Interpreting a Finitary Pi-Calculus in Differential Interaction Nets

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Interpreting a Finitary Pi-Calculus in Differential

Interaction Nets

Thomas Ehrhard, Olivier Laurent

To cite this version:

Thomas Ehrhard, Olivier Laurent. Interpreting a Finitary Pi-Calculus in Differential Interaction Nets.

CONCUR 2007, Sep 2007, Lisboa, Portugal. pp.333-348, �10.1007/978-3-540-74407-8�. �hal-00148816�

Intera tion Nets

ThomasEhrhardandOlivierLaurent

Preuves,Programmes&Systèmes UniversitéDenisDiderotandCNRS

Abstra t. Weproposeandstudyatranslationofapi- al uluswithout sumsnorrepli ation/re ursionintoanuntypedandessentially promotion-freeversionofdierentialintera tionnets.Wedeneatransitionsystem of labeled pro esses and a transition system of labeled dierential in-tera tionnets. Weprovethat ourtranslation from pro essesto netsis a bisimulation between these two transition systems. This shows that dierential intera tion nets are su iently expressive for representing on urren yandmobility,asformalizedbythepi- al ulus.

Introdu tion

Linear Logi proofs[Gir87℄ admit aproof net representation whi h has avery asyn hronousandlo alredu tionpro edure,suggestingstrong onne tionswith parallel omputation. Thisimpressionhasbeenenfor edbytheintrodu tionof intera tionnets andintera tion ombinators byLafontin [Laf95℄.

But the attempts at relating on urren y with linear logi (e.g. [EW97℄, [AM99℄, [Mel06℄, [Bef05℄, [CF06℄ based on[FM05℄...)missed a ru ial feature oftrue on urren y,su hasmodelledbypro ess al ulilikeMilner's

π

- al ulus [Mil93,SW01℄: itsintrinsi non-determinism.Indeed, allknownlogi alsystems had either an essentially deterministi redu tion pro edure  this is the ase of intuitionisti and linearlogi , and of lassi al systems su h asGirard's LC orParigot's

λµ

or anex essivelynon-determiniti one,asGentzen's lassi al sequent al ulusLK,whi h equatesallproofsofthesameformula.

However, many denotational models of the lambda- al ulus and of linear

logi admit some form of non-determinisms (e.g. [Plo76,Gir88℄), showing that a non-deterministi proof al ulus is not ne essarily trivial. The rst author introdu ed su h models, based onve torspa es (see e.g.[Ehr05℄), whi h have a ni e proof-theoreti ounterpart, orresponding to a simple extension of the rulesthatlinearlogi asso iateswiththeexponentials.

Inthisdierentialsetting,theweakeningrulehasamirrorimagerule alled oweakening,andsimilarlyfordereli tionandfor ontra tion,andtheredu tion

rules have the orresponding mirror symmetry. The orresponding formalism

of dierential intera tion nets hasbeenintrodu ed in ajointworkby therst

authorandRegnier[ER06℄

1 . 1

translationofaversionofthe

π

- al ulusinproof-netsforaversionoflinearlogi extended with the o ontra tionrule (as wenow understand). The basi idea onsistsin interpretingtheparallel omposition asa utbetweena ontra tion link (to whi h several outputs are onne ted, throughdereli tion links) and a o ontra tion link, to whi h several promoted re eivers are onne ted. Being promoted,thesere eiversarerepli able,inthesenseofthe

π

- al ulus.Theother fundamental ideaof this translation onsists in using linearlogi polaritiesfor makingthedieren ebetweenoutputs (negative) andinputs(positive),and of imposing astri talternationbetweenthesetwopolarities.Thisallowsto re ast in apolarizedlinearlogi settingatypingsystemforthe

π

- al uluspreviously introdu edbyBerger,HondaandYoshidain[BHY03℄.Thistranslationhastwo features whi h an be onsidered as slight defe ts: it a epts only repli able re eiversand is not really modular (the parallel omposition of twopro esses annotbedes ribedasa ombinationof the orrespondingnets).

Prin ipleofthetranslation. Thepurposeofthepresentpaperisto ontinue this line of ideas, using moresystemati allythe new stru tures introdu ed by dierentialintera tionnets

2 .

