Theoretical Computer Science Focusing and polarization in linear, intuitionistic, and classical logics

22 Pages·2009·1.01 MB·English

TheoreticalComputerScience410(2009)4747–4768
ContentslistsavailableatScienceDirect
Theoretical Computer Science
journalhomepage:www.elsevier.com/locate/tcs
Focusing and polarization in linear, intuitionistic, and classical logics
ChuckLianga,DaleMillerb,∗
aDepartmentofComputerScience,HofstraUniversity,Hempstead,NY11550,USA
bINRIASaclay–Île-de-FranceandLIX/EcolePolytechnique,91128Palaiseau,France
a r t i c l e i n f o a b s t r a c t
Keywords: Afocusedproofsystemprovidesanormalformtocut-freeproofsinwhichtheapplication
Focusedproofsystem of invertible and non-invertible inference rules is structured. Within linear logic, the
Sequentcalculus
focusedproofsystemofAndreoliprovidesanelegantandcomprehensivenormalform
Intuitionisticlogic
forcut-freeproofs.Withinintuitionisticandclassicallogics,therearevariousdifferent
Classicallogic
proofsystemsintheliteraturethatexhibitfocusingbehavior.Thesefocusedproofsystems
have been applied to both the proof search and the proof normalization approaches to
computation.Wepresentanew,focusedproofsystemforintuitionisticlogic,calledLJF,
andshowhowotherintuitionisticproofsystemscanbemappedintothenewsystemby
insertinglogicalconnectivesthatprematurelystopfocusing.WealsouseLJF todesigna
focusedproofsystemLKF forclassicallogic.Ourapproachtothedesignandanalysisof
thesesystemsisbasedonthecompletenessoffocusinginlinearlogicandonthenotionof
polaritythatappearsinGirard’sLCandLUproofsystems.
©2009ElsevierB.V.Allrightsreserved.
1. Introduction
Cut-eliminationprovidesanimportantnormalformforsequentcalculusproofs.Butwhatnormalformscanweuncover
aboutthestructureofcut-freeproofs?Sincecut-freeproofsplayimportantrolesinthefoundationsofcomputation,such
normalformsmightfindarangeofapplicationsintheproofnormalizationfoundationsforfunctionalprogrammingorin
theproofsearchfoundationsoflogicprogramming.
1.1. Aboutfocusing
Andreoli’s focusing proof system for linear logic (the triadic proof system of [1]) provides a normal form for cut-free
proofs in linear logic. Although we describe this system, here called LLF, in more detail in Section 2, we highlight two
aspect of focusing proofs here. First, linear logic connectives can be divided into the asynchronous connectives, whose
right-introductionrulesareinvertible,andthesynchronousconnectives,whoserightintroductionrulesarenot(generally)
invertible. The search for a focused proof can capitalize on this classification by applying (reading inference rules from
conclusion to premise) all invertible rules in any order (without the need for backtracking) and by applying a chain of
invertiblerulesthatfocusesonagivenformulaanditspositivesubformulas.Suchachainofapplications,usuallycalled
a focus, terminates when it reaches an asynchronous formula. Proof search can then alternate between applications of
asynchronousintroductionrulesandchainsofsynchronousintroductionrules.
Asecondaspectoffocusingproofsisthatthesynchronous/asynchronousclassificationofnon-atomicformulasmustbe
extendedtoatomicformulas.Thearbitraryassignmentofpositive(synchronous)andnegative(asynchronous)biastoatomic
∗
Correspondingauthor.
E-mailaddresses:[email protected](C.Liang),[email protected](D.Miller).
0304-3975/$–seefrontmatter©2009ElsevierB.V.Allrightsreserved.
doi:10.1016/j.tcs.2009.07.041
4748 C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768
formulascanhaveamajorimpacton,nottheexistenceoffocusedproofs,buttheshapeoffocusedproofs.Forexample,
considertheHornclausespecificationoftheFibonacciseries:
fib(0,0)∧fib(1,1)∧∀n∀f∀f(cid:48)[fib(n,f)⊃fib(n+1,f(cid:48))⊃fib(n+2,f +f(cid:48))].
( , )
Ifallatomicformulasaregivenanegativebias,thenthereexistsonlyonefocusedproofoffib n f :thisonecanbeclassified
n
asa‘‘backwardchaining’’proofanditssizeisexponentialinn.Ontheotherhand,ifallatomicformulasaregivenapositive
bias,thenthereisaninfinitenumberoffocusedproofsallofwhichareclassifiedas‘‘forwardchaining’’proofs:thesmallest
suchproofisofsizelinearinn.
1.2. Results
Thecontributionsofthispaperarethefollowing.First,weintroduceinSection5anewfocusingproofsystemLJFand
showthatitissoundandcompleteforintuitionisticlogic.NotablefeaturesofLJFarethatitallowsforatomsofdifferentbias
anditcontainstwoversionsofconjunction:whiletheseconjunctionsarelogicallyequivalent,theyareaffectedbyfocusing
differently.Furthermore,inSection6,weshowthatLJFsatisfiescut-elimination.Second,inSection7,weshowhowseveral
otherfocusingproofsystemscanbecapturedinLJF,inthesenseoffullcompleteness(one-to-onecorrespondencebetween
proofsindifferentsystems).Oneshouldnotethatwhiletherearemanyfocusingproofsystemsforintuitionisticlogicin
theliterature,LJFappearstobethefirsttoprovideasingle(intuitionistic)frameworkforcapturingmanyofthem.Third,in
Section8,weuseLJFtoderiveLKF,afocusingsystemforclassicallogic.
1.3. Methodologyandrelatedwork
There is a number of sequent calculus proof systems known to be complete for intuitionistic logic that exhibit
characteristics of focusing. Some of these proof systems fix globally on either forward chaining or backward chaining.
The early work on uniform proofs [2] and the LJT proof system [3] are both backward chaining calculi (all atoms have
negativebias)whiletheLJQ calculus[3,4]selectstheglobalpreferencetobeforwardchaining(allatomshavepositive
bias). The Ph.D. theses of Howe [5] and Chaudhuri [6] also explored various focusing proof systems for both linear and
λ
intuitionistic logics. Less has been published about systems that allow for mixing bias on atoms. While the RCC proof
systemofJagadeesan,Nadathur,andSaraswat[7]isnotafocusingsystemexplicitly,itdoesallowfortwopolaritiesof
atoms(agentsandconstraintsarepositiveandgoalsarenegative)andformixingbothforwardchainingandbackward
chaininginasupersetofthehereditaryHarropfragmentofintuitionisticlogic:forwardchainingisusedtomodelconstraint
propagationandbackwardchainingisusedtomodelgoal-directedsearch.
