Alternation-TradingProofs,LinearProgramming,andLowerBoundsRyanWilliams - PDF document

Alternation-TradingProofs,LinearProgramming,andLowerBoundsRyanWilliamsInstituteforAdvancedStudyAbstractAfertileareaofrecentresearchhasdemonstratedconcretepolynomialtimelowerboundsforsolvingnaturalhardproblemsonrestrictedcomputationalmodels.AmongtheseproblemsareSatisability,VertexCover,HamiltonPath,MOD6-SAT,Majority-of-Majority-SAT,andTautologies,tonameafew.Theproofsoftheselowerboundsfollowacertainproof-by-contradictionstrategy,whichwecall“resource-trading”or“alternation-trading.”Animportantopenproblemistodeterminehowpowerfulsuchproofscanpossiblybe.Weproposeamethodologyforstudyingtheseproofsthatmakesthemamenabletobothformalanalysisandautomatedtheoremproving.Formalizingtheframework,weprovethatthesearchforbetterlowerboundscanoftenbeturnedintoaproblemofsolvingalargeseriesoflinearprogramminginstances.Weimplementasmall-scaletheoremproverandreportsurprisingresults,whichallowustoextractnewhuman-readabletimelowerboundsforseveralproblems.Wealsousetheframeworktoproveconcretelimitationsonthecurrenttechniques. ThismaterialisbasedonworksupportedinpartbyNSFgrantCCR-0122581atCMU,andNSFgrantCCF-0832797atPrincetonUniversity/IAS.Anyopinions,ndingsandconclusionsorrecommendationsexpressedarethoseoftheauthor.Email:ryanw@ias.edu. 1IntroductionThisworkisconcernedwithprovingnewlimitationsoncomputersbyexploitingtheircapabilities.Manyknownlowerboundsfornaturalproblemsfollowapatternthatwecallaresource-tradingscheme.Informallyspeaking,theschemeusesfourbasicsteps:(1)AssumeahardproblemcanbesolvedinnctimewithresourcesR.(Let'sabbreviatetheclassofsuchproblemsasR[nc].)Wewishtoobtainacontradiction.Forexample,R[nc]maybeDTISP[nc;poly(logn)],thesetofproblemssolvableinnctimeandpoly(logn)space,andmaybesatisability(SAT).(2)ProveaSpeedupLemmathat“tradestimeforresources”,whereR[t]isshowntobeinaclassS[o(t)],foramorepowerfulresourceSandpolynomialst.Forexample,S[t]maybetheclassofproblemssolvablebyalternatingmachinesintimet.Nepomnjascii[Nep70]showedthatpoly(logn)spacealgorithmsrunninginnktimecanbesimulatedbyakmachine(usingkalternations)in~O(n)time.(3)ProveaSlowdownLemmathat“tradesresourcesfortime”,whereS[t]isshowntobeinR[td],forsmalld1.Thistypicallyusestheassumptionthat2R[nc].Forexample,ifSAThasannctime,poly(logn)spacealgorithm,then(byastrongformoftheCook-Levintheorem)itfollowsthatNTIME[t]hastc+o(1)time,poly(logt)spacealgorithms,andconsequentlykTIME[t]hastck+o(1)time,poly(logt)spacealgorithms.(4)Combine(2)and(3)toshowC[t]C[t1"],forsome"�0andcomplexityclassC[t],implyingacontradictionwithahierarchytheoremforC.Forexample,ifSAThasannctime,poly(logn)spacealgorithm,then2TIME[t]DTISP[tc2+o(1);poly(logt)]2TIME[tc2=2],wheretherstinclusionholdsby(3)andthesecondholdsby(2).Thiscontradictsthealternatingtimehierarchyifc22.Theaboveisthenp 2"lowerboundofLiptonandViglas[LV99].Thisschemehasbeenappliedinmanysettings,datingbacktothe70's.Apartiallistincludes:TimeVersusSpace.Hopcroft,Paul,andValiant[HPV77]provedthatSPACE[n]*DTIME[o(nlogn)]formultitapeTuringmachines,byprovingthe“speeduplemma”thatDTIME[t]SPACE[t=logt]andinvokingdiagonalization.(Theirresultwasalsoextendedtogeneralmodels[PR81,HLMW86].)DeterminismvsNondeterminismforMultitapeTMs.AcelebratedresultofPaul-Pippenger-Szemeredi-Trotter[PPST83]isthatNTIME[n]=DTIME[n]formultitapeTuringmachines.Thekeycomponentintheproofisthe“speeduplemma”thatDTIME[t]4TIME[t=logt]inthemultitapesetting.DeterministicandNondeterministicSpace-BoundedAlgorithms.Ourmodelisarandomaccessmachineusingsmallspace(n=(logn)c,n1",andno(1)aretypicalvalues).Thetimelowerboundsarefortradi-tionalNP-completeproblemsandproblemshigherinthepolynomialhierarchy[Kan84,For97,LV99,FvM00,FLvMV05,Wil06,Wil08].ThebestknowndeterministictimelowerboundforsolvingSATwithno(1)spacealgorithmsisn2cos(=7)o(1)n1:801[Wil08].TheboundalsoholdsforthecountingproblemMODm-SATwheremisacompositethatisnotaprimepower.Fornondeterministicalgorithmsusingno(1)space,thebestknowntimelowerboundknownfornaturalco-NPproblems(suchasTAUTOLOGY)hasbeennp 2o(1),byFortnowandVanMelkebeek[FvM00].ProbabilisticandQuantumSpace-BoundedAlgorithms.Allenderetal.[AKRRV01]showedthatMaj-Maj-SATrequiresn1+\n(1)timetosolveonunboundederrormachinesthatusen1"space,for"�0.DiehlandVanMelkebeek[DvM06]provedthatfork2,k-QBFrequires\n(nko(1))timewithrandomizedtwo-sidederroralgorithmsusingno(1)space.Viola[Vio07]hasshownthat3-QBFrequiresn1+\n(1)timeonTuringmachineswitharandomaccessinputtapeandtwo-wayread-writeaccesstoarandombittape.VanMelkebeekandWatson[vMW07,vM07]haveshownhowtoadapttheresultofAdlemanetal.[ADH97]thatBQPPPtoextendAllenderetal.toatimelowerboundforsolvingMaj-Maj-SATwithquantumalgorithms.1 GeneralMultidimensionalTMs.Thismodelhasread-onlyrandomaccesstoitsinput,anno(1)read-writestore,andread-writeaccesstoad-dimensionaltapeforaxedd1.Thismodelgeneralizesseveralothers,andisthemostpowerful(andphysicallyrealistic)modelknownwherewestillknownon-trivialtimelowerboundsforSAT.MultidimensionalTMshavebeenstudiedformanyyears;forinstance,[Lou80,PR81,Gri82,Kan83,MS87,LL90,vMR05,Wil06]provedlowerboundsforsolvingproblemsinthismodel,andthebestboundforSATisessentially\n(np (d+2)=(d+1))timeinthed-dimensionalcase.Thelowerboundproofsoftheabovetypehavebeentypicallyadhoc,makingithardtobuildintuitionaboutthem.Onegetsasensethatthespaceofallpossibleproofsmightbedifculttosystematicallystudy.1.1MainResultsWeintroduceamethodologyforreasoningaboutresource-tradingproofsthatisalsopracticallyimplementableforndingshortproofs.Wearguethatforalmostallknownresource-tradinglowerbounds,theproofscanbereformulatedinawaythatthesearchfornewlowerboundsbecomesafeasibleproblemthatcomputerscanhelpattack.1Informally,the“hardwork”inproofscanoftenbereplacedbyaseriesoflinearprogrammingproblems.Furthermoretheframeworkallowsustoprovelimitationsonwhatcanbeproved.Theselimitationsareimportantsincesomecomponentsoftheseproofsdonotrelativizeinsomesense(cf.AppendixA).Inthispaper,thisapproachisappliedinseveralscenarios.Inallcases,theresourcebeing“traded”isalternations,soforthepurposesofthisworkwecalltheproofsalternation-trading.DeterministicTime-SpaceLowerBoundsforSATandBeyond.Aidedbyresultsofacomputerprogram,weshowthatanyalgorithmsolvingSATint(n)timeands(n)spacemusthavet
s\n(n2cos(=7)o(1)).Previously,thebestknownresultwast
s\n(n1:573)[FLvMV05].Ithasbeenconjecturedthatthecurrentframeworksufcedtoprovean2o(1)timelowerboundforSAT,againstalgorithmsusingno(1)space.2Wepresentstrongevidence(fromcomputersearch)thatthebestknownn2cos(=7)o(1)lowerbound[Wil08]isalreadyoptimalfortheframework.Weshowthatitwillbeimpossibletoobtainn2withtheframework,formalizingaconjectureof[FLvMV05].3WealsoprovelowerboundsonQBFk(quantiedBooleanformulaswithatmostkquantierblocks),showingitrequires\n(nk+1k)timeforno(1)spacealgorithms,wherek0:2andlimk!1k=0.4Theseresultsappearalsooptimalforthecurrenttools.NondeterministicTime-SpaceLowerBoundsforTautologies.Adaptingthemethodologyforthisproblem,acomputerprogramfoundaveryshortproofimprovinguponFortnowandVanMelkebeek's8-yearoldbound.Longerproofssuggestedaninterestingpattern.Formalizingit,weprove(onpaper)ann41=3o(1)n1:587timelowerbound,andexperimentssuggestoptimalityfortheframework.Afterlearningofourshortproof,DiehlandVanMelkebeekhaveprovenasimilarresult[DvMW07].Wealsoshowitisnotpossibletoobtainanntimelowerbound,where=1:618:::isthegoldenratio.Thisissurprisingsincewehaveknownforsometime[FvM00]thatanlowerboundisprovablefordeterministicalgorithms.LowerBoundsforMultidimensionalTMs.Herethemethoduncovershighlyregularbehaviorinthebestlowerboundproofs,regardlessofthedimensionofthetape.Studyingtheoutputofatheoremprover,weextractan\n(nr)timelowerboundforthed-dimensionalcase,whererd1istherootofaparticularquintic 1WenotethatcombinatorialargumentssuchasSanthanam'stime-spacelowerboundforSATonmultitapeTuringma-chines[San01]donotfallunderthealternation-tradingparadigm,buttheyarealreadyknowntohavedifferentlimitations.2Icouldnotndanexplicitreferenceforthisconjecture,butIhavereceivedseveralrefereereportsinthepastthatstateit.Alsocf.[LV99]inFOCS'99.3Thatis,weformalizethestatement:“...somecomplexitytheoristsfeelthatimprovingthegoldenratioexponentbeyond2wouldrequireabreakthrough”inSection8of[FLvMV05].4NotetheQBFkresultsappearedintheauthor'sPhDthesisin2007buthavebeenunpublishedtodate.2 pd(x)withcoefcientsdependingond.Forexample,r11:3009,r21:1887,andr31:1372.Again,computersearchsuggeststhisisthebestpossible,andweprovethatitisimpossibletoimprovetheboundford-dimensionalTMston1+1=(d+1)withthecurrenttools.TheabovelowerboundsholdforotherNPandco-NP-hardproblemsaswell,sincetheonlypropertyrequiredisthateverysetinNTIME[n](respectively,coNTIME[n])hassufcientlyefcientreductionstotheproblem.Furthermore,westressthatthisapproachisnotlimitedtotheabovescenarios,andcanbeappliedtotheleagueofproblemsdiscussedinVanMelkebeek'ssurveys[vM04,vM07].Thisworkpromotesanewmethodologyforprovinglowerbounds,whereprospectivelower-boundersformalizetheirproofrules,writeaprogramtotestideasandgenerateshortproofs,thenstudytheresultsandextrapolatenewresults.1.2ReductiontoLinearProgrammingThekeytoourformulationisthatweseparatethediscretechoicesinalowerboundprooffromthereal-valuedchoices.Thediscretechoicesconsistofthesequenceofrulestoapplyinaproof,andwhichcomplexityclassC[t]touseintheproofbycontradiction.Wegiveseveralsimplicationsthatgreatlyreducethenumberofnecessarydiscretechoices,withoutlossofgenerality.Real-valuedchoicescomefromselectingt,aswellasparametersarisingfromruleapplications.Weprovethatoncethediscretechoicesaremade,theremainingreal-valuedproblemcanbeexpressedasaninstanceoflinearprogramming.Thismakesitpossibletosearchfornewproofsviacomputer,anditalsogivesusaformalhandleonthelimitationsoftheseproofs.Onecannoteasilysearchoverallpossibleproofs,asthenumberofdiscretechoicesisstill2n=n3=2forproofsofnlines(proportionaltothenthCatalannumber).Neverthelessitisquitefeasibletosearchoverall20+lineproofs.Thesesearchesalreadyrevealhighlyregularpatterns,indicatingthatcertainstrategieswillbemostsuccessfulinprovinglowerbounds;ineachcasewestudy,theresultingstrategiesaredifferent.Followingthestrategies,weestablishnewlowerboundproofs.Finally,thepatternsalsosuggesthowtoprovelimitationsontheproofsystem.ImportantNote:Intherst12pages,wecanonlybrieydescribetheresultsandtechniques.PleaseseetheAppendicesforbackgroundinformationandmoredetails.2PreliminariesWeassumefamiliaritywiththebasicsofcomplexity,especiallyalternation[CKS81].Weusebig-\nnotationintheinnitelyoftensense,sostatementslike“SATisnotinO(nc)time”areequivalentto“SATrequires\n(nc)time.”Allfunctionsareassumedtobeconstructiblewithintheappropriatebounds.Ourdefaultcomputationalmodelistherandomaccessmachine,butparticularvariantsdonotaffecttheresults.DTISP[t(n);s(n)]istheclassoflanguagesacceptedbyaRAMrunningint(n)timeands(n)space,simultaneously.Forconvenience,wedeneDTS[t(n)]:=DTISP[t(n)1+o(1);no(1)]toavoidnegligibleo(1)factors.Toproperlyformalizealternation-tradingproofs,weintroducenotationforalternatingcomplexityclasseswhichincludeinputconstraintsbetweenalternations.Theseconstraintsarecriticalfortheformalism.Dene(9f(n))bCtobetheclassoflanguagesrecognizedbyamachineNthat,oninputx,writesaf(n)1+o(1)bitstringynondeterministically,copiesatmostnb+o(1)bitszofthetuplehx;yideterministically(inO(nb+o(1))time),thenfeedszasinputtoamachinefromclassC.WerefertothisbehaviorbysayingthattheclassCisconstrainedtonbinput.Dene(9f(n))C:=(9f(n))maxf1;(logf(n))=(logn)gC.Thatis,thedefaultinputlengthisassumedtobeO(f(n)1+o(1)+n1+o(1)).Theclass(8f(n))bCisdenedsimilarly(withco-nondeterminism).Wesaythattheexistentialanduniversalphasesofanalternatingcomputationarequantierblocks,toreectthenotation.Henceamachineoftheclass(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1]withQi23 f9;8gmeansthattheinputtothecomputationstartingattheithquantierblockisoflengthnbi+o(1)foralli=1;:::;k,andtheinputtotheDTScomputationhaslengthnbk+1+o(1).(Ofcourse,therstquantierblockalwayshasaninputoflengthn.)Itisimportanttokeeptrackoftheinputlengthstoquantierblocks,sinceseverallowerboundsrelyonthefactthattheseinputscanbesmallincertaininterestingcases.Forreadersnewtothisarea,westronglyencouragethemtoreadAppendixA,AShortIntroductiontoTime-SpaceLowerBounds.3Time-SpaceLowerBoundsforSATOurstudybeginswithpolynomialtime-spacelowerboundsforNTIME[n]problemssuchasSAT.Weshalldescribetheapproachinsomedetailhere;theothersettingsassumeknowledgeofthissection.Webeginwithaformalizationofthealternation-tradingframework.Alternation-tradingproofsapplyasequenceof“speedup”and“slowdown”lemmasinsomeorder,withthegoalofreachingacontradictionbyatimehierarchytheorem.Weformalizealternation-tradingproofsforDTSclassesasfollows:5Denition3.1Letc�1.Analternation-tradingproofforcisalistofcomplexityclassesoftheform:(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1];(1)wherek0,Qi2f9;8g,Qi=Qi+1,ai�0,andbi1,foralli.(Whenk=0,theclassisdeterministic.)Theitemsofthelistarecalledlinesoftheproof.Eachlineisobtainedfromthepreviouslinebyapplyingeitheraspeedupruleoraslowdownrule.Moreprecisely,iftheithlineis(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1];thenthe(i+1)stlinehasoneofthreepossibleforms:1.(SpeedupRule0)(Qknx)maxfx;1g(Qk+1n0)1DTS[nak+1x],whenk=0andx2(0;ak+1).62.(SpeedupRule1)(Q1na1)b2(Q2na2)