!

?

?

?

!

!

Fig.1. Communi ation area

The rst key de ision we made, guided by the

stru tureofthetypi al o ontra tion/ ontra tion ut intendedto interpretparallel omposition,wasof as-so iating with ea h free name of a pro ess not one, but two free ports in the orresponding dierential intera tion net.Oneofthese ports willhavea

!

-type (positivetype)and willhaveto be onsideredasthe inputport ofthe orrespondingnameforthispro ess, and theother onewill havea

?

-type(negativetype) andwillbe onsideredasanoutput port.

Wedis overedstru tures whi h allowto ombine thesepairsofwires forinterpretingparallel omposi-tion and alled them ommuni ation areas: they are

obtainedby ombiningin a ompletelysymmetri way o ontra tionand

on-tra tion ells.Thereare ommuni ationareasofanyarity (numberofpairsof wires onne tedtoit).The ommuni ationareaofarity

3

anbepi turedasin Figure 1,where o ontra tion ellsarepi tured as

!

-labeledtriangles and on-tra tion ellsas

?

-labeledtriangles. Theports orrespondingto thesamepairs aretheprin ipalportsofantipodi ells.

and thisidenti ation whi hresultsfrom non-determinismdoesnotextendto themultipli ative onne tives:

⊗

and are distin t.

2

One should mention here that translations of the

π

- al ulus into nets of various kinds,subje t tolo al redu tion relations, have beenprovidedby various authors ( f. the workof Laneve, Parrow and Vi toron solo diagrams [LPV01 ℄, of Beara and Maurel [BM05℄,ofMilner onbigraphs [JM04℄,ofMazza[Maz05℄onmultiport intera tion nets et .).But thesesettings haveno lear logi al grounds norsimple

sive typing system (introdu ed by Danos and Regnier in [Reg92℄ and whi h orrespondstotheuntypedlambda- al ulus)foravoidingtheappearan eofnon redu ible ongurations.Thesenetsarenitaryin thesensethat theyuseonly a weak form of promotion. In this setting, we dene a toolbox, a olle tion of nets that we shall ombine for interpreting pro esses, and a few asso iated redu tions,derivedfromthebasi redu tionrulesofdierentialintera tionnets. Weorganizeredu tionrulesofnetsasalabeledtransitionsystem,whose ver-ti esarenets,andwherethetransitions orrespondto dereli tion/ odereli tion redu tion.Thenwedeneapro essalgebrawhi hisapolyadi

π

- al ulus, with-outrepli ationand withoutsums.Wespe ifytheoperationalsemanti sof this

al ulus by means of an abstra t ma hine inspired by the ma hine presented

in [AC98, Chapter 16℄. We dene a transition system whose verti es are the

states ofthis ma hine,andtransitions orrespondto input/output redu tions. Lastwedeneatranslationrelationfromma hinestatestonetsandshowthat thistranslationrelationisabisimulationbetweenthetwotransitionsystems.

1 Dierential intera tion nets

Intera tion netshavebeen introdu ed by Lafont[Laf95℄ asageneralization of linearlogi proofnets.A signature ofintera tion netsis aset ofsymbols,ea h ofthembeinggivenwithanarityandatypingrule.Anetismadeof ells.Ina net,ea h ell

γ

bearsexa tlyonesymbol,andhasthereforeanarity

n

; the ell

γ

must have

n

auxiliary ports (numbered from

1

to

n

) and oneprin ipal port (numbered

0

).Anet analsohavefreeports.Spe ifyingthenet onsistslastin givingitswiring,whi hisapartitionof itsportsin

2

-elementssets(the wires). Typingthenet meansasso iatingaformula ofsomelinearlogi alsystemwith ea hofits orientedwiresin su hawaythat,when reversingthe orientationof the wire, the formula be turned to its orthogonal. Of ourse, the typing rule atta hedtoea h ellofthenetmustalsoberespe tedbythetyping.

Seealso[ER06℄foranintrodu tiontodierentialintera tionnets.

