Paths-based criteria and application to linear logic subsystems characterizing polynomial time

0.49 MB·

by  Matthieu Perrinel

Paths-based criteria and application to linear logic
subsystems characterizing polynomial time
MatthieuPerrinela
aLIP,ENSLyon,46alled’Italie,69007Lyon,FRANCE
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Abstract
a
J Severalvariantsoflinearlogichavebeenproposedtocharacterizecomplexityclasses

5 intheproofs-as-programscorrespondence.Lightlinearlogic(LLL)ensuresapolyno-
mialboundonreductiontime,andcharacterizesinthiswaypolynomialtime(Ptime).
]
O In this paper we study the complexity of linear logic proof-nets and propose three
semantic criteria based on context semantics: stratification, dependence control and
L
nesting. Stratificationaloneentailsanelementarytimebound,thethreecriteriaentail
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s togetherapolynomialtimebound.
c
[ These criteria can be used to prove the complexity soundness of several existing

variants of linear logic. We define a decidable syntactic subsystem of linear logic:
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SDNLL. We prove that the proof-nets of SDNLL satisfy the three criteria, which
v
impliesthatSDNLL is soundfor Ptime. Severalprevioussubsystemsof linearlogic
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1 characterizingpolynomialtime(LLL, mL4,maximalsystemof MS)areembeddedin
4 SDNLL,provingitsPtimecompleteness.
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.
1 1. Introduction
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7 Motivations for a type-system capturing polynomial time. Programming is a notori-
1 ously error-proneprocess. The behaviours of the programs written by programmers
:
v on their first attempt often differ from their expected behaviours. Type systems can
i detectsome of those mistakes so thatprogrammerscan correctthem more easily. In
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thiswork,thepropertyweareinterestedinistimecomplexity:theexecutiontimeofa
r
a programasafunctionofthesizeofitsinput. AtypesystemS enforcingapolynomial
boundonthetimecomplexityofaprogramwouldbeusefulinseveralways:
• In some real-time applications (e.g. car control systems) programs can never
missadeadline,otherwisethewholesystemisafailure.Itisnotenoughtoverify
thatthesystemreactedfastenoughduringtests,weneedanabsolutecertainty.
• For some software, it seems enough to get an empirical estimate of the com-
plexity by running tests. In this case, S could be useful to find the origin of
theslownessobservedduringtests(thisrequiresthetypeinferrertogiveuseful
informationwhenitfailstotypeaterm).
Emailaddress:[email protected](MatthieuPerrinel)
PreprintsubmittedtoInformationandComputation
• Incomplexitytheory,themainmethodtoprovethataproblemisNP-complete,
istodefineapolynomialtimereductionfromanotherNP-completeproblem. If
S iswell-trusted,itcouldbeusedasaspecializedproofassistant:thefactthatthe
reductionistypableinS wouldincreasethetrustintheproof.Moregenerally,S
couldbeusedinanyproofrelyingonacomplexityboundforaprogram[23,29].
Inthiswork,wedefineasubsystemSDNLLoflinearlogicsuchthateveryproof-net
normalizes in polynomial time. This property is called Ptime soundness. And, for
everyfunction f computablein polynomialtime thereexists a SDNLL proof-netG
f
whichcomputes f. ThispropertyiscalledPtimeextensionalcompleteness.
Determiningif a proof-netnormalizesin polynomialtime is undecidable. So for
everysuchsystemS,eitherdeterminingifaproof-netGbelongstoS isundecidable,or
S isnotintensionalcomplete:i.e. thereexistprogramswhichnormalizeinpolynomial
time and are not typable by S. The subsystem SDNLL is in the second case. We
takeinspirationfrompreviousdecidabletypesystemscharacterizingPtime andrelax
conditionswithoutlosing neithersoundnessnordecidability. The moreintensionally
expressiveS is (i.e. the moretermsare typableby S), themore usefulS is. Indeed,
thethreemotivationsforsystemscharacterizingpolynomialtimewedescribedearlier
require S to type programs written by non-specialists: people who may not have a
thoroughunderstandingofS.
Linearlogicandproof-nets. Linearlogic(LL)[13]canbeconsideredasarefinement
ofSystemFwherewefocusespeciallyonhowtheduplicationofformulaeismanaged.