bk(Qknmaxfak;xg)maxfx;bk+1g(Qk+1n0)bk+1DTS[nak+1x];fork�0andanyx2(0;ak+1).3.(SpeedupRule2)(Q1na1)b2


bk(Qknak)bk+1(Qk+1nx)maxfx;bk+1g(Qk+2n0)bk+1DTS[nak+1x];fork�0andanyx2(0;ak+1).4.(SlowdownRule)(Q1na1)b2(Q2na2)


bk1(Qk1nak1)bkDTS[nc
maxfak+1;ak;bk;bk+1g],fork�0.Analternation-tradingproofshows(NTIME[n]DTS[nc]=)A1A2)ifitsrstlineisA1anditslastlineisA2.ThedenitioncomesdirectlyfromthestatementsoftheSpeedupLemma(LemmaA.1)andSlowdownLemma(LemmaA.2)forspace-boundedcomputations.(Notethen0intheSpeedupLemmacorrespondstolognno(1),whichisnegligible.)SpeedupRules0,1,and2canbeeasilyveriedtobesyntacticformulationsoftheSpeedupLemma,wheretheDTSpartofthesped-upcomputationonlyreadstwoguessed 5Thisformalizationhasimplicitlyappearedinpriorwork,butnottothedegreethatweinvestigateinthispaper.6Pleasenotethatthe(k+1)thquantierisn0inordertoaccountfortheO(logn)sizeofthequantier.4 congurations—sotheinputitreadsisdifferentfromtheinputreadbytheinnermostquantierblock.Forinstance,SpeedupRule1holds,since(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1](Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qknx)maxfbk+1;xg(Qk+1n0)bk+1DTS[nak+1](Q1na1)b2(Q2na2)


bk(Qknmaxfak;xg)maxfbk+1;xg(Qk+1n0)bk+1DTS[nak+1]:Rule2isakintoRule1,exceptthatitusesoppositequantiersinitsinvocationoftheSpeedupLemma.TheSlowdownRuleworksanalogouslytoLemmaA.2.Itfollowsthatalternation-tradingproofsaresound.NoteSpeedupRules0and2addtwoquantierblocks,SpeedupRule1addsonlyonequantier,andallthreerulesintroduceaparameterx.Byconsidering“normalform”proofs(denedinthenextsection),wecanshowthatRule2canalwaysbereplacedbyapplicationsofRule1.Foraproof,cf.AppendixC.ForthisreasonwejustrefertotheSpeedupRule,dependingonwhichofRule0orRule1applies.ANormalForm.Deneanyclassoftheformin(1)tobesimple.DeneclassesA1andA2tobecomplemen-taryifA1istheclassofcomplementsoflanguagesinA2.Everyknown(model-independent)time-spacelowerboundforSATshows“NTIME[n]DTS[nc]impliesA1A2”,forsomecomplementarysimpleclassesA1andA2,contradictingatimehierarchy(cf.TheoremA.2).Asimilarclaimholdsfornondeterministictime-spacelowerboundsagainsttautologies(whichproveNTIME[n]coNTS[nc]impliesA1A2),ford-dimensionalmachinelowerboundssolvingSAT(whichproveNTIME[n]DTIMEd[nc]impliesA1A2),andforotherproblems.Wenowintroduceanormalformforalternation-tradingproofs.Weshowthatanylowerboundprovablewithcomplementarysimpleclassescanalsobeestablishedwithanormalformproof.Thissimplicationgreatlyreducesthedegreesoffreedominaproof,aswenolongerhavetoworryaboutwhichcomplementarysimpleclassestochooseforthecontradiction.Denition3.2Letc1.Analternation-tradingproofforcisinnormalformif(a)therstandlastlinesareDTS[na]andDTS[na0]respectively,forsomeaa0,and(b)nootherlinesareDTSclasses.WeshowthatanormalformproofforcimpliesthatNTIME[n]*DTS[nc].Lemma3.1Letc1.Ifthereisanalternation-tradingproofforcinnormalformhavingatleasttwolines,thenNTIME[n]*DTS[nc].Theorem3.1LetA1andA2becomplementary.Ifthereisanalternation-tradingproofPforcthatshows(NTIME[n]DTS[nc]=)A1A2),thenthereisanormalformproofforc,oflengthatmostthatofP.ProofsofLemma3.1andTheorem3.1canbefoundinAppendixD.Theimportantconsequenceisthatweonlyneedtofocusonnormalformproofsinaproofsearch.Fortheremainderofthissection,weassumethatallalternation-tradingproofsunderdiscussionareinnormalform.ProofAnnotations.Differentlowerboundproofscanresultinquitedifferentsequencesofspeedupsandslowdowns.Aproofannotationrepresentssuchasequence.Denition3.3Aproofannotationforanalternation-tradingproofof`linesisthe(`1)-bitvectorAwhereforalli=1;:::;`1,A[i]=1(respectively,A[i]=0)iftheithlineappliesaSpeedupRule(respectively,aSlowdownRule).An(`1)-bitproofannotationcorrespondstoa“strategy”foran`-lineproof.Foranormalformalternation-tradingproofwith`lines,itisnothardtoshowthatitsannotationAmusthaveA[1]=1,A[`2]=0,and5 A[`1]=0.Thenumberofpossibleproofannotationsiscloselyrelatedtothenumberofwell-balancedstringsoverparentheses.RecallthatthekthCatalannumberisC(k)=1 k+12kk
.Awell-knownfactstatesthatthenumberofwell-balancedstringsoflength2kisC(k).Proposition1Let`�3beeven.Thenumberofpossibleannotationsforproofsof`linesisC(`=21).Hencethenumberofpossibleannotationsforproofsof`linesis(2`=`3=2).Notethatanannotationdoesnotdetermineaproofentirely,asthereareotherparameters.(Theproblemofdeterminingoptimalvaluesfortheseparametersistackledinthenextsection.)Toillustratetheannotationconcept,wegivefourexamples.Lipton-Viglas'np 2lowerbound[LV99](fromtheIntroduction)hastheannotation[1;0;0].Then1:6004boundoftheShortIntroduction(cf.AppendixA)correspondsto[1;1;0;0;1;0;0].ThenboundofFortnow-VanMelkebeek[FvM00]isaninductiveproof,correspondingtoaninnitesequenceofannotations.Innormalform,thesequenceis:[1;0;0];[1;1;0;0;0];[1;1;1;0;0;0;0];:::Then2cos(=7)bound[Wil08]hastwostagesofinduction.LetA=1;0;1;0;:::;1;0;0,wherethe`:::'containanynumberofrepetitions.Thesequenceis[A];[1;A;A];[1;1;A;A;A];[1;1;1;A;A;A;A];:::Thatis,theproofperformsmanyspeedups,thenasequenceofmanyslowdown-speedupalternations,thentwoconsecutiveslowdowns,repeatingthisuntilallthequantiershavebeenremoved.3.1TranslationToLinearProgrammingGivena(normalform)proofannotation,howcanwedeterminethebestproofpossiblewithit?Theobstaclesare(a)theruntimesoftherstandlastDTSclassesoftheproofarefreeparameters,and(b)eachapplicationofaSpeedupRuleintroducesaparameterxi.WenowshowhowtoreduceanannotationAandc�1toalinearprogramthatisfeasibleifandonlyifthereisanalternation-tradingproofofNTIME[n]*DTS[nc]withannotationA.Moreprecisely,theproblemofsettingparameterscanbeviewedasanarithmeticcircuitevaluationwherethecircuithasmaxgates,additiongates,andinputgatesthatmultiplytheirinputbyc.Suchcircuitscanbeevaluatedusingalinearprogram(cf.[Der72])thatminimizesthesumofthegatevalues.LetAbeanannotationof`1bits,andletmbethemaximumnumberofquantierblocksinalineofA;notemiseasilycomputedinlineartime.ThetargetLPhasvariablesai;j,bi;j,andxi,foralli=0;:::;`1andj=1;:::;m.Thevariablesai;jrepresenttheruntimeexponentofthejthquantierblockintheclassontheithline,bi;jistheinputexponenttothejthquantierblockoftheclassontheithline,andforalllinesithatuseaSpeedupRule,xiisthechoiceofxintheSpeedupRule.Forexample:IfthekthlineofaproofisDTS[na],thecorrespondingconstraintsareak;1=a;bk;1=1;(8k�0)ak;i=bk;i=0.Ifthekthlineofaproofis(9na0)bDTS[na],thentheconstraintsareak;0=a;bk;1=b;ak;1=a0;bk;1=1;(8k�1)ak;i=bk;i=0:TheobjectiveistominimizePi;j(ai;j+bi;j)+Pixi.TheLPconstraintsdependonthelinesoftheannotation,asfollows.InitialConstraints.Forthe0thand(`1)thlineswehavea0;1a`1;1,aswellasa0;11;b0;1=1;(8k�1)a0;k=b0;k=0;anda`;11;b`;1=1;(8k�1)a`;k=b`;k=0;6 representingDTS[na0;1]andDTS[na`1;0],respectively.The1stlineofaproofalwaysappliesSpeedupRule1,havingtheform(Q1nx)maxfx;1g(Q2n0)1DTS[nax].Sotheconstraintsforthe1stlineare:a1;1=a0;1x1;b1;1=1;a1;2=0;b1;2x1;b1;21;a1;3=x3;b1;3=1;(8k:4km)a1;k=b1;k=0:ThebelowconstraintsetssimulatetheSpeedupandSlowdownRules:SpeedupRuleConstraints.Fortheithlinewherei�1andA[i]=1,theconstraintsareai;11;ai;1ai1;1xi;bi;1=bi1;1;ai;2=0;bi;2xi;bi;2bi1;1;ai;3ai1;2;ai;3xi;bi;3bi1;2;(8k:4km)ai;k=ai1;k1;bi;k=bi1;k1:Theseconstraintsexpressthat


b2(Q2na2)b1DTS[na1]inthe(i1)thlineisreplacedwith


b2(Q2nmaxfa2;xg)maxfx;b1g(Q1n0)b1DTS[nmaxfa1x;1g]intheithline,whereQ1isoppositetoQ2.SlowdownRuleConstraints.FortheithlinewhereA[i]=0,theconstraintsareai;1c
ai1;1;ai;1c
ai1;2;ai;1c
bi1;1;ai;1c
bi1;2;bi;1=bi1;2(8k:2km1)ai;k=ai1;k+1;bi;k=bi1;k+1;ai;m=bi;m=0:Theseexpressthereplacementof


b2(Q1na2)b1DTS[na1]inthe(i1)thlinewith


b2DTS[nc
maxfa1;a2;b1;b2g]intheithline.Thisconcludesthedescriptionofthelinearprogram.TondthelargestcthatstillyieldsafeasibleLP,wecansimplybinarysearchforit.Thefollowingtheoremsummarizestheabovediscussion.Theorem3.2Givenaproofannotationofnlines,thebestpossiblelowerboundprooffollowingtheannota-tioncanbedetermineduptondigitsofprecision,inpoly(n)time.ProofSearchResults.Followingtheaboveformulation,wewroteproofsearchroutinesinMaple.Millionsofproofannotationsweretried,includingallthoseofpreviouswork,withnosuccessbeyondthe2cos(=7)exponent.Thebestlowerboundsfollowedahighlyregularpattern.Fora424lineannotationfollowingthepattern,theoptimalexponentwasonlyintheinterval[1:80175;1:8018).Formoredetails,cf.AppendixE.Oneinterestingsequenceofannotationsfromthepatternis1k11000(10)0(10)20


(10)k0;fork2.Onecanprovethatthissequencecannotyieldanylowerboundbetterthan2cos(=7).Theorem3.3Inthelimit(ask!1),themaximumlowerboundprovablewiththesequenceofannotations1k11000(10)0(10)20(10)30


