitywhenpn�k=2.Sincem=(1+o(1)) mwithhighprobability,choosing m2O(k)nk=2log3n 2givesthedesiredresult. Asanexample,wecanchoose =1 lognandcertifythatOpt(I)1 2+o(1)when m=eOk(nk=2).13 4.2Quasirandomnessandstrongrefutationofanyk-CSPNext,wewillusethealgorithmofTheorem4.1tocertifythataninstanceofCSP(P)isquasirandom.Thiswillimmediatelygiveusastrongrefutationalgorithm.Inordertocertifyquasirandomness,Lemma3.13impliesthatitsucestocertifyeachFouriercoecientofDI;xhassmallmagnitude.Lemma4.3.Let;6=S[k]withjSj=s.Thereisanalgorithmthat,withhighprobability,certiesthatdDI;x(S)2O(s)maxfns=4;p nglog5=2n m1=2forallx2f�1;1gn,assumingalsothat mmaxfns=2;ng.Toprovethislemma,weneedanotherlemmacertifyingthatpolynomialswhosecoecientsaresumsof0-meanrandomvariableshavesmallvalue.Lemma4.4.LetS[k]withjSj=s�0.Let2NandletfwU(i)gU2[n]s;i2[]beindependentrandomvariablessatisfyingconditions(6),(7),and(8)forsomep1 ns=2.ThenthereisanalgorithmthatcertieswithhighprobabilitythatXU2[n]sxUXj=1wU(j)(2O(s)p p
n3s=4log5=2nifs24maxfp p;1g
nlognifs=1.forallx2Rnsuchthatkxk11.TheproofisstraightforwardandwedeferittoSection4.4.ProofofLemma4.3.Withoutlossofgenerality,assume12S.Applyingdenitions,weseethat[DI;x(S)=EzDI;xzS=1 mXU2[n]sXT2[n]kTS=UXc2f1gk1f(T;c)2IgcSxU=1 mXU2[n]sxUXT2[n]kTS=UXc02f1gk�1wS(T;c0):(9)wherewedenewS(T;c0)=1f(T;(1;c0))2Ig(c0)Snf1g�1f(T;(�1;c0))2Ig(c0)Snf1gandrecallthatTSistheprojectionofTontothethecoordinatesinS.ItisclearthatE[wS(T;c0)]=0,Pr[wS(T;c0)6=0]p,andjwS(T;c0)j1.Thereare=2k�1nk�stermsineachsumofwS(T;c0)'sandwecanapplyLemma4.4.Whens=2,wepluginthesevaluesandseethatwecancertifythat[DI;x(S)2O(s)ns=4log5=2n m1=2.Whens=1, mnimpliesthatp1 2andwecancertifythat[DI;x(S)2O(s)p nlogn m1=2.Thelowerboundcanbeprovedinexactlythesamewaybyconsideringtherandomvariables�wS(T;c0). TheexistenceofanalgorithmforcertifyingquasirandomnessfollowsfromLemmas3.13and4.3.Theorem4.5.ThereisanecientalgorithmthatcertiesthataninstanceIFP(n;p)ofCSP(P)is -quasirandomwithhighprobabilitywhen m2O(k)nk=2log5n 2.Since -quasirandomessimpliesthatOpt(I) P+ ,thisalgorithmalsostronglyrefutesCSP(P).Theorem4.6.Thereisanecientalgorithmthat,givenaninstanceIFP(n;p)ofCSP(P),certiesthatOpt(I) P+ withhighprobabilitywhen m2O(k)nk=2log5n 2.14 4.3(;t)-quasirandomnessand (1)-refutationofnon-t-wise-supportingCSPsInthecasethatapredicateisnott-wisesupporting,aweakernotionofquasirandomnesssucestoobtain (1)-refutation.Denition4.7.AninstanceIofCSP(P)is(;t)-quasirandomifDI;xis(;t)-wiseuniformforeveryx2f�1;1gn.Fact3.6showsthatrandominstanceswitheO(n)constraintsare(o(1);t)-quasirandomforalltkwithhighprobability.Lemma4.3directlyimpliesthatwecancertify(;t)-quasirandomnesswhenmeO(nt=2).Theorem4.8.ThereisanecientalgorithmthatcertiesthataninstanceIFP(n;p)ofCSP(P)is( ;t)-quasirandomwithhighprobabilitywhen m2O(t)nt=2log5n 2andt2.Wenowreachthemainresultofthissection,whichstatesthatifapredicateis-farfromt-wisesupporting,thenwecanalmost-refuteinstancesofCSP(P).Theorem4.9.LetPbe-farfromt-wisesupporting.Thenthereisanecientalgorithmthat,givenaninstanceIFP(n;p)ofCSP(P),certiesthatOpt(I)1�+ withhighprobabilitywhen mkO(t)nt=2log5n 2andt2.Wegivetwoproofsofthistheorem.InProof1,thetheoremfollowsdirectlyfromcerticationof( ;t)-quasirandomnessandLemma3.13.Proof1.RunthealgorithmofTheorem4.8tocertifythatIis( =kt;t)-quasirandomwithhighprobability.Bydenition,wehavecertiedthatDI;xis( =kt;t)-wiseuniformforallx2f�1;1gn,.Lemma3.13thenimpliesthatforallxthereexistsat-wiseuniformdistributionD0xsuchthatdTV(DI;x;D0x) .NowdeneDsattobeanarbitrarydistributionoversatisfyingassignmentstoP.SincePis-farfrombeingt-wisesupporting,weknowthatdTV(D0;Dsat)foranyt-wiseuniformdistributionD0.ThetriangleinequalitythenimpliesthatdTV(DI;x;Dsat)� forallx2f�1;1gnandthetheoremfollows. Proof2givesaslightlyweakerversionofTheorem4.9,requiringthestrongerassumptionthat m2O(k)nt=2log5n 2.Itisbasedonthedualpolynomialcharacterizationofbeing-farfromt-wisesupporting.WhileperhapslessintuitivethanProof1,Proof2ismoredirect.ItonlyusestheXORrefutationalgorithmandbypasses[AGM03]'sconnectionbetween(;t)-wiseuniformityand-closenesstoat-wiseuniformdistribution.WewereabletoconvertProof2intoanSOSproof(seeSection6.4),butwedidnotseehowtotranslateProof1intoanSOSversion.Proof2requiresPlancherel'sTheorem,afundamentalresultinFourieranalysis.Theorem4.10(Plancherel'sTheorem).Foranyf;g:f�1;1gk!R,Ez2Uk[f(z)g(z)]=XS[k]bf(S)bg(S):Proof2.SincePis-farfromt-wisesupporting,thereexistsadegree-tpolynomialQthat-separatesP.Thedenitionof-separatingimpliesthatP(z)�(1�)Q(z)forallz2f�1;1gk.Summingoverallconstraints,wegetthatforallx2f�1;1gn,XT2[n]kXc2f1gk1f(T;c)2IgP(xTc)�m(1�)XT2[n]kXc2f1gk1f(T;c)2IgQ(xTc);15 or,equivalently,ValI(x)�(1�)Ez2DI;x[Q(z)].ItthenremainstocertifythatEz2DI;x[Q(z)] .ObservethatEz2DI;x[Q(z)]=Ez2Uk[DI;x(z)Q(z)]=X;6=S[k][DI;x(S)bQ(S);wherethesecondequalityfollowsfromPlancherel'sTheorem.SinceQ�1andE[Q]=0,Q2kandhencejbQ(S)j2kforallS.Tonishtheproof,weapplyLemma4.3tocertifythat[DI;x(S) 22kforallS. WithCorollary3.17,Theorem4.9impliesthatwecan k(1)-refuteinstancesofCSP(P)witheOk(nt=2)constraintswhenPisnott-wisesupporting.Corollary4.11.LetPbeapredicatethatdoesnotsupportanyt-wiseuniformdistribution.Thenthereisanecientalgorithmthat,givenaninstanceIFP(n;p)ofCSP(P),certiesthatOpt(I)1�2�eO(kt)withhighprobabilitywhen m2eO(kt)nt=2log5nandt2.4.4ProofofLemma4.4Recallthestatementofthelemma.Lemma4.4.LetS[k]withjSj=s�0.Let2NandletfwU(i)gU2[n]s;i2[]beindependentrandomvariablessatisfyingconditions(6),(7),and(8)forsomep1 ns=2.ThenthereisanalgorithmthatcertieswithhighprobabilitythatXU2[n]sxUXj=1wU(j)(2O(s)p p
n3s=4log5=2nifs24maxfp p;1g
nlognifs=1.forallx2Rnsuchthatkxk11.TheproofusesBernstein'sInequality.Theorem4.12(Bernstein'sInequality).LetX1;:::;XMbeindependent0-meanrandomvariablessuchthatjXijB.Then,fora�0,Pr"MXi=1Xi�a#exp �1 2a2 PMi=1E[X2i]+1 3Ba!:ProofofLemma4.4.First,wedenevU=Xj=1wU(i):ObservethatthevU'sareindependentandthateachoneisthesumofmean-0,i.i.d.randomvariableswithmagnitudeatmost1.NotingthatPi=1E[wU(i)2]p,wecanuseBernstein'sInequalitytoshowthatthejvUj'sarenottoobigwithhighprobability.Ifs2,Theorem4.1thenimpliesthatthedesiredalgorithmexists.Ifs=1,wearesimplyboundingalinearfunctionover1variables.Weconsidertwocases:Smallpandlargep.16 Case1:p1 4.Choosinga=2slogninBernstein'sInequality,weseethatPr[jvUj2slogn]n�2s.AunionboundoverallUthenimpliesthatPr[anyjvUj�2slogn]n�s.Ifs2,weobservethatPr[vU6=0]p,scalethevU'sdownby2slogn,andapplyTheorem4.1togetthestatedresult.Ifs=1,weobtainthesecondboundbyobservingthatXi2[n]vixiXi2[n]jvij2nlogn:(10)Case2:p�1 4.Weseta=4sp plognandgetthatPr[anyjvUj�4sp plogn]n�sasabove.Ifs2,wecanthendividethevU'sby4sp plognandapplyTheorem4.1.Ifs=1,wegetaboundof4p p
nlogninthesamewayas(10). 5HardnessoflearningimplicationsRecentworkbyDanielyetal.[DLSS14]reducestheproblemofrefutingspecicinstancesofCSP(P)totheproblemofimproperlylearningcertainhypothesisclassesintheProbablyApproximatelyCorrect(PAC)model[Val84].Inthismodel,thelearnerisgivenmlabeledtrainingexamples(x1;`(x1));:::;(xm;`(xm)),whereeachxi2f�1;1gn,each`(xi)2f0;1g,andtheexamplesaredrawnfromsomeunknowndistributionDonf�1;1gnf0;1g.ForsomehypothesisclassHf0;1gf�1;1gnwesaythatDcanberealizedbyHifthereexistssomeh2HsuchthatPr(x;`(x))D[h(x)6=`(x)]=0.InimproperPAClearning,onaninputofmtrainingexamplesdrawnfromDsuchthatDcanberealizedbysomeh2H,andanerrorparameter,thealgo-rithmoutputssomehypothesisfunctionfh:f�1;1gn!f0;1g(notnecessarilyinH)suchthatPr(x;`(x))D[fh(x)6=`(x)].InimproperagnosticPAClearning,theassumptionthatDcanberealizedbysomeh2HisremovedandthealgorithmmustoutputahypothesisthatperformsalmostaswellasthebesthypothesisinH.Moreformally,thehypothesisfhmustsatisfythefollow-ing:Pr(x;`(x))D[fh(x)6=`(x)]minh2HPr(x;`(x))D[h(x)6=`(x)]+.InimproperapproximateagnosticPAClearning,thelearnerisalsogivenanapproximationfactora1andmustoutputahypothesisfhsuchthatPr(x;`(x))D[fh(x)6=`(x)]a