1.1 Presentationof the ells

Our netswill be typed usingatype systemwhi h orrespondsto the untyped lambda- al ulus.This system is based on a single typesymbol

o

(the type of outputs),subje ttothefollowingre ursiveequation

o

= ?o

⊥

_o

.Weset

ι

= o

⊥

, sothat

ι

= !o ⊗ ι

and

o

= ?ι

o

.

Inthepresentsetting,thereareelevensymbols:par(arity

2

),bottom(arity

0

), tensor (arity

2

), one (arity

0

), dereli tion (arity

1

), weakening (arity

0

), ontra tion(arity

2

), odereli tion(arity

1

), oweakening(arity

0

), o ontra tion (arity

2

)and losedpromotion(arity

0

).Wepresentnowthevarious ellsymbols, withtheirtypingrules,in api torialway.Theprin ipalportofa ellislo ated at one of the angles of the triangle representing the ell, the other ports are lo ated on the opposit edge. We put often a bla k dot to lo ate the auxiliary portnumber

1

.

versionsbottom andone areasfollows:

•

o

o

?ι

•

!o

⊗

ι

ι

⊥

o

1

ι

1.1.2 Exponential ells. Theyaretyped a ordingto astri tlypolarized

dis ipline.Herearerstthewhynot ells,whi hare alleddereli tion,weakening and ontra tion:

?

ι

?ι

?

?ι

?

?ι

?ι

?ι

andthenthebang ells, alled odereli tion, oweakening and o ontra tion:

!

o

!o

!

!o

!

!o

!o

!o

1.1.3 Closed promotion ells and the denition of nets. Thenotion

of simple net is then dened indu tively, together with the notion of losed promotion ell.

Given a (non ne essarily simple) net

s

with only one free port

o

s

we introdu ea ell

s

!

!o

.

Asimplenetisatypedintera tionnet,inthesignaturewehavejustdened. A net is a nite formal sum of simple nets having all the same interfa e. Rememberthattheinterfa eofasimplenet

s

isthesetofitsfreeports,together with themappingasso iatingto ea hfreeport thetypeoftheorientedwireof

s

whoseendingpointisthe orrespondingport.

Let

L

bea ountablesetoflabels ontainingadistinguishedelement

τ

(tobe understood astheabsen eof label).Alabeledsimple net isasimplenetwhere all dereli tion and odereli tion ellsare equipped with labels belonging to

L

. Werequiremoreoverthat,iftwolabelso urringinalabelednetareequal,they areequalto

τ

.Allthenetswe onsiderinthispaperarelabeled.Inourpi tures, thelabelsof dereli tionand odereli tion ellswill beindi ated,unless it is

τ

, in whi h asethe( o)dereli tion ellwillbedrawnwithoutanylabel.

2 Redu tion rules

Wedenote by

∆

the olle tionof allsimplenetsand by

N

h∆i

the olle tionof allnets(nitesumsofsimplenetswiththesameinterfa e).

Aredu tionrule is asubset

R

of

∆

× Nh∆i

onsistingofpairs

(s, s

′

₎

where

s

is made of two ells onne tedby their prin ipal ports and

s

′

hasthe same

interfa eas

s

.Thisset anbeniteorinnite.Su harelationiseasilyextended to arbitrary simple nets(

s

R t

if there is

(s

0

, u

1

+ · · · + u

n

) ∈ R

where

s

0

is a subnet of

s

, ea h

u

i

is simpleand

t

= t

1

+ · · · + t

n

where

t

i

is obtainedby repla ing

s

0

by

u

i

in

s

).Thisrelationisextended tonets(sumsofsimplenets):

s

1

+ · · · + s

n

(where ea h

s

i

is simple) is relatedto

s

′

by this extension

R

Σ

if

s

′

= s

′

1

+ · · · + s

′

n

where,forea h

i

,

s

i

R s

′

i

or

s

i

= s

′

i

.Last,

R

∗

isthetransitive losureof

R

Σ

.

2.1.1 Multipli ativeredu tion. Thersttworules on erntheintera tion oftwomultipli ative ellsofthesamearity.

•

•

⊗

?ι

?ι

o

o

;

m

o

o

?ι

⊥

o

;

m

ε

1

where

ε

standsfor theemptysimplenet (not to be onfused withthe net

0 ∈

N

_h∆i

, theempty sum,whi h isnot asimplenet). Thenext tworules on ern theintera tionbetweenabinaryandanullarymultipli ative ell.

