comput.complex.10(2001),117{1381016-3328/01/020117{22$1.50+0.20/0c\rBi - PDF document

comput.complex.10(2001),117{1381016-3328/01/020117{22$1.50+0.20/0c\rBirkhauserVerlag,Basel2001computationalcomplexityREDUCINGTHECOMPLEXITYOFREDUCTIONSManindraAgrawal,EricAllender,RussellImpagliazzo,ToniannPitassi,andStevenRudichAbstract.Webuildontherecentprogressregardingisomorphismsofcom-pletesetsthatwasreportedinAgrawaletal.(1998).Inthatpaper,itwasshownthatallsetsthatarecompleteunder(non-uniform)AC0reductionsareisomorphicunderisomorphismscomputableandinvertiblevia(non-uniform)depth-threeAC0circuits.OneofthemaintoolsinprovingtheisomorphismtheoreminAgrawaletal.(1998)isa\GapTheorem",showingthatallsetscompleteunderAC0reductionsareinfactalreadycompleteunderNC0reduc-tions.Thefollowingquestionswereleftopeninthatpaper:1.Doesthe\gap"betweenNC0andAC0extendfurther?Inparticular,iseverysetcompleteunderpolynomial-timereducibilityalreadycompleteunderNC0reductions?2.Doesauniformversionoftheisomorphismtheoremhold?3.Isdepth-threeoptimal,orarethecompletesetsisomorphicunderiso-morphismscomputablebydepth-twocircuits?Weanswerallofthesequestions.Inparticular,weprovethattheBerman{HartmanisisomorphismconjectureistrueforP-uniformAC0reductions.Moreprecisely,weshowthatforanyclassCclosedunderuniformTC0-computablemany-onereductions,thefollowingthreetheoremshold:1.IfCcontainssetsthatarecompleteunderanotionofreductionatleastasstrongasDlogtime-uniformAC0[mod2]reductions,thentherearesuchsetsthatarenotcompleteunder(evennon-uniform)AC0reductions.2.ThesetscompleteforCunderP-uniformAC0reductionsareallisomor-phicunderisomorphismscomputableandinvertiblebyP-uniformAC0circuitsofdepth-three.3.TherearesetscompleteforCunderDlogtime-uniformAC0reductionsthatarenotisomorphicunderanyisomorphismcomputedby(evennon-uniform)AC0circuitsofdepthtwo.Toprovethesecondtheorem,weshowhowtoderandomizeaversionoftheswitchinglemma,whichmaybeofindependentinterest.(WehaverecentlylearnedthatthisresultisoriginallyduetoAjtaiandWigderson,butithasnotbeenpublished.)Keywords.Isomorphisms;completeness;constant-depthcircuits;Berman{HartmanisConjecture;poweringinniteelds.Subjectclassication.68Q17. 118Agrawaletal.cc10(2001)1.IntroductionMostofthecomputationalproblemsthatariseinpracticeturnouttobecom-pleteforoneofahandfulofcomplexityclasses,evenunderveryrestrictivenotionsofreducibility.Indeed,itwasnotedinBerman&Hartmanis(1977)thatthenaturalcompletesetscanevenbeshowntobeisomorphictoeachotherunderbijectionscomputableandinvertibleinpolynomialtime,andthustheycanbeviewedassimplere-encodingsofeachother.Thisandotherconsidera-tionsledtothefamousBerman{HartmanisConjecture(Berman&Hartmanis1977)thatallNP-completesetsarep-isomorphic.ItwasshowninAgrawaletal.(1998)thataversionofthisconjectureistrue.Moreprecisely,itwasshownthatinNP(andinmostothercomplexityclassesofinterest),allofthesetsthatarecompleteunderAC0reductionsareisomorphictoeachotherunderbijectionscomputableandinvertibleby(non-uniform)depth-threeAC0circuits.Thisisaverynaturalre-statementoftheBerman{HartmanisConjecture,since(a)AC0reductionsarethemostnaturalnotionofreducibilitytoconsiderwhenpresentingcompletesetsforsmallclassessuchasNC1orDSPACE(logn),and(b)allofthewell-knowncompletesetsforNPandothercomplexityclassesarecompleteevenunderAC0reductions.Theworkmentionedaboveleadsustoaskwhether,infact,allsetscompleteforawell-knowncomplexityclass(e.g.,NP)underpolynomial-timereductionsarealreadycompleteunderAC0reductions.(Inregardtothisquestion,itisinterestingtonotethatVeith(1998)showsthatall\succinctlyrepresented"problemsthatarecompleteunderpolynomial-timereductionsarecompleteun-derAC0reductions.Infact,theseproblemsareallcompleteunderprojections,whichareanevenmorerestrictivenotionofreducibility.)Thispossibilitymayseemunlikely,especiallyinlightofthefactthattherearemanyfunctionscom-putableinpolynomialtimethatarenotcomputableinAC0.However,itwasshowninAgrawaletal.(1998)thatallsetscompleteunderAC0reductionsarecompleteunderNC0reductions,inspiteofthefactthattherearemanyfunctionscomputableinAC0thatarenotcomputableinNC0.Inthispaper,wegiveanegativeanswertothisquestionbyshowing:Theorem1.1(\StopGapTheorem").ThereexistsasetthatiscompleteforNPunderDlogtime-uniformAC0[mod2]reductionsbutnotundernon-uniformAC0reductions.Also,byderandomizingtheversionoftheSwitchingLemmausedinAgrawaletal.(1998)weextendtheisomorphismtheoremofAgrawaletal.(1998)toP-uniformAC0reductions: cc10(2001)Reducingthecomplexityofreductions119Theorem1.2.AllsetscompleteforNPunderP-uniformAC0reductionsareisomorphictoeachotherviaisomorphismscomputableandinvertiblebydepth-threeP-uniformAC0circuits.Finally,weshowthattheaboveresultcannotbeimprovedtodepthtwo:Theorem1.3.Thereexisttwosets|bothcompleteforNPunderDlogtime-uniformAC0reductions|suchthatnoisomorphismbetweenthetwosetscanbecomputedbydepth-twonon-uniformAC0circuits.ThisresultimpliesthattheisomorphismscannotbecomputedbyNC0circuitssinceanyNC0circuitcanbeconvertedtoadepth-twoAC0circuit.Thisobservation,coupledwiththefactthatthetwosetsabovearealsocompleteunderu-uniformNC0reductionsforanyreasonablenotionofuniformityu,impliesthattheBerman{HartmanisConjectureisfalseforNC0reductionsforanyreasonablenotionofuniformity.AsinAgrawaletal.