minh2HPr(x;`(x))D[h(x)6=`(x)]+.Danielyetal.reducetheproblemofdistinguishingbetweenrandominstancesofCSP(P)andinstanceswithvalueatleastasaPAClearningproblembytransformingeachconstraintintoalabeledexample.ToshowhardnessofimproperlylearningacertainhypothesisclassinthePACmodel,theydeneapredicatePthatisspecictothehypothesisclassandassumehardnessofdistinguishingbetweenrandominstancesofCSP(P)andinstanceswithndconstraintsandvalueatleastforalld�0.Theythendemonstratethatthesamplecanberealized(orapproximatelyrealized)bysomefunctioninthehypothesisclassiftheCSPinstanceissatisable(orhasvalueatleast).TheyalsoshowthatifthegivenCSPinstanceisrandom,thesetofexampleswillhaveerroratleast1 4(intheagnosticcase1 5)forallhinthehypothesisclasswithhighprobability.Usingthisapproach,theyobtainhardnessresultsforthefollowingproblems:improperlylearningDNFformulas,improperlylearningintersectionsof4halfspaces,andimproperlyapproximatelyagnosticallylearninghalfspacesforanyapproximationfactor.5.1HardnessassumptionsThehardnessassumptionsmadein[DLSS14]arethesameasthosepresentedinSection2.1,exceptforafewminordierences.First,theirmodelxesthenumberofconstraintsratherthantheprobabilitywithwhicheachconstraintisincludedintheinstance.Itiswell-knownthatresultsinonemodeleasilytranslatetotheother.WeincludeaproofinAppendixDforcompleteness.17 Additionally,SRCSPAssumptions1and2purporthardnessofdistinguishingrandominstancesofCSP(P)fromsatisableinstances,evenwhenthealgorithmisallowedtoerrwithprobability1 4overitsinternalcoins.Thealgorithmspresentedintheprecedingsectionsnevererronsatisableinstances;further,theyonlyfailtocertifyrandominstanceswithprobabilityo(1).Asaresult,ourrefutationalgorithmsalsofalsifyweakerversionsofbothSRCSPAssumptions,whereintheallowedprobabilityoferrorisbothlowerandone-sided.Foreachpredicatepresentedin[DLSS14],wefalsifytheappropriateSRCSPassumptionusingthefollowingapproach.ForeachpredicatePandcorresponding�0,wedeneadegree-tpolynomialthat-separatesP.Usingtherefutationtechniquespresentedintheprecedingsections,wededucethateO(nt=2)constraintsaresucienttodistinguishrandominstancesofCSP(P)fromthosethataresatisable(orhavevalueatleast).Inordertosimplifythepresentation,webeginwithsimplerversionsofthepolynomialsandthenscalethemtoattaintheappropriatevaluesof.Thefollowinglemmawillbeofuseforthisscaling.Lemma5.1.ForpredicateP:f�1;1gk!f0;1g,letQ:f�1;1gk!Rbeanunbiasedmultilinearpolynomialofdegreetsuchthatthereexist1�0;00notdependentonzforwhichthefollowingholds:Q(z)1forallz2P�1(1)andQ(z)0forallz2f�1;1gk.Thenthereexistsadegree-tpolynomialQ:f�1;1gk!Rthat1 1�0-separatesP.Proof.DeneQ(z)=Q(z) 1�0.ClearlyQisalsounbiasedandhasdegreet.Thenforallz2P1,Q(z) 1�01 1�0.Similarly,forallz,Q(z) 1�00 1�0=�1�0 1�0+1 1�0=�1+1 1�0. Wenowdemonstratethattheabovecanbeappliedtothepredicatessuggestedin[DLSS14]bydeningseparatingpolynomialsandapplyingTheorem4.95.2Huang'spredicateandhardnessoflearningDNFformulasInordertoobtainhardnessofimproperlylearningDNFformulaswith!(1)terms,Danielyetal.usethefollowingpredicate,introducedbyHuang[Hua13].Huangshowedthatitishereditarilyapproximationresistant;Danielyetal.alsoobservedthatits0-variabilityis (k1=3)[DLSS14].Denition5.2.Letk=+�3
forsomeinteger3.Forz2f�1;1gk,indexzasfollows.Labeltherstbitsofzasz1;:::;z.Theremaining�3
bitsareindexedbyunorderedtriplesofintegersbetween1and.EachT[]withjTj=3isassociatedwithadistinctbitoftheremaining�3
bits,whichisindexedbyzT.WesaythatzstronglysatisestheHuangpredicateiforeveryT=fzi;zj;z`gsuchthatzi;zj;z`aredistinctelementsof[],zizjz`=zfi;j;`g.Additionally,wesaythatzsatisestheHuangpredicateithereexistssomez02f�1;1gksuchthatzhasHammingdistanceatmostfromz0andz0stronglysatisestheHuangpredicate.DeneH:f�1;1gk!f0;1gasfollows:H(z)=1ifzsatisestheHuangpredicateandH(z)=0otherwise.Danielyetal.reducetheproblemofdistinguishingbetweenrandominstancesofCSP(H)with2ndconstraintsandsatisableinstancestotheproblemofimproperlyPAClearningtheclassofDNFformulaswith!(1)termsonasampleofO(nd)trainingexampleswitherror=1=5withprobabilityatleast3 4.HereweshowthatthereexistsapolynomialtimealgorithmthatrefutesrandominstancesofCSP(H)bydemonstratingthatHkdoesnotsupporta4-wiseuniformdistributionandapplyingTheorem4.9.Theorem5.3.Assume9.Thereexistsadegree-4polynomialQ:f�1;1gk!Rthat1 8-separatesH.Consequently,His1 8-farfromsupportinga4-wiseuniformdistribution.18 Proof.Asanotationalshorthand,writezabcforzfia;ib;icg.Dene:[]6f�1;1gk![�5;5]asfollows:(i1;i2;i3;i4;i5;i6;z)=z126z134z235z456+z256z146z345z123+z136z236z145z245+z124z234z356z156+z125z135z346z246:(11)ObservethatforeachmonomialzT1zT2zT3zT4of,foreveryj2[6],P4i=11fTi3jg=2.Further,foreachT[6]withjTj=3,zTappearsexactlyoncein.LetZ6bethesetofallordered6-tuplesofdistinctelementsof[].ForanorderedtupleI,weuse2()todenotemembershipinI.DeneQ:f�1;1gk!Rasfollows.OurnalpolynomialQwillbeascaledversionofQ.Q(z)=avgI2Z6(I;z):ObservethatQdoesnotdependonanyofzf1g;:::zfg.Byconstruction,Qcontainsnoconstantterm,sobQ(;)=0.ClearlyQ(z)�5forallzbecause(11)isalwaysatleast�5.NowwelowerboundthevalueofQonallzthatsatisfyH.Werstshowthatforanyz0thatstronglysatisestheHuangpredicate,Q(z0)=5,thenboundQ(z0)�Q(z)foranyzwithHammingdistanceatmostfromz0.Bydenition,foreachz0Ti,wehavethatz0TiQj2Tiz0j=1.SoforeachmonomialofQ,1 jZ6jz0T1z0T2z0T3z0T4=1 jZ6j4Yi=1Yj2Tiz0j=1 jZ6jYj2T1[T2[T3[T4(z0j)2=1 jZ6j;wherethelastlinefollowsfromthefactthatP41=11fTi3jg=2:Becausethereare5
jZ6jmonomialsinQ,theirsumis5.NowweconsiderthecasewherezdoesnotstronglysatisfytheHuangPredicate,butH(z)=1.Anysingletonindexonwhichzandz0dierwillnotchangethevalueofQ.LetN=fT:zT6=z0Tg.WelowerboundQbycountingthenumberofmonomialsinwhicheachzTappearsandQ(z)5�2 jZ6jXT2NXI2Z1fVzi2Ti2()Ig:ForxedT,thenumberofmonomialscontainingthevariablesofzTisXI2Z1fVzi2Ti2()Ig=120(�3)(�4)(�5)becausethereareexactly120waystopermutethethreeindicesofTinIandtheremaining�3indicesarepermutedintheremaining3positionsofI.SoQ(z)5�240 jZ6j(�3)(�4)(�5)=5�240 (�1)(�2):(12)For9,(12)isatleast5�30 7.ApplyingLemma5.1,thereexistsQ:f�1;1gk!Rthat1 8-separatesH. Fromthisandthefactthat H=2~O(k1=3)�k(see[Hua13]),weobtainthefollowingcorollary.19 Corollary5.4.Forsucientlylargenandk93,thereexistsanecientalgorithmthatre-futesrandominstancesofCSP(H)witheO(n2)constraintswithhighprobability.ThisfalsiesAssumption2.1inthecaseoftheHuangpredicate.Remark5.5.IfweinsteadchoosetoscaleQbyafactorof1 5