(10)k0fork0isthatSATcannotbesolvedinO(n2cos(=7)o(1))timeandno(1)space.Asimilarargumentappliestoallotherannotationsfoundbycomputer.Weareledto:7 Conjecture3.1Thereisnoalternation-tradingproofthatNTIME[n]*DTS[nc],foranyc�2cos(=7).Provingtheaboveconjectureseemscurrentlyoutofreach.Wecangiveapartialresult:Theorem3.4Thereisnoalternation-tradingproofthatNTIME[n]*DTS[n2].AproofisinAppendixF.Atahighlevel,theproofarguesthatanyminimumlengthproofofaquadraticlowerboundcouldbeshortened,givingacontradiction.GoodNews.Despitethebadnewsabove,thetheoremproverdidgiveusenoughinsighttoproveanewlowerboundof\n(n2cos(=7)o(1))onthetime-spaceproductofalgorithmssolvingSAT.Also,theseresultshavealsobeengeneralizedtoquantiedBooleanformulas,leadingtonewlowerbounds.Formore,cf.AppendicesBandM,respectively.4NondeterministicTime-SpaceLowerBoundsforTautologiesTheproblemofprovingnondeterministictime-spacelowerboundsforco-NPhasalsobeenstudied.FortnowandVanMelkebeek[FvM00]provedthatTAUTOLOGYrequires\n(np 2o(1))timeonanondeterministicno(1)spaceRAM.However,sincetheirinitialresult,nofurtherimprovementshadbeenmade.Weshowhowtoextendtheapproachoftheprevioussectiontothisproblem,andndthatthebestproofannotationslookquitedifferent.Heretheapproachturnsouttobesuccessfulinndingnewproofs.4.1TheFrameworkandLinearProgrammingTranslationSimilartotheclassDTS,setNTS[na]:=NTISP[na;no(1)]andcoNTS[na]:=coNTISP[na;no(1)]forbrevity.Asinthepreviouslowerboundsetting,thereareSpeedupandSlowdownrulesthatareappliedinsomewaythatcontradictsatimehierarchy,althoughtherulesaresomewhatdifferenthere.Inthefollowing,letQbeastringofquantierblocks,soQ=(Q1na1)b2


(Qk1nak1).Lemma4.1(Speedup)Forb1,a1,x0,ands0,QbNTISP[na;ns]Qb(9nx+s)maxfb;x+sg(8logn)maxfb;sgNTISP[nax;ns]:Inparticularfors=o(1)wehaveNTS[na](9nx)maxf1;xg(8logn)1NTS[nax].Proof.TheproofisanalogoustoLemmaA.1(theSpeedupLemmaforDTISP).2Lemma4.2(Slowdown)IfTAUTOLOGYisinNTS[nc]then1.Qb(9nak)bk+1NTS[nak+1]QbcoNTS[nc
maxfak;ak+1;bk+1;bg],2.Qb(8nak)bk+1coNTS[nak+1]QbNTS[nc
maxfak;ak+1;bk+1;bg],3.Qb(9nak)bk+1coNTS[nak+1]Qb(9nak)bk+1NTS[nc
maxfak+1;bk+1g],and4.Qb(8nak)bk+1NTS[nak+1]Qb(8nak)bk+1coNTS[nc
maxfak+1;bk+1g].8 Theproofsareomittedandarelefttotheinterestedreader.Toobtaincontradictions,oneusesthealternatingtimehierarchy(TheoremA.2)justasinthedeterministiccase.Theabovelemmasimmediatelyleadtoanaturaldenitionofalternation-tradingproofthatcoNTIME[n]NTS[nc]=)A1A2,forclassesA1andA2.AnotherwaytoyieldacontradictionusesaresultsimilartoLemma3.1,whichshowedthatNTIME[n]DTS[nc]impliesDTS[na]*DTS[na0]fora�a0.Lemma4.3IfcoNTIME[n]NTS[nc]thenNTS[na]*coNTS[na0]fora�a0.Thislemmacanbeusedtomotivateadenitionofnormalformproof,andprovethatanyalternation-tradingproofcanbeconvertedintonormalform.Denition4.1Letc1.Analternation-tradingproofthat(coNTIME[n]NTS[nc]=)A1A2)isinnormalformif(1)A1=NTS[na],A2=coNTS[na0],forsomeaa0,and(2)nootherlinesareNTSorcoNTSclasses.Example.IfTAUTOLOGY2NTS[nc]thencoNTIME[n]NTS[nc+o(1)]byTheoremA.1,soNTS[n2](9n)1(8logn)1NTS[n]byLemma4.1(NTISPSpeedup).ApplyingLemma4.2(NTISPSlowdown)thrice,(9n)1(8logn)1NTS[n](9n)1(8logn)1coNTS[nc](9n)1NTS[nc2]coNTS[nc3].Whenc3p 21:25,NTS[na]coNTS[na0]forsomea�a0,whichcontradictsLemma4.3.Lemma4.4Letc1.Ifthereisanalternation-tradingproofthat(coNTIME[n]NTS[nc]=)A1A2)innormalform,andtheproofhasatleasttwolines,thencoNTIME[n]*NTS[nc].Theorem4.1LetA1andA2besimpleandcomplementary.Ifthereisanalternation-tradingproofPthat(NTIME[n]DTS[nc]=)A1A2),thenthereisanormalformproofforc,oflengthatmostthatofP.Onecanalsodeneproofannotationsforthissetting.Thevectorscorrespondingtovalidannotationschangeduetodifferencesintherules.Forexample,note[1;0;0;0],[1;1;0;0;0;0;0],and[1;0;1;0;0;0;0]arevalidannotationsforthissetting,therstbeingtheannotationfortheaboveexample.FromtheSpeedupandSlow-downLemmasgivenabove,observethattheoperationsonexponentsareagainmax,+,andmultiplicationbyc.Hencethetranslationofannotationstolinearprogrammingfollowsasimilarstrategyasbefore:wedenevariablesai;j,bi;j,xiforalllinesiandpossiblequantierblocksj,replacecomponentsoftheformmaxfa;a0g=a00witha00a,a00a0,thenminimizePai;j+bi;j+xi.4.2ProofSearchResults,aTimeLowerBound,andaLimitationThestructureofgoodlowerboundproofsforthe“TAUTOLOGYversusNTISP”problemturnedouttobedifferentfromthoseforthe“SATversusDTISP”problem.Theprogramuncoveredinterestingnewresults.Forone,FortnowandVanMelkebeek'sp 2lowerboundisnotoptimal;thebest11-lineproofalreadygivesa1:419exponent.Formoredetails,cf.AppendixG.Fromexperiments,wefoundannotationsA1;A2;A3;A4(alloptimalfortheirnumberoflines)withthepropertythatAi+1=[1;Ai;Ai;0].Thisnaturallysuggestsaproofbyinductionwheretheinductionhypothesisisappliedtwice.Wearrivedatthefollowing.Theorem4.2TAUTOLOGYrequiresn3p 4o(1)timefornondeterministicalgorithmsusingno(1)space.TheproofisinAppendixH;itisaninductioncorrespondingtoaninnitesequenceofannotations.Thetheorem'sannotationsandparametersettingsfortherstfourstepsoftheinductionarepreciselythosechosenbythesearchprogramforA1,A2,A3,andA4;inthissense,theformalproofcorrespondswiththebestresults9 fromcomputersearch.AsexperimentsindicatedthatthesequenceA1,A2,A3,etc.isessentiallythebestonecando,webelievethelowerboundisoptimalforthisframework.Conjecture4.1ThereisnoalternationtradingproofthatcoNTIME[n]*NTS[nc],foranyc�3p 4.Someinterestinglimitationcanbeprovedforalternation-tradingproofs.Namely,unlikethecaseoftimelowerboundsforSAT,nogoldenratiolowerboundcanbeachievedinthissetting.TheproofisinAppendixI.Theorem4.3ThereisnoalternationtradingproofthatcoNTIME[n]*NTS[nc],foranyc1:618.5LowerBoundsforMultidimensionalTMsNext,weconsiderlowerboundsforamultidimensionalmachinemodel,whichsubsumesboththesmall-spacerandomaccessmodelandoff-lineone-tapeTuringmachinemodels.Ourapproachyieldsnewlowerboundshereaswell.Torecall,themachinemodelhasaread-only/random-accessinputtape,aread-write/randomaccessstorageofno(1)bits,andad-dimensionaltapethatisread-writewithsequential(two-way)access.5.1TheFrameworkandLinearProgrammingTranslationDeneDTIMEd[t(n)]tobetheclassoflanguagesrecognizedbyd-dimensionalone-tapemachinesinO(t(n))time.Lowerboundproofsforthesemachineshavedifferentstructurefromthersttwo:onespeeduprulesimulatesad-dimensionalmachinebyanondeterministic(orco-nondeterministic)machinewithasmallspacebound.Moreprecisely,letQrepresentastringofquantierblocks,soQ=(Q1na1)b2


