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Sublogarithmic uniform Boolean proof nets ∗ ClémentAubert LIPN–UMR7030,CNRS–UniversitéParis13, 99av.J.-B.Clément,93430Villetaneuse,France AbstractUsingaproofs-as-programscorrespondence,Teruiwasabletocomparetwomodelsof parallelcomputation:Booleancircuitsandproofnetsformultiplicativelinearlogic.Mogbilet.al. gavealogspacetranslationallowingustocomparetheircomputationalpowerasuniformcom- plexityclasses.ThispaperpresentsanoveltranslationinAC0andfocusesonasimplerrestricted notionofuniformBooleanproofnets.Wecanthenencodeconstant-depthcircuitsandcompare complexityclassesbelowlogspace,whichwereoutofreachwiththeprevioustranslations. Introduction Boolean proof nets were introduced by Terui in [8] to study the implicit complexity of proofs nets for Multiplicative Linear Logic [4] comparatively to Boolean circuits. Those two models of paral- lelcomputationweresuccessfullylinkedusingaproofs-as-programsframework,whichmatchesup cut-eliminationinproofnetswithevaluationincircuits. Surprisingly[8]doesnottakeintoaccount uniformity, which guaranteesthattheresources neededtobuilda Boolean circuit isinferiortothe computationalpoweritwilldeliver. [7]and[6]studiedtheBooleanproofnetsinauniformwayand introducedsomenon-determinisminit.AstheirtranslationfromBooleancircuitfamiliestoBoolean proofnetfamiliesisinlogspace(L),itremainedunknowniftheresultswerestillvalidwhenapplied tosublogarithmicclassesofcomplexity,thatistosay AC0andNC1. ByrestrictingtheBooleanproof netsweuse,thispaperoffersanewproofofthecorrespondencebetweencircuitsandproofnetsand extendsittoconstant-depthcircuits. Boolean circuits ([9], section1) and proof nets ([3], section3) are canonical models of parallel computation,butthelatterwasmostlyseenfromtheviewpointofsequentialimplicitcomplexity.To evaluate a proof net is to eliminate its cuts, but to do so with suitable bounds we need to define a parallelelimination. Tryingtoapplythetwousualrulesofrewriting(→ and→ )inparallelleads m a tocriticalpairs,soweareforcedtodefineanewkindofcut(tightening-cut)andanewrewritingrule (→ ).ThesimulationofthisreductionrulebyBooleancircuitsneedsUstConn gatestobemadewith t 2 efficiency. TheproofnetswestudyareforMultiplicativeLinearLogicwithunboundedarity(MLLu,section2) andbecauseofthelinearityofthislogic,weareforcedtokeeptrackofthepartialresultsgenerated by the evaluation that are unused in the result. Boolean proof nets (section4) – as introduced by Terui–haveanexpensivewayofmanipulatingthisgarbage.Inthispaperweintroduceproofcircuits (section5)asarefinementoftheBooleanproofnetsthataresimplertomanipulate. Theyaremade ofpieceswhichtranslategatesandcomposeeasily,sothatwereducethesizeoftheproofnettranslat- ingtheBooleancircuits. Insection6weconcludeourpaperwithourmainresult(theorem2): there existsaconstant-depthreductionfromBooleancircuitfamiliestoproofcircuitfamilies. Soournew frameworkoffersavariantoftheproofsforcomplexityresultsandextendsthemtosmallclassesof complexity,inauniformway. ∗WorkpartiallysupportedbytheFrenchprojectComplice(ANR-08-BLANC-0211-01). Jean-YvesMarion(Ed.):SecondWorkshopon ThisworkislicensedundertheCreativeCommons DevelopmentsinImplicitComputationalComplexity(DICE2011) Attribution-Noncommercial-ShareAlikeLicense. EPTCS75,2012,pp.15–27,doi:10.4204/EPTCS.75.2 