Table Of ContentSublogarithmic 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