2�3+2 22�6�44ratherthansubstituting=9into(12),wecanachieveabetterseparationof=2�3�46 22�6�44.For9,thisexpressionisstrictlyincreasinganditapproaches1 2asgrows.5.3HammingweightpredicatesTheremainingpredicateswewouldliketoexaminearesymmetric,meaningtheyarefunctionsonlyoftheirHammingweights.AgainforeachpredicatePwepresentamultivariatepolynomialthat-separatesPforsome01.EachofthesepolynomialscanalsobewrittenasaunivariatepolynomialontheHammingweightofitsinput,whichwewillusetoshowthateachofthefollowingpolynomials-separatesitspredicatefortheappropriatevalueof.Wegivetheconstructionbelow.Denition5.6.Forz2f�1;1gkwherez=z1;:::;zk,deneSz=Pki=1ziandcallSztheHammingweightofz.NotethatthisisanalogoustothenotionofaHammingweightofavectorinf0;1gk,butdiersinthatitisnotsimplythecountofthenumberof1's.WedeneageneralpredicatethatissatisedwhenSzisatleastsomexedthresholdvalue.Denition5.7.Foralloddkandany2f�k;�k+2;:::;k�2;kg,denethepredicateThrk:f�1;1gk!f0;1gasfollows:Thrk(z)=(1ifSz0otherwiseForexample,MajkisthesameasThr1kandThr�kkisthetrivialpredicatesatisedbyallz2f�1;1gk.Becausethemultilinearseparatingpolynomialswewillusearesymmetric,wepresentatrans-formationtoanequivalentunivariatepolynomialontheHammingweightoftheoriginalinput.Lemma5.8.LetQ:f�1;1gk!Rbeofthefollowingformforsomea;b;c;d2R:Q(z)=aXT[n]jTj=1zT+bXT[n]jTj=2zT+cXT[n]jTj=3zT+dXT[n]jTj=4zT:(13)DeneQu:R!Rasfollows:Q(z)=d 24S4z+c 6S3z+b 2+d 3�dk 4S2z+a+c 6
(�3k+2)Sz�bk 2+dk 24(3k�6):ThenQ(z)=Qu(Sz)forallz2f�1;1gk.Proof.Wecanwrite(13)asfollows:Q(z)=aK1k�Sz 2;k+bK2k�Sz 2;k+cK3k�Sz 2;k+dK4k�Sz 2;k;(14)20 whereKi(;k)=Pij=0(�1)j�i
�k�i�j
denotestheKrawtchoukpolynomialofdegreei[Kra29,KL96].Substituting=k�Sz 2,yieldsthefollowingexpressions.In[KL96]therstthreeexpressionsaregivenexplicitlyandthefourthcanbeeasilyobtainedbyapplyingtheirrecursiveformula.K1k�Sz 2;k=Sz;K3k�Sz 2;k=S3z�(3k�2)Sz 6;K2k�Sz 2;k=S2z�k 2;K4k�Sz 2;k=S4z+(8�6k)S2z+3k2�6k 24:Finally,substitutingtheseexpressionsinto(14)andbysomealgebra,Q(z)=d 24S4z+c 6S3z+b 2+d 3�dk 4S2z+a+c 6
(�3k+2)Sz�bk 2+dk 24(3k�6):(15) Asaconsequence,bychoosingvaluesofa;b;c;andd,wecanworkwithaunivariatepolynomialwhileensuringthatitsmultivariateanalogueisunbiasedandhasdegreeatmost4(degree3whend=0).5.3.1Almost-MajorityandhardnessoflearningintersectionsofhalfspacesDenition5.9.Danielyetal.denethefollowingpredicateinordertoshowhardnessofimproperlylearningintersectionsoffourhalfspaces.I8k= 3^i=0Thr�1k(zki+1:::zki+k)!^: 7^i=4Thr�1k(zki+1:::zki+k)!:Thereductionreliesontheassumptionthatforalld�0,itishardtodistinguishrandominstancesofCSP(I8k)withndconstraintsfromsatisableinstances.BecausetheinputvariablestoeachinstanceofThr�1kabovearedisjoint,itissucienttoshowthateachoftherstfourgroupsofkvariablescannotsupporta3-wiseuniformdistributionandconsequentlyneithercanI8k;therefore,fromTheorem4.9wededucethatthereexistsanecientalgorithmthatrefutesrandominstancesofCSP(I8k)witheO(n3=2)constraintswithhighprobability.Danielyetal.deneapairwiseuniformdistributionsupportedonI8kaswellasapairwiseuniformdistributionsupportedonThr�1k,sot=3isoptimal.Theorem5.10.Assumek5andkisodd.Thereexist=(k)�0whereis (k�4)andadegree-3multilinearpolynomialQ:f�1;1gk!Rthat-separatesThr�1k.Consequently,Thr�1kdoesnotsupporta3-wiseuniformdistribution.Proof.LetQ(z)=(k2�k�1)XT[n]jTj=1zT+(1�k)XT[n]jTj=2zT+(1+k)XT[n]jTj=3zTanddeneQu:R!Rasfollows:Qu(s)=1+k 6s3+1�k 2s2+k2�k�1+1+k 6
(�3k+2)s�(1�k)k 2=1+k 6s3+1�k 2s2+3k2�7k�4 6s�(1�k)k 2:21 ThenbyLemma5.8,forallz2f�1;1gk,Q(z)=Qu(Sz).ItthereforesucestolowerboundQu(s)bothwhens�1andforalls2[�k;k].FirstweshowthatQuismonotonicallyincreasingins.dQu ds=k+1 2s2+(1�k)s+3k2�7k�4 6=1 6h(k�4)�3(s�1)2+2 3+3k
+15�s�3 5
2+53 75i;whichisevidentlypositivefork5.BecauseQismonotonicallyincreasingins,Qu(s)Qu(�k)foralls2[�k;k].Qu(�k)=�k�1 6k3+1�k 2k2�3k2�7k�4 6k�(1�k)k 2=�1 6k4+7k3�13k2�k;(16)=�1 6k(k�2)(k2+9k+5)+9k;(17)whichisclearlynegativefork5.Nowitjustremainstolower-boundQu(s)fors�1.Again,sinceQuismonotonicallyincreasingins,weusethevalueQu(�1):Qu(�1)=�k�1 6+�1�k 2
�3k2�7k�4 6�(1�k)k 2=1:ByapplyingLemma5.1,thereexistsanunbiasedmultilinearpolynomialQ:f�1;1gk!Rofdegree3that6 k4+7k3�13k2�k+6-separatesThr�1k. BecauseVAR0(I8k)isevidently (k)and I8k1 7forallk5,wehavethefollowingCorollary.Corollary5.11.Foroddk5andsucientlylargen,thereexistsanecientalgorithmthatdis-tinguishesbetweenrandominstancesofCSP(I8k)witheO(n3=2)constraintsandsatisableinstanceswithhighprobability.Remark5.12.Thr�13isthesameas3-ORandThr�15isthesameasisthesameas2-out-of-5-SAT,sothisapproachcanbeusedto k(1)-refute3-SATinstancesand2-out-of-5-SATinstanceswitheOk(n3=2)constraints,whichimprovesupontheO(n3=2+)constraintsrequiredforrefutationof2-out-of-5-SATin[GJ02,GJ03].5.4MajorityandhardnessofapproximatelyagnosticallylearninghalfspacesDanielyetal.showthatapproximateagnosticimproperlearningofhalfspacesishardforallap-proximationfactors1basedontheassumptionthatforalld�0andforsucientlylargeoddk,itishardtodistinguishbetweenrandominstancesofCSP(Thr1k)withndconstraintsandinstanceswithvalueatleast1�1 10.ThisisbasedonthefactthatmaxDEzD[Thr1(z)]=1�1 k+1,whereDisapairwiseindependentdistributiononf�1;1gk,andapplyingSRCSPAssumption2.Hereweshowthatforoddk25,Thr1kis0:1-farfromsupportinga4-wiseuniformdistribution.Thevalue0:1isnotsharp,butischosenasacompromisebetweenareasonablylargevalueandareasonablysimpleproof.Theorem5.13.Thereexistsadegree-4multilinearpolynomialQ:f�1;1gk!Rthat0:1-separatesThr1kforalloddk25.22 Proof.LetQ(z)=8 27p kXT[n]jTj=1zT�5 9k3=2XT[n]jTj=3zT+4 3k2XT[n]jTj=4zTandletQu(s)=1 18k2s4�5 54k3=2s3+�1 3k+4 9k2s2+31 54p k�5 27k3=2s+1 6�1 3k(18)=1 543 k2s4�5 k3=2s3+�18 k+24 k2s2+31 p k�10 k3=2s+9�18 k:ThenbyLemma5.8,forallz2f�1;1gk,Q(z)=Qu(Sz).TosimplifyQ,let=sk�1=2.Thenwecanrewrite(18)asfollows:Qu(s)=1 5434�53+��18+24 k
2+�31�10 k
+9�18 k:(19)Firstwelower-boundQu(s)forall2Rusingthefollowingexpression,whichisequivalentto(19)bysomealgebra.Qu(s)=1 54h3(+29 18)2(�22 9)2+383 108�+1832 1149
2�38987378 837621+24 k�(�5 24)2�457 576
i�1 54h3(+29 18)2(�22 9)2+383 108�+1832 1149
2�47+24 k��457 576
i��48 54=�8 9;wherethelastinequalityfollowsfromthefactthatk24andthersttwotermsarealwaysnonnegative.Nextwelower-boundQu(s)fors�0.Qu(s)=1 5434�53+��18+24 k
2+�31�10 k
+9�18 k=1 543(�1 4)2(�25 12)2+41 120((�839 410)2+21507 1344800+27 4+9(�21 10)2�1 8:ApplyingLemma5.1,thereexistsQ:f�1;1gk!RsuchthatQhasdegree4andQ9 73-separatesThr1k. Corollary5.14.Forsucientlylargenandk,thereexistsanecientalgorithmthatdistinguishesbetweenrandominstancesofCSP(Thr1k)witheO(n2)constraintsandinstanceswithvalueatleast0:9withhighprobability.5.5PredicatessatisedbystringswithHammingweightatleast�(p k).Inlightofthefactthatthethresholdbasedpredicatesabovearenot4-wisesupporting,onemayattempttondanotherthreshold-basedpredicate.Hereweshowthatasymmetricthresholdpredicatethatis4-wisesupportingmustbesatisedbyallstringswithHammingweightatleast�p k 2.Furthermore,thereexistsasymmetricthresholdpredicatethatis4-wisesupportingwithathresholdof�(p k)andwesketchitsconstruction.WealsoconsiderthepredicateThr�1 2p kk.Whileitisnotusedin[DLSS14],weshowthatitdoesnotsupporta4-wiseuniformdistributionintheinterestofobtainingatighterboundfortheHammingweightabovewhichanunbiased,symmetricpredicateisnot4-wisesupporting.Thethresholdof�1 2p kisparticularlyinterestinginthatitasymptoticallymatchesthethresholdbelowwhichThrkis4-wisesupporting.23 Theorem5.15.Assumek99andkisodd.Thenthereexistsadegree-4polynomialQ:f�1;1gk!Rthat1 225-separatesThr�1 2p kk.Consequently,Thr�1 2p kkis1 255-farfrom4-wisesupport-ing.Proof.DeneQ:f�1;1gk!RandQu:R!Rasfollows:Q(z)=3 2k�1=2XT[n]jTj=1zT+1 2k�1XT[n]jTj=2zT+2k�3=2XT[n]jTj=3zT+8k�2XT[n]jTj=4zTQu(s)=s4 3k2+s3 3k3=2+��7 4k+8 3k2