In linear logic, the structural rules (contraction and weakening) are only allowed for
formulaeoftheshape!A:
Γ,!A,!A⊢ B Γ ⊢ B
?C ?W
Γ,!A ⊢ B Γ,!A⊢ B
Withthethreefollowingadditionalrules(promotion,derelictionanddigging),lin-
earlogicisasexpressiveasSystemF,sotheeliminationofthecutrule(corresponding
totheβ-reductionofλ-calculus)isnotevenprimitiverecursive.
A ,··· , A ⊢ B Γ, A⊢ B Γ,!!A⊢ B
1 n
!P ?D ?N
!A ,··· ,!A ⊢!B Γ,!A⊢ B Γ, !A⊢ B
1 k
However,becausethestructuralrulesarehandledby5distinctrules,onecanenforce
asubtlecontrolontheuseofressourcesbymodifyingoneofthem.Ifwerestrictsome
ofthoserules, itrestrictstheduplicationofformulae. Forinstance,intheabsenceof
?D and ?N rules, the cut-eliminationnormalizesin elementarytime [10]. The set of
suchproofsisdefinedasElementaryLinearLogic(ELL).
Proof-nets[14]areanalternativesyntaxforlinearlogic,whereproofsareconsid-
ered up-to meaningless commutations of rules. Proof-nets are graph-like structures
wherenodescorrespondtologicalrules. Oneofthereasonsweuseproof-netsinstead
of proof derivationsis that contextsemantics, the main tool we use in this article, is
muchsimplertodefineanduseinproof-nets.
2
Contextsemantics. Contextsemanticsisapresentationofgeometryofinteraction[17,
11] definedbytokenstravelingacrossproof-netsaccordingto somerules. The paths
definedbythosetokensarestable byreductionso theyrepresentthereductionof the
proof-net.Contextsemanticshasfirstbeenusedtostudyoptimalreduction[18].
Recently, it has been used to prove complexityboundson subsystems of System
T[7]andlinearlogic[4, 8]. In[8], DalLagodefinesforeveryproof-netG a weight
W ∈N∪{∞}basedonthepathsofcontextsemanticssuchthat,wheneverGreduces
G
toH,W ≥W +1. ThusW isaboundonthelengthofthelongestpathofreduction
G H G
startingfromG. Thenwecanprovetheoremsoftheshape“wheneverGsatisfiessome
property (for instance if G belongs to a subsystem such as LLL), W satisfies some
G
bound(forinstanceW ≤ P(|G|)withPapolynomialand|G|thesizeofG).”
G
From this point of view, context semantics has two major advantages compared
to the syntactic study of reduction. First, its genericity: some common results can
be proved for different variants of linear logic, which allows to factor out proofs of
complexityresultsforthesevarioussystems. Moreover,theboundsobtainedstandfor
anystrategyofreduction.Onthecontrary,mostboundsprovedbysyntacticmeansare
onlyprovedforaparticularstrategy.Thereareseveraladvantagestostrongbounds:
• Let us suppose we know a strong complexity bound for a system S′. We can
provethesamestrongcomplexityboundonasystemS ifwefindanembedding
φ of S programs in S′ programs such that, whenever t reduces to u in S, φ(t)
reduces to φ(u) in S′ (with at least one step). We use such an embedding in
Section 5.4 to provea strong boundfor λ-terms typedby SDNLL. If we only
had a weak complexity boundfor system S′, we would have to prove that the
reductionfromφ(t) to φ(u)matchesthe reductionstrategyentailingthe bound,
whichisnotalwayspossible.
• Thelanguageswestudyhereareconfluent.However,ifweconsideranextension
oflinearlogicorλ-calculuswithside-effects(suchasλ!R consideredbyMadet
and Amadio in [22]), the reduction strategy influences the result of a program
execution. It is importantthat the programmerunderstandsthe strategy. If the
reduction strategy corresponded to strategies frequently used by programming
languages(suchasleft-to-rightcall-by-value),itwouldnotbeaproblem. How-
ever, in some cases(mL4 for instance[3]), the strategyis ratherfarfetchedand
difficulttounderstandfortheprogrammer.
Our context semantics, presented in Section 2.2, is slightly different from Dal
Lago’scontextsemantics.Inparticular,DalLagoworkedinintuitionnisticlinearlogic,
andweworkinclassicallinearlogic. Sotheresultsof[8]cannotbedirectlyapplied.