16 SublogarithmicuniformBooleanproofnets 1 Booleancircuits Boolean circuits (definition2) are of great interest in the study of complexity, for instance because oftheefficiencyoftheirparallelevaluation. Oneoftheirfeaturesisthattheyworkonlyoninputsof fixedlength,andthatforcesustodealwithfamiliesofBooleancircuits–andtherearisesthequestion ofuniformity(definition4). Definition1(Booleanfunction). An-aryBooleanfunctionfnisamapfrom{0,1}n to{0,1}.ABoolean functionfamilyisasequence f =(fn)n∈NandabasisisasetofBooleanfunctionsandBooleanfunc- tionfamilies.Weset: B0={¬,∨2,∧2}andB1={¬,(∨n)nÊ2,(∧n)nÊ2} TheBooleanfunctionUstConn ,givenininputthecodingofanundirectedgraphGofdegreeatmost 2 2andtwonamesofgatessandt,outputs1iffthereisapathbetweensandt inG. B B Definition2(Booleancircuits). Givenabasis ,aBooleancircuitover withninputsC isadirected acyclic finite and labeled graph. The nodes of fan-in 0 are called input nodes andare labeled with x ,...,x ,0,1.Non-inputnodesarecalledgatesandeachoneofthemislabeledwithaBooleanfunc- 1 n B tion from whose aritycoincides with the fan-in of the gate. There is a unique node of fan-out0 whichistheoutputgate.Weindicatewithasubscriptthenumberofinputs:aBooleancircuitC with ninputswillbenamedC . n ThedepthofaBooleancircuitC d(C )isthelengthofthelongestpathbetweenaninputnode n n andtheoutputgate.Itssize|C |isitsnumberofnodes.WewillonlyconsiderBooleancircuitsofsize n nO(1),thatistosaypolynomialinthesizeoftheirinput. C accepts a word w ≡ w ...w ∈ {0,1}n if C evaluates to 1 when w ,...,w are respectively n 1 n n 1 n assigned to x1,...,xn. A family of Boolean circuits is an infinite sequence C = (Cn)n∈N of Boolean circuits,C acceptsalanguageX ⊆{0,1}∗iffforallw∈X,C|w|acceptsw. WenowrecallthedefinitionoftheDirectConnectionLanguageofafamilyofBooleancircuits,an infinitesequenceoftuplesthatdescribesittotally. Definition3(DirectConnectionLanguage[9]). Given(.)asuitablecodingofintegersandC =(Cn)n∈N B afamilyofBooleancircuitsoverabasis ,itsDirectConnectionLanguage–writtenL (C)–isthe DC setoftuples<y,g,p,b>,suchthatfor|y|=n,wehave: g isagateinC ,labeledwithb∈Bifp=ǫ, n elsebisitspth predecessor. Definition4(Uniformity[1]). AfamilyC issaidtobeDLOGTIME-uniformifthereexistsadetermin- isticTuringMachinewithrandomaccesstotheinputtapethatgivenL (C),nandg outputsintime DC O(log(|C |))anyinformation(position,labelorpredecessors)aboutthegateg inC . n n Despite thefact thata DLOGTIME TuringMachinehasmore computationalpower thanacon- stant-depthcircuit, “a consensushasdevelopedamongresearchersincircuitcomplexitythatthisD- LOGTIMEuniformityisthe‘right’uniformitycondition”forsmallcomplexityclasses[5]. Anyfurther referencetouniformityistobereadasDLOGTIMEuniformity. Definition 5 (ACi, NCi). For all i ∈N, given B a basis, a language X ⊆{0,1}∗ belongs to the class ACi(B)(resp. NCi(B))if X isacceptedbyauniformfamilyofpolynomial-size, logi-depthBoolean circuitsoverB ∪B(resp.B ∪B).Weset ACi(;)=ACi andNCi(;)=NCi. 1 0 ClémentAubert 17 2 MLLu RatherthanusingMultiplicativeLinearLogic(MLL)–whichwouldforceustocomposebinarycon- nectives toobtainsn-aryconnectives – we work with MLLuwhich differsonly on thearitiesof the →− ←− connectivesbutbetterrelatestocircuits. Wewrite A (resp. A)foranorderedsequenceofformulae A ,...,A ,(resp. A ,...,A ). 