Rewritings for Polarized Multiplicative and Exponential Proof Structures

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Rewritings for Polarized Multiplicative and Exponential Proof Structures

Christophe Fouqueré, Virgile Mogbil

To cite this version:

Christophe Fouqueré, Virgile Mogbil. Rewritings for Polarized Multiplicative and Exponential Proof

Structures. Electronic Notes in Theoretical Computer Science, Elsevier, 2008, pre-proceedings version,

203 (1), pp. 109-121. �10.1016/j.entcs.2008.03.037�. �hal-00143929�

Exponential Proof Strutures

(rapport interneLIPN - Février2007)

ChristopheFouqueré

⋆

andVirgileMogbil

⋆

LIPNUMR7030, CNRSUniversitéParis13,

99av.J-BClément,F93430Villetaneuse,Frane

hristophe.fouquerelipn.univ-pa ris13 .fr

virgile.mogbillipn.univ-paris13 .fr

Abstrat. Westudyonditions foraonurrentonstrutionofproof-

netsintheframeworkoflinearlogifollowingAndreoli'sworks.Wedene

spei orretness riteriafor that purpose. We rststudythe multi-

pliativeaseandshowhowtheorretnessriteriongivenbyDanosand

deidableinlineartime,maybeextendedtolosedmodules(i.e.validity

ofpolarizedproofstrutures).Wethenstudytheexponentialase.This

hasnatural appliations in(onurrent) logi programmingas validity

ofpartial proof struturesmay be interpreted intermsofvalidity ofa

onurrentexeutionoflausesinanenvironment.

1 Introdution

Girardin his seminal paper[9℄ gave aparallel syntax for multipliativelinear

logi(MLL)asorientedgraphsalledproof-strutures.LetusreallthataMLL

formulaiseitheranatomiformulaA^{,anegationofanatomiformula,orbuilt}

withabinaryonnetive⊗^{orP.Intheoriginaldenition,a}proof-struturefor MLL isonstrutedbymeansofthefollowingbinarylinks:

⊗^{-link: A B}

A⊗B

⊗ ^P-link:

A B

A^PB

P

axiom-link:

A A^⊥

where every ourrene of formula is a premise of at most one link and is a

onlusionofexatlyonelink.Aorretnessriterionenablesonetodistinguish

sequentializableproof-strutures(thesoalledproof-nets)from"bad"strutures

(that donotorrespond toproofs inthe sequentalulus).After Girard'slong

trip orretness riterion, numerous equivalent properties were found. In par-

tiular,Danosand Regnier[7℄ provedthat swithedproof-strutures should be

trees, where swithing is done by deleting one of the premises of eah P-link.

Danos [6℄showed that it is the asei theproofstruture rewrites to • ⁽⊗ ^is

alledaontratednode):

⋆

(1)

⊗ −→

(2) −→ (3)

P

−→

(4)

P

−→

While a lot of researh has been done on nding eient orretness riteria

forMLL,itstillremainsto studyorretnessriteriainaseofpolarizedproof-

strutures in MLL,and broadenit to theexponentialase. First used byAn-

dreoliinLogiProgramming[1℄andalsoonsideredinGirard'sworks[10℄andin

Laurent'sworks aboutPolarizedLinearLogi[13℄,this oneptofpolarization

allowstoonsiderlusteredstrutures.Reently,polarizedproofstruturesarise

naturallyinlogiprogrammingmodels [24℄.Thebasiobjetsweonsiderare

then proofstrutures with twostrataweallelementary bipolar modules, that

maybeombinedintomodules.Wereallthemultipliativeaseinthefollowing

setion(thereadermayndin[8℄extensiontoopenmodules).Wedeneaor-

retnessriterionthattakesareoftheparallelstrutureofmodules,extending

the Danosriterion.In setion3, weanalyzehowmodules may be generalized

totakeareofexponentials.

2 The multipliative ase

We onsiderin this setion theextension MLLuof MLL with 1 ^{theunit of} ⊗^.

Formulaearegivenas:

F := 1|G

G₁, G₂:=A|A^{⊥ atomi formula or its negation}

|G₁⊗1 |1⊗G₁ |G₁⊗G₂ |G₁^PG₂

LetPS^{n bethedireted graphswhereedgesarelabelledbyformulaeofMLLu}

andbuiltwiththefollowinglinks(n≥1^):

⊗^-link:

A₁⊗ · · · ⊗A_n

⊗ A1 An

P-link:

A1^P. . .^PAn

P

A1 An

axiom-link:

A A^⊥ 1^-link:

1 1

possiblywithedgespendingdownwards.ElementsofPS^{narealledproofstru-}

tures.Formulaelabellingpending edgesare theonlusions ofthe proofstru-

ture,nodeswithpendingedgesarealledonlusion nodes.Labelsonedgesare

omittedwhenlearfrom theontext.