s2+1 2k1=2+2 3k3=2s+3 4�2 kAgain,forsimplicityweset=sk�1=2andobtainthefollowingexpression:Qu(s)=1 34+1 33�7 42+1 2+3 4+1 k�8 32+2 3�2
:(20)Observethatfork99,1 k�8 32+2 3�2
=2 3k�2�1 4
2�49 16��1 48.Wenowlower-boundthevalueofQufors�1 2k1=2,orequivalently,�1 2:Qu(s)=1 34+1 33�7 42+1 2+3 4+1 k�8 32+2 3�2
=1 3��35 29
2�+1 2
�+200 69
+61 12006�+1 2
�+4631 3538
2+1526073 12517444+1 k�8 32+2 3�2
+1 24:Thersttwotermsareclearlynonnegativewhen�1 2,so�1 24�1 48=1 48:WealsoshowthatQu(s)�14 3foralls2R.Qu(s)=1 34+1 33�7 42+1 2+3 4+1 k�8 32+2 3�2
=�+19 9
2��29 18
2+211 486�+397 211
2+195823 8306226+1 k�8 32+2 3�2
�14 3�+19 9
2��29 18
2+211 486�+397 211
2+1079081 365473944�14 3:Therstthreetermsarealwaysnonnegative,soQu(s)�14 3.ApplyingLemma5.1,T�1 2p kkis1 255-farfromsupportinga4-wiseuniformdistribution. Nowwedemonstratethatthereexistsa4-wiseuniformdistributionsupportedonThr1�2p k+1kwhenk=2m�1forsomeintegerm3.Claim5.16.Assumek=2m�1forsomeintegerm3.Thenthereexistsa4-wiseuniformdistributionsupportedonlyonz2f�1;1gksuchthatSz1�2p k+1.Proof.LetCbeabinaryBCHcodeoflengthkwithdesigneddistance2+1andletC?beitsdual.ThentheuniformdistributiononthecodewordsofCis2-wiseuniform[ABI86,MS77];seealso[AS04,Ch16.2].Letc=c1:::ckbeacodewordofC?;whereeachci2f�1;1g.TheCarlitz-Uchiyamabound[MS77,page280]statesthatforallc2C?,kXi=11 2(1�ci)k+1 2+(�1)p k+1:24 Observethatthequantity1 2(1�ci)simplymapscifromf�1;1gtof0;1gsothatwecanwritetheboundtomatchthepresentationin[MS77].Therefore,Sc=kXi=1ci=k�2kXi=11 2(1�ci)k�(k+1)�(2�2)p k+1=�1�(2�2)p k+1:Setting=2,wecanobtain4-wiseuniformityonthisdistributionandeachstringinthesupportofthedistributionhasHammingweightatleast�1�2p k+1: Remark5.17.Inordertoconstructa4-wiseuniformdistributionforanyvalueofk,onecouldsimplyexpresskasasumofpowersof2,constructseparatedistributionsondisjointvariablesasdescribedaboveforeachpowerof2(downtotheminimumlengthforwhichwecanachievedistanceatleast5,afterwhichpointweusetheuniformdistribution,andobtaina4-wiseuniformdistribution.ThetotalHammingweightofavectorsupportedbythisdistributionwouldthenbeatleast�O(p k).6SOSrefutationproofs6.1TheSOSproofsystemWerstdenetheSOSproofsystemintroducedin[GV01].Callapolynomialq2R[X1;:::;Xn]sum-of-squares(SOS)ifthereexistq1;:::;q`2R[X1;:::;Xn]suchthatq=q21+


+q2`.Denition6.1.LetX=(X1;:::;Xn)beindeterminates.Letq1;:::;qm;r1;:::;rm02R[X]andletA=fq10;:::;qm0g[fr1=0;:::;rm0g.Thereisadegree-dSOSproofthatAimpliess0,writtenasA`ds0;ifthereexistSOSu0;u1;:::;um2R[X]andv1;:::;vm02R[X]suchthats=u0+mXi=1uiqi+m0Xi=1viriwithdeg(u0)d,deg(uiqi)dforalli2[m],anddeg(viri)dforalli2[m0].Ifitalsoholdsthatu0;u1;:::;um=0,wewillwriteA`ds=0.Itiswell-knownthatadegree-dSOSproofcanbefoundusinganSDPofsizenO(d)ifitexists[Sho87,Par00,Las00,Las01].Inthissection,wewilltakethesetAtobefx2i=1gi2[n],enforcingthatvariablesare1-valued.Weshowthatwithhighprobabilitythereexistsalow-degreeSOSproofthatapolynomialrepresentingthevalueofaCSPinstanceisclosetoitsexpectation.FormoreinformationontheSOSproofsystemanditsapplicationstoapproximationalgorithms,see,e.g.,[OZ13,Lau09].25 6.2SOScerticationofquasirandomnessAllofourSOSresultsrelyonthefollowingtheorem,whichistheSOSversionofTheorem4.1.Theorem6.2.Fork2andpn�k=2,letfw(T)gT2[n]kbeindependentrandomvariablessuchthatforeachT2[n]k,E[w(T)]=0(21)Pr[w(T)6=0]p(22)jw(T)j1:(23)Then,withhighprobability,fx2i1gi2[n]`2kXT2[n]kw(T)xT2O(k)p pn3k=4log3=2n:ThistheoremwasessentiallyprovenbyBarakandMoitra[BM15].WegiveaproofinAp-pendixA.3.WerstusethistheoremtoshowthatanSOSversionofLemma4.4holds.Lemma6.3.LetS[k]withjSj=s�0.Let2NandletfwU(i)gU2[n]s;i2[]beindependentrandomvariablessatisfyingconditions(6),(7),and(8)forsomep1 ns=2.Then,withhighprobability,fx2i1gi2[n]`2sXU2[n]sxUXj=1wU(j)(2O(s)p p
n3s=4log5=2nifs24maxfp p;1g
nlognifs=1.Proof.WesketchthedierencesfromtheproofofLemma4.4giveninSection4.4.Fors2,thelemmafollowsbyusingTheorem6.2insteadofTheorem4.1.Ifs=1,itsucestoshowthatfx2i1g`2v(i)xijv(i)j:foranyvsincesummingoveralliasin(10)nishestheproof.Ifvi0,observethatjvij�v(i)xi=jv(i)j 2(xi�1)2+jv(i)j 2(1�x2i):Ifv(i)0,weuse(xi+1)2insteadof(xi�1)2. ThelemmaimpliesanSOSversionofLemma4.3.Tomakethisprecise,wedeneaspecicpolynomialrepresentationof[DI;x(S):[DI;x(S)poly=1 mXT2[n]kXc2f1gk1f(T;c)2IgcSxST;wherexST=Qi2SxTi.Notethatthisisapolynomialinthexi's.Wecanshowthesepolynomialsarenottoolarge.Lemma6.4.Let;6=S[k]withjSj=s.Thenfx2i1gi2[n]`2sdDI;x(S)poly2O(k)maxfns=4;p nglog5=2n m1=2fx2i1gi2[n]`2sdDI;x(S)poly�2O(k)maxfns=4;p nglog5=2n m1=2:withhighprobability,assumingalsothat mmaxfns=2;ng.26 Proof.TheproofisessentiallythesameasthatofLemma4.3.Theexpressionweboundinthatproofisexactly[DI;x(S)poly.WeuseLemma6.3insteadofLemma4.4toshowthatthiscanbedoneindegree-2sSOS. BasedonLemma3.13,wewillthinkofLemma6.4asgivinganSOSproofofquasirandomness.Below,weuseittoproveSOSversionsofTheorems4.6and4.9.6.3Strongrefutationofanyk-CSPWenowdenethenaturalpolynomialrepresentationofValI(x):ValpolyI(x)=1 mXT2[n]kXc2f1gk1f(T;c)2Ig0@XS[k]bP(S)cSxST1A;wherexSTisasabove.WecanthengiveanSOSproofstronglyrefutingCSP(P).Theorem6.5.GivenaninstanceIFP(n;p)ofCSP(P),fx2i1gi2[n]`2kValpolyI(x) P+ withhighprobabilitywhen m2O(k)nk=2log5n 2.Proof.Byrearrangingterms,weseethatValpolyI(x)= P+X;6=S[k]bP(S)[DI;x(S)poly:NotethatthisisjustPlancherel'sTheoreminSOS.ThetheoremthenfollowsfromLemma6.4andtheobservationthatPS[k]jbP(S)j2O(k). 6.4 (1)-refutationofnon-t-wisesupportingCSPsTheorem6.6.LetPbe-farfrombeingt-wisesupporting.GivenaninstanceIFP(n;p)ofCSP(P),fx2i=1gi2[n]`maxfk;2tgValpolyI(x)1�+ :withhighprobabilitywhen m2O(k)nt=2log5n 2andt2.Toprovethistheorem,wewillneedtofollowingclaim,whichsaysthatanytrueinequalityinkvariablesoverf�1;1gkcanbeprovedindegree-kSOS.Recallthatthemultilinearizationofamonomialzs11zs22


zskk2R[z1;:::;zk]isdenedtobezs1mod21zs2mod22


zskmod2k,i.e.,wereplaceallz2ifactorsby1.WeextendthisdenitiontoallpolynomialsinR[z1;:::;zk]bylinearity.Claim6.7.Letf:f�1;1gk!Rsuchthatf(z)0forallz2f�1;1gkandletfpolybetheuniquemultilinearpolynomialsuchthatf(z)=fpoly(z)forallz2f�1;1gk.Thenfz2i=1gi2[k]`kfpoly(z)0:27 Proof.Sincef(x)0,thereexistsaBooleanfunctiong:f�1;1gk!Rsuchthatg(z)2=f(z)forallz2f�1;1gk.Letgpolybetheuniquemultilinearpolynomialsuchthatg(z)=gpoly(z)forallz2f�1;1gk.Sincegpoly(z)2=fpoly(z)forallz2f�1;1gk,uniquenessofthemultilinearpolynomialrepresentationoffimpliesthatthemultilinearizationof(gpoly)2isequaltofpoly.Writtenanotherway,wehavethatfz2i=1gi2[k]`kfpoly(z)=gpoly(z)2.Thisimpliesthatfz2i=1gi2[k]`kfpoly(z)0. ProofofTheorem6.6.TheproofisanSOSversionofProof2ofTheorem4.9above.Claim6.7impliesthatforQofdegreeatmosttthat-separatesP,fz2i=1gi2[k]`kQ(z)�P(z)+1�0:Summingoverallconstraints,wegetfx2i=1gi2[n]`kValpolyI(x)�(1�)1 mXT2[n]kXc2f1gk1f(T;c)2Ig0@XS[k]bQ(S)cSxST1A;RearrangingtermsasintheproofofTheorem6.5,weseethattherighthandsideisequaltoXS[k]bQ(S)[DI;x(S)poly:SinceQhasmean0,jQj2kandPS[k]jbP(S)j2O(k).ThetheoremthenfollowsfromLemma6.4. WithCorollary3.17,thetheoremimpliesthatwecan k(1)-refuteanyCSP(P)inSOSwhenPisnott-wisesupporting.Corollary6.8.LetPbeapredicatethatdoesnotsupportanyt-wiseuniformdistribution.GivenaninstanceIFP(n;p)ofCSP(P),fx2i=1gi2[n]`maxfk;2tgValI(x)1�2�eO(kt)+ withhighprobabilitywhen