1

o

;

m

?ι

o

?ι

o

1

!

;

m

!o

ι

⊗

⊥

!o

ι

?

⊥

Soheretheredu tionrule(denoted as

;

m

)hasfourelements.

2.1.2 Communi ation redu tion. Let

R

⊆ L

. Wehavethe following

re-du tionsif

l, m

∈ R

.

?

!

ι

?ι

ι

;

c,R

ι

l

m

Sothe set

;

c,R

isin bije tive orresponden e withthe setof pairs

(l, m)

with

l, m

∈ R

and

l

= m ⇒ l = m = τ

.

2.1.3 Non-deterministi redu tion. Let

R

⊆ L

. We have the following

redu tionsif

l

∈ R

.

?

?

?

?

ι

?ι

?ι

?ι

!

?

₊

l

l

;

nd,R

?ι

l

!

!

!

!

!

?

o

!o

!o

!o

+

l

l

;

nd,R

l

?

!

ι

?ι

l

;

nd,R

0

!

?

o

!o

l

;

nd,R

0

2.1.4 Stru turalredu tion.

?ι

?ι

?ι

?

!

!

!

_;

s

!

!o

!o

!o

?

?

?

_;

s

?ι

?

;

s

ε

s

!

?

?ι

?ι

?ι

;

s

s

!

s

!

s

!

?ι

?

!

_;

s

ε

?

!

?ι

?ι

?ι

?ι

?ι

;

s

!

!

?

?

?

ι

?ι

s

;

b

s

!

l

Observe that the redu tionrules are ompatible with the identi ation of the oweakening ellwith apromotion ell ontainingthe

0

net.Observealso that the only rules whi h do notadmit asymmetri  rule are those whi h involve apromotion ell. Indeed,promotionis theonly asymmetri ruleof dierential linearlogi .

One an he kthatwehaveprovidedredu tionrulesforallpossibleredexes, ompatible withourtypingsystem:foranysimplenet

s

madeoftwo ells on-ne tedthroughtheirprin ipalports,thereisaredu tionrulewhoseleftmember is

s

.This ruleisunique, uptothe hoi eofaset oflabels,but this hoi ehas noinuen eontherightmemberoftherule.

2.2 Conuen e

Theorem1. Let

R, R

′

_{⊆ L}

. Let

R ⊆ ∆ × Nh∆i

be the union of some of the redu tionrelations

;

c,R

,

;

nd,R

′

,

;

m

,

;

s

and

;

b

.Therelation

R

∗

is onuent on

N

h∆i

.

The proof is essentially trivial sin ethe rewriting relation has no riti al pair (see [ER06℄). Given

R

⊆ L

, we onsider in parti ular the followingredu tion:

;

R

= ;

m

∪;

c,{τ }

∪;

s

∪;

b

∪;

nd,R

.Weset

;

d

= ;

∅

(dfordeterministi ) anddenoteby

∼

d

thesymmetri and transitive losureofthisrelation.

Someoftheredu tionruleswehavedened dependonasetoflabels.This dependen e is learly monotone in the sense that the relation be omes larger whenthesetoflabelsin reases.

2.3 A transition systemofsimplenets

2.3.1

{l, m}

-neutrality. Let

l

and

m

bedistin telementsof

L \ {τ }

.We all

(l, m)

- ommuni ation redex a ommuni ation redex whose ( o)dereli tion ells arelabeledby

l

and

m

.Wesaythatasimplenet

s

is

{l, m}

-neutral if,whenever

s ;

∗

_{l,m}

s

′

,noneofthesimplesummandsof

s

′

ontainsan

(l, m)

- ommuni ation redex.

Lemma1. Let

s

beasimplenet.If

s ;

∗

{l,m}

s

′

whereallthesimplesummands of

s

′

are

{l, m}

-neutral,then

s

isalso

{l, m}

-neutral.