1 n n 1 Definition6(FormulaeofMLLu). GivenαaliteralandnÊ2,formulaeofMLLuare: →− ←− A::=α|α⊥| ⊗n(A)| Mn(A) DualityisdefinedwithrespecttoDeMorgan’slaw: (A⊥)⊥ ≡ A →− ←− (⊗n(A))⊥ ≡ Mn(A⊥) ←− −→ (Mn(A))⊥ ≡ ⊗n(A⊥) Asfortherestofthisarticle,considerthatA,BandDwillrefertoMLLuformulae. A[B/D]denotes AwhereeveryoccurrenceofB isreplacedbyanoccurrenceofD.WewriteA[D]ifB=α. Definition7(SequentcalculusforMLLu). AsequentofMLLuisoftheform⊢Γ,whereΓisamultiset offormulae.TheinferencerulesofMLLuareasfollow: ⊢A,A⊥ ax. ⊢Γ1,⊢A1Γ ,...,.Γ.. ,⊗n(⊢→−AΓ)n,An ⊗n 1 n ←− ⊢Γ,A ⊢∆,A⊥ ⊢Γ,A ⊢Γ,∆ cut ⊢Γ,Mn(←A−) Mn DerivationsofMLLuarebuiltwithrespecttothoserules. MLLuhasneitherweakeningnorcontrac- tion, butadmitsimplicit exchangeandeliminatescuts. Theformulae A and A⊥ in therulecut are calledthecutformulae. 3 Proofnets ProofnetsareaparallelsyntaxforMLLuthatabstractawayeverythingirrelevantandonlykeepthe structureoftheproofs. Weintroducemeasures(definition10)ontheminordertostudytheirstruc- tureandcomplexity,andaparalleleliminationoftheircuts(definition11). Definition8(Links). Weintroduceinfigure1threesortsoflinks–•,⊗n andMn –whichcorrespond toMLLurules. Everylinkmayhavetwokindsofports:principalones,indexedby0andwrittenbelow,andaux- iliaryones, indexedby1,...,n andwrittenabove. Theauxiliaryportsareordered,butaswealways representthelinksasinfigure1,wemaysafelyomitthenumbering. Axiomlinkshavetwoprincipal ports,bothindexedwith0,butwecanalwaysdifferentiatethem(forinstancebynamingthem0 and r 0 )ifweneedto. l Remark1. Thereisnosortcut:acutisrepresentedwithanedgebetweentwoprincipalports. 18 SublogarithmicuniformBooleanproofnets ...... ...... • 1 n n 1 M 0 0 N 00 Figure1: ax-link,⊗n-linkandMn-link Definition9(Decoratedderivationsandproofnet). GivenaderivationofMLLu,wedecorateitinthe followingway: – anindexisassociatedtoeveryformula. Theformulaeintroducedbyanaxiomhavethesame properindex,andtheformulaeintroducedbyalogicalrules(⊗n andMn)haveafreshindex, – adescriptionisassociatedtoeverysequent. Therulesgiveninfigure2indicatehowaproofisdecoratedandhowtobuildaproofnet froma description. ThetypeofaproofnetPisΓifthereexistsadecoratedderivationof⊢Γ⊲D(P):aproofnetalways hasseveraltypes,butuptoα-equivalence (renamingoftheliterals)wemayalwaysassumeithasa uniqueprincipaltype.IfaproofnetmaybetypedwithΓ,thenforeveryAitmaybetypedwithΓ[A]. Byextensionwewillusethenotionoftypeofanedge. Thestructuresobtainedbyfollowingthoserulesrespectcriterionofcorrectness.Forinstancetwo portsofasamelinkmaynotbeconnected,aportmaybeconnectedonlyonceandeveryauxiliary portisconnected.Wedonothavetotakeintoaccountpseudonets. Remark2. Thesameproofnet–asitabstractsderivations–maybeinducedbyseveraldescriptions. Conversely,severalgraphs–asrepresentationsofproofnets–maycorrespondtothesameproofnet: wegetroundofthisdifficultybyassociatingtoeveryproofnetasingledrawingamongthedrawings withtheminimalnumberofcrossings betweenedges, forthesakeofsimplicity. Twographsrepre- sentingproofnetsthatcanbeobtainedfromthesamedescriptionaretakentobeequal. Definition10(Sizeanddepthofaproofnet). Thesize|P|ofaproofnetP isthenumberofitslinks. Thedepthofaproofnetisdefinedwithrespecttoitstype: – Thedepthofaformulaisdefinedbyrecurrence: d(α)=d(α⊥)=1 →− ←− d(⊗n(A))=d(Mn(A))=1+max(d(A ),...,d(A )) 1 n – Thedepthd(π)ofaderivationπisthemaximumdepthofcutformulaeinit. – Thedepthd(P)ofaproofnetP is min{d(π)|πcanbedecoratedas ⊢Γ⊲D(P)forsomeΓ} The depth d(P) of a proof net depends on its type, but it is minimal when we consider the principaltypeoftheproofnet. To make themost of thecomputational power of proof nets, we needtoachieve a speed-up in thenumberofstepsneededtonormalizethem. Ifwetryroughlytoreduceinparallelacutbetween twoax-links,wearefacedwithacriticalpair.