Weareinterestedinprovidingaflexibleandunifyingframeworkthatcancollecttogetherimportantaspectsofmanyof
theseproofsystems.Thereareseveralwaystomotivateandvalidatethedesignofsuchasystem.Oneapproachstaysentirely
withinintuitionisticlogicandworksdirectlywithinvertibilityandpermutabilityofinferencerules.Suchanapproachhas
been taken in many papers, such as [2,8,4]. Our approach uses linear logic, with its exponential operators ! and ?, as a
unifyingframeworkforlookingatintuitionistic(andclassical)logic.ThefactthatAndreoli’sfocusedsystemwasdefined
for full linear logic provides us with a convenient platform for exploring the issues around focusing and polarity. We
translateintuitionisticlogicintolinearlogic,thenshowthatproofsystemsforintuitionisticlogicmatchfocusedproofsof
thetranslatedimage(Section3).Acrucialaspectofunderstandingfocusinginintuitionisticlogicisprovidedbyidentifying
thepreciserelationshipbetweenAndreoli’snotionofpolaritywithGirard’snotionofpolarityfoundintheLC[9]andLU[10]
systems(seeSection4).
Another system concerning polarity and focusing is found in the work of Danos, Joinet, and Schellinx [11,12]. Many
techniquesthattheydeveloped,suchasinductivedecorations,areusedthroughoutouranalysis.Ourworkdivergesfrom
η
theirsintheadaptationofAndreoli’ssystem(LLF)asourmaininstrumentofconstruction.TheLK systemof[12]describes
p
focusedproofsforclassicallogic.ItsconnectionstopolarizationandfocusingwerefurtherexploredandextendedbyLaurent,
Quatrini,anddeFalco[13]usingpolarizedproofnets.Itmaybetemptingtospeculatethatthebestwaytoarriveatanotion
ofintuitionisticfocusingisbysimplemodificationstothesesystems,suchasrestrictingthemtosingle-conclusionsequents.
Closerexamination,however,revealsintricateissuesconcerningthisapproach.Forexample,thenotionofclassicalpolarity
appearstobedistinctfromandcontrarytointuitionisticpolarity,especiallyatthelevelofatoms(seeSections4and8).
Resolvingthisissuewouldbecentraltofindingsystemsthatsupportcombinedforwardandbackwardchaining.Although
η
therelationshipbetweenLK andoursystemsisinteresting,wechoseforthisworktoderiveintuitionisticfocusingfrom
p
focusinginlinearlogicasopposedtoclassicallogic.InSection8,weshowhowaclassicalfocusingcalculuscanthenbe
derivedfromthenewintuitionisticsystem.
Focusingproofsystemshavebeenappliedinanumberofsettings.Theearliestworkonfocusinginlinearlogicwas
motivated,inpart,bylogicprogramming:Andreoli’soriginalfocusingpaper[1]wasusedtojustifythedesignoftheLOlogic
programminglanguage[14].ThespecializedproofsystemsfortheLolli[15]andForum[16]logicprogramminglanguages
werejustifiedbythecompletenessofuniformproofs,whichcanbeseenasaparticularkindsoffocusedproofsinwhich
allatomsaregivennegativepolarity.Chaudhuri,Pfenning,andPricein[17]restrictAndreoli’sfocusingproofsystemfor
fulllinearlogictotheintuitionisticfragmentandthenshowthatadoptingdifferentglobalbiasassignmentforatomsleads
C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768 4749
Asynchronousphase
(cid:96)(cid:96)ΨΨ:∆:∆⇑⇑⊥L,L [⊥] (cid:96)(cid:96)ΨΨ:∆:∆⇑⇑FF.........................,......................................................................GG,L,L [...............................................................................................] (cid:96)(cid:96)ΨΨ:,∆F:⇑∆?⇑F,LL [?]
(cid:96)Ψ:∆⇑F,L (cid:96)Ψ:∆⇑G,L (cid:96)Ψ:∆⇑B[y/x],L
[(cid:62)] [&] [∀]
(cid:96)Ψ:∆⇑(cid:62),L (cid:96)Ψ:∆⇑F &G,L (cid:96)Ψ:∆⇑∀x.B,L
(cid:96)Ψ:∆,F ⇑L
[R⇑] providedthatF isnotasynchronous
(cid:96)Ψ:∆⇑F,L
Synchronousphase
(cid:96)Ψ:∆ ⇓F (cid:96)Ψ:∆ ⇓G (cid:96)Ψ:·⇑F
[1] 1 2 [⊗] [!]
(cid:96)Ψ:·⇓1 (cid:96)Ψ:∆ ,∆ ⇓F ⊗G (cid:96)Ψ:·⇓!F
1 2
(cid:96)Ψ:∆⇓F (cid:96)Ψ:∆⇓F (cid:96)Ψ:∆⇓B[t/x]
1 [⊕] 2 [⊕ ] [∃]
(cid:96)Ψ:∆⇓F ⊕F l (cid:96)Ψ:∆⇓F ⊕F r (cid:96)Ψ:∆⇓∃x.B
1 2 1 2
(cid:96)Ψ:∆⇑F
[R⇓] providedthatF iseitherasynchronousoranegativeliteral
(cid:96)Ψ:∆⇓F
IdentityandDeciderules
IfK apositiveliteral: [I ] [I ]
(cid:96)Ψ:K⊥ ⇓K 1 (cid:96)Ψ,K⊥:·⇓K 2
(cid:96)Ψ:∆⇓F (cid:96)Ψ,F:∆⇓F
IfF isnotanegativeliteral: [D ] [D ]
(cid:96)Ψ:∆,F ⇑· 1 (cid:96)Ψ,F:∆⇑· 2
Fig.1.ThefocusedproofsystemLLFforlinearlogic.
toeitherSLD-resolutionorhyper-resolutiononHornclauses.Thepapers[3,11–13]aremotivatedbyfoundationalissuesin
λ λ
functionprogrammingandthe -calculus.Also,Levy[18]presentsfocus-styleproofsystemsfortypinginthe -calculusand
CurienandHerbelin[19](amongothers)havenotedtherelationshipbetweenforwardchainingandcall-by-valueevaluation
andbetweenbackwardchainingandcall-by-nameevaluation.