(1998),allourresultsholdnotjustforNP,butforanyclassclosedunderDlogtime-uniformTC0-computablemany-onereductions.Thepaperisorganizedasfollows.Section2presentsdenitionsfortheclassesofreductionsconsideredinthispaper.InSections3,4,and5weproveTheorems1.1,1.2,and1.3respectively.Andnally,Section6containsadiscussionontheresultsobtainedandfuturedirectionsforresearch.2.BasicdenitionsandpreliminariesWeassumefamiliaritywiththebasicnotionsofmany-onereducibilityaspre-sented,forexample,inBalcazaretal.(1995,1990).Inthispaper,onlymany-onereductionswillbeconsidered.AcircuitfamilyisasetfCn:n2NgwhereeachCnisanacycliccircuitwithnBooleaninputsx1;:::;xn(aswellastheconstants0and1allowedasinputs)andsomenumberofoutputgatesy1;:::;yr.fCnghassizes(n)ifeachcircuitCnhasatmosts(n)gates;ithasdepthd(n)ifthelengthofthelongestpathfrominputtooutputinCnisatmostd(n).AfamilyfCngisuniformifthefunctionn7!Cniseasytocomputeinsomesense.Inthispaper,wewillconsideronlyDlogtime-uniformity(Barringtonetal.1990)andP-uniformity(Allender1989)(inadditiontonon-uniformcircuitfamilies).AfunctionfissaidtobeinAC0ifthereisacircuitfamilyfCngofsizenO(1)anddepthO(1)consistingofunboundedfan-inANDandORandNOTgatessuchthatforeachinputxoflengthn,theoutputofCnoninputxisf(x).Notethat,accordingtothisstrictdenition,afunctionfinAC0must 120Agrawaletal.cc10(2001)satisfytherestrictionthatjxj=jyj=)jf(x)j=jf(y)j.However,theimposi-tionofthisrestrictionisanunintentionalartifactofthecircuit-baseddenitiongivenabove,andithastheeectofdisallowinganyinterestingresultsabouttheclassofsetsisomorphictoSAT(orothercompletesets),sincetherecouldbenoAC0-isomorphismbetweenasetcontainingonlyevenlengthstringsandasetcontainingonlyoddlengthstrings|anditispreciselythissortofindier-encetoencodingdetailsthatmotivatesmuchofthestudyofisomorphismsofcompletesets.ThusweallowAC0-computablefunctionstoconsistoffunctionscomputedbycircuitsofthissort,wheresomesimpleconventionisusedtoen-codeinputsofdierentlengths(forexample,\00"denoteszero,\01"denotesone,and\11"denotesend-of-string;otherreasonableconventionsyieldexactlythesameclassoffunctions).Fortechnicalreasons,wewilladoptthefollowingspecicconvention:eachCnwillhavenk+klog(n)outputbits(forsomek).Thelastklognoutputbitswillbeviewedasabinarynumberr,andtheoutputproducedbythecircuitwillbethebinarystringcontainedintherstroutputbits.ItiseasytoverifythatthisconventionisAC0-equivalenttotheotherconventionmentionedabove,andforusithastheadvantagethatonlyO(logn)outputbitsareusedtoencodethelength.Itisworthnotingthat,withthisdef-inition,theclassofDlogtime-uniformAC0-computablefunctionsadmitsmanyalternativecharacterizations,includingexpressibilityinrst-orderlogicwithf+;;g(Barringtonetal.1990;Lindell1992)1,thelogspace-rudimentaryreductionsofJones(Allender&Gore1991;Jones1975),logarithmic-timeal-ternatingTuringmachineswithO(1)alternations(Barringtonetal.1990)andothers.Thislendsadditionalweighttoourchoiceofthisdenition.TC0istheclassoffunctionscomputedinthiswaybycircuitfamiliesofMAJORITYgatesofsizenO(1)anddepthO(1);NC1andNC0aretheclassesoffunctionscomputedinthiswaybycircuitfamiliesofsizenO(1)anddepthO(logn)(orO(1),respectively),consistingoffan-intwoANDandORandNOTgates.NotethatforanyNC0circuitfamily,thereissomeconstantcsuchthateachoutputbitdependsonatmostcdierentinputbits.TheclassoffunctionsinNC0wasconsideredpreviouslyinHastad(1987).ForacomplexityclassC,aC-isomorphismisabijectionfsuchthatbothfandf1areinC.(Toeliminateunnecessarynotation,wefollowstandardpracticeinignoringthedistinctionbetweenthesetofdecisionproblemsCandtheclosely-relatedsetoffunctions.Thus,forinstance,AC0canbeviewedas1Lindell(1992)showsonlythatthiscoincideswithrst-orderexpressibilityinrstorderlogicwithf+;;;expg,where\exp"denotesexponentiation.However,personalcommu-nicationfromK.ReganandS.Lindellshowsthatexponentiationcanbeeliminated.Fordetails,seeImmerman(1998). cc10(2001)Reducingthecomplexityofreductions121eitherasetoflanguagesorasasetoffunctions,withnoconfusion.)Sinceonlymany-onereductionsareconsideredinthispaper,a\C-reduction"issimplyafunctioninC.Thetheoremsweproveinthispaperholdformostcomplexityclassesthatareofinteresttotheoreticians;werequireonlyclosureunderDlogtime-uniformTC0reductions.(Thatis,ifAisinC,andBisreducibletoAviaamany-onereductioncomputableinTC0,thenBisinC.)Notethatmostcomplexityclasses,suchasNP,P,PSPACE,BPP,etc.,havethisclosureproperty.Infact,inspectionofourproofsshowsthatourresultsholdevenforanyclassCthatisclosedunderreductionscomputedbyDlogtime-uniformthresholdcircuitsofdepthve.(Thenumbervecanprobablybereduced.)WedonotknowhowtoweakentheassumptiontoclosureunderreductionscomputedinACC0;itiseasytoseethatourresultsdonotholdforsomeclassesclosedunderAC0reductions.(Forinstance,thesetsf1gandf1;11garebothhardforAC0underAC0reductions,buttheyarenotisomorphic,andtheyarenothardunderNC0reductions.)Afunctionislength-nondecreasing(resp.length-increasing,length-squaring)if,forallx,jxjjf(x)j(resp.jxjjf(x)j,jxj2jf(x)j);itisC-invertibleifthereisafunctiong2Csuchthatforallx;f(g(f(x)))=f(x).3.ProofofTheorem1.1LetSATbethesetofstringscodingsatisableBooleanformulas(undersomestandardcodingscheme).SATiscompleteforNPunderDlogtime-uniformAC0reductions(and,infact,evenunderprojections).LetPARITYbethesetofallbinarystringswithanoddnumberofones.PARITYisinNP.Theideabehindtheproofisasfollows.WerstdeneafunctionfthatiscomputablebyAC0[mod2]circuits,andisanerror-correctingcodecapableofcorrectinga\large"fractionoferrors.Sincefis1-1,andcomputableinDlogtime-uniformAC0[mod2],SATisAC0[mod2]reducibletof(SAT).Thus,f(SAT)iscompleteforNPunderAC0[mod2]reductions.Assumingf(SAT)isalsocompleteunderAC0reductions(inthatcase,itisalsocompleteunderNC0reductionsbytheGapTheorem(Agrawaletal.1998)),weconsideranNC0reductionofPARITYtof(SAT).Foranyinputlengthn,mostoftheinputbitsofthereductioncircuitcanin\ruenceveryfewoutputbitswhereasthestringsinf(SAT)areveryfarapart.Thus,thiscircuitmustmapallinputsinPARITYtothesameoutput.ThisgivesanAC0circuitforPARITY,acontradiction. 122Agrawaletal.cc10(2001)ItturnsoutthatthestandardReed{Solomoncodecanbeusedtodenefunctionf.Forthepurposeofself-containment,weprovideadescriptionoffunctionf:Inputx,jxj=n.Lety=x10kwithkbeingthesmallestnumbersuchthatjyjisdivisiblebyt(t=O(logn)tobexedlater).Lety=a0a1


