Stream Associative Nets and Lambda-mu-calculus

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Michele Pagani, Alexis Saurin

To cite this version:

Michele Pagani, Alexis Saurin. Stream Associative Nets and Lambda-mu-calculus. [Research Report]

RR-6431, INRIA. 2008, pp.48. �inria-00221221v3�

a p p o r t

d e r e c h e r c h e

9-6399ISRNINRIA/RR--6431--FR+ENG

Thème SYM

Stream Associative Nets and Λ µ-calculus.

Michele Pagani — Alexis Saurin

N° 6431

janvier 2008

Centre de recherche INRIA Saclay – Île-de-France

Mihele Pagani

∗

, Alexis Saurin

†

ThèmeSYMSystèmessymboliques

Équipe-ProjetPARSIFAL

Rapportdereherhe n°6431janvier200848pages

Abstrat: Λµ^{-alulushas beenbuilt as anuntyped extension of Parigot's} λµ^{-alulusin orderto reoverBöhmtheorem whih wasknown tofail in} λµ^-

alulus. An essential omputational feature of Λµ^{-alulus for separationto}

holdistheunrestriteduseofabstrationsoverontinuationsthatprovidesthe

aluluswithaonstrutionofstreams.

BasedontheCurry-HowardparadigmLaurenthasdened atranslationof

λµ^{-alulusin polarized} proof-nets. Unfortunately, this translation annotbe immediately extended to Λµ^{-alulus: the type system on whih it is based}

freezesΛµ^{-alulus'sstreammehanism.}

We introdue stream assoiative nets (SANE), a notion of nets whih is

between Laurent's polarized proof-nets and the usual linear logi proof-nets.

SANEhavetwokindsofO(heneof⊗^{),oneislinearwhiletheotheroneallows}

freestruturalrules(asinpolarizedproof-nets). WeproveonueneforSANE

and givearedutionpreserving enoding of Λµ^{-alulusin SANE, basedona}

newtypesystemintroduedbytheseondauthor. Itturnsoutthatthestream

mehanismatworkin Λµ^{-alulusanbeexplainedbythe}assoiativityofthe twodierentkindsofO^ofSANE.

Atlast,weahieveaBöhmtheoremforSANE.ThisresultfollowsGirard's

programtoput intotheforetheseparationasakeypropertyoflogi.

Key-words: λµ^{-alulus, linear logi, Böhm theorem,} proof-nets, lassial logi,assoiativityinlogi,ontinuations.

∗

PPS,CNRS&UniversitéParisVIIMihele.Paganipps.jussieu.fr

†

LIXParsifal,INRIA&ÉolePolytehniqueSaurinlix.polytehnique.fr

Résumé: LeΛµ^{-alulaétéintroduitommeuneextensionnon-typéedu}λµ^-

aluldeParigot,remanièreàretrouverlapropriétédeséparation(outhéorème

deBöhm)dontonsavaitqu'elleétaitfausseenλµ^{-alul. Unélémentessentiel}

enΛµ^{-alul pourquelaséparationsoitvalide est}l'utilisation sansrestrition d'abstrationsur lesontinuations qui donnent au alul une onstrutionde

streams.

Fondésurleparadigme deCurry-Howard,OlivierLaurentadéniunetra-

dutionduλµ^{-alulusdanslesréseauxdepreuvepolariés.} Malheureusement, ettetradutionnepeutpasêtre étendueauΛµ^{-aluls: lesystème detypage}

surlequelelleestbaséedésativeleméanismedestreamduΛµ^-alul.

Nous introduisons les stream assoiative nets (SANE), une variante de ré-

seauxquisesitueentrelesréseauxpolariséesdeLaurentetlesréseauxhabituels

delalogiquelinéaire. LesSANEontdeuxtypesdeO^(etdonde⊗^{): l'unest}

linéairetandisquel'autreadmetlibrementdesrèglesstruturellesommedans

lesréseauxpolarisés.