Whilethispaperismostlyconcernedwithfirst-orderlogics,webelievethatmostofourresultscanbeextendedto
includesecond-orderquantificationaswell.Althoughthispaper,whichisanextendedversionoftheconferencepaper[20],
ismostlyself-contained,certainproofsarenotpresentedinfulldetailinordertosavespace:missingdetailscanbefound
in[21].
2. Focusinginlinearlogic
Wesummarizethekeyresultsfrom[1]onfocusingproofsforlinearlogic.
Aliteraliseitheranatomicformulaorthelinearnegationofanatomicformula.Alinearlogicformulaisinnegation
normalformifitdoesnotcontainoccurrencesof−◦andifallnegationshaveatomicscope.IfK isliteral,thenK⊥denotes
⊥ ⊥
itscomplement:inparticular,ifK isA thenK isA.
Connectivesinlinearlogicareeitherasynchronousorsynchronous.Theasynchronousconnectivesare⊥,...............................................................................................,?,(cid:62),&,
and∀whilethesynchronousconnectivesaretheirdeMorgandual,namely,1,⊗,!,0,⊕,and∃.Asynchronousconnectives
arethosewheretheright-introductionruleisalwaysinvertible.Formally,aformulainnegationnormalformisofthree
kinds:literal,asynchronous(i.e.,itstop-levelconnectiveisasynchronous),andsynchronous(i.e.,itstop-levelconnectiveis
synchronous).
AsmentionedinSection1.1,theclassificationofnon-atomicformulasasasynchronousorsynchronousispushedto
literalsbyassigningafixedbutarbitrarybiastoatoms:anatomgivenanegativebiasislinkedtoasynchronousbehavior
whileanatomgivenpositivebiasislinkedtosynchronousbehavior.InAndreoli’soriginalpresentationoffocusing[1],all
atomswereclassifiedas‘‘positive’’andtheirnegations‘‘negative’’.GirardmadeasimilarassignmentforLC[9].Inaclassical
setting,suchachoiceispossiblesinceclassicalnegationsimplyflipsbias.Inintuitionisticsystems,however,amorenatural
treatmentistoassignanarbitrarybiasdirectlytoatoms.Thisbiasofatomsisextendedtoliterals:negatinganegativeatom
yieldsapositiveliteralandnegatingapositiveatomyieldsanegativeliteral.
ThefocusingproofsystemLLFforlinearlogic,presentedinFig.1,containstwokindsofsequents.InthesequentΨ:∆⇑L,
the‘‘zones’’Ψ and∆aremultisetsandLisalist.Thissequentencodestheusualone-sidedsequent− ?Ψ,∆,L(here,we
assumethenaturalcoercionoflistsintomultisets).Thissequentwillalsosatisfytheinvariantthatrequires∆tocontain
onlyliteralsandsynchronousformulas.InthesequentΨ:∆ ⇓ F,thezoneΨ isamultisetofformulasand∆isamultiset
ofliteralsandsynchronousformulas,andF isasingleformula.Noticethatthebiasofliteralsisexplicitlyreferredtointhe
[R⇑]andinitialrules:inparticular,intheinitialrules,theliteralontherightofthe⇓mustbepositive.
4750 C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768
Table1
The0/1translationusedtoencodeLJproofsintolinearlogic.
B B1 B0 (B0)⊥
atomQ Q Q Q⊥
true 1 (cid:62) 0
false 0 0 (cid:62)
P∧Q !(P1&Q1) !P0&!Q0 ?(P0)⊥⊕?(Q0)⊥
P∨Q !P1⊕!Q1 !P0⊕!Q0 ?(P0)⊥&?(Q0)⊥
P⊃Q !(?(P0)⊥............................................................................................... Q1) !P1−◦!Q0 !P1⊗?(Q0)⊥
∃xP ∃x!P1 ∃x!P0 ∀x?(P0)⊥
∀xP !∀xP1 ∀x!P0 ∃x?(P0)⊥
Thefollowingtheorem,whichwasprovedin[1],statesthatLLFissoundandcompleteforlinearlogic.
Theorem1. LetFbeaformulaoflinearlogic.TheFisprovableinlinearlogicifandonlyifthesequent·:·⇑FisprovableinLLF.
Itisasimpleconsequenceofthistheoremthatchangestothebiasassignedtoatomsdoesnotaffectprovabilityofa
linearlogicformula:itcan,however,affectthestructureoffocusedproofs.
3. Translatingintuitionisticlogic
Themostwell-knowntranslationofintuitionisticlogicintolinearlogicislikelythe‘‘(·)◦’’translationgivenbyGirardin
[22]:there,theintuitionisticimplicationA⊃Bistranslatedintothelinearlogicformula!A−◦B(atomsarealsotranslatedto
atoms).Unfortunately,thistranslationrulesoutmodelingcertainkindsofproofsasafocusedproof(inparticular,backward
chainingproofs)anditforcesatomstohavenegativepolarity.Forexample,theintuitionisticsequenta,a ⊃ b,b ⊃ c (cid:96) c
I
hastwodifferentproofscorrespondingtoforwardchainingfroma(⊃ Lwitha ⊃ b)andbackwardchainingfromc (⊃ L
withb⊃c).Usingtheabovetranslationonthisintuitionisticsequentyieldsthefocusedsequenta⊥,!a⊗b⊥,!b⊗c⊥;c ⇑·
forwhichthereisonlyonefocusedproof:using[D ]with!b⊗c⊥leadstoaproof(correspondingtobackwardchaining)
2
whileusing[D ]on!a⊗b⊥leadstoafailedproofattempt.Thistranslationofintuitionisticlogicintolinearlogicisclosely
2
relatedtotheLJTproofsystemforbackwardchainingproofsearch(seeSection7).
Thetranslationwewillusetoderiveaunifiedfocusingsystemforintuitionisticlogic(calledLJFinSection5)isinspired
bythepolarizedtranslationofLU.Forthepurposeofprovidingcompletenessproofslateron,wefirstpresentatranslation,
giveninTable1,thatmapsunfocusedLJproofstofocusedlinearlogicproofs.Asiscustomary,intuitionisticnegationis
definedasA⊃false.ThetranslationinducesabijectionbetweenarbitraryLJproofsandLLFproofsofthetranslatedimage
inthefollowingsense.Firstnoticethatthistranslationisasymmetric:theintuitionisticformulaAistranslatedusingA1ifit
occursontheright-sideofanLJsequentandasA0ifitoccursontheleft-side.Sincethistranslationisusedtocapturecut-free
proofs,suchdistinctionsarenotproblematic.Sincetheleft-handsideofasequentinLJwillbenegatedwhentranslatedto
aone-sidedlinearlogicsequent,(B0)⊥isalsoshown.