aswhereeachaihastbits.LetpolynomialY(z)=Psi=0ai
ziwhereeachaiistreatedasanelementofF2t|theniteeldof2telements.Letb1;:::;b2tbeanenumerationofallelementsofF2t.OutputthestringY(b1)


Y(b2t);whereY(bi)isevaluatedoverF2t.Notethatifx6=x0aretwostringsoflengthn,thenf(x)andf(x0)dierinatleast2tsbits(andthusthefractionofthebitpositionsinwhichtheydierisatleast(2ts)=(t2t)).Toseethis,letYandY0betheassociatedpolynomialsasdenedabove.Clearlyf(x)andf(x0)agreeinblockjifandonlyif(YY0)(bj)=0.SinceYY0isanon-zeropolynomialofdegrees,thiscanhappenonlyforsdistinctbj's,andhencef(x)andf(x0)dierinatleast2tsblocksoftbits,whichestablishestheclaim.NumbertshouldbeO(logn)toensurethatfispolynomiallybounded.Itshouldalsobegreaterthanlogs=(logn),sinceotherwisethecodeismeaningless.TofacilitatecomputinginF2t,wechooset=2
3`forthesmallest`suchthat2t2n.Forthischoiceoft,thepolynomialzt+zt=2+1isirreducibleinF2[z]andthusF2[z]=(zt+zt=2+1)=F2t.Withthisvalueoft,thefractionofbitsinwhichtwocodewordsdieris=2tst
2t2t2t1t
2t=12t124lognforlargeenoughn.Letusnowconsiderthecomplexityofcomputingf(x).Consideranypar-ticularoutputbit.ThisbitoftheoutputisoneofthebitsofY(bj)forsomej2t,whereY(z)isthepolynomialPiai
zi,whereeachaiconsistsoft=O(logn)bitsofx.Notethatbijisaconstant,notdependingontheinputx.Itfollowsfromrecentprogressonthecircuitcomplexityofdivision(Hesse2001)thatbijcanactuallybemadeavailableasaconstantinDlogtime-uniformAC0;detailscanbefoundinTheorem3.2below.Thuswecanviewtheconstantsbijasbeing cc10(2001)Reducingthecomplexityofreductions123\hardwired"intothecircuitcomputingf.Next,considerhowtocomputeai
w,wherewisanyt-bitconstant(suchasw=bij).Ofcourse,sinceaiisthebitwisesumofitscomponents,ai
wcanbeexpressedasasumoft=O(logn)termsoftheform0:::0xj0:::0
z,wherexjisoneofthebitsofx.Theterm0:::0xj0:::0
zcanbecomputedby(1)obtaininga2t-bitvectorrepresentingthepolynomialqofdegree2(t1)thatoneobtainsbymultiplyingzby0:::0xj0:::0
z(thisisjustaleftshiftofz,unlessxjis0),andthen(2)nding(bytablelook-up)thet-bitvectorvjsuchthatqisequivalenttovjmodzt+zt=2+1.NotethatvjcanbefoundinDlogtime-uniformAC0,bycheckingifthereexistsavectorv0suchthatqvj=v0
(zt+zt=2+1).Thatis,ai
wcanbeexpressedasumofO(logn)termsoftheformvj,whereeachcomponentofvjiseither0orxj.Thusai
zcanbecomputedbytPARITYgates,eachofwhichiscomputingthesuminF2ofacomponentofthevj's.ThenaloutputPiai
bijcanthusbecomputedbytPARITYgates,eachconnectedtosomeofthePARITYgatescomputingai
bij.ThisestablishesthatfcanbecomputedinDlogtime-uniformAC0[mod2].AcloserexaminationoftheforegoingalgorithmshowsthattheANDandORgatesareusedonlyincomputingcertainconstantsthatdonotdependontheinput,andthateachbitoftheoutputissimplythePARITYofsomeoftheinputbits(andtheseconnectionscanbecomputedinDlogtime-uniformAC0).Thisestablishesthatfiscomputedbyveryuniformdepth-onecircuitsconsistingonlyofPARITYgates.LetS=ff(x):x2SATg.Sincefis1-1,itisareductionfromSATtoS.Sincefiscomputablein(Dlogtime-uniform)AC0[mod2],andSATisNP-completeunderAC0reductions,SisNP-completeunder(Dlogtime-uniform)AC0[mod2]reductions.Suppose,foracontradiction,thatSisNP-completeundernon-uniformAC0reductions.Inparticular,theremustbeanAC0reductionfromPARITYtoS.ByinvokingtheGapTheoremofAgrawaletal.(1998),weseethattheremustbeanNC0reductionfromPARITYtoS.Infact,itisshowninAgrawaletal.(1998)thatthisNC0reductionhasthepropertythatthelengthoftheoutputdependsonlyonthelengthoftheinput.LetthisreductionbecomputedbycircuitfamilyCn.Fixanintegern,andconsiderthecircuitCnthatdenesthereductiononstringsoflengthn.LetCnhavemoutputbits.ThereisaconstantbsuchthateachoutputbitofthecircuitCndependsonatmostbinputbits.Letoibethenumberofoutputbitsthatdependonvariablexi.Thesumofoi'sisthereforeboundedbymb.Henceatmost2b==O(logn)oi'scanbegreaterthanm=2.Inotherwords,atmost2b=inputvariablesin\ruence(i.e.,areinputsto)m=2outputbits.ThuswecansettheseO(logn)additional 124Agrawaletal.cc10(2001)inputvariablesandobtainanNC0circuitfamilyonn0nO(logn)inputvariablesthatalsoreducesPARITYtoS,andhasthepropertythateveryinputvariablein\ruencesfewerthanm=2outputbits.CallthisnewcircuitDn0,andletgbethefunctioncomputedbyit.Considertwostringsx1;x2oflengthn0thatareinPARITYandthatdierinexactlytwolocationsiandj.Weclaimthatg(x1)=g(x2).Otherwiseg(x1)andg(x2)dierinatleastmlocations(sincetheymaptotwodistinctcodewordsinf(SAT)),andthesemlocationsarein\ruencedbyvariablesiandj,incontradictiontotheconstructionofDn0.Sinceanystringxoflengthn0inPARITYcanbeobtainedfrom10n01byasequence10n01=x1;x2;:::;xr=xwherexiandxi+1dierinexactlytwolocations,itfollowsthatthestringsoflengthn0inPARITYcanbecharacterizedasthesetfx:g(x)=g(10n01)g.Thus,wecanconstructacircuitthatrstcomputesg(x)usingDn0andthenchecksequalitywithg(10n01)usingasingle\AND"gateoffan-inm.Thisconstant-depthcircuithassizeO(jCnj)=nO(1)=n0O(1),andsincen0getsarbitrarilylargeasndoes,thiscontradictsthelowerboundsforcomputingparityviaconstant-depthcircuits(Ajtai1983;Furstetal.1984).Examiningtheproof