[8]avoidsthissituationbyusingatighteningreduction whicheliminatesinonestepallthecutsbetweenaxioms.Wecanthensafelyreducealltheothercuts inparallel. ClémentAubert 19 p • p ax. ⊢p:A,p:A⊥⊲ax p P ...... P 1 n {⊢Γ→−i,pi :Ai⊲D(Pi)}1ÉiÉn ⊗n p1 pn ⊢Γ1,...,Γn,s:⊗n(A)⊲tensorsp1,...,pn(D(P1),...,D(Pn)) ...... Ns ⊢Γ,p:A⊲D(P) ⊢∆,q:A⊥⊲D(Q) P Q cut p q ⊢Γ,∆⊲cutp,q(D(P),D(Q)) P p ...... p n 1 ⊢Γ,p :A ,...,p :A ⊲D(P) n ←−n 1 1 Mn ⊢Γ,s:Mn(A)⊲parpn,...,p1(D(P)) ...... s M s EdgesrepresentingΓor∆arenotdrawn,D(P)isoneofthedescriptionoftheproofnetP. Figure2:Fromdecoratedderivationstoproofnets Definition11(Cutsandparallelcut-elimination). Acutisanedgebetweentheprincipalportsoftwo links.Ifoneoftheselinksisanax-link,twocasesoccurs: iftheotherlink is an ax-link,we takethemaximalchain of ax-linksconnectedbytheirprincipal portsanddefinesthissetofcutsasat-cut, otherwisethecutisana-cut. Otherwiseitisam-cutandweknowthatfornÊ2,onelinkisa⊗n-linkandtheotherisaMn-link. Wedefineonfigure3threerewritingrulesontheproofnets.Forr ∈{t,a,m},ifQmaybeobtained fromP byerasingallther-cutsofP inparallel,wewriteP â Q. IfP â Q,P â Q orP â Q,we r t a m write P âQ. To normalize a proof net P is to apply â until we reach a cut-free proof net. â∗ is definedasthetransitivereflexiveclosureofâ. Theorem1(Parallelcut-elimination[8]). EveryproofnetPnormalizesinatmostO(d(P))applications ofâ. So thetime neededtoevaluate a proof netis relative toitsdepth, andcanbe in the worst case linearinitssize–asfortheBooleancircuits. 20 SublogarithmicuniformBooleanproofnets Forall◦∈{(Mn)nÊ2,(⊗n)nÊ2},◦maybe•in→m. • • • • → ... t • → a ...... ...... ...... ...... M → m N Figure3: t-,a-andm-reductions 4 Booleanproofnets In order to compare the complexities of proof netsandof Boolean circuits, we needto define how proof netsrepresentBoolean values(definition12)andBoolean functions(definition13). Tostudy theminauniformframeworkwedefinetheirDirectConnectionLanguage(definition14). Definition12(Booleantype,0and1[8]). Letb andb bethetwoproofnetsoftype 0 1 B=M3(α⊥,α⊥,α⊗α) respectivelyusedtorepresentfalseandtrue: • • D(b0)=parsq,p,r(tensorrp,q(axp,axq)) b0≡ M N • • D(b1)=parsp,q,r(tensorrp,q(axp,axq)) b1≡ M N →− Wewrite b forb ,...,b fori ∈{0,1}. i1 in As we can see, b and b differ on their planarity: descriptions and proof nets exhibit the ex- 0 1 changesthatwerekeptimplicitinderivations. →− Definition13(Booleanproofnets[8]). ABooleanproofnetwithninputsisaproofnetP(p)oftype ⊢p :B⊥[A ],...,p :B⊥[A ],s:⊗1+m(B[A],D ,...,D ) 1 1 n n 1 m →− →− Given b oflengthn,P(b)isobtainedbyconnectingwithcutsp tob forall1≤j ≤n. j ij ClémentAubert 21 →− P(b)â∗Q whereQ isunique,cut-freeandforsomedescriptionsQ ,...,Q describedby 1 n tensor(D(b ),Q ,...,Q )fori ∈{0,1}. i 1 m →− WewriteP(b)→ b . ev. i →− P(p)representsaBooleanfunction fn ifforallw≡i ...i ∈{0,1}n,P(b ,...,b )→ b . 