NousprouvonslaonuenepourSANEetprésentonsunerédutionquipré-

servel'enodageduΛµ^{-aluldansSANE.Cetterédution,fondéesurunnou-}

veausystèmedetypageintroduitparleseondauteur.Ons'aperçoitquelemé-

anismedestreamàl'÷uvreenΛµ^{-alulspeutêtreexpliquépar}l'assoiativité desdeuxtypesdeOdesSANE.

Finalement, onmontre un théorème de Böhm pour lesSANE. Le résultat

suitleprogrammedeGirardvisantàdonneruneplaeléàlaséparationparmi

lespropriétésdessystèmeslogiques.

Mots-lés: λµ^{-alul,logiquelinéaire,théorèmedeBöhm,réseauxdepreuve,}

logiquelassique,assoiativitéenlogique,ontinuations.

Stream Assoiative Nets and

Λ µ

Mihele Pagani ,Alexis Saurin

Λµ^{-alulushasbeenbuiltasanuntypedextensionofParigot's}λµ^-alulus

in order to reover Böhm theorem whih was known to fail in λµ^-alulus.

An essentialomputationalfeatureofΛµ^{-alulusforseparationtohold isthe}

unrestrited use of abstrations overontinuations that provides the alulus

withaonstrutionofstreams.

BasedontheCurry-HowardparadigmLaurenthasdened atranslationof

λµ^{-alulusin polarized} proof-nets. Unfortunately, this translation annotbe immediately extended to Λµ^{-alulus: the type system on whih it is based}

freezesΛµ^{-alulus'sstreammehanism.}

We introdue stream assoiative nets (SANE), a notion of nets whih is

between Laurent's polarized proof-nets and the usual linear logi proof-nets.

SANEhavetwokindsofO(heneof⊗^{),oneislinearwhiletheotheroneallows}

freestruturalrules(asinpolarizedproof-nets). WeproveonueneforSANE

and givearedutionpreserving enoding of Λµ^{-alulusin SANE, basedona}

newtypesystemintroduedbytheseondauthor. Itturnsoutthatthestream

mehanismatworkin Λµ^{-alulusanbeexplainedbythe}assoiativityofthe twodierentkindsofO^ofSANE.

Atlast,weahieveaBöhmtheoremforSANE.ThisresultfollowsGirard's

programtoput intotheforetheseparationasakeypropertyoflogi.

Contents

1 Introdution 4

2 Λµ^{-alulus 7}

2.1 λµ^{-alulus,streamsand}Separation: Λµ^{-alulus . . . . . . . . . 7}

2.2 TypingΛµ^{-alulus. . . . . . . . . . . . . . . . . . . . . . . . . . 9}

3 Streamassoiative nets 12

3.1 Rewritingrules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Corretnessriterion . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Conuenetheorem 25

5 Separation theorem 28

6 Simulation theorem 40

7 Conlusion 46

∗

PPS,CNRS&UniversitéParisVIIMihele.Paganipps.jussieu.fr

†

1 Introdution

Curry-Howard in lassiallogi. Curry-Howardisomorphismstatesaor-

respondenebetweenprogramsandproofs. Basially,itexpresses(i)thatatype

anbeseen asa logialformula, and onversely, and (ii) that a program an

be seenas aproof, s.t. theexeution of theprogramorrespondsto applying

theut-eliminationproedure to theassoiated proof, andonversely. Indeed,

thisorrespondenewasatrst limitedto intuitionistilogion theonehand

andto funtionalprogramming(λ^{-alulus)ontheother hand. Extendingthe}

orrespondenetolassiallogiresultedinstrongonnetionswithontrolop-

eratorsin funtionalprogramminglanguagesasrstnotiedbyGrin[Gri90℄.