,weusedonlythefactsthatPARITY2NP,NPhasacompletesetforAC0[mod2]reductions,andthatNPisclosedunderAC0[mod2]reductions.Generalizing,weget:Theorem3.1.LetRbeaclassoffunctionsclosedundercompositionandcontainingDlogtime-uniformAC0[mod2].LetCbeacomplexityclassclosedunderbothDlogtime-uniformTC0reductionsandR-reductions,andhavingasetthatiscompleteunderR-reductions.ThenCcontainsanR-completesetthatisnotcompleteundernon-uniformAC0reductions.3.1.Poweringinniteelds.TocompletetheproofofTheorem1.1,weneedonlyprovidetheproofthatpoweringinsmallniteeldscanbeperformedinDlogtime-uniformAC0.Theorem3.2.Lett=23`forsome`,wheret=O(logn).Then,giveninput(a;i;b)oflengthO(logn),itcanbedeterminedinDlogtime-uniformAC0ifai=b,whereaandbareelementsofF2t.Weremarkthattherestrictionontismerelysothatwecanbeexplicitaboutourchoiceofanirreduciblepolynomial.InDlogtime-uniformAC0,itispossibletolocatethelexicographically-rstirreduciblepolynomialofdegreet=O(logn),anduseittorepresentF2t,foranychoiceoft. cc10(2001)Reducingthecomplexityofreductions125Proof.ByHesse(2001),itissucienttoshowthataDlogtime-uniformAC0circuitfamilyexiststhattakesasinputthebinaryrepresentationsofrelementsofF2t(wherer=O(logn);calltheseelements(a1;:::;ar))andcomputesQiai.Eachoftheai'scanbeviewedasapolynomialoverF2(takenmodulotheirreduciblepolynomialzt+zt=2+1).Ourapproach,modeledonFrandsenetal.(1994),willbetouseChineseRemainderingovertheringofpolynomialsoverF2.Leth(z)bethepolynomialz2bz,wherebisthesmallestnumbersuchthat2b�rt=O(log2n).AspointedoutinFrandsenetal.(1994),hhas2bdistinctfactorsinF2b,andthuseachoftheirreduciblefactorsofh(callthemh1;:::;hk)hasdegreeboundedbyb=O(logt)=O(loglogn).ByLemma3.3below,inuniformAC0wecantest,foreachshortbitstringrepresentingasmallpolynomialh0,ifh0dividesh.Also,itissimpletocheck,forsuchapolynomialh0,thatnootherpolynomialdividesh0.ThuswecanndtheirreduciblefactorsofhinuniformAC0.LetAbethepolynomialofdegreeO(log2n)thatresultsbytakingtheproductofthepolynomialsrepresenting(a1;:::;ai)inF2[z].ByChineseRe-maindering,Acanberepresentedbythesequence(^A1;:::;^Ak),where^AjistheremainderobtainedwhendividingthepolynomialAbyhj.Similarly,ifweletcai;jbetheremainderofdividingaibyhj,thenwehave^Aj=Qicai;j,wheretheproductistakeninF2[z]=hj.SincethemultiplicativegroupofF2[z]=hjiscyclicandofsizelogO(1)n,itiseasyinDlogtime-uniformAC0tondageneratorofF2[z]=hj,andcomputeatableofdiscretelogsrelativetothisgen-erator.(Toseethis,foreachpotentialgeneratorg,buildagraphwithnodesforgroupelementsandedgesrepresentingmultiplicationbyg.EachnodecanberepresentedwithO(loglogn)bits;thusapathoflengthlogn=loglogncanberepresentedwithO(logn)bits.HenceitiseasyinDlogtime-uniformAC0todetermineifthereisapathoflengthilogn=loglognbetweentwonodes.IteratingthisconstructionO(1)timesallowsonetolookforpathsoflengthlogO(1)n.Thenodegisageneratorifandonlyifthegraphisconnected.)NowtheproductQicai;jcanbecomputedbyaddingthediscretelogs.SinceadditionoflogO(1)nnumberscanbeperformedinDlogtime-uniformAC0,itisclearthatwecancomputetheChineseRemainderrepresentationofA,(^A1;:::;^Ak),inDlogtime-uniformAC0.Tocompletetheproof,weneedonlyshowhowwecanobtainthecoecientsofAfromtheChineseRemainderrepresentation,andthendivideAbyzt+zt=2+1toobtaintherepresentativeelementofF2[z]=(zt+zt=2+1).(Thislatterdivisioncanbeaccomplished,byappealtoLemma3.3.) 126Agrawaletal.cc10(2001)BytheChineseRemainderTheorem,AisequivalenttoPki=1^Aicidimoduloh,whereci=h=hi,anddiisanelementofF2[z]=hisuchthatcidi=1modhi.ByLemma3.3,therepresentationofcicanbecomputedinDlogtime-uniformAC0.SinceF2[z]=hiissosmall,dicanbefoundbybruteforce.Thuswecancomputeeachtermofthesum.Sincethereareonlylogntermsinthesum,andeachcomponentofthetermcanbecomputedbytakingtheparityofO(logn)elements,wecancomputethecoecientsofA,asrequired.Lemma3.3.Letk2N.ThenthereisaDlogtime-uniformAC0circuitfamilythattakesasinputasequenceofcoecientsdeningtwopolynomialsh1andh22F2[z]ofdegreelogkn,andoutputsthesequenceofcoecientsforpolyno-mialsqandrsuchthatthedegreeofrislessthanthedegreeofh2,andsuchthath1=h2
q+r.Thatis,divisionwithpolynomialsofdegreelogO(1)n,andndingremainders,canbeperformedinDlogtime-uniformAC0.Proof.Letm=logkn.Eberly(1989)showsthatdivisionofpolynomialsofdegreemisreducibletotheproblemofcomputingtheproductofmO(1)integers,eachhavingmO(1)bits.EberlyclaimsonlyanNC1-Turingreduction,butanexaminationofhisproofshowsthatitcanbeimplementedasaDlogtime-uniformAC0-Turingreduction.However,itisshowninHesse(2001)thatcomputingtheproductoflogO(1)nintegers,eachoflengthlogO(1)n,canbecomputedinDlogtime-uniformAC0.4.ProofofTheorem1.2OurproofofTheorem1.2resultsfromatechnicalimprovementofoneofthestepsfromtheproofoftheIsomorphismTheoreminAgrawaletal.(1998).Wewillnotrepeattheconstructionhere,butwewillprovideaveryhigh-levelsketchoftheapproachtakenthere.TherearethreemainpartsoftheargumentinAgrawaletal.(1998):agaptheorem(towhichwehavealreadyreferredinthispaper),statingthatsetscompleteunderAC0reductionsarealsocompleteunderNC0reductions,atechnicaltheorem,showingthatallsetscompleteunderNC0reductionsarecompleteundereasilyinvertiblereductionscalledsuperprojections,andanisomorphismtheorem,showingthatallsetscompleteundersuperpro-jectionsareisomorphicunderdepth-threeAC0isomorphisms. cc10(2001)Reducingthecomplexityofreductions127ThetechnicaltheoremandtheisomorphismtheoremofAgrawaletal.(1998)alsoholdintheP-uniformsetting.Thus,inordertoproveTheorem1.2,itsucestoproveaP-uniformversionoftheGapTheorem:Theorem4.1.AllsetshardforNPunderP-uniformAC0reductionsarehardforNPunderP-uniformNC0reductions.WenowoutlinetheproofoftheGapTheoremofAgrawaletal.