1 n i1 in ev. f(w) WemayeasilydefinefamiliesofBooleanproofnetsandlanguageacceptedbyafamilyofBoolean proofnets. Thetensorindexedwithsinthetypeistheresulttensor:itcollectstheresultofthecomputation onitsfirstauxiliaryportandthegarbage–herenamedD ,...,D –onitsotherauxiliaryports. 1 m Definition14(DirectConnectionLanguageforproofnets[7]). GivenP=(Pn)n∈NafamilyofBoolean proofnets,itsDirectConnectionLanguage–writtenL (P)–isthesetoftuples<y,g,p,b>where DC for|y|=n:g isalinkinP ,ofsortbifp=ǫelsethepth premiseofg isthelinkb. n If<y,p,0,b>or<y,b,0,p>belongtoLDC(P),thereisacutbetweenbandpinC|y|. 5 Proofcircuits A proof circuit is a Boolean proof net (fact1) made out of pieces (definition15) which represents Boolean functions, constants or duplicates values. Garbage is manipulated in an innovative way, examples1shouldhelptounderstandthemechanismofcomputation. Fromnowoneveryedgerepresentedby isconnectedonitsrighttoanauxiliaryportnum- beredwithanintegerotherthan1oftheresulttensor:itcarriesapieceofgarbage. Definition15(Pieces). Wepresentinthetable1thesetofpiecesatourdisposal. Entriesarelabeled withe,exitswith s andgarbagewith g. Edgeslabeledb –fork ∈{0,1}–areconnectedtotheedge k labeledsofthepieceb . k Table1:Pieces Weset2Éj Éi. • • • • b ≡ N b ≡ N 0 M 1 M s s • b b g 0 1 • • • • • s DUPL1≡ NEG≡ M M e N e N s 22 SublogarithmicuniformBooleanproofnets Table1:Pieces–continuedfrompreviouspage i t im e s i t im e s b b b b g z 0 }| {0 z1 }| 1{ ...... ...... • s • 1 • s . 2 DUPLi ≡ .. N N • si−1 • s • • i M M e N Ifi =2,theedgeaistheedges. g e 1 1 • b 1 • • • M a gj−1 e2 N b •  1 DISJi ≡ i−3times • M • gi−1 b • 1 e j N  • M • s  ei N g e 1 1 b • 0 • • • gj−1 M a e N 2 • b  0 i−3times • M • gi−1 CONJi ≡ N • ei b0 • • s M N e j ClémentAubert 23 ApiecePwithi Ê0entries,j Ê1exitsandkÊ0garbageisoneofthepieceinthetable1,wherei edgesarelabeledwithe ,...,e , j edgesarelabeledwiths ,...,s andkedgesgototheresulttensor. 1 i 1 j WehaveP∈{b0,b1,NEG,{DUPLi}iÊ1,{DISJi}iÊ2,{CONJi}iÊ2}. TocomposetwopiecesP andP weconnectanexitofP toanentryofP .Itisnotallowedto 1 2 1 2 loop:wecannotconnectanentryandanexitbelongingtothesamepiece. Anentry(resp. anexit)thatisnotconnectedtoanexit(resp. anentry)ofanotherpieceissaidto beunconnected. →− Definition 16 (Proof circuits). A proof circuit C (p) with n inputs and one output is obtained by n composingpiecessuchthatnentriesandoneexitareunconnected. Ifnogarbageiscreatedweadd aDUPL1-piececonnectedtotheunconnectedexittoproducesomeartificially.Thenweaddaresult tensorwhosefirstedgeisconnectedtotheexit–whichisalsotheoutputoftheproofcircuit–andthe otherstothegarbage. Wethenlabeleveryunconnectedentrieswithp ,...,p : thosearetheinputs 1 n oftheproofcircuit. →− →− Given b oflengthn,C (b)isobtainedbyconnectingwithcutsp tob forall1≤j ≤n. n j ij Fact1. EveryproofcircuitisaBooleanproofnet. Proof. Weprovethisfactwithacontractibilitycriterion[2],byinductionontheheightofthepieces oftheproofcircuit(countedasthenumberofpiecesfromtheconsideredpiecetotheresulttensor). Asaproofcircuitcanalwaysbetypedwith ⊢B⊥[A ],...,B⊥[A ],⊗1+m(B[A],D ,...,D ) 1 n 1 m itisaBooleanproofnet. This fact establishes that proof circuits normalize and output a value, and that it is possible to