Howevermosttheoremsof[8]havecorrespondentsinourframework,withquitesim-
ilar proofs. This is why we omit the proofs of most of the results of this section,
completeproofscanbefoundin[26].
Our approach. Contrary to previous works, we do not directly define a linear logic
subsystem. First, we define semantic criteria forbiddingbehaviourswhich can result
in non-polynomialcomplexity. We definerelations→ on boxes(specialsubtermsof
proof-nets) such that B → C means that ”the number of times B is copied depends
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SDNLL Decidable
Undecidable?
L4
0 MS QBAL
L4
LLL SLL BLL
Figure1: Stateoftheart
onthenumberoftimesC iscopied”. Moreprecisely,wedefinethreerelations( −−−>>>,
... and ///)representingdifferentkindsofdependence. Theacyclicityofthesethree
relationsensuresa boundon the numberof times everybox is copiedso a boundon
thelengthofnormalizationsequences.
Then(inSection5),wedefineStratifiedDependencecontrolNestedLinearLogic
(SDNLL),asubsystemoflinearlogicsuchthat −−−>>>, ... and ///areacycliconevery
proof-netof SDNLL. This entails a bound on the length of normalization for every
SDNLL proof-net. The relations ( −−−>>>, ... and ///) are based on the paths of con-
text semantics. We use the syntactic restrictionsto define invariantsalong the paths,
proving that if B → C then the ”types” of B and C are such that we cannot have
C → B.Finally,inSection5.4,wetransformSDNLLintoatype-systemSDNLL for
λ
λ-calculus,whichenforcesapolynomialboundonβ-reduction. Thistranformationis
similartothetransformationofLLL[15]intoDLAL[5].
PreviouspolynomialtimesubsystemsofLinearlogic. Therealreadyexistseveralsub-
systems of linear logic characterizing polynomial time. The first such subsystem is
BLL[16],whichenforcesPtimesoundnessbylabellingof!modalitiesbypolynomi-
als. However, given a proof-netG, determiningif G is in BLL (or its generalization
QBAL[9])seemsundecidable.Thus,theydonotfitinourapproach.
ThefirstdecidablesystemwasLLL[15]whichisdefinedastheproof-netsofELL
such that the contextshave at most one formula in every !P rule1. A decidable type
systemDLALforλ-calculuswasinspiredbyLLL[5,1].
BaillotandMazzageneralizedELLwithasubsystemL3 oflinearlogiccharacter-
izingelementarytime[3]. ThentheydefinedmL4 andmL4,characterizingpolynomial
0
time, based on L3 in the same way as LLL is based on ELL. In a separate direction,
RoversiandVercellialso extended LLL with MS2 [28]. Thosethreesystemsareob-
tained by decoratingformulae with labels and adding local constraints on the labels.
mL4,mL4andMS aretriviallydecidableonproof-nets:givenaproof-netGthereexist
0
onlyafinitenumberofwaystolabeltheformulaeofG. Onecantryeverypossibility
andcheckwhetherthelabelsverifytheconstraints. LafontdefinedSLL[20],another
subsystemoflinearlogiccharacterizingpolynomialtime.Thissystemdoesnotcontain
LLL,andnoneoftheabovegeneralizationsofLLLcontainsSLL.
Figure1 summarizesthe state ofthe art. Thereis an arrowfromthe system S to
the system T if there is a canonical embedding of S in T. The arrow between MS
1Tokeepsomeexpressivity,Girardaddsanewmodality§.
2Whichisasetofsystemratherthanauniquesystem.
4
andSDNLL isdottedbecausethe embeddingisonlydefinedforoneofthemaximal
systems of MS. In [26], we define SwLL (based on the the ideas of this article) in
whichonecanembedSDNLL,SLL,andeveryMS polynomialsubsystem.
Thispaperextendsapreviouswork[24]by: providinganon-trivialnestingcondi-
tion,definingasyntacticsubsystembasedonthesemanticcriteria,andprovidingmost
of the proofs (in [24] the proofs are only sketched). More details, and the technical
proofsomittedinthispapercanbefoundinPerrinel’sthesis[26].