(1998)inordertoidentifythenon-uniformstep.LetAbeahardsetforNPunderAC0reductions.WeneedtoshowthatanysetBinNPhasanNC0reductiontoA.Forthis,rstaversionB0ofBisdenedwithalotofredundancy:correspondingtoastringxinB,B0hasmanystringswitheachsuchstringyhavingjxjblocksofbitssuchthattheithbitofxequalstheparityoftheithblockofbitsofy(thisisnottheexactdenitionofB0asinAgrawaletal.(1998)butcapturestheessentialidea).ThesetB0alsoisinNPandthereforethereisanAC0reduction,givenbycircuitfamilyfCmg,ofB0toA.ItisimplicitinFurstetal.(1984)andAjtai(1983)(andadetailedproofisprovidedinAgrawaletal.(1998))thatarandomrestriction2oftheinputvariablesofcircuitCmtransformsit(withhighprobability)toanNC0circuitonatleastmbitsforsome�0.Ifwedividetheinputintonblocksofequallengthwithn=m=2,asimpleprobabilitycalculationshowsthatintherandomrestriction,withhighprobability,eachoftheseblockswouldhaveatleastthreeunsetbits.Fixarestrictionmthathasboththeaboveproperties.Modifythisrestrictionasfollows:ineachblock,setallbutoneoftheunsetbitsinsuchawaythatparityofallthesetbitsintheblockbecomeszero.(Sinceeachblockstartswithatleastthreeunsetbits,wecanalwaysdothis|actuallytwounsetbitssuceforthisbutthreeunsetbitsareneededfortechnicalreasonsinAgrawaletal.(1998).)Letthismodiedrestrictionbe0m(clearly,0mtransformsCmtoanNC0circuitonnbits).WenowdeneareductionofBtoB0as:givenx,jxj=n,output0m(x)whichisthestringconstructedfrom0mbyllingintheithbitofxintotheunsetbitoftheithblockof0m.BythedenitionofB0,x2Bi0m(x)2B0.Also,thisreductionhastrivialcircuitcomplexity|itisjustaprojection.AcompositionofthiscircuitwithCmyieldsanNC0circuitwhichreducesBtoA,completingtheproof.Intheaboveconstruction,althoughthecircuitcomputingthereductionofBtoB0istrivial,itisnon-uniformsinceitrequiresa\good"randomre-strictionm.Inthelemmabelow,weshowhowtocomputesucharestriction2Arandomrestrictionhereleaveseachbitunsetwithprobability1=m12,andsetsitto1or0withprobability12(11=m12)each. 128Agrawaletal.cc10(2001)inpolynomial-timegiventhecircuitCm.NowifthecircuitfamilyfCmgisP-uniform,theentireconstructionbecomesP-uniform,provingtheuniformversionoftheGapTheorem.Lemma4.2.ForanyAC0reductioncomputedbyafamilyfCmgofcircuits,thereexistsana2Nsuchthat,foralllargemoftheformr2a,thereisarestrictionmsuchthatmtransformsCmintoanNC0circuit,andmassignstoatleastthreevariablesineachblockoflengthr2a1.Furthermore,mcanbecomputedintimepolynomialinmiffCmgisP-uniform.TheremainderofthissectionisdevotedtoprovingLemma4.2.4.1.Derandomizingtheswitchinglemma.Inthissectionweprovideaproofthattheswitchinglemmacanbecarriedoutfeasibly.Moreprecisely,givenacircuitCofdepthd,andsizeS=nk,withn=rmunderlyingvariablesarrangedintorblocks,eachofsizem=ra(adependsondandk),thereexistsarestrictiontothevariablessuchthateachoutputbitofCddependsonlyonaconstantnumberofvariables,andeachoftherblockshasatleastr2variablesleftunset.Furthermore,wegiveauniformalgorithmforndingintimepolynomialinthesizeofC.Theswitchinglemmastatementandproofthatwewillfollowisasimpli-cationofthatduetoFurst,SaxeandSipserbutwithtwoadditionalcompli-cations:(1)weneedtotaketheblocksintoaccountand(2)weneedtogivepolynomial-timealgorithmsforndingtherestrictions.LetCbeadepthd,sizeS=nkcircuit.Itwillbeconvenientforustoconsideramodiedclassofcircuits,consistingofusualANDandORgates,butateach\input"gatetothecircuitweinsteadattachadecisiontree;thecircuitreceivesasinputthevalue(0or1orxi)thatisreachedbyqueryingtheinputbitsspeciedbythedecisiontreeandproceedingtoaleafofthedecisiontree.Thusanordinarycircuitcorrespondstothecasewhereweusedecisiontreesofheightzero.WewillassumewithoutlossofgeneralitythatCisarrangedintodalternatinglevelsofANDandORs,andattheleavesofthecircuitareconstant-depthdecisiontreesofheightc1.Theconstantc1willbechosentobesucientlylargeasafunctionofk,wherenk=Sisthesizeoftheoriginalcircuit.Theproofwillproceedindsteps.Atstepone,wewillapplyc1successiverestrictionsinordertoreplacethebottomlevelsof(ANDsofconstant-depth-c1decisiontrees)by(constant-depth-c2decisiontrees),orsimilarly,inordertoreplacethebottomlevelsof(ORsofconstant-depth-c1decisiontrees)by(constant-depth-c2decisiontrees).Ingeneralinstepi,wewillbeapplyingcirestrictionsinordertoreplacethebottomlevelsofANDs cc10(2001)Reducingthecomplexityofreductions129andORsofconstant-depth-cidecisiontreesbydepth-ci+1decisiontrees.Thusafterdsteps,thetotalnumberofrestrictionsappliedwillbec1+


+cd,withcirestrictionsatstepi,fordsteps.Theunderlyingvariableswillalwaysbegroupedintorblocks,wheretheblocksizewillbem=m1atthestart.Afterapplyingonerestriction,wewillstillhaverblocks,andexactlym1=41variableswillremainunsetwithineachblock.(Ifarestrictionisappliedtorblocks,eachofsizem,thentherestrictionwillconsistofarstpartwhererm1=2variablesarechosenuniformlytobesetto,andwiththeconditionthatnoblockwillhavesizelessthanm1=4,andthenasecondclean-uppartwherewesetadditionalvariablessothateachblockwillhaveuniformsizem1=4.)Thusafteronestep,therewillberblocks,eachofsizem2=m1=4c11,andnallyafterdsteps,therewillberblocks,eachofsizemd+1=m1=4c1+