Theliberaluseof!inthistranslationthrottlesfocusinginthefollowingsense.Incontrasttounfocusedsequentcalculus
whereinferencerules areappliedindependentlyof eachother,focused proofsystemsorganizea proofintoalternating
phasesofasynchronousandsynchronousrules.Focusingontheformula!P,however,forcestheimmediatereleaseofthe
focusandreturnsthesequenttoanunfocusedsequent.Furthermore,thetranslationofeveryformulaB1issynchronousor
atomicattheouter-mostlayer.Thesecharacteristicsofthetranslationensuresthatafocusedproofisalwaysreturnedto
anunfocusedsequentaftertheintroductionofeachcorrespondingintuitionisticconnective,thusmimickinganunfocused
sequentcalculus.
Although nominally a multiset, the unbounded context of a LLF sequent is, in fact, treated additively. In mapping
intuitionisticlogictolinearlogic,theleft-handsidecontextsofsequentsarealsotreatedadditively.Thecontextsnever
decreasefromconclusiontopremise.WewillnotbeabletodirectlyaccountforcertainknownoptimizationsforLJproofs:
forexample,theleftintroductionruleforimplicationcanbeoptimizedsothattheintroducedimplicationismaintainedin
onepremisebutnottheother(c.f.,[23]).However,asourtranslationwillremainfaithfultointuitionisticprovability,one
canexpectotherrefinementstoremainadmissibleinthenewintuitionisticsequentcalculusthatwederive.Wetherefore
considersuchoptimizationsasanorthogonalissue.
Fig.2providesasequentcalculusproofsystemforintuitionisticlogicinwhichtheleft-hand-sideofthesequentarrowis
asetofformulas.Ineachleft-introductionrule,theprincipalformulaisassumedtoalsobeaside-formula:inthisway,the
principleformulaisretainedinallpremises.ThisvariantofLJbarescloseresemblancestothe‘‘G3i’’calculus[24]particularly
sincecontractionandweakeningarenotexplicitinferencerules.InFig.2,theadditiveversionof∧Lisusedinsteadofthe
followingmultiplicativerule
A,B,Γ (cid:96) R
A∧B,Γ (cid:96)I R ∧L.
I
C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768 4751
A,Γ (cid:96) A ID, Aatomic false,Γ (cid:96) R falseL Γ (cid:96) true trueR
I I I
A,Γ (cid:96) R Γ (cid:96) A Γ (cid:96) B
A ∧iA ,ΓI (cid:96) R ∧L ΓI(cid:96) A∧BI ∧R
1 2 I I
A,Γ (cid:96) R B,Γ (cid:96) R Γ (cid:96) A
A∨IB,Γ (cid:96) R I ∨L Γ (cid:96) AI ∨iA ∨R
I I 1 2
Γ (cid:96) A B,Γ (cid:96) R Γ,A(cid:96) B
AI⊃B,Γ (cid:96) RI ⊃L Γ (cid:96) A⊃I B ⊃R
I I
A∀[tx/Ax,],ΓΓ(cid:96)(cid:96)IRR ∀L ΓΓ(cid:96)(cid:96)I∀AyA ∀R ∃AyA,,ΓΓ(cid:96)(cid:96)I RR ∃L ΓΓ(cid:96)(cid:96)I A∃[tx/Ax] ∃R
I I I I
Fig.2.Aproofsystembasedon‘‘G3i’’forintuitionisticlogic.In∀Rand∃L,yisnotfreeintheconclusion.
Such variations are common in intuitionistic calculi. One of the results of this paper is the further clarification of these
variations.Wewilllaterdiscusswhenoneversionof∧mightbepreferabletotheotherandeventuallyofferaproofsystem
thatincludesboth.
Thefollowingpropositionrelatesfocusedproofsunderthe0/1translationwithintuitionisticproofs.NotethatsinceR1
isalwayssynchronousoratomic,(cid:96)(Γ0)⊥ :⇑R1isinterchangeablewith(cid:96)(Γ0)⊥ :R1 ⇑.
Proposition2. Let(Γ0)⊥bethemultiset{(D0)⊥ |D∈Γ}.Thefocusedproofsof(cid:96)(Γ0)⊥ :⇑R1areinbijectivecorrespondence
withtheproofsofΓ (cid:96) R.
I
Proof. Abijectivemappingbetweenproofscanbedescribedbyasimplerecursiononstructureofproofs.Weshowthe
followingrepresentativecases.
1. ((D ∧D )0)⊥ =(!D0&!D0)⊥ =?(D0)⊥⊕?(D0)⊥ ∈ (Γ0)⊥:
1 2 1 2 1 2
(cid:96)(Γ0)⊥,(D0)⊥ :R1 ⇑
i [?]
(cid:96)(Γ0)⊥ :R1 ⇑?(D0)⊥
i [R⇓] Γ,D (cid:96) R
(cid:96)(Γ0)⊥ :R1 ⇓?(D0i)⊥ [⊕] ←→ Γ,D ∧iDI (cid:96) R ∧L
1 2 I
(cid:96)(Γ0)⊥ :R1 ⇓?(D0)⊥⊕?(D0)⊥
1 2 [D ]
(cid:96)(Γ0)⊥ :R1 ⇑ 2
2. (G ∧G2)1 =!(G1&G1).
1 1 2
(cid:96)(Γ0)⊥;⇑G1 (cid:96)(Γ0)⊥;⇑G1
1 2 [&]
(cid:96)(Γ0)⊥ :⇑G1&G1 Γ (cid:96) G Γ (cid:96) G
(cid:96)(Γ0)⊥ :⇓!(G111&G212) [[!D] ] ←→ ΓI(cid:96)I1G1∧GI2 2 ∧R
(cid:96)(Γ0)⊥ :!(G1&G1)⇑ 1
1 2
3. ((G⊃D)0)⊥ =(!G1 −◦!D0)⊥ =!G1⊗?(D0)⊥ ∈ (Γ0)⊥:
(cid:96)(Γ0)⊥,(D0)⊥ :R1 ⇑
[?]