(Qknak1).Lemma5.1(DTIMEdtoDTISP)LetQk+12f9;8g.Thenforall0saandb1,QbDTIMEd[na]Qb(Qk+1nas)maxfas;bgDTISP[na;nds].TheproofofLemma5.1guessesashortcrossingsequencethatdividesthetapeintonsblocks,eachofwhichcanbesimulatedinO(nds)space,cf.[MS87,vMR05].Weomititsproofhere.Lemma5.1letsususetheSpeedupLemmaforspace-boundedmachines(LemmaA.1)toproveclassinclu-sions.AnotherSlowdownLemmaisrequired;itsprooffollowsthelinesofearlierresults.Again,letQbeastringofquantierblocks.Lemma5.2(SlowdownforDTIMEd)SupposeNTIME[n]DTIMEd[nc].Thenforai;bi1,eak+1,Qbk(Qk+1nak)bk+1DTIMEd[nak+1]QbkDTIMEd[nc
maxfbk;bk+1;ak;ak+1g],andQbk(Qk+1nak)bk+1DTISP[nak+1;ne]QbkDTIMEd[nc
maxfbk;bk+1;ak;ak+1g]:Wealsouseastandardtimehierarchytheorem:Lemma5.3Fora&#x-278;&#x.223;a0,DTIMEd[na]*DTIMEd[na0].Example.In1983,Kannan[Kan83]provedthatNTIME[n]*DTIME1[n4p 3=2],usingaweakerversionofLemma5.1.Reproducinghisargument:DTIME1[n3=2](9n)DTISP[n3=2;n1=2]byDTIME1toDTISP(Lemma5.1)(9n)(8logn)DTISP[n;n1=2]bytheSpeedupLemmaforDTISP(LemmaA.1)(9n)DTIME1[nc]bytheSlowdownLemmaforDTIMEd(Lemma5.2)DTIME1[nc2]bySlowdownforDTIMEd10 Acontradictionfollowsfromcp 3=2andLemma5.3.ButSAT2DTIME1[nc]impliesNTIME[n]DTIME1[nc+o(1)](TheoremA.1),soSATcannotbesolvedinO(np 3=2")timeona1-DTM.ThisistheSATlowerboundprovedbyVanMelkebeekandRaz[vMR05].Correspondingnotionsofalternation-tradingproofsandannotationscanbedenedhereaswell.ThesimpleclassesherehaveeitheraDTISP[t;s]orDTIMEd[t]phaseattheendoftheirdescriptions.TherearetwopossibleSpeedupLemmasforaclasswithaDTISPphase:oneintroducesonlyasinglequantier,andanotherintroducestwo.However,justasbefore,wecanprovethesecondSpeedupisunnecessary.Hencetheproofannotationsforthissettingcanbebitvectors,forthetworulesapplicableateachstep.IfthedeterministicclassisaDTIMEdclass,a“speedup”meansthatweapplytheDTIMEdtoDTISPLemma.IfthedeterministicclassisDTISP,a“speedup”meansthatweapplytheSpeedupLemma.Forexample,theannotationforaboveexampleis[1;1;0;0].Thestructureofvalidproofannotationschangesaccordingly.OnecanalsodeneanormalformproofthatbeginswithDTIMEd[na]andendswithDTIMEd[na0],wherea0a.SuchproofsimplylowerboundsduetoLemma5.3.Everyalternation-tradinglowerboundproofagainstDTIMEdhasacorrespondingnormalformproof,byanargumentanalogoustoTheorem4.1.Finally,thetranslationtolinearprogrammingissimilar,exceptthatnewvariablessiareintroducedforthespaceexponentoftheDTISPclassinlinei(forallrelevanti).Thesesiareadditionalfreeparameters.5.2TimeLowerBoundandProofLimitationThebestannotationfound(foralldimensionsd)wasthe66lineannotation[1;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;1;1;0;0];whichleadstoproofsthatNTIME[n]isnotinanyofDTIME1[n1:3009],DTIME2[n1:1875],andDTIME3[n1:1343].Infact,allthebestannotationsfoundhadtheform[111(011)k0(011)`00].(MoredetailsareinAp-pendixJ.)Theannotationssuggestaproofwhereonehasaninductivelemmacapturingthe(011)behavior,thenappliesthelemmatwicetoaproofofsixmorelines.Thisstrategyleadsto:Theorem5.1SAT=2DTIMEd[nc],forallcrdwhererd1isarootofpd(x)=(2d+1)(d+1)2x52(d+1)(2d+1)x4d2x3+2(d+1)x2((2d+1)(d+1)2+1)x+d(d+1):TheproofisinAppendixK.ItestablishesaninductiveSpeedupLemmathatefcientlysimulatesDTIMEdin2TIMEassumingNTIME[n]DTIMEd[nc],thenappliesthelemmatwiceinthelowerboundproof.Weagainconjecturethattheabovelowerboundisoptimalforalternation-tradingproofs.Wecanprovethatnon1+1=dlowerboundispossibleford-dimensionalTMs.TheproofisinAppendixL.Theorem5.2ThereisnoalternationtradingproofthatSAT=2DTIMEd[nc],foranyc1+1=d.6DiscussionWeintroducedamethodologyforreasoningaboutlowerboundsinthealternation-tradingframework.Thisgivesanelegantandgeneralwaytoattacklowerboundproblemsviacomputer,andletsusestablishconcretelimitationsonknowntechniques.Wenowhaveabetterunderstandingofwhatthesetechniquescanandcannotdo,andatoolforaddressingfutureproblems.Previously,theproblemofsettingparameterstogetagoodlowerboundwasanhighlytechnicalexercise.Thisworkshouldreducetheloadonfurtherresearch:onceanew11 speeduporslowdownlemmaisfound,oneonlyneedstondtherelevantlinearprogrammingformulationtobeginunderstandingitspower.Weendwithtwoopenproblems.1.Establishtightlimitationsforalternation-tradingproofs.Thatis,showthatthebestpossiblealternation-tradingproofsmatchtheoneswehaveprovided.Empiricalresultsaresometimesmetwithskepticism,soitiscriticaltoverifythelimitationswithformalproof.Wehavemanagedtoprovenon-triviallimita-tions,anditseemslikelythattheideasinthosecanbeextended.2.Discoveringredientsthataddsigncantlytotheframework.Herethereareseveralpossibleavenues.Oneistondnewseparationresultsthatleadtonewcontradictions.AnotheristondimprovedSpeedupand/orSlowdownLemmas.TheSlowdownLemmasarethe“blandest”oftheingredients,inthattheyarethemostelementary(andtheyrelativize).Forinstance,itmaybepossibletogiveanbetterSpeedupLemmabyprovingthatDTSiscontainedinnitelyofteninafasteralternatingtimeclass,anduseanalmost-everywheretimehierarchy[GHS91,ABHH93]toobtainacontradiction.Finally,combinatorialmethodshaveledtoseveralimpressivetime-spacelowerbounds.Forexample,Ajtai[Ajt02]andBeameetal.[BJS01]haveproventime-spacelowerboundsforbranchingprograms;GurevichandShelah[GS88]gaveaprobleminNTISP[n;logn]butnotDTISP[n1+a;nb])whenb+2a1=2.Isitpossibletoincorporatecombinatorialmethodsintothealternation-tradingframework?7AcknowledgementsIamgratefultoDietervanMelkebeek,RyanO'Donnell,ManuelBlum,andStevenRudichfortheirinvaluablefeedbackonmyPhDthesis,whichincludedsomepreliminaryresultsonthiswork.IalsothankScottAaronsonforusefuldiscussionsaboutirrelativization,andanonymousrefereesfortheircomments.References[ADH97]L.Adleman,J.DeMarrais,andM.Huang.Quantumcomputability.SIAMJournalonComputing26:1524-1540,1997.[ABHH93]E.Allender,R.Beigel,U.Hertrampf,andS.Homer.Almost-EverywhereComplexityHierarchiesforNondeterministicTime.Theor.Comput.Sci.115(2):225–241,1993.[AKRRV01]E.Allender,M.Koucky,D.Ronneburger,S.Roy,andV.Vinay.Time-SpaceTradeoffsintheCountingHierarchy.InProceedingsofIEEEConferenceonComputationalComplexity(CCC),295–302,2001.[Ajt02]M.Ajtai.DeterminismversusNondeterminismforLinearTimeRAMswithMemoryRestrictions.J.ComputerandSystemSciences65:2–37,2002.[BJS01]P.Beame,T.S.Jayram,andM.E.Saks.Time-SpaceTradeoffsforBranchingPrograms.J.ComputerandSystemSciences63(4):542–572,2001.[CKS81]A.K.Chandra,D.Kozen,andL.J.Stockmeyer.Alternation.JACM28(1):114–133,1981.[Coo88]S.A.Cook.ShortPropositionalFormulasRepresentNondeterministicComputations.InformationProcessingLetters26(5):269-270,1988.12 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GuidetotheAppendicesAppendixAisashortintroductiontothetechniquesusedinthisworkandrelatedones.AppendixBprovesthatforanytimetandspacesSATalgorithm,ts=\n(nc)forallc2cos(=7).AppendixCprovesthatSpeedupRule2isredundant,i.e.,itcanbesimulatedbySpeedupRule1intheproofsystem.AppendixDprovessomepropertiesthatallowustorestrictourselvestoconsideringnormalformproofs.AppendixEanalyzesresultsofacomputerproofsearchforSATtime-spacelowerbounds.AppendixFprovesthatnoquadratictimelowerboundforSATintheno(1)spacesettingcanbeprovedwithalternation-tradingproofs.AppendixGanalyzesresultsofacomputerproofsearchforTautologytime-spacelowerbounds(onnondeterministicmachines).AppendixHprovesan3p 4timelowerbound,basedonresultsofAppendixF.AppendixIprovesthatannn1:618timelowerboundforTautologyisnotpossible.AppendixJanalyzesresultsofacomputerproofsearchforSATlowerboundsonmultidimensionalTMs.AppendixKproveslowerboundsbasedontheresultsofAppendixI.AppendixLprovesthatann1+1=dtimelowerboundforSATond-dimensionalTMsisnotpossiblewithalternation-trading.AppendixMprovesnewlowerboundsforQBFk,inspiredbytheresultsofcomputersearches.15 AAShortIntroductiontoTime-SpaceLowerBoundsHerewegiveabriefoverviewofthetoolsthathavebeenusedtoprovetime-spacetradeofflowerbounds.Wefocusondeterministictimelowerboundsforsatisabilityforalgorithmsusingno(1)space,astheotherrelevantlowerboundproblemsuseanalogoustoolsandtheno(1)spacecaseisespeciallysimpletoworkwith.ItisknownthatsatisabilityofBooleanformulasinconjunctivenormalform(SAT)isacompleteproblemundertightreductionsforasmallnondeterministiccomplexityclass.DeneNQLasnondeterministicquasi-lineartime,i.e.NQL:=[c0NTIME[n
(logn)c]=NTIME[n
poly(logn)]:TheoremA.1(Cook[Coo88],Schnorr[Sch78],Tourlakis[Tou01],Fortnowetal.[FLvMV05])SATisNQL-complete,underreductionsinquasi-lineartimeandO(logn)spacesimultaneously,forbothmultitapeandrandomaccessmachinemodels.Moreover,eachbitofthereductioncanbecomputedinO(poly(logn))timeandO(logn)spaceinbothmachinemodels.LetC[t(n)]representatimet(n)complexityclassunderoneofthethreemodels:deterministicRAMusingtimetandto(1)space,co-nondeterministicRAMusingtimetandto(1)space,d-dimensionalTuringmachineusingtimet.TheabovetheoremimpliesthatifNTIME[n]*C[t],thenSAT=2C[t],modulopolylogarithmicfactors.CorollaryA.1IfNTIME[n]*C[t(n)],thenthereisak�0suchthatSAT=2C[t(n)
(logt(n))k].HencewewanttoproveNTIME[n]*C[nc]foralargeconstantc�1.Forthepurposesoftimelowerboundsforsmallspacealgorithms,weworkwithC[nc]=DTS[nc]=DTISP[nc;no(1)].VanMelkebeekandRaz[vMR05]observedthatasimilarcorollaryholdsforanyproblemsuchthatSATreducestounderhighlyefcientreductions,e.g.VERTEXCOVER,HAMILTONPATH,3-SAT,andMAX-2-SAT.Itfollowsthatidenticaltimelowerboundsholdfortheseproblemsaswell.Speedups,Slowdowns,andContradictions.GiventhatourgoalistoproveNTIME[n]*DTS[nc],howcanwegoaboutthis?Inanalternation-tradingproof,weassumethatNTIME[n]DTS[nc]andattempttoestablishacontradiction,byapplyingtwolemmasinsuchawaythatatimehierarchyisviolated.Onelemma(calledthe“speeduplemma”)takesaDTS[t]classandplacesitinanalternatingclasswithruntimeo(t);theother(calledthe“slowdownlemma”)takesanalternatingclasswithruntimetandplacesitinaclasswithonelessalternationandruntimeO(tc).LemmaA.1(SpeedupLemma)Leta1,e0and0xa.ThenDTISP[na;ne](Q1nx+e)maxf1;x+eg(Q2logn)maxf1;egDTISP[nax;ne];forQi2f9;8gwhereQ1=Q2.Inparticular,DTS[na](Q1nx)maxf1;xg(Q2logn)1DTS[nax]:16 Proof.LetMbearandomaccessmachineusingnatimeandnespace.TogetasimulationofMhavingtype(9nx+e)maxf1;x+eg(8logn)maxf1;egDTISP[nax;ne],thesimulationN(x)existentiallyguessesasequenceofcongurationsC1;:::;CnxofM(x).Itthenappendstheinitialcongurationtothebeginningofthesequenceandtheacceptingcongurationtotheendofthesequence.ThenN(x)universallyguessesai2f0;:::;nxg,erasesallcongurationsexceptCiandCi+1,thensimulatesM(x)startingfromCi,acceptingifandonlyiftheCi+1isreachedwithinnaxsteps.Itiseasytoseethatthesimulationiscorrect.Theinputconstraintsonthequantierblocksaresatised,sinceaftertheuniversalguess,theinputisonlyx,Ci,andCi+1,whichisofsizen+2nenmaxf1;eg+o(1).2TheSpeedupLemmadatesbacktoworkofNepomnjascii[Nep70]andKannan[Kan84].Notethatintheabovealternatingsimulation,theinputtothenalDTISPcomputationislinearinn+ne,regardlessofthechoiceofx.Thisisasurprisinglysubtlepropertythatisexploitedheavilyinalternation-tradingproofs.TheSlowdownLemmaisthefollowingfolkloreresult:LemmaA.2(SlowdownLemma)Leta1,e0,a00,andb1.IfNTIME[n]DTISP[nc;ne],thenforbothQ2f9;8g,(Qna0)bDTIME[na]DTISP[nc
maxfa;a0;bg;ne
maxfa;a0;bg]:Inparticular,ifNTIME[n]DTS[nc],then(Qna0)bDTIME[na]DTS[nc
maxfa;a0;bg]:Proof.LetLbeaproblemin(Qna0)bDTIME[na],andletAbeanalgorithmsatisfyingL(A)=L.Onaninputxoflengthn,Aguessesastringyoflengthna0+o(1),thenfeedsannb+o(1)bitstringztoA0(z),whereA0isadeterministicalgorithmthatrunsinnatime.SinceNTIME[n]DTISP[nc;ne]andDTISPisclosedundercomplement,bypaddingwehaveNTIME[p(n)][coNTIME[p(n)]DTISP[p(n)c;p(n)e]forpolynomialsp(n)n.ThereforeAcanbesimulatedwithadeterministicalgorithmB.SincethetotalruntimeofAisna0+o(1)+nb+o(1)+na,theruntimeofBisnc
maxfa;a0;bg+o(1)andthespaceusageissimilar.2Thenalcomponentofanalternation-tradingproofisatimehierarchytheorem,themostgeneralofwhichisthefollowing,provablebyasimplediagonalization.TheoremA.2(AlternatingTimeHierarchy)Fork0,forallQi2f9;8g,a0i�ai1,andb0ibi1,(Q1na1)b2