Inpartiular,λµ^{-alulus[Par92℄wasintroduedbyMihelParigotasanexten-}

sionofλ^{-alulusisomorphitoan}alternativepresentationoflassialnatural dedution(knownasfreededution)in whihoneanenodeusualontrolop-

eratorsandinpartiulartheall/operator.

Polarizedlinear logi. BasedontheextensionoftheCurry-Howardisomor-

phismtolassiallogi,LaurentdenesatranslationofParigot'sλµ^-alulusin

polarizedlinearlogi: avariantoflinearlogi(LL),allowingfreestruturalrules

onnegativeformulas[Lau02℄. Laurent'stranslationenlargestheomparisonbe-

tweenLLandusualλ^{-alulus,startedfromGirard[Gir87℄,Danos[Dan90℄and}

Regnier[Reg92℄. Inpartiular polarized LLprovidesalassof proof-nets(the

graph-theoretialrepresentationof LL proofs)orrespondingto the λµ^-terms,

soshedingnewlightinto theomputationofλµ^-alulus.

λµ^{-alulus and} Separation. λµ^{-alulusbeameone of themoststandard}

waystoexaminelassiallambda-aluli. Asaresult,thealulushasbeenmore

andmorestudied andmorefundamental questionsarose. Amongthem, oneof

themostimportantisseparation. Thebestknownexampleofseparationresult

isBöhm'stheorem forthepureλ^{-alulus[B68℄: if}t, t^{′ aretwodistintlosed} βη^{-normalterms,thenthereexistterms}u1, . . . , un^,suhthat(t)u1. . . un →β x

and(t^′)u1. . . un→βy^{. Thisresulthasonsequenesbothatthesemantiallevel}

aswellasatthesyntatialone: ontheonehanditentailsthatamodelofthe

λ^{-alulusannotidentify twodierent}βη^{-normalformswithoutbeingtrivial;}

onthe other hand it establishes abalane betweensyntatial onstrutsand

β^{-redution: anydiereneinthestrutureofa}βη^{-normalformimpliesadier-}

eneinthevalueofthat normalform onsuitablearguments. In2001David &

PyaddressedthequestionofseparationtoParigot'sλµ^{-alulusandtheygavea}

negativeanswerbyexhibitingaounter-example[DP01℄. Inapreviousworkof

2005,theseondauthorintroduedanextensiontoλµ^-alulus,Λµ^-alulus,for

whih heouldprovethat separationholds [Sau05℄. Λµ^{-alulusisfairly lose}

tostandardpresentationsofλµ^{-alulus(see[dG94,dG98℄forinstane),butis}

denitelyadierentalulus. Inpartiular,anessentialomputationalfeature

ofΛµ^{-alulusforseparationtoholdisthe}unrestriteduseofabstrationsover ontinuationsthatprovidesthealuluswithaonstrutionofstreams.

The logi of Λµ^{-alulus. We pursue an} investigation of the logi behind

Λµ^{-alulus. Ourfeelingis thattherulesof lassiallogi imposesatoostrit}

disipline over the use of streams: in Parigot's λµ^{-alulusstreams represent}

onlyhannelsthroughwhih termsanbesent,thesehannelsanbeplugged

toeahother,theyanbeexhanged,buttheydonotreallyommuniatewith

the termsin theourse ofa omputation. Streams andterms live in dierent

worlds,inpartiulartheformeronesarenotrstlassitizensintheearlyver-

sionsofλµ^{-alulus. Wethinkthatthe}Curry-Howardisomorphismatthebase ofParigot'sλµ^{-alulusrestritstoomuhthe}omputationalpowerofstreams, a onsequeneof whih is thefailure of theseparationproperty,as provedby

David &Py. If we forget theCurry-Howardisomorphisms and startto build

morefreelytheprogramsinΛµ^{-alulus,thenwegetbaktheseparationprop-}

ertyandinthesametimewemoveawayfromlassiallogi.