(cid:96)(Γ0)⊥ :⇑G1 (cid:96)(Γ0)⊥ :R1 ⇑?(D0)⊥
[!] [R⇓] Γ (cid:96) G Γ,D(cid:96) R
(cid:96)(Γ0)⊥ :⇓!G1 (cid:96)(Γ0)⊥ :R1 ⇓?(D0)⊥ [⊗] ←→ ΓI,D⊃G(cid:96) RI ⊃L
(cid:96)(Γ0)⊥ :R1 ⇓!G1⊗?(D0)⊥ I
[D ]
(cid:96)(Γ0)⊥ :R1 ⇑ 2
4. (D⊃G)1 =!(!D0 −◦G1)=!(G1 ............................................................................................... ?(D0)⊥):
(cid:96)(D0)⊥,(Γ0)⊥ :⇑G1
[?]
(cid:96)(cid:96)(Γ(Γ0)0⊥)⊥:⇑:⇑GG11.........................,......................................................................?(?D(D0)0⊥)⊥ [...............[..............................................!..................................]] ←→ ΓD,(cid:96)ΓD(cid:96)⊃I GG ⊃R
(cid:96)(Γ0)⊥ :⇓!(G1 ............................................................................................... ?(D0)⊥) [D ] I
(cid:96)(Γ0)⊥ :!(G1 ............................................................................................... ?(D0)⊥)⇑ 1
4752 C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768
F Fq(right) Fj(left)
atomC C C
false 0 0
A∧B Aq⊗Bq !Aj⊗!Bj
A∨B Aq⊕Bq !Aj⊕!Bj
A⊃B (!Aj −◦Bq)⊗1 Aq −◦!Bj
Fig.3.Thej/qtranslationforLJQ(cid:48).
5. (false0)⊥ =0⊥ =(cid:62) ∈ (Γ0)⊥:
[(cid:62)]
(cid:96)(Γ0)⊥ :R1 ⇑(cid:62)
(cid:96)(Γ0)⊥ :R1 ⇓(cid:62) [[RD⇓]] ←→ Γ,false(cid:96)I R falseL.
(cid:96)(Γ0)⊥ :R1 ⇑ 2
IneachcasethefocusarrowsoftheremainingpremisesoftheLLFproofsalwayspointupwards(⇑),indicatingatermination
offocus.ThetranslationensuresthatallformulasG1aresynchronousoratomic,andthus(cid:96)(Γ0)⊥ :⇑G1isprovableifand
onlyif(cid:96) (Γ0)⊥ : G1 ⇑isprovable(viathereactionruleR ⇑).Thustheinductivehypothesisappliestotheremaining
premises.EveryrightruleofLJisinitiatedbyaD decisionruleandeveryleftruleisinitiatedbyaD rule.
1 2
Thegreatlyconstrainedchoicesofinferencerulesinthefocusedproofsystemcanbeusedtoestablishthatthismapping
isbijective. (cid:3)
Animportantcharacteristicofthe0/1translationisthatthebiasofatomsdoesnotaffectthestructureofproofs:biasof
atomsplayagreaterroleintranslationsthatinsertfewer!’s.Wealsonotethatthistranslationprocedureremainsbijective
eveninthepresenceof0.Evenaproofsuchas
false(cid:96) A false,B(cid:96) C
falsIe,A⊃B(cid:96) CI ⊃L
I
hasexactlyonecorrespondingfocusedproof.The!exponentialforcescontextstobesplitinthe‘‘rightway’’,preventing
occurrencesofmultiple-conclusionsequentsevenwhen0ispresentinthecontext(seecase3above).Dealingwith0will
causesomeprobleminothertranslations(see,forexample,Lemma5).
The0/1translationformsastartingpointinestablishingthecompletenessofotherproofsystems.Thesesystemscanbe
(cid:48)
seenasinducedfromalternativetranslationsofintuitionisticlogic.Consider,forexample,theLJQ proofsystempresented
in[4].AtranslationtoLLFproofsforthissystemisgiveninFig.3.ThelinearimplicationP −◦ Q isdefinedastheusual
P⊥ ............................................................................................... Q.Allatomsmustbegivenpositivebiasforthistranslation.Noticethatthetranslationforimplicationontheleft-hand
sideisoftheformA −◦ !B,whichiscomplementarytoGirard’soriginaltranslation:!A −◦ B.The‘‘⊗1’’deviceisawayto
controlthestructureoffocusingproofswithoutaffectingprovability.Anotherconsequenceofhavingonlypositiveatoms
isthatconjunctionistranslatedusing⊗insteadof&.WeshallreturntoLJQ(cid:48) inSection7,inthecontextofageneralized
focusingframeworkforintuitionisticlogic.
(cid:48)
Withminorchanges,Girard’soriginaltranslationofintuitionisticlogicin[22]inducesthecomplementtoLJQ called
LJT [3](whichwasderivedfromLKT[11]).Asalreadynoted,allatomsmustbegivennegativebiasforthistranslation.For
amoreextendedtreatmentofLJT,see[21].
(cid:48)
GivenatranslationsuchasthatofLJQ,onecangiveacompletenessproofforthesystemusinga‘‘grandtour’’through
linearlogicasfollows:
1. Showthataproofunderthe0/1translationcanbeconvertedintoaproofunderthenewtranslation.Thisusuallyfollows
fromcut-eliminationinlinearlogic.
(cid:48)
2. Defineamappingbetweenproofsinthenewsystem(suchasLJQ)andLLFproofsofitstranslation.
3. ShowsoundnessofthenewsystemwithrespecttoLJ.Thisisusuallytrivial.The‘‘tour’’isnowcomplete,sinceproofsin
LJmaptoproofsunderthe0/1translation.
/
Fig.4furtherillustratesthestrategy.Thelabell rrepresentsthetranslationforsomearbitraryintuitionisticsequentsystem,
whichisindicatedby(cid:96) .Thearrowontheleft-hand-sidedepictsProposition2;thetoparrowdepictsStep1,theright-hand-
O
sideruledepictsStep2,andthebottomarrowdepictsStep3.
AnintuitionisticsystemthatcontainsatomsofbothpositiveandnegativebiasisλRCC[7].Twospecialcasesofthe⊃L
rulearedistinguishedinvolvingE ⊃ DforpositiveatomE andG ⊃ AfornegativeatomA.Eachrulerequiresthatthe
complementaryatom(Eontheleft,Aontheright)ispresentwhenapplied,thusterminatingonebranchoftheproof.One
cantranslatethesespecialcasesusingformsE−◦!D(cid:48)and!G(cid:48)−◦A,respectively,inlinearlogic.Thestrategyoutlinedabove
canthenbeusedtonotonlyproveitscompletenessbutalsoextenditwithmoreaggressivefocusingfeatures.