2. LinearLogicandContextSemantics
2.1. LinearLogic
Linearlogic(LL) [13] canbe consideredas a refinementofSystem F[12] where
wefocusespeciallyonhowtheduplicationofformulaeismanaged. Inthisworkwe
useneithertheadditives(⊕and&)northeconstants. Thisfragmentisusuallynamed
MultiplicativeExponentialLinearLogicwithQuantifiers(abbreviatedbyMELL ).To
∀
simplify notations, we will abusively refer to it as Linear Logic (abreviated by LL).
ThesetF ,definedasfollows,designsthesetofformulaeoflinearlogic.
LL
F =X | X⊥ |F ⊗F |F `F |∀X.F |∃X.F |!F |?F
LL LL LL LL LL LL LL LL LL
We define inductively an involution ( )⊥ on F , which can be considered as a
LL
negation: (X)⊥ = X⊥, (X⊥)⊥ = X, (A ⊗ B)⊥ = A⊥ ` B⊥, (A ` B)⊥ = A⊥ ⊗ B⊥,
(∀X.A)⊥ =∃X.A⊥,(∃X.A)⊥ =∀X.A⊥,(!A)⊥ =?(A⊥)and(?A)⊥ =!(A⊥).
Linearlogicisusuallypresentedasasequentcalculus(asintheintroduction). In
thisarticle,wewillconsideranalternativesyntax:proof-nets[14].
Definition1. ALLproof-netisagraph-likestructure,definedinductivelybythegraphs
of Figure 2 (G and H being LL proof-nets). Every edge e is labelled by β(e) ∈ F
LL
satisfyingtheconstraintsofFigure2. Thesetofedgesiswritten~E .
G
Aproof-netisagraph-likestructure,whoseedgesarenotlabelled,definedinduc-
tivelybythegraphsofFigure2(GandHbeingproof-nets).TheconstraintsofFigure2
onlabelsarenottakenintoaccount.
Forthefollowingdefinitions,wesupposedfixedaproof-netG.
Directed edges. The edges in the definition of proof-nets (the elements of E~ ) are
G
directed. We willoftenneedtoconsidertheirinvertededges: forany(l,m)(theedge
from l to m), we denote its inverted edge (m,l) (the edge from m to l) by (l,m). We
definetheset E as ~E ∪{e|e ∈ ~E }. In LL proof-nets,we extendthe labellingβ( )
G G G
fromE~ toE byβ(e)=β(e)⊥.
G G
Premisesandconclusions. Foranynoden,theincomingedgesofn(inE~ )arenamed
G
the premises of n. The outgoingedges of n (in E~ ) are named the conclusionsof n.
G
Someedgesarenotthepremisesofanynode.SuchedgesaretheconclusionsofG.
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ax G H
A A⊥
A cut A⊥
G H G G G
A B A B A[B/X] A
⊗
A⊗B ` ∃ ∀
A`B ∃X.A ∀X.A
G
A1 An B G G ?W G G
?P ?P !P ?A ?A ?A A ??A
?C ?D ?N
?A1 ?An !B ?A ?A ?A
Figure2: Constructionof LLproof-nets. Inthe∀rule, X cannotbefreeintheother
conclusionsofG.
··· ··· ··· ··· ax ··· ···
A A⊥[B/X] A[B/X] A⊥[B/X] A A cut A
a cut b cut
∀ ∃
cut
∀X.A cut ∃X.A⊥
······ ······
A B A⊥ B⊥ cut AaBb eA⊥fB⊥
`ab ⊗ef cut cut
A`B A⊥⊗B⊥
c cut d
Figure3: Non-exponentialcut-eliminationsteps. Forthe∀/∃step,[B/X]takesplace
onthewholenet.
Boxes. TherectangleofFigure2withthe?Pand!Pnodesiscalledabox. Formallya
boxisasubsetofthenodesoftheproof-net.Wesaythattheedge(m,n)∈E~ belongs
G
toboxBifn∈ B,inthiscase(n,m)alsobelongstoboxB.
Letuscall BtheboxinFigure2. Thenodelabelled!PistheprincipaldoorofB,
itsconclusioniswrittenσ(B),andisnamedtheprincipaledgeofB.The?Pnodesare
theauxiliarydoorsofbox B. Theedgegoingoutofthei-th auxiliarydooriswritten
σ(B)and is namedan auxiliaryedge of B. Thedoorsof box B are consideredin B,
i
theyareexactlythenodeswhichareinBbutwhoseconclusionsarenotinB.