m2eO(kt)nt=2log5nandt2.7DirectionsforfutureworkItwouldbeinterestingtoshowanalogousecientrefutationresultsformodelsofrandomCSP(P)inwhichliteralsarenotused.Thiswouldallowforresultson,say,refutingq-colorabilityforrandomk-uniformhypergraphs.Wegiveasimpleresultonrefutingq-colorabilityofrandomhypergraphsinAppendixC,butitfollowsfromrefutationofbinaryCSPsandperhapsastrongerresultcouldbeprovenbystudyingCSPswithlargeralphabets.Forsomepredicates(e.g.,monotoneBooleanpredicates),randomCSPinstancesaretriviallysatisablewhentherearenoliterals.Howeverforsuchpredicatesonecouldconsidera\Goldreich[Gol00]-style"modelinwhicheachconstraintisrandomlyeitherPor:Pappliedtokrandomvariables.Additionally,itwouldbegoodtoinvestigatewhetherourrefutationalgorithmscanbeextendedfromthepurelyrandomCSP(P)settingtothe\smoothed"/\semi-random"settingofFeige[Fei07],inwhichthemconstraintsscopesareworst-caseandonlythenegationpatternforliteralsisrandom.28 Feigeshowedhowtoecientlyrefuterandom3-SATinstanceswithmeO(n3=2)constraintseveninthismodel.Anothervaluableopendirectionwouldbetoshoreuptheknownproof-complexityevidencesuggestingthate(nt=2)constraintsmightbenecessarytorefuterandomCSP(P)whenPisnott-wisesupporting.ThenaturalquestionhereiswhethertheSOSlowerboundof[BCK15]canbeextendedfromnon-pairwiseuniformsupportingandm=O(n)constraints,tonon-t-wiseuniformsupportingandm=O(nt=2�)constraints.(Ofcourse,itwouldalsobegoodtoeliminatethepruningstepfromtheirrandominstances.)Onemightalsoinvestigatethemorerenedquestionofwhether,forPthatare-farfromt-wisesupporting,onecanimproveon-refutationwhentherearemeO(nt=2)constraints.Followupworkontheveryinterestingpaper[FKO06]ofFeige,Kim,andOfekalsoseemswarranted.Recallthatitgivesanondeterministicrefutationalgorithmforrandom3-SATwhenmO(n1:4)(aswellasasubexponential-timedeterministicalgorithm).Thisraisesthequestionofwhetherthereexistpolynomial-sizerefutationsforrandomCSP(P)instancesthatareneverthelesshardtondeciently.Finally,wesuggesttryingtorehabilitatethehardness-of-learningresultsin[DLSS14],givenournewknowledgeaboutwhatrandomCSP(P)instancesseemhardtorefute.Asmentioned,thefollowupworkofDanielyandShalev-Shwartz[DSS14]showshardnessofPAC-learningDNFswith!(logn)termsbasedontheveryreasonableassumptionthatrefutingrandomk-SATrequiresnf(k)constraintsforsomef(k)=!(1).SubsequentworkbyDaniely[Dan15]showshardnessofapproximatelyPAC-learninghalfspacesassumingthatrefutingrandomk-XORishardbothwhenm=ncp 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AProofofTheorem4.1WerestateTheorem4.1:Theorem4.1.Fork2andpn�k=2,letfw(T)gT2[n]kbeindependentrandomvariablessuchthatforeachT2[n]k,E[w(T)]=0(24)Pr[w(T)6=0]p(25)jw(T)j1:(26)ThenthereisanecientalgorithmcertifyingthatXT2[n]kw(T)xT2O(k)p pn3k=4log3=2n:(27)forallx2Rnwithkxk11withhighprobability.Theproofofthistheoremconstitutestheremainderofthissection.ItwilloftenbeconvenienttoconsiderT2[n]ktobe(T1;T2)2[n]k1[n]k2withk1+k2=k.Insuchsituations,wewillwritew(T)=w(T1;T2).Forintuition,thereadercanthinkofthespecialcaseofw(T)2f�1;0;1gforallTandy2f�1;1gn.Undertheseadditionalconstraints,PT2[n]kw(T)xTisOpt(I)�1 2forarandomk-XORinstanceIsowearecertifyingthatarandomk-XORinstancedoesnothavevaluemuchbiggerthan1 2.A.1TheevenaritycaseWhenkiseven,wecanthinkofPT2[n]kw(T)xTasaquadraticform:XT2[n]kw(T)xT=XT1;T22[n]k=2w(T1;T2)yT1yT2;(28)whereyU=xU.WegivetwomethodstocertifythatthevalueofthisquadraticformisatmostOk(p pn3k=4logn).TherstmethodusesanSDP-basedapproximationalgorithmandworksonlyforx2f�1;1gn.Thesecondmethodusesideasfromrandommatrixtheoryandworksforanyxwithkxk11.ApproximationalgorithmsapproachIfx2f�1;1gn,wecanapplyanapproximational-gorithmofCharikarandWirth[CW04]forquadraticprogramming.Theyprovethefollowingtheorem:TheoremA.1.[CW04,Theorem1]LetMbeanynnmatrixwithalldiagonalelements0.Thereexistsanecientrandomizedalgorithmthatndsy2f�1;1gnsuchthatE[y�My] 1 lognmaxx2f�1;1gnx�Mx:ByMarkov'sInequality,thisstatementholdswithprobabilityatleast1=2.WecanrunthealgorithmO(logn)timestogetahighprobabilityresult.ToapplyTheoremA.1,weseparateoutthediagonaltermsof(28),rewritingitasXT16=T22[n]k=2w(T1;T2)yT1yT2+XU2[n]k=2w(U;U)y2U:(29)35 WecancertifythateachofthetwotermsinthisexpressionisatmostO(p pn3k=4logn).Fortherstterm,wewillneedthefollowingclaim.ClaimA.2.Withhighprobability,itholdsthatXT1;T22[n]k=2w(T1;T2)yT1yT2O(p pn3k=4):ThisfollowsfromapplyingBernstein'sInequality(Theorem4.12)forxedyandthentakingaunionboundoverally2f�1;1gnk=2.Usingtheclaim,weseethatTheoremA.1allowsustocertifythatthevalueofthersttermin(29)isatmostO(p pn3k=4logn).Wewillusethenextclaimtoboundthesecondtermof(29).ClaimA.3.Withhighprobability,itholdsthatXU2[n]k=2jwU;UjO(p pn3k=4):SincethejwTj1andPr[wT6=0]p,theclaimfollowsfromtheChernoBound.Thesecondtermof(29)isupperboundedbyPU2[n]k=2jwU;UjandwecancomputethisquantityinpolynomialtimetocertifythatitsvalueisatmostO(p pn3k=4).RandommatrixapproachObservethat(28)isy�ByforamatrixBindexedbyU2[n]k=2sothatBU1;U2=w(U1;U2).Theny�BykBkkyk2.Tocertifythaty�Byissmall,wecomputekBk.WeneedtoshowthatkBkissmallwithhighprobability.First,notethatkBkisequaltothenormofthe2nk=22nk=2symmetricmatrix~B=0BB�0Forexample,thisappearsas(2.80)in[Tao12].Theuppertriangularentriesof~Bareindependentrandomvariableswithmean0andvarianceatmostpbythepropertiesofthewS's.Wecanthenapplyastandardboundonthenormofrandomsymmetricmatrices[Tao12].PropositionA.4.[Tao12,Proposition2.3.13]LetMbearandomsymmetricmatrixnnwhoseuppertriangularentriesMijwithijareindependentrandomvariableswithmean0,varianceatmost1,andmagnitudeatmostK.Then,withhighprobability,kMk=O(p nlogn
maxf1;K=p ng):Let~B0=1 p p~B.Theuppertriangularentriesof~B0areindependentrandomvariableswithmean0,varianceatmost1,andmagnitudeatmost1=p p.ApplyingPropositionA.4to~B0showsthatkBk=Oknk=4p plogn
maxn1;1 p pnk=4owithhighprobability.Sincekyk11byassumption,kyk2nk=2and(28)isatmostO(kp pn3k=4logn)withhighprobabilitywhenpn�k=2.36 A.2TheoddaritycaseFixanassignmentx2[�1;1]n.Fori2[n],themonomialscontainingxicancontributeatmostWi:=PT2[n]k�1w(T;i)xTtotheobjectiveifxiissetoptimally.ByCauchy-Schwarz,XT2[n]kw(T)xTXi2[n]Wip ns Xi2[n]W2i;(30)soitsucestoboundPi2[n]W2i.Wewillwritethisasaquadraticpolynomialandthenbounditusingspectralmethods:Xi2[n]W2i=XT;U2[n]k�1Xi2[n]w(T;i)w(U;i)xTxU=XT01;T02;U01;U022[n]k�1 2Xi2[n]w(T01;U01;i)w(T02;U02;i)x(T01;T02)x(U01;U02):(31)Denethenk�1nk�1matrixAindexedby[n]k�1:A(i1;i2);(j1;j2)=(P`2[n]w(i1;j1;`)w(i2;j2;`)if(i1;j1)6=(i2;j2)0otherwise;(32)wherewehavedividedtheindicesofAinto2blocksofk�1 2coordinateseach.Denex k�12R[n]k�1sothatx k�1(T)=xT.Then(31)isequalto(x k�1)�Ax k�1+XT;U2[n]k�1 2(xT)2(xU)2Xi2[n]w(T;U;i)2:(33)ThersttermisatmostkAknk�1sincethevariablesarebounded.WecancomputekAktocertifythis.Withhighprobability,kAkisnottoobig.LemmaA.5.Letk3andpn�k=2.Letfw(T)gT2[n]kbeindepedentrandomvariablessatisfyingconditions(6),(7),and(8)above.LetAbedenedasin(32).Withhighprobability,kAk2O(k)pnk=2log3n:Wecanthereforecertifythatthersttermis2O(k)pn3k=2�1log3n.WewillprovethelemmainAppendixA.4.Thesecondtermof(33)isatmostPT2[n]kw(T)2.WecaneasilycomputethisandtheChernoBoundimpliesthatitsvalueisatmostpn3k=2�1withhighprobability.Sofar,withhighprobabilitywecancertifythatPi2[n]W2i=2O(k)pn3k=2�1log3n.Pluggingthisboundinto(30)concludestheproof.RemarkA.6.ItwouldhavebeenmorenaturaltohavewrittenPi2[n]W2i=(x k�1)�A0x