Ourinteresthereisnottheconstructionofindividualsystemsbutthebuildingofaunifyingframeworkforfocusingin
intuitionisticlogic.Suchataskrequiresacloserexaminationofpolarityanditsconnectiontofocusing.
C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768 4753
0/1 l/r
(cid:96)⇑ (cid:45)(cid:96)⇑
(cid:54) (cid:54)
(cid:63) (cid:63)
(cid:96) (cid:27) (cid:96)
I O
Fig.4.Grandtourthroughlinearlogic.
4. Permeableformulasandtheirpolarity
Inordertoconstructageneralfocusingschemeforintuitionisticlogic,thenon-linear(exponential)aspectsofproofs
needspecialattention,especiallyinlightofthefactthatthe[!]rulestopsabottom-upconstructionoffocusedapplication
ofsynchronousrules(thearrow⇓intheconclusionflipsto⇑inthepremise).
For our purposes here, a particularly flexible way to deal with the exponentials in the translations of intuitionistic
formulasisviathenotionofpermeationthatisusedinLU[10].Inparticular,thereareessentiallythreegradesofpermeation.
TheformulaBisleft-permeableifB≡!B,isright-permeableifB≡?B,andneutralotherwise.Withinsequentcalculusproofs,
aformulaisleft-permeableifitadmitsstructuralrulesontheleftandright-permeableifitadmitsstructuralrulesonthe
right.Anexampleofaleft-permeableformulais∃x!A.Allleft-permeableformulasaresynchronousandallright-permeables
areasynchronous.IntheLUsystem,boththeleftandrightsidesofsequentscontaintwozones—onethattreatsformulas
linearlyandonethatpermitsstructuralrules.Aleft-permeable(resp.,right-permeable)formulaisallowedtomovebetween
bothzonesontheleft(right).Inaddition,LUintroducesatomsthatareinherentlyleftorright-permeableorneutral:one
cansimulateLUin‘‘regular’’linearlogicbytranslatingleft-permeableatomsAas!Aandright-permeableonesas?A.
Topreservethefocusingcharacteristicsofpermeableatomsaspositivelyornegativelybiasedatoms,weusethefollowing
asymmetricaltranslation.Thesuperscript−1indicatestheleft-sidetranslationand+1indicatestheright-sidetranslation:
P−1 =!PandP+1 =P,forleft-permeableatomP.
N−1 =NandN+1 =?N,forright-permeableatomN.
B−1 =B+1 =B,forneutralatomB.
The!ruleofLLFcausesalossoffocusinallcircumstances:themainreasonweuseasymmetrictranslationsisthatthey
caninsertfeweroccurrencesof!.Thetranslationofpositiveatomsabovepreservespermeationontheleftwhileallowingfor
focusontheright.Thatis,left-permeableatomscannowbeinterpretedmeaningfullyaspositivelybiasedatomsinfocused
proofs,andduallyforright-permeableatoms.Furthermore,thepermeationofpositiveatomsis‘‘one-wayonly:’’theycannot
beselectedforfocusagainoncetheyenterthenon-linearcontext.Inotherwords,thischaracteristicofthefocusingsequent
calculusallowsasynchronousformulastobefullydecomposed,uptoandincludingatoms.
Intuitionisticlogicusestheleft-permeableandneutralformulasandatoms.LUdefinesatranslationforintuitionistic
logicsothatallsynchronousformulasareleft-permeable.
Thefinalelementofintuitionisticpolarityisthatneutralatomsshouldbeassignednegativebiasinfocusedproofs.Neutral
atomsthatareintroducedintotheleftcontext(e.g.bya⊃Lrule)mustimmediatelyendthatbranchoftheproofinan
identityrule.Otherwise,theuniquestoupislostwhenmultiplenon-permeableatomsaccumulateinthelinearcontext.
TheLUandLLFsystemsserveasaconvenientplatformfortheunifiedcharacterizationofpolarityandfocusinginall
threelogics.Wecannowunderstandtheterminologyof‘‘positive’’and‘‘negative’’formulasineachlogicasfollows:
Linearlogic: Positive formulas are synchronous formulas and positively biased neutral atoms. Negative formulas are
asynchronousformulasandnegativelybiasedneutralatoms.
Intuitionisticlogic: Positiveformulasareleft-permeableformulas(includingleft-permeableatoms).Negativeformulasare
asynchronousneutralformulasandnegativelybiasedneutralatoms.
Classicallogic: Positiveformulasareleft-permeableformulas.Negativeformulasareright-permeableformulas.
5. TheLJFsequentcalculus
TheasymmetricaltranslationofatomsunifiesLU’snotionofpolaritywiththepositive/negativedualityofLLFproofs.We
cannowuseaLU-basedtranslationofintuitionisticlogicthatwillinduceanew,focusedsequentcalculusforintuitionistic
logic,onethatisevenmoresensitivetothepolarityofformulasthanLLF.
4754 C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768
P+1 =P,P−1 =!P forpositiveatomP
N+1 =N−1 =N fornegativeatomN
true+1 =true−1 =1 false+1 =false−1 =0
(P∧+Q)−1 =P−1⊗Q−1 (A∧+B)+1 =A+1⊗B+1
(P∧+N)−1 =P−1⊗!N−1
(N∧+P)−1 =!N−1⊗P−1 (A∧−B)−1 =A−1&B−1
(N∧+M)−1 =!N−1⊗!M−1 (A∧−B)+1 =A+1&B+1
(P∨Q)−1 =P−1⊕Q−1 (A∨B)+1 =A+1⊕B+1
(P∨N)−1 =P−1⊕!N−1
(N∨P)−1 =!N−1⊕Q−1
(N∨M)−1 =!N−1⊕!M−1
(P ⊃B)+1 =P−1−◦B+1 (A⊃B)−1 =A+1−◦B−1
(N ⊃B)+1 =!N−1−◦B+1
(∃xP)−1 =∃xP−1 (∃xA)+1 =∃xA+1
(∃xN)−1 =∃x!N−1
(∀xA)−1 =∀xA−1 (∀xA)+1 =∀xA+1
Fig.5.The−1/+1translationofintuitionisticlogic(−1forleft,+1forright).Here,P,Qrepresentpositiveformulas;N,Mrepresentnegativeformulas;
andA,Brepresentarbitraryformulas.