The number of boxes containing an element (box, node or edge) x is its depth
written∂(x).∂ isthemaximumdepthofanedgeofG. ThesetofboxesofGisB .
G G
Cut-elimination. isarelation→ on(LL)proof-netswhichisrelatedtotheβ-reduction
cut
ofλ-calculus.Figures3and4describetherulesofcut-elimination.
Lemma2. [14]Proof-netsandLLproof-netsarestableundercut-elimination.
6
G′ ··· G′ ···
A1 Ak AB A⊥ A1 Ak A bA⊥
?aP1 ?aPk !aP ?bD cut ?aD1 ?aDk a cut
c1?A1 ck?Ak c !Acut d?A⊥ c1?A1 ck?Ak
G′ H′ G′ H′
cut
A1 Ak A A⊥ Bl B A1 Ak A A⊥ B2 Bl B
?aP1 ?aPk !aP ?bP1 ?bPl !bP ?aP1 ?aPk a cut b1b?2P ?bPl !bP
c1?A1ck?Ak c !Acut d1?A⊥dl?Bl d!B c1?A1ck?Ak d2?B2dl?Bld!B
G′
A1 Ak A
?aP1 ?aPk !aP ?W cut ?W ?W
d1?A1dk?Akc !Acut d?A⊥ d1?A1 dk?Ak
G′ ··· G′
A1 Ak A ??A⊥ A1 Ak A
B B
?aP1 ?aPk !aP ?Nf cut ?aP1 ?aPk !aP i
c1?A1ck?Ak c!A cut d?A⊥ ?bP?A1 ?bP?Ak !bP!A Be ···
1 k ??A⊥
??A1 ??Ak !!A f
?eN1 ?eNk c cut
?A ?A
c 1 c k
1 k
G′ ··· ··· G′l G′r ··· ···
?aP1A1 ?aPkAk !aPAB ?A⊥ ?Cfg?A⊥cut ?aPl1Ak ?aPlkA1 !aPlABl ?aPr1Ak ?aPrkA1 !aPrABr
c1?A1ck?Akc!A cut d?A⊥ ?Ab1l1 ?Abklk cl!A cut ?Af⊥ !cAr cut?Ag⊥
?C b?Ar11?C b?Arkk
?A ?A
1 k
c c
1 k
Figure4: Exponentialcut-eliminationsteps
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2.2. DefinitionofContextSemantics
A common method to prove strong bounds on a rewriting system is to assign a
weight W ∈ N to each term G such that, if G reduces to H, W > W . In LL,
G G H
the !P/?C step makes the design of such a weight hard: a whole box is duplicated,
increasing the number of nodes, edges, cuts,... The idea of context semantics is to
defineW as|A |withA theedgeswhichappearduringreduction:edgesofanetG
G G G k
suchthatG → G → ··· → G 3. We cannoticethatwheneverG → H,we
cut 1 cut cut k cut
have A ⊆ A : if e ∈ A thene isanedgeofa proof-netH with H → H ···H
H G H k cut 1 k
soG → H → H ···H . Moreover,thisinclusionisstrictbecausethepremisesof
cut cut 1 k
thecutreducedbetweenGandH areinA butnotinA . Thus,|A |≥|A |+24.
G H G H
However, such a definitionof W would be impractical: provinga boundon W
G G
doesnotseemeasierthandirectlyprovingaboundonthenumberofreductionsteps.
ThesolutionofcontextsemanticsistoconsiderforeveryedgeeofE ,thesetCan(e)
G
of residues of e: the elements of A “coming” from e. For instance, in the leftmost
G
proof-netofFigure5,theresiduesofeare{e,e ,e ,e ,e }. ThesetCan(E )ofevery
1 2 3 4 G
edge residue is contained in A but is not always equal to it (e.g. the premises of
G
the two cuts in the middle proof-netof Figure 5 are not residues of any edge of the
leftmost proof-netso they are in A but not in Can(E ). Nonetheless, we still have
G G
|Can(E )|≥|Can(E )|+1wheneverG → H,sothelengthofanypathofreduction
G H cut
beginningbyGisatmost|Can(E )|(Theorem11).