k�1forA0suchthatA0T;U=Pi2[n]w(T;i)w(U;i).However,kA0kcouldbetoolargebecauseofthecontributionofthesecondtermin(33).Weusetheadditionalassumptionthatkxk11togetaroundthisissue.RemarkA.7.WehavedenedAsothatA(i1;i2);(j1;j2)=P`2[n]w(i1;j1;`)w(i2;j2;`),notAi;j=P`2[n]w(i;`)w(j;`).Thereducesthecorrelationamongentriesw(b;c)andw(b;c0)forc6=c0.Intuitively,Alooksmorelikearandommatrixwithindependententries,sowecanbounditsnormusingthetracemethod.SeetheproofofLemmaA.5inAppendixA.4.37 A.3AnSOSversionInthissection,wewillprovetheSOSversionofTheorem4.1.Theorem6.2.Fork2andpn�k=2,letfw(T)gT2[n]kbeindependentrandomvariablessuchthatforeachT2[n]k,E[w(T)]=0Pr[w(T)6=0]pjw(T)j1:Then,withhighprobability,fx2i1gi2[n]`2kXT2[n]kw(T)xT2O(k)p pn3k=4log3=2n:Ratherthanwritingoutthefullproof,wewillindicatethesmallchangesrequiredtoconverttheaboveproofofTheorem4.1intoSOSform.EvenarityTherandommatrixprooffortheevencasecaneasilybeconvertedintoanSOSproofwithdegreek.WhenO(kp pnk=4logn)I�B0,thereexistsamatrixMsuchthatM�M=O(kp pnk=4logn)I�B.ThenO(kp pnk=4logn)kyk2�y�By=(My)�(My)=XT2[n]k=20@XU2[n]k=2MT;UyU1A2sofx2i1gi2[n]`kXT2[n]kw(T)xTO(kp pn3k=4logn):OddarityAcoupleofadditionalissuesariseintheoddcase.Firstofall,thesquarerootin(30)isnoteasilyexpressedinSOS,soweinsteadprovethesquaredversion0@XT2[n]kw(T)xT1A22O(k)n3k=2log3n:(34)Byasimpleextensionof[OZ13,Fact3.3],(34)implies(27)inSOS:FactA.8.X2b2`2Xb:Proof.1 2b(b2�X2)+1 2b(b�X)2=b 2�1 2bX2+b 2�X+1 2bX2=b�X: Secondly,wedonotknowhowtoprovetheCauchy-Schwarzinequality(30)inSOS.However,O'DonnellandZhoushowthataverysimilarinequalitycanbeprovedinSOS[OZ13,Fact3.8]:FactA.9.`2YZ1 2Y2+1 2Z2:38 UsingthisfactinsteadofCauchy-Schwarztoprovethesquaredversionof(30),wecanfollowtheargumentabovetoshowthatfx2i1gi2[n]`2k0@XT2[n]kw(T)xT1A2nx k�1�Ax k�1+nXT2[n]kw(T)2:ThenormboundcanbeproveninSOSexactlyasintheevencase.A.4ProofofLemmaA.5WerestatethedenitionofthematrixAandthestatementofthelemma.LemmaA.5.Letk3andpn�k=2.Letfw(T)gT2[n]kbeindepedentrandomvariablessatisfyingconditions(6),(7),and(8)above.LetAbethe[n]k�1[n]k�1indexedby[n]k�1thatisdenedasfollows:A(i1;i2);(j1;j2)=(P`2[n]w(i1;j1;`)w(i2;j2;`)if(i1;j1)6=(i2;j2)0otherwise:Thenwithhighprobability,kAk2O(k)pnk=2log3n:Theproofcloselyfollowstheargumentsof[COGL04,Lemma17]and[BM15,Section5].Bothproofsusethetracemethod:ToboundthenormofasymmetricrandommatrixM,itsucestoboundE[tr(Mr)]forlarger.Fornon-symmetricmatrices,wecaninsteadworkwithMM�.Inourparticularcase,wehavethefollowing.ClaimA.10.IfE[tr((AA�)r)]nO(k)2O(r)r6rp2rnkr,thenkAk2O(k)pnk=2log3nwithhighprobability.Proof.ObservethatkAk2rtr((AA�)r).ByMarkov'sInequality,Pr[kAkB]E[tr((AA�)r)] B2r.Wegettheclaimbyplugginginr=(logn)andsettingconstantsappropriately. RemarkA.11.Wecangetarbitrarilysmall1=poly(n)probabilityoffailure:ThisproofshowsthatkAkK2O(k)pnk=2log3nwithprobabilityatmostn�logK.Inthetheremainderofthissection,wewillboundE[tr((AA�)r)].LemmaA.12.UndertheconditionsofLemmaA.5,E[tr((AA�)r)]nO(k)2O(r)r6rp2rnkrwithhighprobability.Proof.RecallthatweindexAbyelementsofnk�1dividedintotwoblocksofk�1 2coordinateseach.First,notethattr((AA�)r)=Xi1;:::;i2r2[n]k�1 2(AA�)(i1;i2);(i3;i4)(AA�)(i3;i4);(i5;i6)


(AA�)(i2r�1;i2r);(i1;i2):ExpandingthisoutusingthedenitionofAandsettingwT=w(T),wegetthattr((AA�)r)=Xwi1;j1;`1wi2;j2;`1wi3;j1;`2wi4;j2;`2


wi2r�1;j2r�1;`2r�1wi2r;j2r;`2r�1wi1;j2r�1;`2rwi2;j2r;`2r;39 wherethesumisover`1;:::;`2t2[n]andi1;:::;i2t;j1;:::;j2t2[n]k�1satisfying(is;js)6=(is+1;js+1)for1s2r�1(35)(is+2;js)6=(is+3;js+1)for1s2r�3(36)(i1;j2r�1)6=(i2;j2r):(37)Let bethesetofall(i1;:::;i2r;j1;:::;j2r)2([n]k�1 2)4rsatisfying(35),(36),and(37).ThenforJ2 andL=(`1;:::;`2r)2[n]2r,denePJ;L=wi1;j1;`1wi2;j2;`1wi3;j1;`2wi4;j2;`2


wi2r�1;j2r�1;`2r�1wi2r;j2r;`2r�1wi1;j2r�1;`2rwi2;j2r;`2r:(38)LetjJj=jfi1;:::;i2r;j1;:::;j2rgjbethenumberofdistinctelementsof[n]k�1 2inJanddenejLj=jf`1;:::;`2rgjsimilarly.WethenhaveE[tr((AA�)r)]=XJ2 XL2[n]2rE[PJ;L]=4rXa=12rXb=1XJ2 jJj=aXL2[n]2rjLj=bE[PJ;L]:Toboundthissum,wewillstartbyboundingE[PJ;L].Wewillneedtwoclaims.ClaimA.13.Thenumberofdistinctwi;j;`factorsinPJ;Lisatleast2jLj.Proof.Foreach`2L,(38)showsthatPJ;Lcontainsapairoftheformwis;js;`wis+1;js+1;`orwis+2js;`wis+3;js+1;`.SinceJ2 ,weknowthat(is;js)6=(is+1;js+1)or(is+2;js)6=(is+3;js+1),soeachofthesepairsmusthavetwodistinctwi;j;`factors.Wethenhaveatleast2jLjdistinctwi;j;`factors. ClaimA.14.Thenumberofdistinctwi;j;`factorsinPJ;LisatleastjJj�2.Proof.ConsiderlookingoverthefactorsofPJ;Lfromlefttorightintheorderof(38)untilwehaveseenallelementsofJ.Therstpairwi1;j1;`1wi2;j2;`1containsatmostfourpreviously-unseenelementsofJ.Everysubsequentpairoffactorswis;js;`swis+1;js+1orwis+2js;`s+1wis+3;js+1;`s+1inPJ;LsharestwovariablesofJwithitsprecedingpair.EachsuchpaircanthencontainatmosttwonewelementsofJ.Afterseeinguwi;j;`'s,wehavethereforeseenatmost4+2�u�2 2
distinctelementsofJ.TogetalljJjelementsofJ,wemusthaveseenatleastjJj�2wi;j;`'sandthesemustbedistinct. SincePr[wi;j;`6=0]p,E[PJ;L]p#fdistinctwi;j;`factorsinPJ;Lg.ItthenfollowsthatE[PJ;L]pmaxf2jLj;jJj�2g:Thetwoclaimsalsoimplytwootherfactswewillneedbelow.ClaimA.15.IfjLj�r,thenE[PJ;L]=0.Proof.WewillshowthatifjLj�r,thereisanwi;j;`factorinPJ;Lthatoccursexactlyonce.SinceE[wi;j;`]=0,thisprovestheclaim.AssumeforacontradictionthatjLj�randeverywi;j;`factoroccursatleasttwice.Sincethereareatleast2jLjdistinctwi;j;`'s,theremustbeatleast4jLj�4rtotalwi;j;`'s.However,lookingat(38),PJ;Lhasatmost4rwi;j;`factors. 40 ClaimA.16.IfjJj�2r+2,thenE[PJ;L]=0.Thiscanbeprovedinexactlythesamemanner.Next,observethatthenumberofchoicesofJwithjJj=aisatmostna(k�1) 2a4rna(k�1) 2(4r)4r.ThenumberofchoicesofLwithjLj=bisatmostnbb2rnb(2r)2r.Alltogether,wecanwriteE[tr((AA�)r)]2r+2Xa=1rXb=1210rr6rna(k�1) 2+bpmaxf2b;a�2g:Weboundeachtermofthesum.ClaimA.17.na(k�1) 2+bpmaxf2b;a�2gnkr+k�1p2r:Proof.If2b�a�2,na(k�1) 2+bpmaxf2b;a�2gn(2b+2)(k�1) 2+bpmaxf2b;a�2g=nk�1(nkp2)b:If2ba�2,na(k�1) 2+bpmaxf2b;a�2gna(k�1) 2+a 2�1pa�2=nk�1(nkp2)a=2�1:Recallthatweassumednkp21.Sincea2r+2andbr,theclaimfollows. Toconclude,observethatE[tr[(AA�)r]]2r+2Xa=1rXb=1210rr6rnkr+k�1p2rnO(k)2O(r)r6rp2rnr RemarkA.18.Ifwedidnothaveconditions(35),(36),and(37),wewouldonlyhavebeenabletoshowthatjLj2r.ThiswouldhaveledtoaweakerboundofO(p n).BExtensiontolargeralphabetsB.1PreliminariesCSPsoverlargerdomainsWebeginbydiscussingCSPsoverdomainsofsizeq�2.WeprefertoidentifysuchdomainswithZq,soourpredicatesareP:Zkq!f0;1g.TheextensionsofthedenitionsandfactsfromSection3.1arestraightforward;theonlyslightlynonobviousnotionisthatofaliteral.Wetakethefairlystandard[Aus08]denitionthataliteralforvariablexiisanyxi+cforc2Zq.Thustherearenowqkpossible\negationpatterns"cforaP-constraint.WedenotebyFq;P(n;p)thedistributionoverinstancesofCSP(P)inwhicheachoftheqknkconstraintsisincludedwithprobabilityp;theexpectednumberofconstraintsistherefore m=qknkp.WehavethefollowingslightvariantofFact3.6.FactB.1.LetIFq;P(n;p).Thenthefollowingstatementsholdwithhighprobability.1.m=jIj2 m
1Oq logn m.2.Opt(I) P
1+Oq logq P
n m.3.IisOq qklogq