WefirstmakeanotheradjustmentontheLUtranslationofintuitionisticconjunction.Insteadofusing&or⊗depending
onthepolaritiesofthesubformulas,weintroducetwoversionsoftheintuitionisticconjunction:∧+and∧−(Danosetal.used
similarconnectives[12]).Theseconnectivesareequivalentinintuitionisticlogicintermsofprovabilitybutdifferintheir
impactonthestructureoffocusedproofs.1 Theuseoftwoconjunctionsmeansthatthetop-levelstructureofaformula
completely determines its polarity. Using two conjunctives provides some flexibility in capturing different approaches
to focusing. For example, Chaudhuri [6, Section 6.4] provides a treatment of conjunction that interprets occurrences of
conjunctionsinanasynchronousphaseasasynchronousandinasynchronousphaseassynchronous:suchatreatmentcan
becapturedbymapping∧explicitlytoeither∧−or∧+dependingthestructureoftheformulainwhichtheyoccur.
Polarityinintuitionisticlogicisdefinedasfollows.
Definition3. Atoms in LJF are arbitrarily divided between those that are positive and those that are negative. Positive
formulasareofthefollowingforms:positiveatoms,true,false,A∧+B,A∨Band∃xA.Negativeformulasareamongnegative
atoms,A∧−B,A⊃Band∀xA.
Notice that the classification of atoms as positive and negative is done on possibly open atomic formulas since
eigenvariablesoftheproofmayappearfreeinthem.Inthemostgeneralapproachtoassigningbiastoatoms,itmightwell
bethecasethatanatomandasubstitutioninstanceofanatomcanbeofdifferentpolarities.Itwouldseemthatthestability
ofbiasassignmentunder(first-order)substitutionisanaturalexpectationbutitisnotarequirementforthedescriptionof
focusedproofs.Weshallassumesuchstability,however,intheproofofcut-eliminationinSection6.Formoreaboutpossible
approachestoassigningbiastoatomicformulas,see[25,26].
Ourfulltranslationofintuitionisticlogicextendsthe−1/+1translationofatoms.Itremainsasymmetricalandeliminates
nearlyalloccurrencesof!ontheright-handside.ThetranslationisshowninFig.5.Allpositiveformulasaretranslatedinto
left-permeableformulasontheleft-handside.Thepositiveformulasofintuitionisticlogic(andclassicallogic)canbegiven
lineartreatmentsontheleft.
The+1/−1translationinducestheLJFsequentcalculusdisplayedinFig.6.SequentsinLJFcanbeinterpretedasfollows:
1. Thesequent[Γ],Θ −→ Risanunfocusedsequent.Here,Γ andΘ arebothmultisetsandΓ containsonlynegative
formulasandpositiveatoms.ThesymbolRdenoteseithertheformulaRorthe‘‘bracketed’’formula[R].Endsequents
ofLJFproofsusuallyhavetheform[],Θ −→R.
2. Thesequent[Γ]−→[R]isaspecialcaseoftheprevioussequentinwhichΘisempty(and,hence,notwritten)andR
isoftheform[R].Suchsequentsdenotetheendoftheasynchronousphase:proofsearchcontinueswiththeselectionof
afocus.
1 Thereadermaynoticethattherecanalsobetwoversionsoftrue.Itispossibletousethelinearconstant(cid:62)torepresenttrue,butthisisunnecessary
sincefalse⊃falsecanbeusedinstead.Insistingondualversionsofallconstantsismoresuitableforclassicallogic(seeSection8).
C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768 4755
3. Thesequent[Γ]−→A [R]representsleft-focusingontheformulaA.Provabilityofthissequentisrelatedtotheprovability
ofΓ,A(cid:96) R.
I
4. Thesequent[Γ]− →representsright-focusingontheformulaA.Provabilityofthissequentisrelatedtoprovabilityof
A
thesequentΓ (cid:96) A.
I
Intuitionistic focused proofs are more structured than LLF proofs since the polarities of intuitionistic logic observe
stronger invariances. While the non-linear context of LLF sequents can contain both synchronous and asynchronous
formulas, the translation of an intuitionistic sequent into a LLF sequent is such that positive formulas, which are
asynchronousontheleft,areplacedinthelinearcontext(becauseP−1isalwaysequivalentto!P−1forpositiveP).
LikeLLF,akeycharacteristicofLJFistheassignmentofpolaritytoatoms.Toillustratetheeffectoftheseassignmentson
thestructureoffocusedproofs,considerthesequenta,a ⊃ b,b ⊃ c (cid:96) cwherea,bandcareatoms.Thissequentcanbe
provedeitherbyforwardchaining throughtheclausea ⊃ b,orbackwardchaining throughtheclauseb ⊃ c.Assumethat
atomsaandbareassignedpositivepolarityandthatcisassignednegativepolarity.Thisassignmenteffectivelyadoptsthe
forwardchainingstrategy,reflectedinthefollowingLJFproofsegment(here,Γ isthemultiset{a,a⊃b,b⊃c}):
I
[b,Γ]− → Ir [b,Γ]−→c [c] l
b ⊃L
[b,Γ]−b⊃→c [c]
Lf
[b,Γ]−→[c]
[]
[Γ],b−→[c] l
R
[Γ]− → Ir [Γ]−→b [c] l .
a ⊃L
[Γ]−a⊃→b [c]
Thepolaritiesofaandc donotfundamentallyaffectthestructureoftheproofinthisexample,whileassigningnegative
polaritytoatombrestrictstheprooftousethebackwardchainingstrategy:
I
[Γ]− → Ir [Γ]−→b [b] l
a ⊃L
[Γ]−a⊃→b [b]
Lf
[Γ]−→[b]
[]
[Γ]−→b r I
[Γ]− → Rr [Γ]−→c [c] l
b ⊃L
[Γ]−b⊃→c [c] .
Theavailabilityofboth∧− and∧+ offersimportantdesignchoicesforproofsearchstrategies.Forexample,afocused
proofsearchofthesequent
[a,a⊃b,b⊃c,c ⊃d,c ⊃e]−→d∧−e,
requirestheright-sideasynchronous∧−tobedecomposedimmediatelyintodande.Thus,thederivationofcfromthefirst
threeformulasinthecontextcannotbesharedbetweenthesesubgoals.Usingd∧+ewilldelaythedecompositionofthe
goaluntilitisselectedforfocusatanothertime,makingaforwardchainingstrategypossible(and,hence,ccanbeadded
totheleft-contextbeforeprocessingtheconjunctiveright-hand-side).Ontheotherhand,usingasynchronousformulason
theright-handsideisconsistentwiththeuniformproofmodeloflogicprogramminginterpreters[2].