StreamAssoiativeNEts: fromtherulesoflassiallogitothelogi

of Λµ^{-alulus rules. This}turning-pointindues ahangeoftheenodingof

λµ^-alulusintoproof-nets: indeedLaurent'stranslationisbasedontheCurry- Howardisomorphismwithlassiallogi. Wefollowanotherdiretion,in order

tohaveanenodingofΛµ^{-aluluswhihismorefaithfulltothestreambehavior}

at thebase oftheseparationproperty. Webelievethat itisbydepartingfrom

therulesoflassiallogithatwewill understandthereallogiofΛµ^-alulus

rules.

Wethusdeneanewlassofnets,StreamAssoiativeNEts(SANE).SANE

lies in betweenusual linearlogiproof-netsand polarizedproof-nets: wehave

twokindsofO(anddually of⊗^{),oneomingfromLL(assoiatedwiththe}λ^-

variables)andtheotheroneomingfrompolarizedLL(andassoiatedwiththe

µ-variables). Theessentialingredientistheassoiativitypropertybetweenthese twokindsofmultipliatives,whihmakespossibletheommuniationbetween

streamsandλ^{-variablesmuh inthesamewayasfstruledoesin} Λµ^-alulus.

Better be in SANE to study Λµ^{-alulus. The} orrespondene between

Λµ^{-alulusandSANEwillallowfor}onsiderabletransfersoftehnologiesbe- tweenthetwodomains,inpartiularproof-netswillprovidepowerfulgeometri-

alabstrationsandadeeplysymmetrialframeworkaswellasstrongdualities.

Inadditiontoaner-grainedstudyoftheredutionrulesofΛµ^{-alulus(asem-}

phasized byour simulationresult), SANE redutions will provide Λµ^-alulus

with a notionof expliit substitution. Moreover,SANE should help studying

therelationshipsofΛµ^{-aluluswithother}ontinuation-basedaluli.

Proof-nets with separationproperty. SANE havebeendesigned in order

to studyΛµ^{-alulus,butseparationpropertyplaysakeyrolein thetheoryof}

SANE.AsinLudis[Gir01℄whereGirardhoseseparationtobearequirement

for his elementary objets, the designs, the nets we introdue in the present

work have been designed with separation property to be at the heart of the

theory,muhinthesamewayasonuenedoes.

Struture ofthe Paper. Thefollowingsetionisdediatedto ashort intro-

dutiontoParigot'sλµ^{-alulustoseparationrelatedtopisandto}Λµ^-alulus.

A new type systemfor Λµ^{-alulus is provided whih servesas abasis to de-}

ne, in setion 3,the pureStream assoiativenets,their redutions,statethe

orretness riterionfor SANE and proveanoriginal strong normalizationre-

sult of s,r,a ^{whih impliesthestrong} normalizationof exponentialredution

in SANE 1

. The followingsetion is dediated to proving onuene of SANE

beforegoingtothequestionoftheseparationpropertyinsetion5. Finally,we

simulateΛµ^{-alulusinSANEin setion6.}

1

ThisgivesasaorollarytheSNoftheimpliitexpliitsubstitutionsystem

2

Λ µ

2.1

λµ

Λ µ

David & Py ounter-example to Separation in λµ^{-alulus. In their}

2001 paper [DP01℄, David & Py addressed the question of separation prop-

ertyinλµ^{-alulusbyexhibitinga}ounter-exampletoseparation,theλµ^-term W =λx.µα.[α]((x)µβ.[α](x)U0 y)U0 ^withU0=µδ.[α]λz1.λz2.z2^{. Separation}

propertyfails in this setting beausethere is no way to put thevariable y ⁱⁿ

headposition. Thekeypointisthattheentireappliativeontextinwhihthis

term is plaed is transmitted through µα^{to subterms; asa onsequene, the}

usualtehnique(whihonsistsinbuildingaontextthatshallexplorethepart

ofthetermwewant)annotbeapplied.