+cd1.Wewillnowdescribeonestep.Assumethatthebottomlevelsubcircuitshavetheform:ANDofdepth-c1decisiontrees.TheneachsuchsubcircuitcanbeexpressedasanANDofsize-c1ORs.LetS1;:::;Sqbethesetof(polynomiallymany)ANDsofsize-c1ORs.Therststepproceedsinc1stagesasfollows.Instage1,wewillndarestrictionsuchthatforeachi,Sidhasapartialdecisiontreeofconstantdepthc01,andwhereeachleafislabeledbyeitheraconstant,orbyanANDofsize-(c11)ORs.TherestrictionisobtainedbyusingAlgorithmA.Stagejisthesameasstage1,butnowthesetofformulasunderconsideration(theSi's)arethenon-constantformulaslabelingtheleavesofthedecisiontreesthathavebeencreatedthusfar.Afterstagej,wehavecreatedpartialdecisiontreesfortheoriginalSi's,wherenowtheleavesofthetreearelabeledeitherbyconstantsorbyANDsofsize-(c1j)ORs.Foreachstage,weuseAlgorithmAtondtherestriction.Finallyafterc1stages,wehavedecisiontreesfortheoriginalSi'swhereallleavesarelabeledbyconstants.Afteronestep,wehavegonefromadepth-dsize-Scircuitwithrm1un-derlyingvariables,arrangedintorblocks,whereeachblockhassizem1,toadepth-(d1)size-Scircuit,wherenowthenumberofunderlyingvariablesisrm2,againarrangedintorblocks,andwhereeachblockhassizem2.Itiseasytoseethatnowthebottomlevelconsistsofdecisiontreesofdepthc01c1,whichwillbechosentobeatmostc2.Afterrepeatingthisfordsteps,eachoutputgateoftheoriginalcircuitwillberepresentedbyadepth-cd+1decisiontreeontheremainingvariables.Attheend,therewillbermd+1remainingvariables,againconsistingofrblocks,eachcontainingmd+1variables.Wewillnowdenetherelationshipsbetweenthevariousparameters.First,c1=O(k),andforalli2,ci=8ci1.Foralli1,c0i=6ci.Thus,foreachi,wehavec0icici+1asrequired.Initiallytherearerm1variables.One 130Agrawaletal.cc10(2001)restrictionwillsetallbutm1=41variablesperblock.Thusafteronerestriction,therearerm1=41variablesremaining,andafteronestep,therearerm2variablesremaining(m2variablesperblock),wherem2=m1=4c11.Thenumberofvariablesremainingafterdstepsismd+1=m1=4c1+


+cd1m1=5cd1.Recallthatinitially,therearerblocks,eachofsizem1=raforsomea,andwewantittobethecasethatafterallrestrictionsaresuccessfullyappliedtoreducethecircuit,thenalblocksize,md+1,isatleastr2.Itshouldbeclearthatacanbechosentobesucientlylarge(dependingonkandd)suchthatthisholds.WewillnowdescribeAlgorithmA.4.2.AlgorithmA.Theinputtothisalgorithmisacollectionofpolynomi-allymanyformulasQ1;:::;Qq,whereeachQiisanANDofsize-cORs.(Oralternatively,eachQiisanORofsize-cANDs.Thiscaseishandledduallysowewillnotconsiderithere.)Therearermunderlyingvariables,arrangedintoblocksb1;:::;br,whereeachblockhassizem.(Thevalueofmwillberaforaappropriatelychosen.Thus,misapolynomialinr.)Theoutputisarestrictionsuchthat:(1)assignsexactlym1=4'stoeachblock,andallothervariablesaresetto0and1;(2)foreachQi,wecanconstructadepth-c0decisiontreeforQidsuchthattheleavesofthedecisiontreearealllabeledbyeitheraconstant,orbyanANDofsize-(c1)ORs.WefollowtheusualconventionandrefertoanORofliteralsasaclause.GivenaQi,wedeneasetMaxset(Qi)ofclausesasfollows.Firstndthelex-icographicallyrstsetofclausesinQithatarevariable-disjoint.Ifthenumberofclausesinthissetisgreaterthanflogm(forasuitablychosenconstantfwhosevaluewillbexedlater),thenletMaxset(Qi)bethelexicographicallyrstflogmoftheseclauses.(SojMaxset(Qi)jflogm.)WedividetheQi'sintotwodisjointsets:First,fn1;:::;nsg,thenarrowformulas,arethoseQi'ssuchthatjMaxset(Qi)jflogm.Secondly,fw1;:::;wtg,thewideformulas,arethoseQi'ssuchthatjMaxset(Qi)j=flogm.Wewillndarestrictionsettingallbutrm1=2variablessuchthat:(1)assignsatleastm1=4'stoeachblock;(2)foreachni,thenumberofunderlyingliteralsinMaxset(ni)thataresettobyisatmostc0;and(3)foreachwj,atleastoneclauseinMaxset(wj)issetto0by.Oncewehavefoundsucharestriction,wesetadditionalvariablesinordertosetallbutexactlym1=4variablesperblock.Secondly,foreachwj,wecan cc10(2001)Reducingthecomplexityofreductions131createthetrivialdecisiontreeforwjdlabeledby0.Thirdly,foreachni,wecancreateadepthc0decisiontreefornidbyqueryingthe'dvariablesinMaxset(ni)d.Byproperty(2),thereareatmostc0suchvariables.Oncethesehaveallbeenqueried,weareleftateachleafwitheitheraconstantorwithanANDofsize-(c1)ORs,sinceanyotherclauseintersectsatleastonevariableofMaxset(ni),andallvariablesinMaxset(ni)havebeenset.Thefollowingthreelemmasshowthatforsuitablechoicesoftheparameters,sucharestrictionexists.Lemma4.3.Letfb1;:::;brgbeapartitionoftheunderlyingrmvariablesintorblocks.LetBibetheeventthatblockbihaslessthanm1=4'safterisapplied.ThenPiPr[Bi]1=4,wheretheprobabilityisoverallrestrictionssettingexactlyrm1=2variablesto.Proof.Letpbetheprobabilitythataparticularelementis'd.Thenp=1=pm.Letthesizeofbibemandletl=m1=41.Thenwehave,foralllargem,Pr[Bj]=lXi=0jbjjipi(1p)jbjjilXi=0empmil(mlepm)2m1=4:SummingupoverallBjshowsthatthetotalprobabilityisatmost1=4.Wewillapplytheabovelemmarepeatedly,forsmallerandsmallervaluesofm.However,foreachapplication,mwillbeequaltom1forsomeverytiny,whichwillbeequaltorfor=l,andthustheaboveprobabilitywillalwaysbelessthan1/4.Lemma4.4.Considerthesetfs1;:::;sSgwhereeachsiisacollectionofatmostcflogmliterals,whereSisapolynomialinm,andwherethesi'sarepairwisedisjoint.(Foragivennarrowformulani,siisthesetofvariablesthatunderlytheclausesinMaxset(ni);sincetherearefewerthanflogmclausesinMaxset(ni),thetotalnumberofvariablesinsiisatmostcflogm.)LetNibetheeventthatsihasmorethanc0'safterisapplied.