2.3.2 The transition system. We dene a labeled transition system

D

L

whose obje ts are simplenets, and transitions are labeled by pairs of distin t elementsof

L \ {τ }

.Let

s

and

t

besimplenets,wehave

s

lm

−→ t

ifthefollowing

holds:

s ;

∗

{l,m}

s

1

+ s

2

+ · · · + s

n

where

s

1

is a simple net whi h ontains an

(l, m)

- ommuni ationredex(withdereli tionlabeledby

m

and odereli tion labeledby

l

)andbe omes

t

whenoneredu esthisredex,andea h

s

i

(for

i >

1

) is

{l, m}

-neutral.

3.1 Compound ells

3.1.1 Generalized ontra tion and o ontra tion. A generalized

on-tra tion ell or ontra tion tree is asimplenet

γ

(withoneprin ipal port and anitenumberofauxiliaryports)whi hiseither awireoraweakening ellor a ontra tion ellwhose auxiliaryports are onne tedto the prin ipal port of other ontra tiontrees,whose auxiliaryports be ometheauxiliaryports of

γ

. Generalized o ontra tion ells( o ontra tiontrees)aredeneddually.

Weusethesamegraphi alnotationsforgeneralized( o) ontra tion ellsas for ordinary ( o) ontra tion ells, with a 

∗

 in supers ript to the 

!

 or 

?

 symbolstoavoid onfusions.Observethatthere areinnitelymanygeneralized ( o) ontra tion ellsofanygivenarity.

3.1.2 Thedereli tion-tensorandthe odereli tion-par ells. Let

n

be

anon-negativeinteger.We dene an

n

-ary ellasfollows.It will be de orated bythelabelofitsdereli tion ell(if dierentfrom

τ

).

?⊗

!o

!o

?ι

⊗

⊗

⊗

1

?

!o

!o

!o

ι

?ι

•

•

•

•

=

. . .

l

l

Thenumberoftensor ellsinthis ompound ellisequalto

n

.Onedenesdually

the

!

ompound ell.

3.1.3 The prex ells. Now we andene the ompound ellswhi h will

playthemain roleintheinterpretationofprexesofthe

π

- al ulus.Thanksto theabovedened ells,alltheorientedwiresofthenetsweshalldenewillbear type

?ι

or

!o

. Therefore we omit types and drawall wires with an orientation orrespondingtothe

?ι

type.

The

n

-ary input ell andthe

n

-ary output ell aredenedas

!

?⊗

?⊗

!

•

•

. . . . . . . . .

=

l

l

?

!

!

?⊗

•

•

. . . . . . . . .

=

l

l

with

n

pairsofauxiliaryports.

Prex ells are labeled by thelabel arriedby their outermost dereli tion-tensoror odereli tion-par ompound ell,ifdierentfrom

τ

,theother odereli tion-parordereli tion-tensor ompound ellsbeingunlabeled(thatis,labeledby

τ

).

3.1.4 Transistorsandboxedidentity. Inordertoimplementthe

sequen-tiality orrespondingtosequen esofprexesinthe

π

- al ulus,weshallusethe unaryoutputprex elldenedaboveasakindoftransistor,thatis,asakind

Thisideaisstronglyinspiredbythetranslationofthe

π

- al ulusinthe al ulus ofsolos 3 .

?⊗

⊥

o

•

Fig.2.Identity Theseswit heswillbe losedbyboxedidentity ells,

whi h are the unique use we make of promotion in the

presentwork.Let

I

betheidentity netof Figure2. Thenweshallusethe losedpromotion elllabeledby

I

!

:

I

!

.

3.2 Communi ation tools

3

Fig.3. Area of or-der3

3.2.1 The ommuni ation areas. Let

n

≥ −2

. We

dene a family of nets with

2(n + 2)

free ports, alled ommuni ationareasoforder

n

,that weshalldrawusing

re tangles with beveled angles. Figure 3 shows how we

pi turea ommuni ationareaoforder

3

.

A ommuni ationareaoforder

n

ismadeof

n

+ 2

pairs of

(n + 1)

-ary generalized o ontra tionand ontra tion ells

(γ

+

1

, γ

1

−

), . . . , (γ

n+1

+

, γ

n+1

−

)

,with,forea h

i

and

j

su h that

1 ≤ i < j ≤ n + 2

,awirefromanauxiliaryportof

γ

+

i

toanauxiliaryport of

γ

−

j

andawirefromanauxiliaryportof

γ

−

i

to anauxiliaryport of

γ

+

j

. Sothe ommuni ationareaoforder

−2

istheemptynet

ε

,and ommuni a-tionareasoforder

−1

,

0

and

1

arerespe tivelyoftheshape

?