Reovering Separation in λµ^{-alulus: relaxing impliit}(underlying) typing onstraints. What we doby introduing Λµ^{-alulusis preisely to}

bemoreliberal withtheonstrutionoftermsinorderto providethealulus

with more appliative ontexts and retrieve the ability to realize the needed

explorationpaths. Inpartiular,Parigot'sλµ^{-alulussyntax hasaonstraint}

ofnamingatermrightbeforeitisµ^{-abstrated(termshavetheform}µα.[β]^_)

whih anatually beseen asatyping onstraintdiretly built in the syntax

of the untyped alulus. Λµ^{-alulus is basially the result of removing this}

onstraint. By doing so, we obtain a alulus whih is lose to de Groote's

presentation of λµ^{-alulus but it is not equivalent to this alulus sine de}

Groote's presentation also ontains a typing onstraint whih is built in the

syntax,namelytheǫ^{rulethatisabsentfrom}Λµ^-alulus2.

Λµ^{-aluluswasintroduedin [Sau05℄asanuntyped extensionofParigot's} λµ^{-alulus in whih separation holds. Given twoinnite disjoint sets} Vt ^(of

term variables,denoted byx, y, z . . .^{) and}Vs ^{(of streamvariables, denoted by}

α, β, γ . . .^),Λµ^{-alulusisdened bythefollowinggrammar:}

t, u...::= x|λx.t| (t)u|µα.t|(t)α

Anabstration isatermofshapeλx.t^orµα.t ^{andanappliation is aterm}

of shape(t)u ^or(t)α^{. We referto the appliation of an abstration asa ut.}

There arefour kindsofutsin Λµ^{-alulusasshownin gure1:} (T)T^, (T)S^, (S)T^,(S)S^.

Λµ^{-alulusredutions.}

Cutsoftype(T)T ^and (S)S ^{areredexesforthefollowingrules:}

(λx.t)u →βT t[u/x] ⁽¹⁾

(µα.t)β →βS t[β/α] ⁽²⁾

Bututsoftype(S)T ^and(T)S ^{arenotredexesfortheserules.}

2

Theresultofthe ǫ^ruleinΛµ^{-aluluswouldatuallybetoanelmultiplestreamab-}

(T)T : (λx.t)u (T)S: (λx.t)α

(S)T : (µα.t)u (S)S : (µα.t)β

Figure1: Cutsin Λµ^-alulus.

⊲(λx.t)u −→βT t[u/x]

⊲ λx.(t)x −→ηT t

⊲(µα.t)β −→βS t[β/α]

⊲ µα.(t)α −→ηS t

⊲ µα.t −→fst λx.µβ.t[(U)xβ/(U)α]

Proviso:

Inη,fst^, x6∈F Vt(t)^;in ηs^,α6∈F Vs(t)

Figure2: Λµ^{-alulusredutionrules}

The following fst^{-rule relates term variables with stream variables, it is a}

way to aess therst term of thestream and it will allowto redue the last

twotypesofuts:

µα.t →fst λx.µβ.t[(U)xβ/(U)α] ⁽³⁾

ˆ Indeedthefst^{-rulemakesreduibletheutsoftype}(S)T^:

(µα.t)u→fst (λx.µβ.t[(U)xβ/(U)α])u→βT µβ.t[(U)uβ/(U)α]

ˆ aswellasthoseoftype(T)S^{,wheneversubtermsofalosedterm:}

µβ. . . .(λx.t)β· · · →f stλx.µβ. . . .(λx.t)xβ· · · →βλx.µβ. . . .(t)β . . .