representBooleanfunctionswiththem. Familiesofproofcircuitsandacceptationofalanguagebyafamilyofproofcircuitsaredefinedas usual. Examples1. Wepresentbrieflytwoexamplesofcomputation: thenormalizationofapieceb con- 0 nectedtoaNEG-piece(figure4,page24)andhowtheconditionalworks(figure5,page25). Condi- tionalis thecore of thecomputation, we findthispatternin everypiece except NEG andthecon- stants. Remark 3. TocomposetwoproofcircuitsC andC ,we remove theresult tensorofC ,identifythe 1 2 1 unconnected exit of C with the selected input of C , and recollect all the garbage with the result 1 2 tensorofC .Wethenlabeltheunconnectedentriesanewandobtainaproofcircuit. 2 Definition 17 (PCCi (resp. mBNi, [7])). A language X ⊆ {0,1}∗ belongs to the class PCCi (resp. mBNi) if X is accepted by a polynomial-size, logi-depth uniformfamily of proof circuits (resp. of Booleanproofnets). If we add "UstConn -pieces" to the set of pieces, we may easily define PCCi(UstConn ) and 2 2 remarkthatforalli ∈N,PCCi ⊆PCCi(UstConn ). [8]provesthatthereexistsBooleanproofnetsof 2 constant-depthandpolynomialsizethatrepresentUstConn ,sowedonotdefinemBNi(UstConn ) 2 2 becauseitwouldbeeasytoprovethatthisclassisequaltomBNi foralli ∈N. Remark4. Byfact1wehavetriviallythatforalli ∈N,PCCi ⊆mBNi. →− →− →− Lemma1. ForallproofcircuitC (p)andall b, thecutsatmaximumdepthinC (b)arebetween n n theentryofapieceandavalue(aconstantb orb ,oraninputb forsome1≤j ≤n). 0 1 ij 24 SublogarithmicuniformBooleanproofnets • • • • • • • • → • • m N N M M M N (cid:21) a • • • • ≡ N N M M Figure4:b connectedtoNEGnormalizestob 0 1 →− Proof. ForeverypieceP ofC (b)anycutconnectinganentryisalwaysofdepthsuperiororequal n tothemaximaldepthofthecutsconnectingtheexits.ThecutsofP thatdonotconnectanentryor anexitofapiecearealwaysofdepthinferiororequaltocutsconnectingtheentries. Thedepthsofthecutformulaeslowlyincreasefromtheexittotheentry,andastheentriesthat arenotconnectedtootherpiecesareconnectedtovalues,thislemmaisproved. 6 Results ByusingourproofcircuitsweproveanewtheinclusionsbetweenACi andlogicalclassesofcomplex- ityandextendthisinclusiontosublogarithmicclassesofcomplexity. Definition18(Problem:TranslationfromACi toPCCi). Input: L (C)forC afamilyofBooleancircuitsinACi. DC Output: L (C)forC afamilyofproof circuitsin PCCi, such thatfor DC →− →− alln∈N,forall b ≡b ,...,b , C (b)→ b iffC (i ,...,i ) i1 in n ev. j n 1 n evaluatesto j. Theorem2. Foralli ∈N,translationfrom ACi toPCCi belongstoAC0. Proof. ThetranslationfromC toC isobvious,itreliesoncoding: foreveryn,afirstconstant-depth circuit associatetoeverygateofC thecorrespondingpiecesimulatingitsBoolean function. Ifthe n fan-outofthisgateisk >1,aDUPLk-pieceisassociatedtotheexitofthepiece,andthepiecesare connectedasthegates.TheinputnodesareassociatedtotheinputsofC .Asecondconstant-depth n circuit recollects the only free exit and the garbage of the pieces and connects them to the result tensor. The composition of these two Boolean circuits produces a constant-depth Boolean circuit thatbuildsproofcircuits. It is easy to check thatCONJk, DISJk and NEG represent ∧k, ∨k and ¬ respectively. DUPLk duplicates a value k times, b and b represent 0 and 1 by convention. The composition of these 0 1 piecesdoesnotraiseanytrouble:C effectivelysimulatesC oneveryinputofsizen. n n