∗

!

∗

!

∗

?

∗

!

∗

?

∗

!

∗

?

∗

?

∗

!

∗

?

∗

!

∗

3.2.2 Identi ation stru tures. Let

n, p

∈ N

and let

f

: {1, . . . , p} →

{1, . . . , n}

beafun tion.An

f

-identi ation net isastru turewith

p

+ n

pairs of free ports (

p

pairs orrespond to the domain of

f

and, in our pi tures, will beatta hedto thenon beveledside oftheidenti ation stru ture,and

n

pairs orrespondtothe odomainof

f

,atta hedtothebeveledsideofthestru ture) asinFigure4(a).Su hanetismadeof

n

ommuni ationareas,andonthe

j

'th area, the

j

'th pair of wires of the odomain is onne ted, as well asthe pairs of wires of index

i

of the domain su h that

f

(i) = j

. For instan e, if

n

= 4

,

p

= 3

,

f

(1) = 2

,

f

(2) = 3

and

f

(3) = 2

,a orrespondingidenti ationstru ture is madeof four ommuni ationareas,twoof order

−1

,oneof order

0

andone oforder

1

,asin Figure4(b).

3

It is shown in[LV03 ℄ that one anen ode the

π

- al ulussequentiality indu edby prexnestinginthe ompletelyasyn hronoussoloformalism:theideaofsu h trans-lationsistoobservethat,inasolopro esslike

P

= νy (u(x, y) | y(. . . )) | Q

,therst solomustintera tbeforethese ondonewiththeenvironment

Q

.

1

. . .

. . .

f

p

n

1

(a)Notation

−1

1

0

−1

(b)Example

;

∗

s

f

g

g

◦ f

. . .

. . .

. . .

. . .

. . .

( )Redu tion

Fig.4.Identi ationstru tures

3.3 Usefulredu tions.

3.3.1 Aggregationof ommuni ation areas. Oneoftheni eproperties

of ommuni ationareasisthat,whenone onne tstwosu hareasthroughapair ofwires,onegetsanother ommuni ationarea;ifthetwoareasareofrespe tive orders

p

and

q

,theresultingareaisoforder

p

+ q

,seeFigure5.

p

+ q

. . . . . .

p

q

;

∗

s

. . . . . . Fig.5.Aggregation

3.3.2 Composition of identi ation stru tures. In parti ular, we get

theredu tionofFigure 4( ).

3.3.3 Port forwarding in a net. Let

t

beanetand

p

beafreeportof

t

.

Wesaythat

p

isforwardedin

t

ifthereisafreeport

q

of

t

su hthat

t

isofone ofthetwofollowingshapes:

?

∗

p

q

. . .

· · ·

. . .

!

∗

p

q

· · ·

. . . . . .

3.3.4 Forwardingofdereli tionsand odereli tionsin ommuni ation

areas. Thefollowingredu tionshowsthatdereli tions and odereli tions an

meetea hother,when onne tedtoa ommon ommuni ationareas.Let

l, m

∈

L

, then

?

!

!

?

?

!

!

t

i

?

r

r

′

;

∗

_{l,m}

· · ·

p

+ 1

p

+

N

X

i=1

· · ·

· · ·

l

m

l

m

l

m

where

N

isanon-negativeinteger(a tually,

N

= (p + 1)

2

)and, inea hsimple net

t

i

,bothports

r

and

r

′

areforwarded.

3.3.5 Generalforwarding. Let

l

∈ L

.Thefollowingmoregeneralbutless

informativepropertywillalsobeused:onehas

?

. . .

u

i

r

. . .

?

∗

?

_;

∗

_{l}

N

X

i=1

l

. . . . . .

p

l

where in ea hsimplenet

u

i

, theport

r

is forwarded(see 3.3.3).Of ourseone also hasadual redu tion(where thedereli tion is repla ed by a odereli tion, andthegeneralized ontra tionbyageneralized o ontra tion).