Thefollowingrulesdenesextensionalequivalenes(withtheusual proviso

x /∈F VT(t)^andα /∈F VS(t)^):

λx.(t)x →ηT t ⁽⁴⁾

µα.(t)α →ηS t ⁽⁵⁾

Λµ^{-alulusredutionrulesaresummarizedin gure2.}

InΛµ^-alulus,µ^{anbeseenasanabstrationoverstreamsofterms3.}

For instane, while λx.λy.λz.((z)(t)xy)(t^′)xy ^{may dupliate twoterms passed}

3

Streamsasrst-lassitizensareonsequenesofmoreextensionalityinΛµ^-alulusthan

inλµ^{-alulus,duetothefatthatitispossibletousethe}extensionalityrulesη^andηs^where

λµ^{-alulussyntaxforbidstodoso,forinstane:} µα.(t)β→ηµα.(λx.(t)x)β^.

V arT

Γ, x:T ⊢x :T |∆

Γ, x:T ⊢t :T^′|∆

AbsT

Γ⊢λx.t :T → T^′|∆

Γ⊢t :T → T^′|∆ Γ⊢u :T |∆

AppT

Γ⊢(t)u :T^′|∆

Γ⊢t :⊥|∆, α:A Γ⊢µα.t :A|∆ µAbs

Γ⊢t :A|∆, α:A Γ⊢(t)α :⊥|∆, α:A µApp

Figure3: Λµ^{-alulusClassialTypeSystem.}

throughx^andy^,Λµ^-termµα.µβ.λz.((z)(t)αβ)(t^′)αβ^{andupliatetwostreams}

ofterms,these streamsbeingforinstaneappliedthroughtheappliativeon-

text: []t1. . . tkγu1. . . ulδ^.

Comparedtoλµ^{-aluluswheretheeetof}µ^{isonlytoredirettheompu-}

tation ow, inΛµ^{-alulus,oneanmanageto dealwith streamsasrst-lass}

itizens: forinstane,µα.µβ.λx.λy.x^{isatermthaterasestwostreamsofterms}

andreturnsthebooleanvaluetrue. Aspreviouslysaid,Λµ^{-alulushasbeende-}

signedinordertoreovertheseparationproperty. Theoriginalounter-example

toseparationbyDavid&Py[DP01℄,W^{,issolvedbythefollowing}Λµ^-ontext:

C = []Px0x1α0α1α ^where P = λz0, z1.µγ.λu.((u)µβ.z1)z0^: C(W) →^⋆ y ^(see

[Sau05℄ formoredetails).

2.2 Typing

Λ µ

Typing Λµ^{-alulus as} λµ^{-alulus. Oneouldthink of typing} Λµ ^usinga

standardtypesystemforlassiallambda-aluliasshowningure3. However,

thisapproahisnotsatisfatoryonsideringourmotivationsindeveloppingthe

new alulus, that is from the point of view of separation. Indeed, the main

struturesusedin[Sau05℄inordertoobtainseparationwouldnotbetypablein

thesystemofgure3andforveryfundamentalreasons. Anytermoftheform

µα.λx.t^{wouldbeuntypablewhereasthisisthetypialtermusedintheproofof}

separationforΛµ^{-alulus. Infat,thetypingsystemoriginally introduedin}

ordertoonnetthealuluswithfreededution[Par92℄preiselyforbidssuh

terms: λx.t^isaλ^{-abstratedtermandthusshallbeofan}→^{-typewhereasthe}

fat that itis µ^{-abstratedthroughstream variable}α^{fores thetermto beof}

type⊥^whihisinompatible(seeruleµ^{Absin gure3).}

Making streams rst-lass itizens in the typed setting. The stream

mehanismthatwasusedintheuntypedalulusinordertoobtainseparation

is thus desativatedwhen lassialtypesare reintrodued. Weshalllook fora

variantofthistypesystemthat wouldreetintypesthestreamonstrution.

In partiular, sine µ ^{is seen as a stream} abstration, one might think of a funtional type for streams: if the term t ^{is of type} T ^{when stream} α ^{is of}

stream typeS^,then µα.t^{would beofthetypeofastream funtionfrom} S ^to T ^{(that wewrite} S ⇒ T^{). Weanthusthink ofthefollowingtyping rulesfor}