Proposition1. Letπ^{beaproofstrutureof} PS^n,π^{isaproofnet (i.e.sequen-}

tializable) iπ→^∗•^:

(1) ⊗ −→ (2)

−→

(3)

1

−→

(4)

P

−→

(5)

P −→ ^{(6) P} −→

Theproofofthepropositionfollowsfromthestandardoneon binaryproof

struturesforMLL[6℄,andthefollowingremarks:⊗^{andPareassoiativeand}

ommutative,the 1^{-ary P onnetive isby onventionthe identity,} 1 ^{is aunit}

for⊗^.

Werstgivethedenitionofanelementarybipolarmodule (EBM)andgive

theorrespondenewithproofstrutures. Wethendeneamoduleastheom-

position of EBMs. A module is orret if theorresponding proof struture is

sequentializable.

Denition1 (EBM). An EBM M ^{is given by anite set} H(M) ^{of proposi-}

tional variables (alledhypotheses) h_i ^{andanon emptynite set}C(M)^varying

over k ^{of nite sets} Ck(M) ^of propositional variables (alled onlusions) c^j_k^.

Variables aresupposed pairwise distint.

1

Theset of propositional variables ap-

pearingin M ^isnotedv(M)^.Equivalently, oneandene itasadireted graph with labelled pending edges and two kinds of nodes, one positive pole under a

non-emptynite setof negativepoles:

c^j₁¹ c^j_K^K

hi

The setof pendingedgesof anEBM M ^{isalled theborder} b(M)^.

TheproofstrutureorrespondingtoanEBMisgivenbythefollowingtrans-

formationonpoles.Theonversetransformationrequires thedenition ofBMs

dened later.

ifCk(M) =∅^: → ^{1 ,if}Ck(M)6=∅^:

c^j_k^k

→ ^P

z }| { c^j_k^k⊥

hi

→ ^⊗

z}|{

hi

An EBM M ^{may be} equivalently dened as a(type) formula t(M) ^{in the}

dual language of MLLu(reall that A ⊸B = A^{⊥ P}B^): t(M) = (N

ih_i)⊸ (k(N

jkc^j_k^k))^{, where weuse theonventionthat} kF_k =N

kF_k =F₁ ^when

thedomainofk^{isofardinal1,andifthedomainof}i^isempty,(N

ih_i)⊸C=C

andifthedomainofj_k ^forsomek^isempty,(N

jkc^j_k^k) =⊥^{.Howeverthereader}

should arethat thissupposesabilateralsequentalulus,although thelogial

1

Thisrestritionistakenforsimpliity.Theframeworkanbegeneralizedifweon-

sidermultisets(ofhypothesesandonlusions)insteadofsets,andaddasrequired

are ofspeialinterest:An EBM isinitial (resp.nal) ifitsset ofhypothesesis

empty(resp.itssetofonlusionsisempty).AnEBMistransitoryifitisneither

initial nornal. Initial EBMs allowto delareavailable resoures,thoughnal

EBMs stop part of a omputation by withdrawing a whole set of resoures.

TransitoryEBMs arealleddenitelausesin standardlogiprogramming.

Denition2 (BM). A bipolar module (BM) M ^{is dened with hypotheses} H(M)^,onlusionsC(M)^,andtypet(M)^{,indutively inthe following way:}

An EBMisaBM.

LetM ^beaBM,andN ^beanEBM,letI=C(M)∩H(N)^{,theiromposition}

wrttheinterfaeI^,M◦IN ^{isaBMwiththemultisetofhypotheses}H(M)∪ (H(N)−I)^{,themultisetofonlusions} (C(M)−I)∪ C(N)^{,the type}t(M)⊗ t(N) ^andvariables v(M)∪v(N)^.

Theinterfaewillbeomittedwhenitislearfromtheontext.Notethatthe

interfaemaybeempty. Thetranslationfrom proof struturesof PS^{n to BMs}

isgivenbythetwofollowingrules, plusrules notexpliitedheredueto lakof

spaethat takeareofpolarityandtheonstant1^:

P

⊗ z }| {α

−→

P

p^⊥ p

⊗ z}|{α

wherep^{isafreshatomiformula}

⊗ z}|{

hi

1 1

P

z }| { c^j₁^1⊥

P

z }| { c^j_K^K⊥

−→

c^j₁¹ c^j_K^K

hi

ConsideringBMs inplae ofproof struturesforMLLuhasvaluableonse-

quenesin termsofsimpliityoforretnessriteriaasoneantakeareofthe

bipolestrutureofBMsmorediretlythanitistheasewithabinarystruture.