bk(Qknak)bk+1DTS[nak+1]*(R1na01)b02


b0k(Rkna0k)b0k+1DTS[na0k+1];whereRi2f9;8gandRi=Qi.Remark1Arealternation-tradingproofsrelativizing?TheSlowdownLemmarelativizes,buttheSpeedupLemmadoesnotrelativizeinmostoraclemodels,forthesimplereasonthattheoriginalmachinerunslongerthanthe(sped-up)hostmachine,andcanthereforeasklongerqueries.Thisistypicallythecase.Forexample,theproofthatNTIME[n]=DTIME[n]isnon-relativizing,sinceapowerfulenoughoraclemakesthetwoclassesequal.Therefore,weconsideralternation-tradingproofstobeinthatrareclassofnon-relativizingandnon-naturalizinglowerbounds(butacknowledgethatourbeliefisnotunanimouslyheld).17 TwoInstructiveExamples.Inordertounderstandalternation-tradingproofs,itisnecessarytoconsidersomeexamples.Theartbehindtheirconstructionconsistsofndingthepropersequenceofrulestoapply,andtherightsettingsoftheparameterxintheSpeedupLemma.1.InFOCS'99,LiptonandViglasprovedthatSATcannotbesolvedbyalgorithmsrunninginnp 2"timeandno(1)space,forall"�0.Theirproofcanbesummarizedasfollows:byTheoremA.1,theassumptionthatthereissuchaSATalgorithmimpliesthatNTIME[n]DTS[nc]withc22.Then(9n2=c2)(8n2=c2)DTS[n2=c2](9n2=c2)DTS[n2=c](SlowdownLemma)DTS[n2](SlowdownLemma)(8n)(9logn)DTS[n](SpeedupLemma,withx=1).But(9n2=c2)(8n2=c2)DTS[n2=c2](8n)(9logn)DTS[n]contradictsTheoremA.2.Infact,onecanshowthatifc2=2,westillhaveacontradictionwithNTIME[n]DTS[nc],sothe"canberemovedfromthepreviousstatementandstatethatSATcannotbesolvedinnp 2timeandno(1)exactly.72.Improvingonthepreviousexample,onecanshowSAT=2DTS[n1:6004].IfNTIME[n]DTS[nc]andp 2c2,thenbyapplyingtheSpeedupandSlowdownLemmasappropriately,onecanderive:DTS[nc2=2+2](9nc2=2)(8logn)DTS[n2](9nc2=2)(8logn)(8n)(9logn)DTS[n]=(9nc2=2)(8n)(9logn)DTS[n](9nc2=2)(8n)DTS[nc](9nc2=2)DTS[nc2](9nc2=2)(9nc2=2)(8logn)DTS[nc2=2]=(9nc2=2)(8logn)DTS[nc2=2](9nc2=2)DTS[nc3=2]DTS[nc4=2]Whenc2=2+2&#x-3.2;≦c4=2(whichhappensifc1:6004),wehaveDTS[na]DTS[na0]forsomea&#x-3.2;≦a0.NoticethatwedonotknowifDTS[na]*DTS[na0]whena0&#x-3.2;≦a,asthespaceboundsonbothsidesoftheinequalityarethesame.Howeveronecanstillshowbyatranslationargument(similartothefootnote)thateitherDTS[na]*DTS[na0]orNTIME[n]*DTS[nc],concludingtheproof.Whilethesecondexampleismorecleverinstructure,amoreinterestingfactisthattheproofwasfoundbyacomputerprogram.By“found”,wemeanthattheprogramappliedtheSpeedupandSlowdownLemmasinpreciselythesameorderandthesameparametersettings,havingonlyminimumknowledgeoftheseLemmasalongwithawaytocheckthevalidityoftheparameters.Moreover,theprogramveriedthattheaboveisthebestpossiblealternation-tradingproofthatappliestheSpeedupandSlowdownLemmasforatmost7times.Aprecisedenitionof“alternation-tradingproof”isgiveninSection3. 7SupposeNTIME[n]DTS[nc]and2TIME[n]2TIME[n1+o(1)].Therstassumption,alongwiththeSpeedupandSlowdownLemmas,impliesthatforeverykthere'saKsatisfying2TIME[nk]NTIME[nkc]KTIME[n].ButthesecondassumptionimpliesthatKTIME[n]=2TIME[n1+o(1)].Hence2TIME[nk]2TIME[n1+o(1)],whichcontradictsthetimehierarchyfor2TIME.18 BNewLowerBoundontheTime-SpaceProductforSAT8UsingLemmaA.1initsfullgenerality,itispossibletoadaptourlinearprogrammingframeworktoprovetimelowerboundsforSATforanyxedspaceboundn,where2(0;1).Tryingarangeofvaluesfor,wefoundthatforeachofthem,theoptimalannotationsfortheno(1)spacesettingalsoappearedtobeoptimalforeveryspaceboundn.Thefollowingtablegivestime-spacepairsforwhichourtheoremproverhasshownthatnoSATalgorithmcansatisfybothtimeandspacerequirementssimultaneously. Time Space n1:06 n:9 n1:17 n:75 n1:24 n:666 n1:36 n:5 n1:51 n:333 n1:58 n:25 n1:7 n:1 n1:75 n:05 Basedonthistable,itisnaturaltoconjecturethatthetime-spaceproductforanyalgorithmsolvingSATisatleast\n(n2cos(=7))\n(n1:801),andtheproductisminimizedwhenthespaceisassmallaspossible.Inthebelow,weestablishtheconjecture.Tothebestofourknowledge,thepreviouslybestknownboundonthetime-spaceproductwasonly\n(n1:573)[FLvMV05].Whiletheproofannotationsinthebelowareanalogoustothe2cos(=7)bound(assuggestedbytheexperiments),theparametersettingsintheproofrelyagreatdealonourstudyofthetheoremprover'soutput.TheoremB.1Lett(n)ands(n)beboundedabovebypolynomials.AnyalgorithmsolvingSATintimetandspacesrequirest
s=\n(n2cos(=7)")forall"�0.Proof.SupposeSATissolvedintimet=ncandspaces=nd,withc+d2cos(=7).Ofcoursewemusthavec1,andsod12cos(=7)1.ByTheoremA.1,itfollowsthatNTIME[n]DTISP[nc+o(1);nd+o(1)].Denethesequencesc1:=2d,ck+1:=c+ck c+d,andd1:=d,dk+1:=d
ck+1 c.Itiseasytoseethatckcforallk,andthatthesequencesfckgandfdkgaremonotonenondecreasingforc+d2andd1.Firstweprovethatforallk,DTISP[nck;ndk](9n1+o(1))(8logn)DTISP[n1+o(1);nd+o(1)]:(2)BytheSpeedupLemma,DTISP[nc1;nd1+o(1)]=DTISP[n2d;nd+o(1)](9n1+o(1))(8logn)DTISP[n;nd+o(1)]:Fortheinductivestep,wehavethefollowing(subtle)seriesofinclusions:DTISP[nck+1;ndk+1+o(1)](9n1+o(1))(8logn)DTISP[nck+1(1dk+1);ndk+1+o(1)](Speedup)=(9n1+o(1))(8logn)DTISP[nck=c;ndk+1+o(1)](def.ofck+1,&ckc)(9n1+o(1))DTISP[nck+o(1);ndk+o(1)](Slowdown&def.ofdk)(9n1+o(1))(8logn)DTISP[n1+o(1);nd+o(1)](byinduction): 8Aearlyversionoftheworkinthissectionwasreportedintheauthor'sPhDthesisin2007.19 Thesequencefckgconvergestoc1=c=(c1+d),hencefdkgconvergestod1=d=(c1+d).(Notewehavecc1+d,sinced1.)Thereforeforallc0c1,d0d1,DTISP[nc0;nd0](9n1+o(1))(8logn)DTISP[n1+o(1);nd+o(1)]:(3)Noteifc2ckforanyk,wealreadyobtainacontradiction:forsufcientlylarget,wehaveNTIME[tck=c]DTISP[tck;tdk](9t1+o(1))(8logn)DTISP[t1+o(1);td+o(1)]NTIME[tc];wheretherstinclusionfollowsfromSlowdown,thesecondfromeq.(2),andthethirdfromSlowdown.Fromthesecontainments,onecanderiveatimehierarchystylecontradiction,alongthelinesofLemma3.1.Thereforewemayassumethatc2&#x-278;&#x.223;ckforallk.Equation(3)canbecombinedwithanotherinductiveargumenttoproduceacontradiction.Inparticular,forallkand`,DTISP[nc`kd`+Pki=1(c2=c`)i;nd`](9n(c2=c`)k)(8logn)DTISP[n(c2=c`)k+o(1);nd(c2=c`)k+o(1)]:(4)Theproofofequation(4)isverysimilarinstructuretoLemma6.8in[Wil08].Forcompleteness,wegivethesequenceofinclusionstoderiveit.Whenk=1,forarbitrary`wehaveDTISP[nc`+(c2=c`)d`;nd`](9nc2=c`)(8logn)1DTISP[nc`;nd`](Speedup)(9nc2=c`)(8logn)1(8n1+o(1))(9logn)DTISP[n1+o(1);nd+o(1)](Eq.(2))(9nc2=c`)(8n1+o(1))DTISP[nc+o(1);nd+o(1)](Slowdown)(9nc2=c`)DTISP[nc2+o(1);ndc+o(1)](Slowdown;notec`csocc2=c`)(9nc2=c`)(9nc2=c`)(8logn)DTISP[nc2=c`+o(1);ndc2=c`+o(1)](Eq.(2))=(9nc2=c`)(8logn)DTISP[nc2=c`+o(1);ndc2=c`+o(1)]:Fortheinductivestep,wehaveDTISP[nc`kd`+Pki=1(c2=c`)i;nd`](9n(c2=c`)k)(8logn)DTISP[nc`(k1)d`+Pk1i=1(c2=c`)i+o(1);nd`](Speedup)(9n(c2=c`)k)(8logn)(8n(c2=c`)k1)(9logn)DTISP[n(c2=c`)k1+o(1);nd(c2=c`)k1+o(1)](Induction)(9n(c2=c`)k)(8logn)(8n(c2=c`)k1)DTISP[nc(c2=c`)k1+o(1);nd(c2=c`)k1+o(1)](Slowdown)(9n(c2=c`)k)DTISP[nc2(c2=c`)k1+o(1);ndc(c2=c`)k1+o(1)](Slowdown)(9n(c2=c`)k)(9n(c2=c`)k)(8logn)DTISP[n(c2=c`)k+o(1);nd(c2=c`)k+o(1)](Eq.(2))=(9n(c2=c`)k)(8logn)DTISP[n(c2=c`)k+o(1);nd(c2=c`)k+o(1)]Now,forsufcientlylarget(n)nwehaveNTIME[t(c`kd`+Pki=1(c2=c`)i)=c]DTISP[tc`kd`+Pki=1(c2=c`)i+o(1);nd`](Slowdown)(9t(c2=c`)k)(8logn)DTISP[t(c2=c`)k+o(1);nd(c2=c`)k+o(1)](Eq.(4))(9t(c2=c`)k)DTISP[tc(c2=c`)k+o(1);nd(c2=c`)k+o(1)](Slowdown)NTIME[tc(c2=c`)k+o(1)]:20 Acontradictionwiththenondeterministictimehierarchyfollows,whenc`kd`+kXi=1(c2=c`)ic2(c2=c`)k:(5)Finally,weclaimthatwhenc+d2cos(=7),inequality(5)holdsforsufcientlylargekand`.Thetheoremfollowsimmediatelyfromthisclaim.Toseewhytheclaimistrue,weanalyzethecasewherekand`growunboundedlyandusethefactthattheunderlyingsequencesaremonotonenondecreasing.Rewritetheinequalityintotheformc`kd` (c2=c`)k+kXi=1(c2=c`)ikc2:(6)Ask!1,thersttermontheLHSvanishessincec2&#x-278;&#x.223;c`forall`.Thesecondtermconvergesto1 1c`=c2.Nowas`!1,c`!c1=c=(c1+d).Henceinthelimit,inequality(5)becomes0+1 11 c(c1+d)c2()1c2c2 c2c+dc()c2c+dcc2(c2c+dc)c2()c1+dc(c2c+dc)c()2c1+dc3c2+dc2()0c3c2+dc22c+1d:Itremainstoshowthatthebivariatepolynomialp(x;y)=x3x22x+1+y(x21)isgreaterthan0overallpoints(x;y)wherex+y&#x-278;&#x.223;2cos(=7).Wheny=0,p(x)=0overtherangex2[1;1]preciselywhenx=2cos(=7),andp(x)&#x-278;&#x.223;0forallx2cos(=7).Butforally&#x-3.2;≦0andx&#x-3.2;≦1,theresultingpolynomialstrictlydominatesp(x;0)andwealsohavep(x;y)&#x-3.2;≦0foranyx+y2cos(=7).2CSpeedupRule2isRedundantToprovethiswerstneedalemmarelatingtheai'sandbi'sofanalternatingclass.DenitionC.1Aclass(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1]isorderlyifitiseither(a)aDTS[na]class,or(b)foralli=1;:::;k,aibi+1.LemmaC.1SupposeA1isorderly.Theneveryalternation-tradingproofbeginningwithA1consistsofonlyorderlyclasses.Proof.Inductiononthenumberoflines.Thebasecaseistrivial.Theinductionhypothesisisthatthe`thline,(Q1na1)b2(Q2na2)


bk(Qknak)bk+1DTS[nak+1];isorderly.Forthe(`+1)thline,weconsidertherulesinturn:(SpeedupRule0)Clearly(Qknx)maxfx;1g(Qk+1n0)1DTS[nak+1x]isorderly.(SpeedupRule1)Supposethelineis(Q1na1)b2(Q2na2)


bk(Qknmaxfak;xg)maxfx;bk+1g(Qk+1n0)bk+1DTS[nak+1x]:Thenakbk+1bytheinductionhypothesis,somaxfak;xgmaxfx;bk+1g,1bk+1,thustheclassisorderly.21 (SpeedupRule2)Thiscaseisclear,asthelineis:(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1nx)maxfx;bk+1g(Qk+2n0)bk+1DTS[nak+1x]:(SlowdownRule)Obvious,giventhehypothesis.Thisconcludesallthecases.2LemmaC.2LetA1beorderly.Foreveryalternation-tradingproofthatNTIME[n]DTS[nc]=)A1A2,thereisanotheralternation-tradingproofofthesameimplicationthatdoesnotuseSpeedupRule2.Proof.ConsideraproofPthatappliesSpeedupRule2atsomeline.ThelinehastheformA=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1nx)maxfx;bk+1g(Qk+2n0)bk+1DTS[nak+1x]:Weconsidertwocases:1.Ifxak,thenxbk+1byLemmaC.1.ByapplyingSpeedupRule2,oneobtainsA=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1nx)maxfx;bk+1g(Qk+2n0)bk+1DTS[nak+1x]=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1nx)bk+1(Qk+2n0)bk+1DTS[nak+1x]:IfweinsteadapplySpeedupRule1withx0=x,theclassisB=(Q1na1)b2(Q2na2)


bk(Qknmaxfak;x0g)maxfx0;bk+1g(Qk+1n0)bk+1DTS[nak+1x0]=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1n0)bk+1DTS[nak+1x]:ThenbyapplyingSpeedupRule1withx0=0,theaboveclassisin(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1n0)bk+1(Qk+2n0)bk+1DTS[nak+1x]:ItisclearthatBA:everyparameterinBisatmostthecorrespondingparameterinA.ThusanyinclusionderivedwithRule2couldonlybemadestrongerbyapplyingRule1twiceinstead.2.Ifxak,thenSpeedupRule2givesA=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1nx)maxfx;bk+1g(Qk+2n0)bk+1DTS[nak+1x]:SpeedupRule1withx0=akgivesB=(Q1na1)b2(Q2na2)


bk(Qknmaxfak;x0g)maxfx0;bk+1g(Qk+1n0)bk+1DTS[nak+1x0]:=(Q1na1)b2(Q2na2)


bk(Qknak)bk+1(Qk+1n0)bk+1DTS[nak+1ak]:whereweusedthefactthatx0=akbk+1(LemmaC.1).ApplyingSpeedupRule1againwithx0=xak,Biscontainedin(Q1na1)b2


bk(Qknak)bk+1(Qk+1nmaxfxak;1g)maxfxak;bk+1g(Qk+2n0)bk+1DTS[nak+1x]:Again,observeBAinthiscase,andeveryparameterinBisatmostthecorrespondingparameterinA.Thiscompletestheproof.2Asaconsequence,SpeedupRule2isnotnecessaryfornormalformproofs.TheoremC.1Foreveryalternation-tradingproofofNTIME[n]DTS[nc]=)A1A2innormalform,thereisanotheralternation-tradingproofofthesamethatdoesnotuseSpeedupRule2.Proof.ByLemmaC.1,everynormalformalternation-tradingproofisorderly.SobyLemmaC.2,thereisanequivalentalternation-tradingproofthatdoesnotuseSpeedupRule2.222 DProofsofNormalFormPropertiesHereweestablishLemma3.1andTheorem3.1,whichshowthatitsufcestoconsideralternation-tradingproofswritteninnormalform:Lemma3.1Letc1.Ifthereisanalternation-tradingproofforcinnormalformhavingatleasttwolines,thenNTIME[n]*DTS[nc].Proof.LetPbeanalternation-tradingproofforcinnormalform.Weconsidertwocases.Supposea�a0.Inthiscase,NTIME[n]DTS[nc]impliesDTS[na]DTS[na]forsome�0.Bytranslation,DTS[na]DTS[na]impliesDTS[na2=(a)]DTS[na]DTS[na];andDTS[na
(a=(a))i]DTS[na]foralli0.Since�0,thisimpliesDTS[nL]DTS[na]forallLa.Therefore,ifNTIME[n]DTS[nc]thenforallLa,NTIME[nL]DTS[nLc]DTS[na]coNTIME[na];acontradictiontothetimehierarchy(TheoremA.2).Supposea=a0.LetAbealineinPwithapositivenumberofalternations.(SuchalinemustexistsincePhasatleasttwolines.)TheproofPprovesthatNTIME[n]DTS[nc]impliesDTS[na]ADTS[na0],soA=DTS[na].SinceDTS[na]isclosedundercomplement,A=A0;(7)whereA0isthecomplementofA.Withoutlossofgenerality,assumeA=(9n)BandA0=(8n)B0forsome�0andcomplementaryclassesBandB0.Clearly,A0=(8n)A0andA=(9n)A:(8)NowconsidertheclassDTS[ndk e]DTS[nk],forarbitraryk1.BytheSpeedupLemma(LemmaA.1)andthefactthatDTS[n"]A0forsome"�0,DTS[nk]DTS[ndk e](9n)(8n)