G
ToboundW ,wecharacterizeedgeresiduesbycontextsemanticspaths,simulating
G
cut-elimination. Thosepathsaregeneratedbycontextstravellingacrosstheproof-net
accordingtosomerules. Thepathsofcontextsemanticsina proof-netG areexactly
the paths which are preserved by cut-elimination (such paths are called persistent in
the literature [11]). Computing those paths is somehow like reducing the proof-net.
Proving boundson the number of residues thanks to those paths rather than proving
boundsdirectlyonthereductionofferstwoadvantages:
• Complex propertieson proof-nets, which may be hard to manipulate formally,
aretransformedintoexistence(orabsence)ofpathsofacertainshape.
• ForeveryG → H,wehaveW > W . Thus,thelengthofanynormalization
cut G H
sequenceisboundedbyW . Theboundsobtainedin thispaperdonotdepend
G
onthereductionstrategy.
Torepresentlistsweusethenotation[a ;··· ;a ]. Torepresentconcatenation,we
1 n
use@:[a ;··· ;a ]@[b ;··· ;b ]isdefinedas[a ;··· ;a ;b ;··· ;b ]and.represents
1 n 1 k 1 n 1 k
“push”([a ;··· ;a ].bisdefinedas[a ;··· ;a ;b]).|[a ;··· ;a ]|refersto j,thelength
1 n 1 n 1 j
ofthelist. IfXisaset,|X|isthenumberofelementsofX.
A contextisa pair ((e,P),T)composedof a potentialedge(e,P)representingan
edgeresidue(eisadirectededgeoftheproof-net)andatraceT usedtoremembersome
informationaboutthe beginningof the path. Thisinformationis necessaryto ensure
thatthepathsarepreservedbycut-elimination.Thefollowingdefinitionsintroducethe
componentsofpotentialedgesandtraces.
3Weidentifyedgeswhichareunaffectedbyreduction:inFigure5,konlycountsonceininAG.
4Wecannotdeducethat|AG|>|AH|becausetheymightbebothinfinite.
8
ax g ax h ax ax ax ax
ax ` `
e` C f?C ⊗i e1`C1 e2`C2 aaxx 5 !Pe3 !Pe4
!P ?W ?P !P !P !P ?C ⊗
a a a h ⊗
B j d k 1 B1 2 B2 i
!P ?C !P ?W !P ?P !P !P
b cut c cut cut k k
Figure5:Cut-eliminationofaproof-net.
ThelanguageSigofsignaturesisdefinedbyinductionbythefollowinggrammar:
Sig=e|l(Sig)|r(Sig)|p(Sig)|n(Sig,Sig)
A signature correspondsto a list of choices of premises of ?C nodes, to designate a
particularresidueofabox. Thesignaturer(t)means: “Ichoosetherightpremise,and
inthenext?C nodesIwillusettomakemychoices”. Theconstructionn(t,u)allows
toencapsulatetwosequelsofchoicesintoone.Itcorrespondstothediggingrule(!!A⊢
B !A ⊢ B, representedby the ?N node in proof-nets)which “encapsulates” two !
modalitiesintoone. Thep(t)constructionisadegeneratedcaseofthenconstruction.
Intuitively,p(t)correspondston(∅,t).
Apotentialisalistofsignatures:asignaturecorrespondstotheduplicationofone
box,butanelementiscopiedwheneveranyoftheboxescontainingitiscutwitha?C
node. ThesetofpotentialsiswrittenPot. Foreveryedgee ∈ E ,wedefinePot(e)as
G
{(e,P)|P∈ Potand|P|=∂(e)}suchpairsarenamedpotentialedges.
Potentialsareusedtorepresentresidues.Forinstance,theresiduesofeinFigure5,
(e, e , e , e and e ) are respectively represented by the potential edges (e,[e;e]),
1 2 3 4
(e,[l(e);e]),(e,[r(e);e]),(e,[r(e);l(e)])and(e,[r(e);r(e)]).
A trace element is one of the following symbols: `,` ,⊗,⊗ ,∀,∃,!,? with t
l r l r t t
a signature. A trace elementmeans“I have crosseda node with this label, fromthat
premiseto itsconclusion”. Atrace isa non-emptylist oftraceelements. Thesetof
traces is Tra. A trace is a memoryof the path followed, up to cut-eliminations. We
definedualsoftraceelements: `⊥ = ⊗, !⊥ =?,... andextendthenotiontotracesby
l l t t
[a ;··· ;a ]⊥ =[a⊥;··· ;a⊥].