(I.e.,Niisthebadeventthatthenarrowformulanidoesnothaveproperty(2)above.)ThenaslongasSmc0=4,PiPr[Ni]1=4.Proof.Letrmbetheoriginalnumberofvariables,andletm0=rpmbethenumberof'dvariablesin.Thenp=m0=m=1=pmistheprobabilitythataparticularvariableissettobyarandom.WewillrstgetanupperboundonPr[Ni].TheexpectednumberofelementsinsisettoisjsijpO(logm)p. 132Agrawaletal.cc10(2001)Theprobabilitythattherearemorethanc0'sinsiisatmostjsijc0pc0ejsijpc0c0=ecflogmc0pmc0(flogm=pm)c0:SincethereareSmanysi's,thetotalprobabilityoffailureisatmost(flogm=pm)c0Smc0=4S=4forsucientlylargem.Thus,aslongasSmc0=4,theoverallprobabilityisatmost1/4.Wewillapplytheabovelemmarepeatedly.Atthestartofeachstepi,Snk,andattheendofstagejinstepi,Swillbeboundedby2jc0ink.Thusinallcaseswhereweapplythelemma,Swillbeboundedby2ci+1nk2ci+1m2k1.Atstepi,mwillbeequaltomi,cwillbeequaltoci,andc0willbeequaltoc0i.Forourchoicesofparameters(c0i=6ciandmi�m1=5ci1),ourconditionthatSbelessthanmc0=4thusholdsif2ci+1m2k1m6=5ci=41,whichholdsforalllargem1.Lemma4.5.Letfw1;:::;wSgbeANDsofsize-cORs,whereforeachwi,jMaxset(wi)j=flogm.(Thewi'sarethewideformulas.)Lettheunderlyinguniversebeofsizerm.Foragivenwi,letWibethebadeventthatnoclauseinMaxset(wi)issettozero.ThenifSe(flogm)=4c=4,thenPiPr[Wi]1=4.(Thatis,withprobabilityatmost1=4,arandomrestrictionsettingrm1=2variablestohasthepropertythatforsomewi,noclauseinMaxset(wi)issetto0by.)Proof.Foragivenwi,lets1;:::;sflogmdenotetheunderlying(disjoint)clausesinMaxset(wi).Wewillrstshowthatforagivenpolynomialp(m)thereexistsanf(dependingonp(m)andc)suchthatPr[Wi]1=(4p(m)):Pr[Wi]=flogmYj=1Pr[sjisnotallzero]=flogmYj=1(1Pr[sjisallzero])=flogmYj=11rmrpm2rmcflogmYj=1(1(1=4)c)=(1(1=4)c)flogme(flogm)=4c:Sincethetotalnumberofwi'sisS,thetotalprobabilitythatsomewidoesnothaveaclauseinMaxset(wi)thatissetto0,isatmost1=4,byourchoiceofparameters. cc10(2001)Reducingthecomplexityofreductions133Onceagain,wewillbeapplyingtheabovelemmarepeatedlyforvariousvaluesofcandm.Atstepi,misequaltomi�m1=5ci1m1=5cd1,andinallcasesccd.Inallapplications,wewillpicktheconstantftobeequaltocd+2.AsinouranalysisofLemma4.4,inallapplicationsSwillbeboundedby2cd+1m2k1.Hence,S2cd+1m2k12cd+1e2klogm1e(cd+2=20cd)logm1=e(flogm1=5cd1)=4cdeflogm=4c:Wenowwanttoobtainagoodusingthemethodofconditionalproba-bilities(Alon&Spencer1992).Wewillobtainbychoosingoneelementatatimetobeset.Thatis,werstchooseoneofthermvariablesandsetitto1or0;equivalentlywechooseoneofthe2rmliteralsandsetitto1.Thenwechooseoneoftheremaining2(rm1)literalsandsetitto1.Theprocessterminatesafterwehavesetatotalofrmrm1=2variables.Thealgorithmforndingproceedsasfollows.First,foreachofthe2rmliteralsl,wecalculatethefollowingquantitiesexactly:Pr[Bijl],Pr[Njjl]andPr[Wkjl],wherePr[Bijl]istheprobabilityofeventBi,overaran-domlychosen,giventhatliterallissetto1.Eachofthesequantitiescanbecalculatedexactlyinpolynomialtime.Wechoosealiteralltobesetto1suchthatthesumPiPr[Bijl]+PjPr[Njjl]+PkPr[Wkjl]isminimized.Bythethreelemmasabove,PiPr[Bi]+PjPr[Nj]+PkPr[Wk]isatmost3=4.Thusitfollowsthatforsomevariablelwedoatleastaswellas3=4.(Therearetwoargumentstoseethatthisfollows:(1)youcanviewthesamplespaceofpossible'saslargerthantheoriginalone,wherethermrm0setvariablesareordered,andthendothefollowingcalculationsrelativetothisenlargedsamplespace.Inthiscase,theconditionslareindependentsowhenwedotheabovesumoverall2rmconditionslwegetexactlythesamenumberastheoriginalunconditionalsum.Or(2)workintheoriginalsamplespaceofpossible's.Inthiscase,theconditionalspacesgivenlarenotindepen-dent,buttheyarecompletelysymmetricsotheaveragingargumentisstillvalid.)Werepeatthisargumentrmrm1=2times,ateachpointconditioninguponthesetHofvariablessetthusfar.Attheend,weareguaranteedtohaveobtainedagoodrestrictionsincewemaintainthattheconditionalprobabilityisalwaysnogreaterthantheoriginalprobabilitywhichislessthan1.ItremainstoshowhowtoexactlycalculatethequantitiesPr[BijH],Pr[NjjH],andPr[WkjH],whereHisacollectionofatmostrmrm1=2variablesthathavebeenset.LetAbeasetofsizea;letHbeasetofhvariablesthathavebeenset;letjA\Hj=d;letrmbetheoriginaluniversesize,andletrm0bethenumber 134Agrawaletal.cc10(2001)of'safterhasbeenapplied.ThentheprobabilitythatAdhasmorethanl's,giventhateveryvariableinHhasalreadybeensetto0or1,isaXi=l+1adi
rmah+drm0i
rmhrm0
:ThisquantityisusedtocalculateexactlyPr[NjjH],andaverysimilarformulacanbeusedtocalculatePr[BijH].CalculatingPr[WkjH]exactlyismorework.Consideraparticularwk,andlets1;:::;st,t=flogm,de-notetheflogmdisjointclausesinMaxset(wk),eachconsistingoftheORofatmostcdisjointliterals.RecallthatWkistheeventthatnoneofthesi'saresetto0by.Inorderforthistohappen,eachsimusthaveatmostjsij1ofitsliteralssetto0,andtheremainingliteralsinsicanbesettoeitheror1.Wecalculatethisquantitystraightforwardlybycon-sideringallpossiblesubsetsxiandyiofsi,wherexiisthesetofatmostjsij1literalsinsisetto0,andyiisthesubsetofremainingliteralsinsisetto1.Whiledoingthecalculation,wehavetokeeptrackofwhichofthesepossibilitiesareactuallynotvalidduetothefactthatHhasalreadybeenset.LetI(x1;y1;:::;xflogm;yflogm;H)beanindicatorrandomvariablethatoutputs1iftheassignmentgivenbysettingallliteralsinthexi'stozero,andsettingallliteralsintheyi'stoone,isconsistentwiththeassign-mentH.Alsoletb=jH\(s1[:::[st)j.WecancomputePr[WkjH]asA=(rmrmrm0
2rmrm0),whereAisgivenbythesumoverallx1;y1;:::;xt;ytofthefollowingquantity,wherethexi'sandyi'ssatisfy:x1s1,jx1jjs1j1,y1s1,x1\y1=;,...,xtst,jxtjjstj1,ytst,xt\yt=;:I(x1;y1;:::;xt;yt;H)rmjs1j


jstjh+brmrm0jx1jjy1j


jxtjjytjh+b2rmrm0jx1jjy1j