n m-quasirandom.41 FourieranalysisoverlargerdomainsLetUqistheuniformdistributionoverZq.Wecon-siderthespaceL2(Zq;Uq)offunctionsf:Zq!Requippedwiththeinnerproducthf;gi=EzUq[f(z)g(z)]anditsinducednormkfk2=EzUq[f(z)2]1=2.Fixanorthonormalbasis0;:::;q�1suchthat0=1.NowletL2(Zkq;Ukq)bethespaceoffunctionsf:Zkq!R,whereUkqistheuniformdistributionoverZkqandwehavetheanalogousinnerproductandnorm.Then,for2Zkq,dene:Zkq!Rsuchthat(x)=Yi2[k]i(xi):Thesetfg2ZkqformsanorthonormalbasisforL2(Zkq;Ukq)[Aus08,Fact2.3.1]andwecanwriteanyfunctionf:Zkq!Rintermsofthisbasis:f(x)=X2Zkqbf()(x):OrthonormalityonceagaingivesusPlancherel'sTheoreminthissetting:TheoremB.2.hf;gi=X2Zkqbf()bg():For2Zkq,denesupp()=fi2[k]ji6=0gandjj=jsupp()j.Thenwedenethedegreeofftobemaxfjjjbf()6=0g.Notethatthisisthedegreeoffwhenitiswrittenasapolynomialinthea'sfora2Zq.Givenak-tupleTand2Zkq,weuseT()todenotethejj-tupleformedbytakingtheprojectionofTontothecoordinatesinsupp().Similarly,useT( )todenotethe(k�jj)-tupleformedbytakingtheprojectionofTontocoordinatesin[k]nsupp().See[O'D14,Aus08]formorebackgroundonFourieranalysisoverlargerdomains.B.2ConversiontoBooleanfunctionsTomoreeasilyapplyouraboveresults,wewouldliketorewriteafunctionf:Zkq!RasaBooleanfunctionfb:f0;1gk0!Rforsomek0.Itwillactuallybemoreconvenienttodenefbonasubsetoff0;1gk0.Inparticular,considertheset k=fv2f0;1g[k]ZqjPa2Zqv(i;a)=18i2[k]g.NotethereisabijectionbetweenZkqand k:Forz2Zkq,((x))(i;a)=1fzi=ag.Intheotherdirection,givenv2 kset�1(v)i=Pa2Zqa
v(i;a).Forafunctionf:Zkq!R,wecanthendeneitsBooleanversionfb: k!Rasfb(v)=X2Zkqf()Yi2[k]v(i;i);Observethatf(z)=fb((z))forz2Zkq.Also,notethatiff(z)=g(z)forallz2Zkq,fb=gboverallRkbyconstruction.fbisamultilinearpolynomialanditsdegreeisdenedinthestandardway.Thedegreeoffisdenedasintheprevioussection.ClaimB.3.Thedegreeoffbisequaltothedegreeoff.42 Proof.Abbreviatesupp()ass()anddenotesupp()'scomplementwithrespectto[k]ass( ).Applyingthedenitionandwritingf'sFourierexpansion,weseethatfb(v)isequaltoX2ZkqX2Zkqbf()()Yi2[k]v(i;i)=X2Zkqbf()X02Zjjq(0)jjYi=1v(s()i;0i)X002Zk�jjqk�jjYi=1v(s( )i;00i):NowobservethatX002Zk�jjqk�jjYi=1v(s( )i;00i)=k�jjYi=1Xa2Zqv(s( )i;a)=1bytheassumptionthatv2 k.Thedegreeoffbisthereforejj. B.3QuasirandomnessandstrongrefutationToprovequasiandomnessandstrongrefutationresultsforCSPsoverlargeralphabets,weproceedexactlyasinthebinarycase.Weusedthet=kcaseofLemma3.13(theVaziraniXORLemma[Vaz86,Gol11])tocertifyquasirandomnessforbinaryCSPs.AgeneralizationofthiscaseholdsforAbeliangroups[Rao07,Lemma4.2].LemmaB.4.LetGbeanAbeliangroupandletUGbetheuniformdistributionoverG.Also,letfg2GbeanorthonormalbasisforL2(G;UG)andletD:G!RbeadistributionoverG.IfbD()forall2G,thendTV(D;UG)1 2jGj3=2.ViewingtheinduceddistributiondensityDI;x()asafunctionofx2Znqforxed2Zkq,wewillconsiderDbI;y(): n!R.Asbefore,wecancertifythatDbI;yhassmallFouriercoecients.LemmaB.5.Let2Zkqsuchthat6=0andjj=s.ThereisanalgorithmthatwithhighprobabilitycertiesthatdDbI;y()qO(k)maxfns=4;p nglog5=2n p mforally2f0;1g[n]Zqwhen mmaxfns=2;ng.Proof.TheproofisessentiallyidenticaltotheproofofLemma4.3.Wehighlightthedierences.Firstofall,wecanwritedDbI;y()=Xx2Znq[DI;x()Yi2[n]y(i;xi)=1 mXT2[n]kXc2Zkq1f(T;c)2IgXx2Znq(xT+c)Yi2[n]y(i;xi):Sinceonlydependsoncoordinatesinsupp(),wecanrearrangeandusethefactthatPa2Zqyi;a=1togetdDbI;y()=1 mX2ZjjqXU2[n]jjjjYi=1y(Ui;i)XT2[n]kT()=Uw;(T);wherew;(T)=Pc2Zkq1f(T;c)2Ig(+c()).ObservethatE[w;(T)]=0andPr[w;(T)6=0]qkp.Sincekk=1,observethattheCauchy-SchwarzInequalityimpliesthatjjqk=2forall.Thenjw;(T)jq3k=2foralland.Forevery,wecanthenapplyLemma4.4justasintheproofofLemma4.3. 43 Thesetwolemmasthenimplythelargeralphabetversionsofthequasirandomnesscerticationandstrongrefutationresultsabove.TheoremB.6.ThereisanecientalgorithmthatcertiesthataninstanceIFq;P(n;p)ofCSP(P)is -quasirandomwithhighprobabilitywhen mqO(k)nk=2log5n 2.TheoremB.7.Thereisanecientalgorithmthat,givenaninstanceIFq;P(n;p)ofCSP(P),certiesthatOpt(I) P+ withhighprobabilitywhen mqO(k)nk=2log5n 2.B.4Refutationofnon-t-wisesupportingCSPsWewillshowthatthedualpolynomialcharacterizationofbeingfarfromt-wisesupportingdescribedinSection3.4generalizestolargeralphabets.Weextendthedenitionsoft-wisesupportingand-separatingpolynomialstotheZqcaseinthenaturalway.LemmaB.8.ForP:Zkq!f0;1gand01,thereexistsapolynomialQ:Zkq!Rofdegreeatmosttthat-separatesPifandonlyifPis-farfromsupportingat-wiseuniformdistribution.Proof.TheproofusesthefollowingduallinearprogramsexactlyasintheproofofLemma3.16.minimizeXz2Zkq(1�P(z))D(z)(39)s.t.Xz2ZkqD(z)(z)=qkbD()=082Zkq0jjt(40)Xz2ZkqD(z)=1D(z)08z2Zkq maximizes.t.X2Zkq0jjtc(S)(z)1�P(z)�8z2Zkq: ToproveLemma3.16,weneededtoshowinthebinarycasethatfeasiblesolutionstotheprimalLP(1)weret-wiseuniform.Wenowarguethattheconstraint(40)isasucientconditionfort-wiseuniformityofDintheq-arycase.ForadistributionDoverZkqandS[k],deneDStobethemarginaldistributionofDon(Zkq)S,i.e.,DS(z)=Pz02Zkq;z0S=zD(z0).Weneedtoshowthat(40)impliesthatDS=UjSjqforallS[k]with1jSjt.FixsuchanSandletjSj=s.Considerthebasisfg2Zsq.LemmaB.4impliesthatitsucestoshowthatEzUsq[DS(z)(z)]=0forall2Zsq.ObservethatEzUsq[DS(z)(z)]=Ez0Ukq[D(z0)(z0)]for2Zkqsuchthati=ifori2Sandi=0otherwise.SincejSjt,weknowthatjjtand(40)impliesEz0Ukq[D(z0)(z0)]=0.Therestoftheproofisexactlyasinthebinarycase. 44 Wecanagainusetheseseparatingpolynomialstoobtainalmost-refutationforpredicatesthatare-farfromt-wisesupporting.TheoremB.9.LetPbe-farfrombeingt-wisesupporting.Thereexistsanecientalgorithmthat,givenaninstanceIFq;P(n;p)ofCSP(P),certiesthatOpt(I)1�+ withhighprobabilitywhen mqO(k)nt=2log5n 2andt2.TheproofisessentiallyidenticaltoProof2ofTheorem4.9.Corollary4.11alsoextendstolargeralphabets.CorollaryB.10.LetPbeapredicatethatdoesnotsupportanyt-wiseuniformdistribution.Thenthereisanecientalgorithmthat,givenaninstanceIFq;P(n;p)ofCSP(P),certiesthatOpt(I)1�2�eO(qtkt)withhighprobabilitywhen m2eO(qtkt)nt=2log5nandt2.ThisfollowsdirectlyfromTheoremB.9andthefollowingextensionofCorollary3.17tolargeralphabets.CorollaryB.11.SupposeP:Zkq!f0;1gisnott-wisesupporting.Thenitisinfact-farfromt-wisesupportingfor=2�eO(qtkt).TheproofisessentiallyidenticaltotheproofofCorollary3.17:ObservethattheLP(39)hasatmostqtktvariablesandproceedexactlyasbefore.B.5SOSproofsHerewegiveSOSversionsofourrefutationresultsforlargeralphabets.CertifyingFouriercoecientsaresmallTogiveanSOSproofthatFouriercoecientsofDbI;yaresmall,weagainneedtodeneaspecicpolynomialrepresentationofdDbI;y().dDI;y()poly=1 mXT2[n]kXc2Zkq1f(T;c)2IgX2Zjjq(+c())jjYi=1y(T()i;i):LemmaB.12.Let06=2Zkqwithjj=s.Thenfy(i;a)21gi2[n]a2Zq`maxf2s;kgdDI;y()polyqO(k)maxfns=4;p nglog5=2n m1=2fy(i;a)21gi2[n]a2Zq`maxf2s;kgdDI;y()poly�qO(k)maxfns=4;p nglog5=2n m1=2:withhighprobability,assumingalsothat mmaxfns=2;ng.Proof.IntheproofofLemmaB.5,wecertifythatjdDI;y()jissmallbycertifyingthatdDI;y()polyissmall.TheproofofLemmaB.5reliesonlyonLemma4.4;wecanreplacethiswithitsSOSversionLemma6.3. RemarkB.13.Westatedthelemmawiththeweakersetofaxiomsfy(i;a)21gi2[n];a2Zq.Sincey(i;a)2=y(i;a)impliesy(i;a)21indegree-2SOS,thelemmaholdswiththeaxiomsfy(i;a)2=y(i;a)gi2[n];a2Zqaswell.45 Strongrefutationofanyk-CSPFromourSOSproofthattheFouriercoecientsdDbI;y()aresmall,wecangetSOSproofsofstrongrefutationforanyk-CSP.Todothis,weneedtodeneaspecicpolynomialrepresentationofValbI(y)foraninstanceIofCSP(P):ValI(y)poly=1 mXT2[n]kXc2Zkq1f(T;c)2IgX2ZkqP(+c)Yi2[k]y(Ti;i):TheoremB.14.GivenaninstanceIFq;P(n;p)ofCSP(P),fy(i;a)2=y(i;a)gi2[n]a2Zq[8:Xa2Zqy(i;a)=19=;i2[n]`2kValI(y)poly P+ withhighprobabilitywhen mqO(k)nk=2log5n 2.Proof.First,usetheFourierexpansionofPtowriteValI(y)poly=1 mXT2[n]kXc2Zkq1f(T;c)2IgX2ZkqX2ZkqbP()(+c)Yi2[k]y(Ti;i):ForeachT2[n]k,c2Zkq,and2Zkq,wehaveatermoftheformP2Zkq(+c)Qi2[k]y(Ti;i).Notethatonlydependsonthecoordinatesinsupp().Wecanthenwritethisas0B@X2Zjjq(+c())jjYi=1y(T()i;i)1CA0@k�jjYi=1Xa2Zqy(T( )i;a)1AUsingtheaxiomsPa2Zqyi;a=1,thesecondtermisequalto1andwehave8:Xa2Zqy(i;a)=19=;i2[n]`kX2Zkq(+c)Yi2[k]y(Ti;i)=X2Zjjq(+c())jjYi=1y(T()i;i):SummingoverallT,c,and,weobtainthefollowing.8:Xa2Zqy(i;a)=19=;i2[n]`kValI(y)poly=1 mXT2[n]kXc2Zkq1f(T;c)2IgX2ZkqbP()X2Zjjq(+c())jjYi=1y(T()i;i):Thisisequalto P+X06=2ZkqbP()dDI;y()poly:SincejP(z)j1andj(z)jqO(k),P2ZkqjbP()jqO(k).WecanthenapplyLemmaB.12foreachtocompletetheproof. 