(9n)(8n)| {z }dk=eA0:Applyingequations(7)and(8),wehave(9n)(8n)


(9n)(8n)A0=(9n)(8n)


(9n)A0=(9n)(8n)


(9n)A=(9n)(8n)


A=


=(9n)(8n)A0=(9n)A0=(9n)A=A:23 ThereforeDTS[nk]A,foreveryk1.HenceNPDTS[nO(1);no(1)]A.ButbyapplyingaslowdownstepforanitenumberoftimestoA,thereisanalternation-tradingproofthatADTS[nK]foraconstantK.ItfollowsthatNPADTS[nK]coNTIME[nK],contradictingthetimehierarchy(TheoremA.2).SoNTIME[n]*DTS[nc]inthiscaseaswell.2Theorem3.1LetA1andA2becomplementary.Ifthereisanalternation-tradingproofPforcthatshows(NTIME[n]DTS[nc]=)A1A2),thenthereisanormalformproofforc,oflengthatmostthatofP.Proof.Consideranalternation-tradingproofPforc,writtenasP=A1;C1;:::;Ck;A2:DenethedualproofP'byP0=A2;:C1;:::;:Ck;A1;wherethenotation:CdenotestheuniquecomplementarysimpleclassforC,i.e.every`8'inCisreplacedwith`9',andvice-versa.NotethatP0isanalternation-tradingproofifandonlyifPisone.SincethequantiersoftherstandlastlineofParedifferent,notetheretheremustbealineCi=DTS[na]forsomea.SupposethereisonlyonedeterministicclassinP;callitCi.ThenP00=Ci;Ci+1;:::Ck;A2;:C1;:::;:Ciisalsoanalternation-tradingproof,obtainedbypiecingtogethertheappropriatelinesfromPandP0.However,Ci=:Ci,sinceDTS[na]isclosedundercomplement.HenceP00isinnormalform:itsrstandlastlinesareDTSclasses,andnointermediateclassisaDTSclass.Supposetherearek2differentDTSclassesinP.WritePasP=A1;:::;DTS[na1];:::;DTS[na2];:::;:::;DTS[nak];:::;A2:Therearetwocases:-Ifthereisani2[k]satisfyingaiai+1,wearedone:letP00tobethesequenceoflinesfromDTS[nai]andDTS[nai+1],andthisisinnormalform.-Ifaiai+1foreveryi,thensetP00=DTS[nak];:::;A2;:::;DTS[na1],wheretheclassesintherst“:::”inP00aretakendirectlyfromP,andtheclassesinthesecond“:::”inP00areobtainedbytakingthelinesA2;:::;DTS[na1]inP0.P00isinnormalformsinceak&#x-278;&#x.223;a1.2EExperimentalResults:Time-SpaceLowerBoundsforSATWewroteaprogramthatgivenaproofannotationgeneratestherelevantlinearprogramminginstance,solvesit,thenprintstheproofinhuman-readableform.Forproofannotationsexceeding100lines,weusedthe24 lp solvepackagetosolvethecorrespondinglinearprogram.9Wealsowroteheuristicsearchroutinesthattrytoderivenewproofsfromoldones.Oneprogramstartswithaqueueofannotations,pullstheheadofthequeue,andthentriesallpossiblewaystoaddatmostfourbitstotheannotation.Iftheresultinglowerboundfromthenewannotationincreases,thenewannotationisaddedtothequeue.Interestingly,thissimplestrategygeneratedalltheoptimallowerboundsthatwerefoundbyexhaustivesearch,andmore.Firstweveriedallthepreviouslyknownlowerbounds,suchasthen2cos(=7)bound.Insomecases,wefoundbettersettingsoftheparametersthanhadbeenfoundinthepast,butnoproofbetterthann2cos(=7).Thenwesearchedthespaceofproofannotations,lookingforinterestingpatterns.Forallevenk=2;:::;26,weexhaustivelysearchedoverallvalidproofannotationswithklines.Thebestproofannotationsforeachkaregiveninthebelowtable.Fork�26wehavenotexhaustivelysearchedallproofs,butinsteadusedaheuristicsearchasdescribedabove;theserowsofthetablearemarkedwithanasterisk.Forrowswithmultipleannotations,wecheckedtheannotationstotwomoredecimalplacestofurtherverifythattheobtainedlowerboundsarelikelythesame.The
ofarowisthedifferencebetweentheexponentofthatrowandtheexponentofthepreviousrow. #Lines BestProofAnnotation(s) L.B.
4 [1;0;0] 1.4142 0 6 [1;0;1;0;0] 1.5213 0.1071 [1;1;0;0;0] 8 [1;1;0;0;1;0;0] 1.6004 0.0791 10 [1;1;0;0;1;0;1;0;0] 1.633315 0.032915 [1;1;0;1;0;0;1;0;0] [1;1;1;0;0;0;1;0;0] 12 [1;1;1;0;0;1;0;0;1;0;0] 1.6635 0.0302 14 [1;1;1;0;0;1;0;0;1;0;1;0;0] 1.6871 0.0236 16 [1;1;1;0;0;1;0;1;0;0;1;0;1;0;0] 1.699676 0.012576 [1;1;1;0;1;0;0;1;0;0;1;0;1;0;0] [1;1;1;1;0;0;0;1;0;0;1;0;1;0;0] 18 [1;1;1;1;0;0;1;0;0;1;0;0;1;0;1;0;0] 1.7121 0.0125 20 [1;1;1;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;0] 1.7232 0.0111 22 [1;1;1;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] 1.7322 0.0090 24 [1;1;1;1;0;0;1;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] 1.737851 0.005651 [1;1;1;1;0;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] [1;1;1;1;1;0;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] 26 [1;1;1;1;1;0;0;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] 1.7437 0.005849 28* [1;1;1;1;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0] 1.7491 0.0054 30* [1;1;1;1;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0;1;0;1;0;1;0;0] 1.7537 0.0046 32* [1;1;1;1;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0;1;0;1;0;1;0;1;0;0] 1.7577 0.0040 34* [1;1;1;1;1;0;0;1;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0;1;0;1;0;1;0;1;0;0] 1.760632 0.002932 [1;1;1;1;1;0;1;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0;1;0;1;0;1;0;1;0;0] [1;1;1;1;1;1;0;0;0;1;0;0;1;0;1;0;0;1;0;1;0;1;0;0;1;0;1;0;1;0;1;0;0] Weobservethattheproofsproducedbytheannotationsinthetablehavestrongsimilaritiestothoseinthe2cos(=7)lowerbound.Forexample,thebest14-lineproof(provingan\n(n1:6871)lowerbound)lookslike:0,DTS[nˆ5.275587925]1,(Enˆ1.853485593)(Anˆ1.)DTS[nˆ3.422102331]2,(Enˆ1.853485593)(Anˆ1.422102331)(Enˆ1.)DTS[nˆ2.000000001] 9Thelp solvepackageisanopensourcesimplex-basedlinearprogrammingsolver.ItismaintainedbyacommunityonYahooGroups:http://groups.yahoo.com/group/lp solve.25 3,(Enˆ1.853485593)(Anˆ1.422102331)(Enˆ1.000000001)(Anˆ1.000000000)DTS[nˆ1.]4,(Enˆ1.853485593)(Anˆ1.422102331)(Enˆ1.000000001)DTS[nˆ1.687100000]5,(Enˆ1.853485593)(Anˆ1.422102331)DTS[nˆ2.846306408]6,(Enˆ1.853485593)(Anˆ1.423153204)(Enˆ1.000000000)DTS[nˆ1.423153204]7,(Enˆ1.853485593)(Anˆ1.423153204)DTS[nˆ2.401001771]8,(Enˆ1.853485593)DTS[nˆ4.050730087]9,(Enˆ1.853485593)(Anˆ1.000000000)DTS[nˆ2.197244494]10,(Enˆ1.853485593)DTS[nˆ3.706971186]11,(Enˆ1.853485593)(Anˆ1.000000000)DTS[nˆ1.853485593]12,(Enˆ1.853485593)DTS[nˆ3.127015544]13,DTS[nˆ5.275587925]Lookingcloselyatthetable,thereisastrongcorrelationbetweenlaterrowsofthetableandearlierones.Forexample,thereisatieforbestannotationat10,16,24,and34lines,amongthreeannotationsthatdifferonlyinthreeoftheirbits.Todevelopagreaterunderstandingofwhatishappening,letusintroducesomeabbreviationsintheannotation.Whereanannotationcontainsthestring(10)k0,weputthesymbolk,fork1.Whereanannotationcontainsthestring11000,wejustput0.Thefollowingtableemerges: #Lines BestProofAnnotation(s) L.B.
4 1 1.4142 0 6 2 1.5213 0.1071 0 8 12 1.6004 0.0791 10 112 1.633315 0.032915 121 101 12 11111 1.6635 0.0302 14 11112 1.6871 0.0236 16 11122 1.699676 0.012576 11212 11012 18 1111112 1.7121 0.0125 20 1111122 1.7232 0.0111 22 1111123 1.7322 0.0090 24 1111223 1.737851 0.005651 1112123 1110123 26 111111123 1.7437 0.005849 28* 111111223 1.7491 0.0054 30* 111111233 1.7537 0.0046 32* 111111234 1.7577 0.0040 34* 111112234 1.760632 0.002932 111121234 111101234 Foranoptimalannotationthatendswithanon-zerok,alongeroptimalannotationcanbeobtainedbyaddingeitherakork+1totheend,anda1atthebeginning.(Thereareofcoursesomerestrictions–therearenomorethanthreeconsecutive1's,nomorethantwoconsecutive2's,etc.)Whilewedonotyetknowhowtoprovethatallofthebestproofsmusthavethisbehavior,itseemsextraordinarilyunlikelythatthispatterndeviatesatsomelaterpoint.26 Thetablesuggeststhatweexamineproofannotationsoftheform1


101234


.Unfortunatelytheseannotationsdonotleadtoanimprovement.Toillustrate,forthe424lineproofannotationdenotedby1111111111111111101234


171819;experimentswithlp solverevealedthattheoptimalexponentisonlyintheinterval[1:80175;1:8018).Theseresults(alongwiththefactthatanyannotationintheaboveformprovablycanyieldnobetterthana2cos(=7)exponent,cf.Theorem3.3)pointstronglytotheconjecturethatthereisnoalternation-tradingproofthatNTIME[n]*DTS[nc],foranyc�2cos(=7)1:8019.FProofofTheorem3.4:NoQuadraticLowerBoundIntheirpapershowingthatSATcannotbesolvedinO(n)timeandno(1)space,Fortnowetal.[FLvMV05]writethat“somecomplexitytheoristsfeelthatimprovingthegoldenratioexponentbeyond2wouldrequireabreakthrough.”Herewegiveaformalproofofthissentiment.Althoughtheproofissimple,webelieveitisimportantasaformalizationofafolkloreconjecture.Theorem3.4Thereisnoalternation-tradingproofofNTIME[n]*DTS[n2].Proof.(Sketch)Supposethereissuchaproof,andletAbeaminimumlengthannotationinnormalformforit.WeclaimthatAcanbemadeshorter,yettheresultingLPisstillfeasibleiftheoriginalLPwasfeasible.Firstweobservethateverynormalformannotationcontainsasequence1;0.Normalformannotationscanbeputin1-1correspondencewithstringsofbalancedparenthesesoftheform(x),wherexisannon-emptybalancedparenthesesstring.Therstspeedupinaproofcorrespondsto((,asitintroducestwoquantiers,allotherspeedupapplicationscorrespondtoa(,andaslowdowncorrepondstoa).Forexample,(())correspondsto[1;0;0].Sincethereisalwaysanadjacentparenthesespair()inanystringofbalancedparentheses,theremustalsobesomeoccurrenceof1;0inavalidproofannotation.Ifthis1;0canberemovedfromAwithoutchangingthefeasibilityoftheunderlyinglinearprogram,theclaimisproved.Thetwolinesintheproofcorrespondingtothesequence1;0(includingthepreviousline)havetheform:


bk1(Qk1nak1)bk(Qknak)bk+1DTS[nak+1](9)


bk1(Qk1nak1)bk(Qknmaxfx;akg)maxfx;bk+1g(Qk+1logn0)bk+1DTS[nak+1x](10)