Denition3 (Corretness (wrt sequentialization)). Let M ^{be aBM,} M

isorretif the orresponding proof strutureinPS^{n is}sequentializable.

SequentializationmeansthatthereexistsaformulaC^{builtwiththeonne-}

tives⊗^{andP, andthevariables}C(M)^{suh thatthesequent}H(M), t(M)⊢C

is provable in Linear Logi. Let us briey interpret EBMs and BMs in terms

ofomputation. AnEBM hasthefollowingoperationalbottom-upreading:be-

inggiveninsomeontextamultisetofhypotheses(dataforthepositivepole),

the EBM triggersone (linear) eah of the negative poles, these last have to

beusedin separateontexts.TriggeringanEBM,that isomposing itwithan

existing BM, is nothingelse but doingaresolution stepin logiprogramming.

termorretthekindofmoduleswespeakof,e.g.-orretwhenthemoduleis

losed,o-orretinthegeneralsetting.

A losed module is a BM without any pending edges, i.e. with the sets of

hypothesesandonlusionsempty.Corretnessoflosedmodulesmaybetested

either in termsof provability in asequentalulus orby meansof orretness

riteriaforproofstrutures.Inthefollowing,weonsidertheorretnessriteria

ofDanos[6℄using aontrationrelationandexplainedin theprevioussetion,

andalsotheonegivenbyDanosandRegnier[7℄thatusesswithings:letπ^bea

proofstruturewithbinarylinksandS(π)^{thesetof(swithed)graphsobtained}

fromπ^{byremovingexatlyonepremiseedgeforeahPlink,}π^isaproofneti

eahgraph in S(π) ^{is ayliand onneted.One}generalizes this denition to

n^{-aryonnetivesbyintroduing}generalizedswithes:eahn^{-aryPonnetive}

indues n ^{swithed graphs. One still an dene swithed} proof-strutures and a riterion generalizing Danos-Regnier orretness riterion on PS^{n: a proof}

struture π ^{is a proof net i thegraphs in} S(π) ^{are ayli and onneted. A}

losed module M ^{is DR-orretif theproofstruture} M^{∗ assoiated to}M ^isa

proofnetwrtthepreviousriterion.WeabusivelyrefertothemoduleM ^instead

oftheorrespondingproofstrutureM^{∗inthefollowing,speakingofforinstane}

swithed moduleinsteadofswithed proofstruture.Weimmediatelyhavethe

followingpropositionasaorollaryoftheDanosandRegnierriteriontheorem:

Proposition2. LetM ^{bealosedmodule,}M ^is-orretiM ^isDR-orret.

Wegivebelowa(bigstep)redutionrelationthattakesareofthefoaliza-

tionproperty.ThoughaDanos-likerelationwouldredueeahsteponevariable,

ourformulationusesasawholethestrutureofamodulethankstofoalization.

Thefoalizationpropertystatesthatasequentisprovableithereexistsaproof

suhthatdeompositionofthepositivestratumofformulaeisdoneinonestep.

Consideringbipolarmodules,itmeansthatonemaydenearedutionrelation

suhthateahstepreduesonepositive-negativepairofnodes.

Proposition3 (Stability). Let M ^and N ^{be two losed modules suh that} M ։N^,M ^is-orretiN ^{is-orret(seeFig. 1).}

Proof. Oneandeneafuntionfromtheswithedstruturesofthemoduleon

the left of the relation onto the swithed strutures assoiated to the module

on therightsuh that a swithed struture from the left is ayli(resp. on-

neted) itheorrespondingswithed struturefromthe rightisayli(resp.

onneted).

Theorem1 (-orretness). Alosedmodule M ^is-orretiM →→^∗

⊥∪

▽

.