46 SOSrefutationofnon-t-wisesupportingCSPsTheoremB.15.LetPbe-farfrombeingt-wisesupporting.Then,givenaninstanceIFq;P(n;p)ofCSP(P),y(i;a)2=y(i;a) i2[n]a2Zq[fy(i;a)y(i;b)=0gi2[n]a6=b2Zq[8:Xa2Zqy(i;a)=19=;i2[n]`maxfk;2tgValI(y)poly1�+ :withhighprobabilitywhen mqO(k)nt=2log5n 2andt2.Toprovethistheorem,weneedaversionofClaim6.7forlargeralphabets.ClaimB.16.Letf:Zkq!Rsuchthatf(z)0forallz2Zkqandletfb(v)=P2Zkqf()Qi2[k]v(i;i).Thenv(i;a)2=v(i;a) i2[k]a2Zq[fvi;avi;b=0gi2[k]a6=b2Zq`kfb(v)0:Proof.Sincef(z)0forallz2Zkq,thereexistsafunctiong:Zkq!Rsuchthatg2(z)=f(z)forallz2Zkq.Wethenwritegb(v)=P2Zkqg()Qi2[k]v(i;i).Usingv(i;a)2=v(i;a),itfollowsthatgb(v)2=X2Zkqg()2Yi2[k]v(i;i)+X06=002Zkqg(0)g(00)Yi2[k]v(i;0i)v(i;00i)Thersttermisequaltofb().Forthesecondterm,notethateachoftheproductsQi2[k]v(i;0i)v(i;00i)mustcontainfactorsv(i;a)v(i;b)witha6=bsince06=00.Wehavetheaxiomv(i;a)v(i;b)=0,sothesecondtermis0.Thenfb=(gb)2andtheclaimfollows. Withthisclaim,theproofofthetheoremexactlyfollowsthatofTheorem4.9.ProofofTheoremB.15.ClaimB.16impliesthatv(i;a)2=v(i;a) i2[k]a2Zq[fv(i;a)v(i;b)=0gi2[k]a6=b2Zq[8:Xa2Zqv(i;a)=19=;i2[k]`kPb(v)�(1�)Qb(v)Summingoverallconstraints,wegetthatA`kmValI(y)poly�m(1�)XT2[n]kXc2Zkq1f(T;c)2IgX2ZkqQ(+c)kYi=1y(i;i)whereA=y(i;a)2=y(i;a) i2[n]a2Zq[fy(i;a)y(i;b)=0gi2[n]a6=b2Zq[nPa2Zqy(i;a)=1oi2[n].UsingtheFourierexpansionofQ,weseethattheright-handsideoftheinequalityisXT2[n]kXc2Zkq1f(T;c)2IgX2ZkqX2ZkqbQ()(+c)kYi=1y(Ti;i)JustasintheproofofTheoremB.14,wecanrewritethisindegree-kSOSasXT2[n]kXc2Zkq1f(T;c)2IgX2ZkqbQ()X2Zjjq(+c())jjYi=1y(T()i;i):47 WethenrearrangetogetX06=2ZkqbQ()dDI;y()poly:SinceE[Q]=0andQ�1,weknowthatjQjqO(k)andthereforejbQ()jqO(k).WecanthenapplyLemmaB.12foreachtocompletetheproof. CCertifyingthatrandomhypergraphshavesmallindependencenumberandlargechromaticnumberFirst,werecallsomestandarddenitions.LetH=(V;E)beahypergraph.WesaythatSisanindependentsetofHifforalle2E,itholdsthate=2S.Theindependencenumber(H)isthenthesizeofthelargestindependentsetofH.Aq-coloringofHisafunctionf:V![q]suchthatf�1(i)isanindependentsetforeveryi2[q].Thechromaticnumber(H)isthethesmallestq2Nforwhichthereexistsaq-coloringofH.WedeneH(n;p;k)tobethedistributionovern-vertex,k-uniform(unordered)hypergraphsinwhicheachofthe�nk
possiblehyperedgesisincludedindependentlywithprobabilityp.Let mbetheexpectednumberofhyperedgesp�nk
.Coja-Oghlan,Goerdt,andLankausedCSPrefutationtechniquestoshowthefollowingresults[COGL07]:TheoremC.1.(Coja-Oghlan{Goerdt{Lanka[COGL07,Theorem3]).ForHH(n;p;3),thereisapolynomialtimealgorithmcertifyingthat(H)nwithhighprobabilityforanyconstant&#x-431;0when m�n3=2ln6nand m=o(n2).TheoremC.2.(Coja-Oghlan{Goerdt{Lanka[COGL07,implicitinSection4]).ForHH(n;p;4),thereisapolynomialtimealgorithmcertifyingthat(H)nwithhighprobabilityforanyconstant&#x-278;0when mOn2 4.TheoremC.3.(Coja-Oghlan{Goerdt{Lanka[COGL07,Theorem4]).ForHH(n;p;4),thereisapolynomialtimealgorithmcertifyingthat(H)�withhighprobabilityforconstantwhen mO(4n2).Wegeneralizetheseresultstok-uniformhypergraphs:TheoremC.4.ForHH(n;p;k),thereisapolynomialtimealgorithmcertifyingthat(H)withhighprobabilitywhen mOkn5k=2log3n 2k,assumingthatn3=4logn.TheoremC.5.ForHH(n;p;k),thereisapolynomialtimealgorithmcertifyingthat(H)�withhighprobabilitywhen mOk�2knk=2log3n
,assumingthatn1=4 logn.Theproofsaresimpleextensionsofthek=3andk=4casesfrom[COGL07].WewillrstproveTheoremC.4usingTheorem4.1andthiswillalmostimmediatelyimplyTheoremC.5.ProofofTheoremC.4.RecallthatTheorem4.1dealswithk-tuples,notsetsofsizek.Itiseasytoexpressahypergraphintermsofk-tuplesratherthansetsofsizek.ForasetSandt2Z0,recallthenotation�St
=fTSjjTj=tg.Foreachpossiblehyperedgee2�[n]k
,weassociateanarbitrarytupleTefromamongthek!tuplesin[n]kcontainingthesamekelements.TodrawfromH(n;p;k),weincludeeachTeindependentlywithprobabilitypandincludeallotherT2[n]kwithprobability0.48 ForT2[n]k,wedenetherandomvariablew(T)asfollows:w(T)=(p�1fe2EgifT=Teforsomee2�[n]k
0otherwise.Letx2f0;1gnbetheindicatorvectorofanindependentsetIsothatxT=1ifTIandxT=0otherwise.First,observethatXT2[n]kw(T)xT=pXS2([n]k)xS�Xe2([n]k)1fe2Egxe=pjIjk;wherethesecondtermis0becauseIisanindependentset.Thew(T)'ssatisfyconditions(6),(7),and(8)andkxk11,soTheorem4.1implieswecancertifythatpjIjk=XT2[n]kw(T)xT2O(k)p pn3k=4log3=2nwithhighprobability.Simplifying,weseethatwecancertifyjIjOk n5=4log3 2kn m1 2k!andplugginginthevalueof mfromthestatementofthetheoremcompletestheproof. ProofofTheoremC.5.ForacoloringofahypergraphH,eachcolorclassisanindependentsetofH.If(H),thenthereexistsacolorclassofsizeatleastn andtherefore(H)n .Wecanthencertifythat(H)n usingTheoremC.4. DSimulatingFP(n;p)withaxednumberofconstraintsThesettingof[DLSS14]xesthenumberofconstraintsinaCSPinstance,whereasthemodeldescribedinSection3includeseachpossibleconstraintintheinstancewithsomeprobabilityp.Hereweshowthatresultsfromoursettingeasilyextendtothatof[DLSS14]bygivinganalgorithmthatsimulatesthebehaviorofourmodelwhenthenumberofconstraintsisxed.RecallthataninstanceIFP(n;p)isgeneratedasfollows.ForeachS2[n]kandeachc2f�1;1gk,constraint(c;S)isincludedwithprobabilityp,sotheexpectednumberofconstraintsisp
(2n)k.Inthemodelwherethenumberofconstraintsisxed,theinstanceisguaranteedtohavemdistinctconstraintsforsomevalueofm.TheinstanceJischosenuniformlyfromallsubsetsoff�1;1gk[n]kwithsizeexactlymTheoremD.1.SupposethereexistsanecientalgorithmRthat,onagivenCSPinstanceIFP(n;p),forallppmin,certiesthatOpt(I)forsome01withhighprobability.ThenthereexistsanecientalgorithmAthatcertiesthatarandominstanceJofCSP(P)withconstraintshasOpt(J)+2��1ln
1=2withhighprobabilitywhen1���1ln
1=2(2n)kpmin.49 Algorithm1 AlgorithmA1:p (1�d)(2n)�k.2:drawmBinomial�p;(2n)k
3:ifm�orm(1�2d)then4:return\fail."5:I J6:fori=m+1:::do7:RemovearandomconstraintfromIchosenuniformly8:RunRonI9:ifRcertiesthatOpt(I)then10:return\Opt(J)+2d."11:else12:return\fail." Proof.Onarandominstancewithconstraints,wecangenerateaninstanceIthatsimulatesthisbehaviorbychoosinganappropriatevalueforp,drawingmBinomial�p;(2n)k
andthendiscarding�moftheconstraints.Forbrevity,letd=��1ln
1=2:Algorithm1describesthebehaviorofA.Thefractionofremovedconstraintsisatmost2d,soevenifalloftheremovedconstraintswouldhavebeensatised,theircontributiontoOpt(J)isatmost2d.Consequently,Awillneverincorrectlyoutput\Opt(J)+2d."Furthermore,theprobabilityoffailingtorefuteaninstancewithvalueatmost1�+2dduetoexitingatstep2isok;t(1).Wetreatmasasumof(2n)kindependentBernoullivariableswithprobabilitypanddenoteE[m]by m.ApplyingaChernoboundyieldsthefollowing.Pr[m�]=Pr[m� m=(1�d)]=Pr[m� m(1+d 1�d)]exp��1ln(1�(ln)�1=2) 3exp(�(ln))=1=poly():Similarly,Pr[m(1�2d)]=Pr[m m(1�d 1�d)]exp(�(ln))=1=poly():If(1���1ln
1=2)(2n)kpmin,thenppminandRwillbeabletocertifyOpt(I)withhighprobability. 50