bk1(Qk1nak1)bk(Qknmaxfx;akg)maxfx;bk+1gDTS[nmaxfc(ak+1x);cx;cbk+1g](11)Everyparameterintheclass(11)isatleastthecorrespondingparameterintheclass(9),exceptforpossiblytheruntimeoftheDTScomputation.Henceifak+1c(ak+1x),orak+1cx,orak+1cbk+1,then1;0couldberemovedwithoutchangingthefeasibilityoftheLP.However,ifbothak+1�c(ak+1x)andak+1�cx,then2ak+1�c(ak+1x)+cx,acontradictionwhenc2.2GExperimentalResults:LowerBoundsforNondeterministicAlgorithmsSolvingTautologiesBelowisatableofresultsfoundbyexhaustivesearchovervalidannotations.27 #Lines BestProofAnnotation(s) L.B. 5 [1,0,0,0] 1.323 8 [1,1,0,0,0,0,0] 1.380 [1,0,1,0,0,0,0] 11 [1,1,0,0,0,1,0,0,0,0] 1.419 14 [1,1,1,0,0,0,0,0,1,0,0,0,0] 1.433 [1,1,0,1,0,0,0,0,1,0,0,0,0] [1,1,0,0,0,1,1,0,0,0,0,0,0] [1,1,0,0,0,1,0,1,0,0,0,0,0] 17 [1,1,1,0,0,0,1,0,0,0,0,1,0,0,0,0] 1.445 [1,1,0,0,0,1,1,0,0,0,1,0,0,0,0,0] 20 [1,1,1,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0] 1.455 [1,1,1,0,0,0,1,0,0,0,0,1,0,1,0,0,0,0,0] [1,1,1,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0] [1,1,0,1,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0] 23 [1,1,1,0,0,0,1,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0] 1.465 Asonecansee,thestructureofgoodlowerboundproofsforthe“TAUTOLOGYversusNTISP”problemturnsouttobedifferentfromthoseforthe“SATversusDTISP”problem.Takenfromtheprogram,thebest11-lineproofreads:0,NTS[nˆ4.788956584]1,(Enˆ2.369956583)(Anˆ1.)NTS[nˆ2.419]2,(Enˆ2.369956583)(Anˆ1.)(Enˆ1.419)(Anˆ1.)NTS[nˆ1.]3,(Enˆ2.369956583)(Anˆ1.)(Enˆ1.419)(Anˆ1.)coNTS[nˆ1.419]4,(Enˆ2.369956583)(Anˆ1.)(Enˆ1.419)NTS[nˆ2.013561000]5,(Enˆ2.369956583)(Anˆ1.)coNTS[nˆ2.857243059]6,(Enˆ2.369956583)(Anˆ2.378351877)(Enˆ1.)coNTS[nˆ1.181167036]7,(Enˆ2.369956583)(Anˆ2.378351877)(Enˆ1.)NTS[nˆ1.676076023]8,(Enˆ2.369956583)(Anˆ2.378351877)coNTS[nˆ2.378351877]9,(Enˆ2.369956583)NTS[nˆ3.374881314]10,coNTS[nˆ4.788956584]Notehowlargerannotationsarecomposedofsmallerones:forexample,[1;1;1;0;0;0;0;0;1;0;0;0;0]is[1;A7;A4;0],whereA4andA7areoptimalannotationsforfourandsevenlines.Inparticular,observethatthreeoptimalannotationsfromthetablehaveadistinctivepattern,namely[1;0;0;0];[1;1;0;0;0;1;0;0;0;0];and[1;1;1;0;0;0;1;0;0;0;0;1;1;0;0;0;1;0;0;0;0;0]:Thepatternsuggeststhatwelookforaninductiveproofwhereaninductionhypothesisisappliedtwiceintheinductivestep.Thenextannotationinthepatternwouldbe[1;1;1;1;0;0;0;1;0;0;0;0;1;1;0;0;0;1;0;0;0;0;0;1;1;1;0;0;0;1;0;0;0;0;1;1;0;0;0;1;0;0;0;0;0;0];a47-lineannotationthatgivesa1:49exponent.Aheuristicsearchfoundtheabove47-lineannotation,andnootherannotationsfound(withatmost47lines)attainedalowerboundofthatquality.Formanylinenumbers`,heuristicsearchfoundalargenumberof`-lineproofannotationsthatachievethesamelowerbound.Forexample,thereareeightsuchannotationsof26lines.Eachoptimalannotationfoundcouldbewrittenasaconcatenationofsmalleroptimalannotationsalongwithanadditional1and0.28 HProofofTheorem4.2:TheNondeterministicTime-SpaceLowerBoundToproveTheorem4.2,weuseaninductivelemma.Foragivenconstantc1,denethesequencex1:=1,x2:=c,xk:=c3(xk1)2=(Pk1i=1xi).LemmaH.1IfcoNTIME[n]NTS[nc]andc2(xk)2Pki=1xi,thenforallk2,NTS[nPki=1xi](9nxk)(8nxk)coNTS[nxk].Inthefollowing,wedonotspecifytheinputstoquantierblocks,exceptwhereabsolutelynecessaryfortheargument.Proof.Fork=2wehaveNTS[nc+1](9nc)(8logn)NTS[n](9nc)(8logn)coNTS[nc].Notethatc2(xk)2Pki=1xiimpliesxk+11.ByinductionwehavethatNTS[nPk+1i=1xi](9nxk+1)(8logn)1NTS[nPki=1xi](Speedup)(9nxk+1)(8logn)1(9nxk)(8nxk)coNTS[nxk](byInductionHypothesis)(9nxk+1)(8logn)1(9nxk)NTS[ncxk](Slowdown)(9nxk+1)(8logn)1coNTS[nc2xk](Slowdown)(9nxk+1)(8logn)1(8nc2(xk)2=(Pixi))(9nc2(xk)2=(Pixi))NTS[nc2(xk)2=(Pixi)](InductionHypothesis,andAssumption)(9nxk+1)(8logn)1(8nc2(xk)2=(Pixi))coNTS[nc3(xk)2=(Pixi)](Slowdown)(9nxk+1)(8nxk+1)coNTS[nxk+1]:2Notethelemmaindeedappliesitsinductionhypothesistwice,assuggestedbyexperiments.Thelowerboundprooffortautologiescannowbederived.ProofofTheorem4.2.AssumecoNTIME[n]NTS[nc].Suppose`isthesmallestintegersatisfyingc2(x`)2P`i=1xi.Notethatc2(x2)2=c4c+1=x1+x2forc&#x-3.2;≦1:2207,whichweknowholdsduetoFortnowandVanMelkebeek.Therefore`3.ByLemmaH.1andtheSlowdownLemma,foreveryk`wehaveNTS[nPki=1xi](9nxk)(8nxk)coNTS[nxk]coNTS[nc2xk]:(12)Denethesequencesk:=(Pki=1xi)=xk=1+Pk1i=1xi=xk.Byinductiononecanshowthatsk=1+(sk1)2=c3,andthatthissequenceisincreasing.Theinclusion(12)saysthatacontradictionisobtainedwithLemma4.3whenc2sk.Henceifc2s`1,wehaveacontradictionwithLemma4.3(theNTSversuscoNTShierarchy).However,weknowthatc2x`P`i=1xi=x`andc2x`1P`1i=1xi=x`1,byourchoiceof`.Takentogether,thetwoinequalitiesareequivalenttotheconditionx`+1cx`.Withalgebramanipulationandthefactthatc3&#x-377;&#x.223;c+1(whichholdsforc&#x-377;&#x.223;1:33),onecanshowthatthisconditionimpliesc2s`1.Hencenosuch`exists.Nowsupposeinsteadthatc2(xk)2Pki=1xi,forallk.Theninclusion(12)holdsforallk.Forwhichccanweobtainc2sk,forsufcientlylargek?Eitherthesequencefskgisunbounded(inwhichcasewearedone,astheinequalityholdsforallc)orithasalimitpoint.Inthelattercase,wehaves1=1+s21=c3.Thepolynomialp(x)=1+x2=c3xhasrootsx=c
(c2=2p (c44c)=2).Whenc=41=3,thisrootisimaginary,therefores1wouldbeimaginary,acontradiction.Itfollowsthatc41=3.229 INoGoldenRatioLowerBoundforSolvingTautologiesWithNondetermin-ismHereweprovethatthereisnoalternationtradingproofthatcoNTIME[n]*NTS[nc],foranyc1:618,thegoldenratio.Proof.(Sketch)SupposethereisaproofthatcoNTIME[n]*NTS[nc]withc,andletAbeanormalformannotationforit,ofminimumlength.First,observethateveryvalidannotationcontainsthesequence1;0;0init,forifeveryoccurrenceof1;0wasfollowedbya1,theproofcouldnotpossiblybeinnormalform.(Inparticular,whenaspeedupruleandslowdownruleareappliedtoasimpleclassA,theresultingclassA0hasmorequantiersthanA,inthissetting.)Therefore1;0;0mustoccursomewhereintheproof.Next,weshowthatanysubsequence1;0;0canberemovedfromA,andtheresultingLPwillstillbefeasiblefortheconstantc.Thisimpliesacontradiction.Considerthefourlinesinaprospectiveproofcorrespondingtothesequence1;0;0,whereweincludethelinebeforethethreerulesareapplied.Therstlineisoneoffourpossibilities:


b0(9na0)bNTS[na];


b0(8na0)bcoNTS[na];


b0(9na0)bcoNTS[na];or


b0(8na0)bNTS[na]:Thersttwocasesaresymmetrictoeachother,asarethelasttwocases,soitsufcesforustoconsider


b0(9na0)bNTS[na]and


b0(9na0)bcoNTS[na].Intherstcase,thefourlineshavetheform:


b0(9na0)bNTS[na](13)


b0(9nmaxfa0;xg)maxfb;xg(8logn)bNTS[nax](14)


b0(9nmaxfa0;xg)maxfb;xg(8logn)bcoNTS[nmaxfc(ax);cbg](15)


b0(9nmaxfa0;xg)maxfb;xgNTS[nmaxfc2(ax);c2b;cxg](16)Observethateachparameterinclass(16)isatleastthecorrespondingparameterinclass(13),exceptforpossiblytheruntimeoftheNTScomputation.However,ifanyoneofac2(ax),ac2b,oracxhold,thentheabovelinescanberemovedfromtheproof,andtheoptimalassignmenttotheparameterswouldonlybelarger.Sosupposea�c2(ax),a�cb,anda�cx.Thenc2aa+c2xa+ca,implyingthatc2(1+c),orc(c1)1.Forc,thisisacontradiction.Onecanarguesimilarlyforthesecondcase.Therethefourlineshavetheform:


b0(9na0)bcoNTS[na](17)


b0(9na0)b(8nx)maxfx;bg(9logn)bcoNTS[nax](18)


b0(9na0)b(8nx)maxfx;bg(9logn)bNTS[nmaxfc(ax);cbg](19)(9na0)b(8nx)maxfx;bgcoNTS[nmaxfc2(ax);c2b;cxg](20)Usinganargumentsimilartotheabove,class(20)containsclass(17)whenc,soremovingtheabovelinescanonlyimprovetheoptimumsettingoftheparameters.230 JExperimentalResults:LowerBoundsforMultidimensionalTMsSolvingSATFor1-dimensionalmachines,asummaryoflowerboundsfoundbytheLP-basedtheoremproverisgiveninthebelowtable.Unliketheprevioustwocases,theoptimalboundsattainedbyoptimalproofshavenon-monotonicbehavior(withrespecttolength)atrst.Perhapssurprisingly,thetablelooksthesameforthe2-dimensionaland3-dimensionalcases,albeitwithsmallerlowerboundexponents. #Lines BestProofAnnotation(s) L.B. 5 [1,1,0,0] 1.224 6 [1,1,0,1,0] 7 [1,1,1,0,0,0] 1.201 8,9 [1,1,0,1,1,0,0],[1,1,0,1,1,0,1,0] 1.262 10 [1,1,1,0,0,1,1,0,0] 1.261 11,12 [1,1,0,1,1,0,1,1,0,0],[1,1,0,1,1,0,1,1,0,1,0] 1.274 13 [1,1,1,0,0,1,1,0,1,1,0,0] 1.277 14,15 [1,1,0,1,1,0,1,1,0,1,1,0,0],[1,1,0,1,1,0,1,1,0,1,1,0,1,0] 1.278 16,17 [1,1,1,0,1,1,0,0,1,1,0,1,1,0,0],[1,1,1,0,1,1,0,0,1,1,0,1,1,0,1,0] 1.287 19 [1,1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,0] 1.292 25 [1,1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0] 1.297 28 [1,1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,0,1,1,0,1,1,0,1,1,0,0] 1.298 AsubsetoftheoptimalannotationshavetheformA=[1(110)k0(110)`0];forintegersk;`.(Infactthosethatdonotcanbewritteninthisway.)Inotherwords,00occursexactlytwice.Mightitbethatforlongerproofsthereareoptimalannotationswiththreeoccurrencesof00?Asbefore,weusedaheuristicsearchtoinvestigate.Thesearchuncoveredmoreinterestingannotations,butallofthebesthadtheformofAabove.Forinstance,thebest25lineproofwas:0,DTIME1[nˆ1.751958454]1,(Enˆ1.)DTISP[nˆ1.751958454,nˆ.7519608720]2,(Enˆ1.040108911)(Anˆ1.)DTISP[nˆ1.463810415,nˆ.7519608720]3,(Enˆ1.040108911)(Anˆ1.)(Enˆ1.)DTISP[nˆ1.215771287,nˆ.7519608720]4,(Enˆ1.040108911)(Anˆ1.)DTIME1[nˆ1.577881470]5,(Enˆ1.040108911)(Anˆ1.)DTISP[nˆ1.577881470,nˆ.5778814720]6,(Enˆ1.040108911)(Anˆ1.)(Enˆ1.)DTISP[nˆ1.155762944,nˆ.5778814720]7,(Enˆ1.040108911)(Anˆ1.)DTIME1[nˆ1.5]8,(Enˆ1.040108911)(Anˆ1.)DTISP[nˆ1.5,nˆ.5]9,(Enˆ1.040108911)(Anˆ1.)(Enˆ1.)DTISP[nˆ1.,nˆ.5]10,(Enˆ1.040108911)(Anˆ1.)DTIME1[nˆ1.297844000]11,(Enˆ1.040108911)DTIME1[nˆ1.684399048]12,(Enˆ1.040108909)DTISP[nˆ1.684399048,nˆ.6442901394]13,(Enˆ1.040108909)(Anˆ1.)DTISP[nˆ1.288580278,nˆ.6442901394]14,(Enˆ1.040108909)DTIME1[nˆ1.672376183]15,(Enˆ1.040108909)DTISP[nˆ1.672376183,nˆ.6322672739]16,(Enˆ1.040108909)(Anˆ1.)DTISP[nˆ1.264534548,nˆ.6322672739]31 17,(Enˆ1.040108909)DTIME1[nˆ1.641168576]18,(Enˆ1.040108909)DTISP[nˆ1.641168576,nˆ.6010596669]19,(Enˆ1.040108911)(Anˆ1.)DTISP[nˆ1.202119332,nˆ.6010596669]20,(Enˆ1.040108911)DTIME1[nˆ1.560163362]21,(Enˆ1.040108911)DTISP[nˆ1.560163362,nˆ.5200544533]22,(Enˆ1.040108908)(Anˆ1.)DTISP[nˆ1.040108908,nˆ.5200544533]23,(Enˆ1.040108908)DTIME1[nˆ1.349899105]24,DTIME1[nˆ1.751958454]KProofofTheorem5.1:Thed-DimensionalTMLowerBoundHereweprovethatSATcannotbesolvedbyaTuringmachinewithrandomaccesstoitsinputandsequentialaccesstoad-dimensionaltape,inO(nr)time,whererd1isarootofthepolynomialpd(x)=(2d+1)(d+1)2x52(d+1)(2d+1)x4d2x3+2(d+1)x2((2d+1)(d+1)2+1)x+d(d+1):Ascorollaries,SAT=2DTIME1[n1:3009],SAT=2DTIME2[n1:1887],andSAT=2DTIME3[n1:1372].BeforeweproveTheorem5.1,werstgiveaninductivelemma.Letc1,anddenethesequencee1:=(d+2)=(d+1),ek+1:=1+ek c(d+1).LemmaK.1SupposeNTIME[n]DTIMEd[nc].Thenforallk1,DTIMEd[nek](9n)(8logn)DTISP[n;nd=(d+1)]:Proof.Whenk=1,DTIMEd[nek](9n)DTISP[n(d+2)=(d+1);n:5](9n)(8logn)DTISP[n;nd=(d+1)];byLemma5.1(DTIMEdtoDTISP)andLemmaA.1(DTISPSpeedup),respectively.Fortheinductivestep,DTIMEd[n1+ek c(+1)](9n)DTISP[n1+ek c(+1);nek c(+1)](DTIMEdtoDTISP)(9n)(8logn)DTISP[nek=c;nek c(+1)](Speedup)(9n)DTIMEd[nek](DTIMEdSlowdown)(9n)(9n)(8logn)DTISP[n;nd=(d+1)]=(9n)(8logn)DTISP[n;nd=(d+1)];wherethelastcontainmentholdsbyinduction.2Notetheproofannotationsforthederivationsintheabovelemmahavetheform(110)k111.CorollaryK.1Forall"�0andc1,ifNTIME[n]DTIMEd[nc]thenDTIMEd[n1+1 c(+1)1"](9n)(8logn)DTISP[n;nd=(d+1)].Proof.Fore1+1 c(d+1)1,wehavee1+e c(d+1)1.Thesequencesk=1+sk1 c(d+1)1convergesto=1+1 c(d+1)1forallc1.(Notee=1+e=(c(d+1))impliese=1+1 c(d+1)1.)Soforanye,bysettinge=(d+1)=(d+2)andobservingDTIMEd[n(d+1)=(d+2)](9n)(8logn)DTISP[n;nd=(d+1)],onecanapplyLemmaK.1aconstantnumberoftimestogetthatthesamecontainmentholdsforDTIMEd[ne].232 Intuitively,thecorollarysaysthataswemakestrongerassumptionsabouthowquicklySATcanbesolvedonad-dimensionalone-tapeTM,thenwecanplacemoreofDTIMEd[nO(1)]in(9n)(8logn)DTISP[n;nd=(d+1)],whenc(d+2)=(d+1).Wecannowprovethelowerbound.ProofofTheorem5.1.Leta1beaparameter.ThenDTIMEd[na](9n)DTISP[na;nd(a1)](DTIMEdtoDTISP)(9nx+d(a1))x+d(a1)(8logn)1DTISP[nax;nd(a1)](Speedup)wherexisaparametersatisfyingcx+d(a1)1.BySpeedup,theaboveclassisin(9nx+d(a1))x+d(a1)(8n(1d(a1))+d(a1))1(9logn)1DTISP[nax(1d(a1));nd(a1)]=(9nx+d(a1))x+d(a1)(8n1)1(9logn)1DTISP[nax(1d(a1));nd(a1)](9nx+d(a1))x+d(a1)(8n1)1DTIMEd[nc(ax(1d(a1)))](Slowdown);assuming(forthemoment)that1d(a1)0.Supposethataandxsatisfyc(ax(1d(a1)))=c((d+1)(a1)x)1+1 c(d+1)1",forsome"�0.ApplyingCorollaryK.1,theaboveiscontainedin(9nx+d(a1))(8n)(9logn)DTISP[n1;nd=(d+1)](9nx+d(a1))(8n)DTISP[nc;nd=(d+1)](Slowdown)(9nx+d(a1))x+d(a1)DTIMEd[nc2];(Slowdown)sincecx+d(a1).Nowsupposeaandcsatisfyc2=(x+d(a1))=1+1=(c(d+1)1)".ThenCorollaryK.1canbeappliedagain,obtainingtheclass(9nx+d(a1))(8n)DTISP[n(x+d(a1));n(x+d(a1))
+1](9nx+d(a1))DTIMEd[nc(x+d(a1))](Slowdown)DTIMEd[nc2(x+d(a1))]:(Slowdown):Settinga=c2(x+d(a1))yieldsacontradictionwithLemma5.3.Observethataproofannotationfortheabovehastheform[1(110)k0(110)`0].Theanalysisintroducedthreeparameters(c,a,x)alongwiththreeequationstosatisfy:a=c2(x+d(a1));c(axd(a1)+1)=1+1 c(d+1)1;c2 x+d(a1)=1+1 c(d+1)1:Addingtheconstraintthatc1,asolutiontothissystemalsosatisestheconstraintscx+d(a1)1(andtherefore1d(a1)0)thataroseintheanalysis.Withsubstantialalgebraicmanipulation,onecanshowthatc1satisfyingtheequationsistheuniqueroot(greaterthan1)ofthequinticpd(x)=(2d+1)(d+1)2x52(d+1)(2d+1)x4d2x3+2(d+1)x2((2d+1)(d+1)2+1)x+d(d+1):Foranyrc,wecannda,x,and"&#x-509;&#x.223;0satisfyingr(d(a1)x)=1+1=(r(d+1)1)",a=r2(x+d(a1)),andr2=(x+d(a1))=1+1=(r(d+1)1)".Thiscompletestheproof.2LNon1+1=(d+1)LowerBoundforSolvingSATWithd-DimensionalMachinesProof.(Sketch)Proofbyminimalcounterexample.Supposethereisanalternation-tradingproofthatNTIME[n]*DTIMEd[nd]withc1+1=(d+1),andletAbeanormalformannotation.Wemayas-sumethatjAj&#x-509;&#x.223;4,otherwisetheonlyannotationis[1;1;0;0]whichweknowdoesnotyieldtheresult.First,33 weprovethatinthissetting,everysequence0;1;0inanormalformannotationcanbereplacedwithjust0;0.Afteraslowdown,thedeterministicportionofaclassisDTIMEd,therefore0;1;0producesthelines:


b0(9na0)bDTIMEd[na](21)


b0(9na0;asg)maxfb;asgDTISP[na;nds](22)


b0DTIMEd[nmaxfcb;cacs;ca0;ca;cdsg];(23)butthisgivesnoimprovementoverapplyingthesequence0;0sincecacsca.Sowithoutlossofgenerality,every1inanannotationcanbeassumedtooccuradjacenttoanother1.Oncewehaveremoved0;1;0,theproofannotationscanbeplacedin1-1correspondencewithstringsofbalancedparenthesesoftheform(x),asinTheorem3.4.Everyrunofk1'scorrespondstok1openparentheses,andevery0correspondstoaclosedparenthesis.Hencethereisasequence1;1;0;0inaproofannotation,asthiscorrespondstothesubstring()),whichmustoccurinastringoftheform(x)wherexisnottheemptystring.Finally,weclaimthatifc(d+2)=(d+1),then1;1;0;0canbereplacedwithjust0,correspondingto).Toprovethis,oneexaminestheoutcomeoffourlineswheretwospeedupsandtwoslowdownsareapplied,thenarguesthatwhenc(d+2)=(d+1),theresultingconstantsarenobetterthanthecasewhereoneslowdownisapplied.Therearetwocasestoanalyze:onewheretherstlinehasaDTISPclassandonewheretherstlinehasaDTIMEdclass.ThereasoningfollowsthestyleofTheorems3.4andIbutismuchmoretechnicalinnature,soweomititsderivationhere.2MLowerBoundsforQBFk10AsrstobservedbyFortnowandVanMelkebeek[FvM00],thealternation-tradingschemeforlowerboundsagainstnondeterminismextendsnaturallytolowerboundsagainstalternatingcomputations.SinceAP=PSPACE[CKS81],itfollowsthatATIME[n]*DTISP[nk;no(1)]foreveryk1.11SowealreadyhaveapolynomialtimelowerboundforthequantiedBooleanformulaproblem(QBF)inthesmallspacesetting.HowlargecanlowerboundsforquantiedBooleanformulasbe,whenthenumberofquantierblocksisaxedconstant?DeneQBFktobetheproblemofsolvingaQBFwithkquantierblocks(i.e.decidingthetruthofkandksentencesinrst-orderBooleanlogic).BuildingonFortnowandVanMelkebeek[FvM00]whoprovedthatQBFkrequires\n(nk")timeonno(1)-spacemachines,weprovetimelowerboundsforQBFkoftheform\n(nk+1"k)onthesamemodel,wheref"kgisadecreasingsequencesuchthatlimk!1"k=0.WeusethefactthatQBFkis“robustlycomplete”intheappropriatesense,thenshowkTIME[n]*DTS[nc]forcertainc�kbyprovingaseriesofclasscontainments.LetusrecallthecompletnessresultofFortnowetal.:TheoremM.1(Fortnow-Lipton-VanMelkebeek-Viglas[FLvMV05])Forallk1,QBFkisrobustlycom-pleteforkQL[kQL.Inparticular,thereisaquasi-linearreductionfromanarbitrarylanguageintheclasstoQBFk,whereanarbitrarybitofthereductioncanbecomputedinpolylogarithmictime.WecanmodifytheLPframeworkforSATlowerboundstoobtainasimilarLPframeworkforQBFklowerbounds:onlytheSlowdownRulediffers,asitsapplicationremovestwoquantiersinsteadofjustone.Doing 10Apreliminaryversionoftheworkinthissectionwasreportedintheauthor'sPhDthesisin2007.Alsocf.[vM07],Section4.2,foranoverviewoftheresult.11Otherwise,SPACE[n]ATIME[n2]SPACE[no(1)],contradictingthespacehierarchytheorem.34 so,wewroteaprogramforprovingQBFktimelowerbounds,whichproducedproofsthatcloselyresembledthebelowargument.TheoremM.2Forallk1,QBFkrequires\n(nc)timeonno(1)spaceRAMs,wherec3=kc22c+k0.Notethisresultgeneralizesthe2cos(=7)lowerboundforSAT.TheremainderofthissectionprovesThe-oremM.2,whichwaspartlyinspiredbysomeshortproofsgeneratedbyourtheoremprover.Themaintoolweuseisthefollowing.TheoremM.3(ConditionalSpeedupforthePolynomialHierarchy)IfkTIME[n]DTS[nc]forsomec&#x-3.2;≦k,thenforalldsatisfyingcdc ck,DTS[nd]k+1TIME[n1+o(1)]\k+1TIME[n1+o(1)].Proof.SimilartotheproofoftheConditionalSpeedupTheoremin[Wil08].WeshowthatDTS[nd]k+1TIME[n1+o(1)]\k+1TIME[n1+o(1)]impliesDTS[n1+dk=c]k+1TIME[n1+o(1)]\k+1TIME[n1+o(1)].Thisprocessconvergeswhend=1+dk=c,ord=c=(ck).TheSpeedupLemma(LemmaA.1)impliesthatDTS[n1+dk=c](9n)(8logn)DTS[ndk=c;no(1)]:Applyingspeedupforkmoretimes,DTS[n1+dk=c](9n)(8logn)(8nd=c)


(Qnd=c)| {z }k1(:Qlogn)DTS[nd=c]forsomeQ2f9;8g,where:QisoppositetoQ.SincekTIME[n]DTS[nc],(9n)(8nd=c)


(Qnd=c)(:Qlogn)| {z }kDTS[nd=c](9n)DTS[nd]:Finally,sinceDTS[nd]k+1TIME[n1+o(1)]\k+1TIME[n1+o(1)],(9n)DTS[nd]k+1TIME[n1+o(1)]:AnanalogousargumentimpliesDTS[n1+dk=c]k+1TIME[n1+o(1)].2TheoremM.4IfkTIME[n]DTS[nc],thenforall`1anddsatisfyingcdc=(ck),DTS[nd+P`i=1c2 dki]k+1TIME[nc2 dk`+o(1)]\k+1TIME[nc2 dk`+o(1)]:Proof.Inductionon`.Thecase`=0isimmediate,bytheprevioustheorem.Fortheinductivestep,supposeDTS[nd+P`i=1c2 dki]k+1TIME[nc2 dk`+o(1)].First,theSpeedupLemmaimpliesDTS[nd+P`+1i=1c2 dki](9nc2 dk`+1)(8logn)DTS[nd+P`i=1c2 dki];wheretheinputtotheDTSparthaslengthn+2no(1).Bytheinductionhypothesis,theaboveiscontainedin(9nc2 dk`+1)(8logn)k+1TIME[nc2 dk`+o(1)]:35 ApplyingkTIME[n]DTS[nc]tothekpartofthek+1TIMEclass,theaboveliesin(9nc2 dk`+1)(8logn)(8nc2 dk`)DTS[ncc2 dk`]:Ifck,wealreadyhaveacontradiction,becausekTIME[nk]DTS[nkc]kTIME[nc](thisfollowsfromapplyingtheSpeedupLemma,ktimes).Ifck,theaboveclassiscontainedin(9nc2 dk`+1)(8logn)(8nc2 dk`)kTIME[nc k
c2 dk`+o(1)]=(9nc2 dk`+1)kTIME[nc k
c2 dk`+o(1)]:Note(c2 dk)`+1c k
(c2 dk)`,becausedc.InvokingtheassumptionkTIME[n]DTS[nc]againresultsintheclass(9nc2 dk`+1)DTS[nc2 k
c2 dk`]:Finally,sinced(c2 dk)`+1=c2 k
(c2 dk)`,TheoremM.3applies,andtheaboveclassisin(9nc2 dk`+1)k+1TIME[nc2 dk
c2 dk`+o(1)]=k+1TIME[nc2 dk`+1+o(1)]:Ananalogousargumentprovesthecontainmentfork+1TIME[nc2 dk`+1+o(1)].2LetK`=d+P`i=1c2 dki,for`1.Weclaim(theproofisnothard)thatc2 dk`K`1dk c2"`;forasmallconstant"`�0satisfyinglim`!1"`=0.Wededucethechain:k+1TIME[nK`](9nK`)DTS[ncK`](9nK`)kTIME[n(c=k)K`]DTS[n(c2=k)K`]k+1TIME[n(c2=k)K`(1dk c2"`)]:ForsufcientlylargeK`,acontradictionisreachedwhenc2=k(1(dk)=(c2))1.Recallingthatdc=(ck),theconditionsimpliestopk(c)=c3=kc22c+k0.Forconcretebounds,cf.Table1.Astheevidencesuggests,atleastonerootofthepolynomialpkgraduallyapproachesk+1askincreasesunboundedly;hencethelowerboundexponentforQBFkapproachesk+1.Proposition2limk!1pk(k+1)=0.Inparticular,forallk,pk(k+11=k)0andpk(k+1)&#x-3.2;≦0.Proof.Algebraicmanipulationgivespk(k+1)=1=k&#x-3.2;≦0andpk(k+11=k)=3=k311=k1=k21=k41=k40,forallk1.236 Problem TimeLowerBoundExponent SAT n1:801 QBF2 n2:903 QBF3 n3:942 QBF4 n4:962 QBF10 n10:991 QBF100 n100:999902 Table1:TimelowerboundsforQBFkonsmallspaceRAMs.37