Proof. As the redution rules are stable wrt orretness, it remains to prove

thataorretnon-terminallosedmodule M ^{analwaysberedued.Wedene}

a partial relation on negative poles: a negative pole is smaller than another

z }| { α

β

z }| { γ

δ

−→→

z }| { α

z }| { γ

β δ

α

β z }| {

γ

δ

−→→

α β z }| {

γ

δ

Fig.1.Bigstepredutionrelation.

the bottom of the positive pole and the seond negative pole is linked to the

top of the positive pole.We onsider the transitive losure of this relation. If

maximal negativepoles donotexist then there exists at least oneyle in the

module alternatingpositiveandnegativepoles. Weanthendeneaswithing

funtiononthemodule(hoosingtheorretlinksfornegativepoles)suhthat

the swithed module has a yle. Hene ontradition. So let us onsider one

of themaximalnegativepole, andthe orrespondingpositivepole. Weremark

that suh a negative pole has no outoming links (the module is losed and

thenegativepole ismaximal).If thepositivepole hasothernegativepoles, we

an omit themaximal negativepole by neutrality. Otherwise,letus study the

inomingnegativepoles.Ifthereisnosuhinominglink,thenM ^{istheterminal}

module. If eah inoming negative pole hasat least onelink going to another

positivepole, then one andene a swithing funtion using for eah of these

negativepolesoneofthelinksthatdoesnotgotothepositivepoleweonsidered

rst.Henetheswithedmoduleisnotonneted(therearenooutgoinglinks).

Hene ontradition. So there exists at least one inoming negative pole with

thewhole set of links assoiatedto the positivepole:therst ruleapplies and

wearenished.

Notethatthisproofextensivelyusesthebipolarnatureofmodules.Moreover,

the proof may havebeengivenonsidering minimal poles in plae of maximal

poles, and for eah proof only oneof the two redution rules is suient and

neessary!Finally,thesametehniqueGuerrini[11℄usedforDanosriterionmay

beappliedheretogetalinearalgorithm.Studyingorretnessofopenmodules

is a neessarystep towards the speiationof alogi programming language

based on bipolar modules. We detailed in another paper the extension of the

3.1 Multipliativeexponentiallinear logi (MELL)

Addingexponentialstothelanguageobviouslyinreasesitsexpressivity:itallows

for representing reusable resoures. In linear logi, the 'of ourse' modality !

has this main property: !A ⊸ A⊗ · · · ⊗A^{. Tehnially, three operations are}

neessary:ontration,derelitionandweakening.Therstoperationstatesthat

!A ^{is dupliable.Derelition allowsto onsider the lassialformula} !A ^asthe

linear one A^{. The last operation states that} !A ^{may be forgotten. The dual}

modality 'why not' ? ^{may be} interpreted in the following way: ?A^{⊥ waits for}

the'lassial' resoure!A^{. Thispromotion operationis moreomplexthanthe}

other operations: in terms ofproofnets, orretnessis assuredif a'box' in the

proofnetharaterizestheontext(andthisontexthastobeorretbyitself).

Entriesof suhaboxaregivenbyone! ^andasetof?^.

From MELLuto ?-EBMs. ThetranslationfromformulaeofMELLtomod-

ulesisnotaseasyasitiswithoutexponentials.WeonsideranextensionMELLu

ofMELLwiththeneutralelement1^for⊗^{,formulaearebuiltfromthefollowing}

grammar:

F := 1 |G

G₁, G₂:=A |A^⊥ |G₁⊗1|1⊗G₁ |G₁⊗G₂ |G₁^PG₂ |?G1 |!G1

Convertingfromformulaetomodulesrequirestheuseofpolarizationandfoal-

ization.Foalizationallowstoonsidern^{-aryonnetives.Formulaearepolarized}

negativelyorpositivelyaordingtotheirmainonnetives,onsideringonve-

nientlythatvariablesA, B, . . . ^{arepositivewhereastheirnegations}A^⊥, B^⊥, . . .

are negative. A preise study of the exponential onnetives leads to the a-

knowledgmentthat exponentialonnetiveshange thepolarityof formulae:if

A^{isapositiveformula,}?A^{isnegativewhereas}!A^{⊥ ispositive.Heneexponen-}

tialonnetivesmay be splitinto twoparts:!A^⊥ = ↓♯A^{⊥ and}?A= ↑♭A^{. The}

shiftonnetives↓^and↑^{dothehangingof}polarities.Theintrodutionofshift onnetivesmaybegeneralizedalsotothelinearasewheneverthereisahange

ofpolarity.Thetwomodalities♭^and♯^expressexponentiality.