3.3.6 Redu tionof prexes. Let

l, m

∈ L

.Ifwe onne tan

n

-aryoutput

prexlabeledby

m

to a

p

-ary inputprexlabeledby

l

,weobtainanetwhi h redu esby

;

c,{l,m}

toanet

u

whi hredu esby

;

∗

{τ }

to

0

if

n

6= p

andtosimple wires,inFigure6(a),if

n

= p

.

3.3.7 Transistor triggering. Aboxed identity onne tedto the prin ipal

portof aunary output ellusedasatransistor turnsitintoasimplewireas in Figure6(b).

•

•

. . .

!

?

. . . . . .

m

l

;

c,{l,m}

u ;

∗

∅

(a)Prexesintera tion

I

!

?

•

;

∗

_∅

(b)Transistortriggering Fig.6.Prexredu tion

4 A polyadi nitary

π

- al ulus and its en oding

Thepro ess al uluswe onsiderisafragmentofthe

π

- al uluswherewehave suppressedthefollowingfeatures:sums,repli ation,re ursivedenitions,mat h andmismat h. Thisdoesnotmeanthat dierentialintera tionnets annot in-terpret these features

4

. Let

N

bea ountable set of names. Ourpro essesare dened bythefollowingsyntax.Weusethesamesetoflabelsasbefore.



nil

istheemptypro ess.

 If

P

1

and

P

2

arepro esses,then

P

1

| P

2

isapro ess.

 If

P

isapro essand

a

∈ N

,then

νa

· P

isapro esswhere

a

isbound.  If

P

isapro ess,

a, b

1

, . . . , b

n

∈ N

,thenames

b

i

beingpairwisedistin tand

if

l

∈ L

,then

Q

= [l]a(b

1

. . . b

n

) · P

isapro ess(prexedbyaninputa tion, whosesubje tis

a

andwhoseobje tsarethe

b

i

s;thename

a

isfreeandea h

b

i

isboundin

Q

andhen e

a

isdistin tfromea h

b

i

).

 If

P

is a pro ess,

a, b

1

, . . . , b

n

∈ N

and

l

∈ L

, then

[l]ahb

1

. . . b

n

i · P

is a pro ess(prexedbyanoutputa tion,whosesubje tis

a

andwhoseobje ts arethe

b

i

s).This onstru tiondoesnotbindthenames

b

i

,andonedoesnot requirethe

b

i

stobedistin t. Thename

a

anbeequaltosomeofthe

b

i

s. Thepurposeofthislabelingofprexesistodistinguishthevariouso urren es ofnamesassubje tofprexes.Theset

FV

(P )

offreenamesofapro ess

P

and the

α

-equivalen erelationonpro essesaredenedintheusualway.

4

distin t labels,allthese labelsbeingdierentfrom

τ

. If

P

is alabeledpro ess,

L(P )

denotesthesetofitslabels.Allthepro esseswe onsiderinthispaperare labeled.

4.1 Anexe ution model

Rather than onsidering arewriting relationon pro esses asoneusually does, weprefertodeneanenvironmentma hine,similartothema hineintrodu ed in [AC98,Chapter 16℄

5 .

Anenvironment isafun tion

e

: Dom e → Codom e

betweennitesubsetsof

N

.A losure isapair

(P, e)

where

P

isapro essand

e

isanenvironmentsu h that

FV

(P ) ⊆ Dom(e)

.Asoup isamultiset

S

= (P

1

, e

1

) · · · (P

N

, e

N

)

of losures (denoted bysimplejuxtaposition). Theset

FV

(S)

offree namesof asoup

S

is theunion ofthe odomains ofthe environmentsof

S

.The soup

S

is labeledif all the

P

i

s are labeled, with pairwise disjoint sets of labels. A state is a pair

(S, L)

where

S

isasoup and

L

isaset of names(the nameswhi h haveto be onsideredaslo altothestate)andweset

FV

(S, L) = FV(S) \ L

.

Thestate

(S, L)

islabeledifthesoup

S

islabeled.Allthestateswe onsider are labeled. One denes the set

L(S, L)

of all labels of the state

(S, L)

asthe disjointunionofthesets oflabelsasso iated tothepro essesofthe losuresof

S

.