jxtjjytjh+b:Sincejsijc,thereareatmost2cvaluesforthevariablesxiandyi.Thusthetotalnumberoftermsintheabovesummationisboundedby2ct=2cflogm,whichispolynomialinm.Toseethattheentirealgorithmispolynomialtime,notethatthenum-berofiterationsoftheabovealgorithmisrmrm0,andeachiterationtakestimepolynomialinm.Furthermore,theentireprocedureforndingispoly-nomialtime,sinceweapplytheabovealgorithmforaconstantnumberofstages,andateachstagethenumberofformulasunderconsiderationisalsopolynomial. cc10(2001)Reducingthecomplexityofreductions1355.ProofofTheorem1.3LetAbeanyAC0-completesetforNP.DenetwosetsbasedonA:E=fx:(x=10yandy2A)or(10isnotaprexofx)g,F=fx:x=yz,andz=y,andy2Ag.(Here,ydenotesthebinarystringthatisthebitwisecomplementofy.)ItisobviousthatbothofthesesetsareAC0-completeforNP.Nowsupposethatthesetwosetsareisomorphictoeachotherunderisomorphismh,com-putedbyadepth-twoAC0circuitfamily.LetfDngbethefamilyofdepth-twoAC0circuitscomputinghthatreducesFtoE.Werstobservethat01E.Therefore,h1(01)F.Sinceh1canblowupthesizeonlypolynomially,thereexistsapolynomialpsuchthatforanynthereisanumbermp(n)suchthattheseth1(01n)\2mcontainsatleast2n=p(n)strings.Choosealargeenoughn,andthecorrespondingmasabove.ConsiderthecircuitD2m.AssumethatD2mhasORgatesatthebottomlevel,andANDgatesatthetop.Observethatif,onaninputx,therst(i.e.,leftmost)outputbitofD2miszero,thenh(x)2E,andthereforex2F,which,inturn,meansthatx=yyforsomestringyoflengthm.Letthisrstoutputbitbedenotedby`.Thesubcircuitcomputing`isanANDofORs.Thus,`canbewrittenas`=c1^


^crwhereeachciisadisjunctionofliterals.Wenowclaimthateachcimustcontainallthe2minputvariables.Supposenot.Letcjbeadisjunctionnotcontainingallthevariables.Setallthevariablesoccurringincjtomakeitevaluatetofalse.Therefore,`=0.Thisimpliesthatx=yyforsomeyasnotedabove.However,sincenotallbitsofxareset,wecanassigntheunsetbitsavaluesoastohavex=2F.Contradiction.Therefore,eachofcimustcontainallthevariables.Now,`wouldbezeroforexactlyroftheinputstringswhererisboundedbyapolynomialinn.However,atleast2n=p(n)stringsmustbemappedbyD2mtostringsbeginningwithazero.Sincenwaschosentobelargeenough,thisisacontradiction.AsimilarargumentcanbegivenforthecasewhenD2misanORofANDsusingthesecondbitoftheoutputofD2m(wheneverthisbitis1,theinputmustbelongtoF).Therefore,thereisnodepth-twoAC0circuitfamilythatcomputesanisomorphismbetweenEandF. 136Agrawaletal.cc10(2001)6.ConclusionsAlthoughTheorem1.1showsthatnotallsetscompleteunderAC0[mod2]re-ductionsareAC0-isomorphic,itisnaturaltowonderiftheyareallAC0[mod2]-isomorphic,orifthereissomeothersortofGapTheoremthatstillawaitsdiscovery.Inthisregard,itisworthnotingthatthesetsconstructedintheproofoftheStopGapTheoremare,infact,allAC0[mod2]-isomorphictoSAT.(Sketchofproof:Thesetsweconstructareallcompleteunderreductionscom-putablebydepth-onecircuitsconsistingentirelyofparitygates.Reductionsofthissortaretrivialtoinvert:Ifthestringyisgiven,andwewanttoseeifthereisanxsuchthatf(x)=y,thentheconditionsonthexiformasystemoflinearequationsintheyj,andinfacteachxiistheparityofsomesubsetoftheyj.Thuswesimplyndwhatthexiwouldhavetobeiftheymaptoy,andthendoafewconsistencychecks.AtthispointthetechniquesofAgrawaletal.(1998)canbeusedtobuildtheisomorphisms.)Itisnotclearhowtoextendthisobservationtohandlesetscompleteunder(PARITYofAND)or(ANDofPARITY)reductions.Weespeciallycallattentiontothefollowingproblems:1.DoestheBerman{HartmanisConjectureholdforAC0[mod2]reductions?Thatis,areallofthesetsthatarecompleteunderAC0[mod2]reductionsisomorphicunderAC0[mod2]isomorphisms?2.Assumingtheexistenceofafunctionthatisone-wayinaverystrongaveragecasesense,isitpossibletoconstructacounter-exampletotheoriginalBerman{HartmanisConjecture?3.IsthereanyclassCsuchthatDlogtime-uniformAC0-completesetsforCareallDlogtime-uniformAC0-isomorphic?Veryrecently,Agrawalhasprovidedaverystrongarmativeanswertoquestion3:EveryclassCclosedunderDlogtime-uniformTC0reductionshasthisproperty.Moreprecisely,Agrawal(2001b)improvesourTheorem1.2toreplacetheP-uniformityconditionbyL-uniformity,andthenthisisimprovedfurtherinAgrawal(2001a)toachieveDlogtime-uniformity.AcknowledgementsWeacknowledgehelpfulconversationswithO.Goldreich,J.Laerty,M.Ogi-hara,D.vanMelkebeek,R.Pruim,M.Saks,D.Sivakumar,WilliamHesse,DavidMixBarrington,andD.Spielman. cc10(2001)Reducingthecomplexityofreductions137ApreliminaryversionofthisworkappearedinProc.29thACMSymposiumonTheoryofComputing(STOC1997).Partoftherstauthor'sresearchwasdonewhilevisitingtheUniversityofUlmunderanAlexandervonHumboldtFellowship.TheresearchofthesecondauthorwassupportedinpartbyNSFgrantsCCR-9509603,CCR-9734918,andCCR-0104823.TheresearchofthethirdauthorwassupportedbyNSFAwardsCCR-92-570979andCCR-0098197,bySloanResearchFellowshipBR-3311,andUSA-IsraelBSFGrant97-00188.TheresearchofthethirdauthorwassupportedbyNSFAwardCCR-94-57782,andUSA-IsraelBSFGrant95-00238.ReferencesM.Agrawal(2001a).Therst-orderisomorphismtheorem.InProc.21stFoundationsofSoftwareTechnologyandTheoreticalComputerScienceConference(FST&TCS),LectureNotesinComput.Sci.,Springer,toappear.M.Agrawal(2001b).TowardsuniformAC0-isomorphisms.InProc.16thIEEEConferenceonComputationalComplexity,13{20.M.Agrawal,E.Allender&S.Rudich(1998).Reductionsincircuitcomplexity:Anisomorphismtheoremandagaptheorem.J.Comput.SystemSci.57,127{143.M.Ajtai(1983).11formulaeonnitestructures.Ann.PureAppl.Logic24,1{48.E.Allender(1989).P-uniformcircuitcomplexity.J.Assoc.Comput.Mach.36,912{928.E.Allender&V.Gore(1991).Rudimentaryreductionsrevisited.Inform.Pro-cess.Lett.40,89{95.N.Alon&J.Spencer(1992).TheProbabilisticMethod.Wiley.J.Balcazar,J.Daz&J.Gabarro(1995,1990).StructuralComplexityTheoryIandII.Springer.D.A.M.Barrington,N.Immerman&H.Straubing(1990).OnuniformitywithinNC1.J.Comput.SystemSci.41,274{306.L.Berman&J.Hartmanis(1977).OnisomorphismanddensityofNPandothercompletesets.SIAMJ.Comput.6,305{322.W.Eberly(1989).Veryfastparallelpolynomialarithmetic.SIAMJ.Comput.18,955{976. 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