/
ToprovethecorrectnessofLJFwefollowthe‘‘grandtour’’strategy.Firstweshowthataproofunderthe0 1translation
canbetransformedtoaproofunderthe+1/−1translation.Let(cid:96) representsequentsintheunfocusedlinearlogicsequent
LL
calculus.
Proposition4. LetΓ,Rconsistofamultisetofintuitionisticformulas.LetΓ−1consistofallpositiveformulasinΓ−1andΓ−1
2 1
containallnegativeformulasinΓ−1.If!Γ0 (cid:96) R1isprovablethen!Γ−1,Γ−1 (cid:96) R+1isalsoprovable.
LL 1 2 LL
Proof. For every positive intuitionistic formula P, the linear logic formula P−1 is left permeable (in particular, P−1 is
provabilityequivalentto!P−1).WenowarguethatforeveryintuitionisticformulaF,wehaveF1 (cid:96) F+1and!F−1 (cid:96) !F0.
LL LL
Theproofisbymutualinductiononthesizeofformulas.Weshowthefollowingrepresentativecases.
1. !(P∨N)−1 (cid:96) !(P∨N)0:
LL
P−1 (cid:96) !P−1 !P−1 (cid:96) !P0
LL LL
Cut
P−1 (cid:96) !P0 !N−1 (cid:96) !N0
LL ⊕R LL ⊕R
P−1 (cid:96) !P0⊕!N0 !N−1 (cid:96) !P0⊕!N0
LL LL ⊕L
P−1⊕!N−1 (cid:96) !P0⊕!N0
LL !L
!(P−1⊕!N−1)(cid:96) !P0⊕!N0
LL !R
!(P−1⊕!N−1)(cid:96) !(!P0⊕!N0) .
LL
4756 C.Liang,D.Miller/TheoreticalComputerScience410(2009)4747–4768
DecisionandReactionRules
[[NN,,ΓΓ]]−−→→N [[RR]] Lf [Γ[Γ]]−−→P→[P] Rf [Γ[Γ],]P−−→P→[R[]R] Rl [[ΓΓ]]−−→→N Rr
N
[C,Γ],Θ −→R [Γ],Θ −→[D]
[] []
[Γ],Θ,C −→R l [Γ],Θ −→D r
InitialRules
,
[P,Γ]− → Ir, atomicP [Γ]−→N [N] Il atomicN
P
IntroductionRules
[Γ],Θ −→R
[Γ],Θ,false−→R falseL [Γ],Θ,true−→R trueL [Γ]− → trueR
true
[[ΓΓ]]A−−1∧A→→−iA2[[RR]] ∧−L [Γ],Θ[Γ−],→ΘA−→[ΓA],∧Θ−B−→B ∧−R
[Γ],Θ,A,B−→R [Γ]− → [Γ]− →
[Γ],Θ,A ∧+B−→R ∧+L [ΓA]−A∧+B→B ∧+R
[Γ],Θ,[AΓ−],→Θ,RA∨[BΓ−],→Θ,RB−→R ∨L [Γ[Γ]−]−A1A∨iA→2→ ∨R
[Γ]−A[→Γ]−A⊃→[BΓ[]R]−→B [R] ⊃L [Γ[Γ],],ΘΘ−,→A−A→⊃BB ⊃R
[Γ[Γ],],ΘΘ,,∃AyA−−→→RR ∃L [Γ[Γ]]−−A∃[tx/Ax→]→ ∃R [[ΓΓ]]A−−∀[t→→x/Ax][[RR]] ∀L [Γ[Γ],]Θ,Θ−−→→∀AyA ∀R
Fig.6.TheIntuitionisticSequentCalculusLJF.Here,Pispositive,Nisnegative,Cisanegativeformulaorpositiveatom,andDapositiveformulaornegative
atom.Otherformulasarearbitrary.Also,yisnotfreeinΓ,Θ,orR.
HerePispositiveandNnegative,andtheproofmustappealtothepermeabilityofP−1.Theremainingpremisesfollow
frominductivehypotheses.
2. !(A⊃B)−1 (cid:96) !(A⊃B)0:weprovethiscaseintwostages,joinedbyacut:
LL
!(AAA+++111(cid:96)−−LL◦◦ABB+−−111,)A,BA+−+111(cid:96)(cid:96)(cid:96)LLLLLLBB−B−1−11−!!L◦LL !B−1 (cid:96)LL !B0 !!AAA111(cid:96)(cid:96)(cid:96)LLLLLL!AAA+++111 !!−LR◦L
!(A+1−◦B−1),!A+1 (cid:96)LL B−1 !R !A+1−◦!B−1,!A1 (cid:96)LL !B0 −◦R
!(A+1−◦B−1),!A+1 (cid:96) !B−1 !A+1−◦!B−1 (cid:96) !A1−◦!B0
LL −◦R LL !L
!(A+1−◦B−1)(cid:96) !A+1−◦!B−1 !(!A+1−◦!B−1)(cid:96) !A1−◦!B0
LL !R LL !R
!(A+1−◦B−1) (cid:96) !(!A+1−◦!B−1) !(!A+1−◦!B−1) (cid:96) !(!A1−◦!B0)
LL LL
Cut
!(A+1−◦B−1) (cid:96) !(!A1−◦!B0) .
LL
Thepremisesareeithertriviallyprovableorfollowfrominductivehypotheses.
Othercasesaresimilar.Thepropositionholdsbycut-eliminationinthelinearsequentcalculus. (cid:3)
ThenextstepinthecompletenessproofconstructsamappingbetweenLJFproofsandLLFproofsofitsimageinlinear
logic.Thepossiblepresenceof0causessomeadditionaldifficultyandslightlycompromisestheperfectbijectionbetween
proofs.
Lemma5. Ifthereisnoproofof!Γ−1 (cid:96) 0andif!Γ−1 (cid:96) ∆+1then∆containsexactlyoneformula.
LL LL
Proof. Proceedbycontradiction.Assumethereissuchaproofwithanon-singletonright-handsideandconsidersucha
proofofminimalheight.Byinspectingthepremisesofeachpossibleinferencefigureforthefinalruleoftheproof,wesee