Weonsider aslightlydierentversionof apolarized systemasit wasde-

signedby Boudes[5℄ orLaurent[13℄:thesystemLL

pol

givenby Laurenttakes

areofmultipliativeaswellasadditiveonnetiveswhereatomiformulaeare

always exponentialized. Following our motivations, our languagen^{MELLpol is}

restritedto themultipliativeasefor simpliityand atomiformulaemay be

linear orexponential. Finallyweuse n^{-ary onnetivesand the}deomposition of exponentialsis expliit. The grammar for n^{MELLpol is givenin the follow-}

ing way where the set of formulae is expliitlysplit into positive (P, . . .^{) and}

negative(N, . . .^)formulae(A^{is apositiveatomiformula):}

P :=N

i∈Iρ_i|♭(N

i∈Iρ_i) ρ :=A | ↓N

N:=^Pk∈K ν_k|♯(^Pk∈K ν_k) ν :=A^⊥ | ↑P

Wekeepasonventionthata1^{-arytensoristheidentityanda}0^-arytensoris

thetensorunit 1^{.Moreover,oneanremarkthatdening}1^as ↓♯⊤^,where⊤^is

theneutralfortheadditiveonnetive &^{,isoherentwithoursettingandmay}

beusefulextending ourframework to additives.Nevertheless,in the following,

thestandardrulefor1^{isimpliitlyaddedtothealuli.Oneandenea}n^-ary

foalizedsequentalulus(A^{isanatomiformula)asinFig.2.Sequentsontain}

a distinguished plae between⊢ ^and ; ^{, they are in one of the two following}

forms:⊢ ; Γ ^or⊢N ; Γ ^whereN ^{isanegativenonatomiformulaand}Γ ^isa

multisetofpositiveformulaeoratominegativeformulae.Thesequentalulus

is designedsuh that,beginningwith thedistinguishedplaeempty,searhfor

proofsonsistsofrepeatingthedeomposition ofapositiveformulafollowedby

thedeompositionofnegativeformulae(neessarilysubformulaeofthepositive

formula just deomposed), until applying axioms. Note that exponential rules

are aspossibleintegrated to linearrules to quotient thesearhspae (e.g. the

axiom rule inludes (♭w^{), (}♭⊗^{) manages (}♭c^{)). The following} translation (−)⁻

from MELLu to n^{MELLpol is suh that if} F ^{is aMELLu formula,} ⊢MELLu F ^is

provablei⊢_nMELLpolF⁻;^isprovable:

1⁺=1 A⁺=A (F1⊗F2)⁺=F₁⁺⊗F₂⁺ (!F)⁺= ↓♯F⁻ F⁺= ↓F^−otherwise A^⊥−=A^⊥ (F1^PF2)⁻=F₁^−PF₂⁻ (?F)⁻= ↑♭F⁺ F⁻= ↑F^+otherwise

⊢ ; A^⊥, A, ♭Ξ (axiom)

⊢1, ♭Ξ (1)⊢ ; Γ, A, ♭Ξ ⊢ ; A^⊥, ∆, ♭Ξ

⊢ ; Γ, ∆, ♭Ξ (cut)

. . . ⊢Ni; Γi, ♭Ξ . . . ⊢ ; Aj, ∆j, ♭Ξ . . .

⊢ ; ^N_i∈I↓NiN

j∈JAj, Γ1, . . . , Γ_|I|, ∆1, . . . , ∆_|J|, ♭Ξ (⊗) . . . ⊢Ni; ♭(N

i∈I↓NiN

j∈JAj), Γi, ♭Ξ . . . ⊢ ; ♭(N

i∈I↓NiN

j∈JAj), Aj, ∆j, ♭Ξ . . .

⊢ ; ♭(N

i∈I↓NiN

j∈JAj), Γ1, . . . , Γ_|I|, ∆1, . . . , ∆_|J|, ♭Ξ (♭⊗)

⊢ ; P1, . . . , P_|I|, A^⊥₁, . . . , A^⊥_|J|, Γ

⊢^Pi∈I ↑Pi^Pj∈J A^⊥_j ; Γ (^P)⊢ ; P1, . . . , P_|I|, A^⊥₁, . . . , A^⊥_|J|, ♭Γ

⊢♯(^Pi∈I ↑Pi^Pj∈JA^⊥_j) ; ♭Γ (♯^P)

Fig.2.n^{-arysequentalulusfor}n^MELLpol(0^-arytensoris1^).

Thenalsteponsistsinatteningn^{MELLpolformulaetogetmodules.Bipo-}

lar modules werepreviously obtainedby adding atomi formulaebetween two

strata(sayfromnegativetopositive):letP₁, P₂ ^{bepositiveformulae,}N ^aneg-

ative formula, ⊢ P₁⊗(N ^P P₂) ^{is provable i} ⊢ P₁ ⊗(N ^P Z^⊥), Z ⊗P₂ ^is

provable,where Z ^{is afresh (positive) atomi formula. Howeverthis priniple}

annot be fully applied when exponentialsour:try to atten the(provable)

sequent⊢ A^{⊥ P} ↑♭(B⊗C), A⊗ ↓♯(B^{⊥ P}C^⊥)^{. This an be overomeby al-}