1 k 1 k
A context is a tuple ((e,P),T) with (e,P) a potential edge and T ∈ Tra. It can
beseenasastateofa tokenthattravelsaroundthenet. Itislocatedonedgee(more
preciselyitsresiduecorrespondingtoP)andcarriesinformationT aboutitspasttravel.
ThesetofcontextsofGiswrittenCont . Weextendthemapping( )⊥ oncontextsby
G
((e,P),T)⊥=((e,P),T⊥).
The nodes define two relations and ֒→ on contexts (Figure 6). For any rule
C D presented in Figure 6, we also define the dual rule D⊥ C⊥. We define
7→as theunionof and֒→. In otherwords, 7→is the smallestrelationon contexts
includingeveryinstance of rulesin Figure6 togetherwith everyinstance of their
dualsandeveryinstanceofthe֒→rule.
The rules are sound: if ((e,P),T) 7→ ((f,Q),U), then ∂(e) = |P| iff ∂(f) = |Q|.
Thoserelationsaredeterministic. Inparticular,ifC = ((e,P),T.?)withethepremise
t
9
g ax h e g
e cut f f∀ h∃
((e,P), T) ((f,P), T) ((e,P), T) ((f,P), T.∀)
((g,P), T) ((h,P), T) ((g,P), T) ((h,P), T.∃)
a b e f e g h
` ⊗ ?D ?C
c g f i
((a,P), T) ((c,P), T.`l) ((e,P), T) ((f,P), T.?e)
((b,P), T) ((c,P), T.`r) ((g,P), T.?t) ((i,P), T.?l(t))
((e,P), T) ((g,P), T.⊗l) ((h,P), T.?t) ((i,P), T.?r(t))
((f,P), T) ((g,P), T.⊗ )
r
e g
g ?P !P
f h
?N
h ((e,P.t), T) ((f,P), T.?)
t
((g,P), T.?t1.?t2) ((h,P), T.?n(t1,t2)) ((g,P.t), T) ((h,P), T.!t)
((g,P), [?]) ((h,P), [? ]) ((f,P), [!]) ֒→ ((h,P), [!])
t p(t) t t
Figure6: Rulesofthecontextsemantics
ofa?N nodeorC = ((σ(B),P),T.!),thecontextDsuchthatC 7→ Ddependsonthe
i t
size of T: there is a rule in the case T = [] and another in the case T , []. Let us
noticethat isinjective(Lemma3). Itisnotthecaseforthe7→relation.Indeed,ifB
isa boxwithtwoauxiliarydoorsthen,foreverypotentialPandsignaturet, wehave
((σ (B),P),[!])7→((σ(B),P),[!])and((σ (B),P),[!])7→((σ(B),P),[!]).
1 t t 2 t t
Lemma3. IfC DandC DthenC =C
1 2 1 2
Finally, we canobservethatforeverysequence((e ,P ),T ) ((e ,P ),T )
1 1 1 2 2 2
···((e ,P ),T ), the sequence of directed edges e ,··· ,e is a path (i.e. the head of
n n n 1 n
e is the same node as the tail of e ). The ֒→ relation breaks this property as it is
i i+1
non-local: itdealswithtwonon-adjacentedges. The7→-pathsrepresentthereduction
of a proof-netbecause they are stable along reduction. For example, the path in the
firstproof-netofFigure5:
((e,[r(e);l(e)]),[` ])7→((a,[r(e)]),[` ;! ])7→((b,[]),[`;! ;! ])7→
r r l(e) r l(e) r(e)
((c,[]),[` ;! ;! ])7→((d,[]),[` ;! ;! ])7→((f,[e]),[` ;! ])7→
r l(e) r(e) r l(e) e r l(e)
((g,[e]),[` ;! ])7→((h,[e]),[` ;! ])7→((i,[e]),[` ;! ;⊗ ])7→((k,[]),[`;! ;⊗ ;! ])
r e r e r e r r e r e
becomes the path ((e ,[e;e]),[` ]) 7→ ((h,[e]),[` ;! ]) 7→ ((i,[e]),[` ;! ;⊗ ]) 7→
3 r r e r e r
((k,[]),[` ;! ;⊗ ;! ])inthethirdproof-netofFigure5.
r e r e
10