4.1.1 Canoni al form of a state. We say that a pro ess is guarded if

it starts with an input prex or an output prex. We say that a soup

S

=

(P

1

, e

1

) · · · (P

N

, e

N

)

is anoni al ifea h

P

i

isguarded,andthatastate

(S, L)

is anoni alifthesoup

S

is anoni al.Onedenesarewritingrelation

;

can

whi h allowstoturnastateintoa anoni alone.

((nil, e)S, L) ;

can

(S, L)

((νa · P, e)S, L) ;

can

((P, e[a 7→ a

′

])S, L ∪ {a

′

})

((P | Q, e)S, L) ;

can

((P, e)(Q, e)S, L)

where, in the se ond rule,

a

′

_{∈ N \ (L ∪ Codom(e) ∪ Codom(S))}

. One shows

easily that, up to

α

- onversion, this redu tion relation is onuent, and it is learly strongly normalizing. We denote by

Can

(S, L)

the normal form of the state

(S, L)

forthis rewritingrelation. Observethat if

(S, L) ;

can

(T, M )

then

FV

_{(T, M ) ⊆ FV(S, L)}

.

4.1.2 Transitions. Next, one denes a labeled transition system

S

L

. The

obje tsof this systemare labeled anoni alstatesand the transitions, labeled

5

Thereason for this hoi eis that the rewritingapproa husesanoperation whi h onsists inrepla ing anameby anothernameinapro ess. The orresponding

op-(([l]a(b

1

. . . b

n

) · P, e)([m]a

′

hb

1

′

. . . b

′

n

i · P

′

, e

′

)S, L)

lm

−→ Can((P, e[b

1

7→ e

′

(b

′

1

), . . . , b

n

7→ e

′

(b

′

n

)])(P

′

, e

′

)S, L)

if

e(a) = e

′

_(a

′

₎

.Observethatif

(S, L)

lm

−→ (T, M )

then

FV

(T, M ) ⊆ FV(S, L)

. 4.2 Translationof pro esses

Sin ewedonotworkuptoasso iativityand ommutativityof ontra tionand o ontra tion,itdoesnotmakesensetodenethistranslationasafun tionfrom pro essesto nets.Forea h repetition-free list ofnames

a

1

, . . . , a

n

, wedene a relation

I

a

1

,...,a

n

from pro esseswhosefreenamesare ontainedin

{a

1

, . . . , a

n

}

tonets

t

whi hhave

2n + 1

freeports

a

ι

1

, a

o

1

, . . . , a

ι

n

, a

o

n

and

c

asinFigure7(a). Theadditionalport

c

willbeusedfor ontrollingthesequentialityofthe redu -tion,thankstotransistors.Redu ingthetranslationofapro esswillbepossible only when a boxed identity ell will be onne ted to its ontrol port. This is ompletely similarto theadditional ontrol freenameinthe translationofthe

π

- al ulusinsolos,in [LV03℄

6 . Clearly,if

P

and

P

′

are

α

-equivalent,then

P

I

a

1

,...,a

n

s

i

P

′

_I

a

₁

,...,a

n

s

.

4.2.1 Empty pro ess. Onehas

nil

I

b

1

,...,b

n

t

if

t

isasinFigure7(b).

4.2.2 Namerestri tion. Onehas

νa

· P I

b

1

,...,b

n

t

i

t

isasinFigure7( ), with

s

satisfying

P

I

a,b

1

,...,b

n

s

.

4.2.3 Parallel omposition. One has

P

1

| P

2

I

b

1

,...,b

n

t

ithe simplenet

t

is as in Figure 7(d),where

P

1

I

b

1

,...,b

n

t

1

,

P

2

I

b

1

,...,b

n

t

2

and

γ

1

, . . . , γ

n

are ommuni ationareasoforder

1

.

4.2.4 Inputprex. Let

l

∈ L

.Assumethat

a, b

1

, . . . , b

n

, c

1

, . . . , c

p

are pair-wisedistin t namesand let

Q

= [l]a(b

1

. . . b

n

) · P

. Onehas

Q

I

a,c

1

,...,c

p

t

ifall thefreenamesof

P

are ontainedin

a, b

1

, . . . , b

n

, c

1

, . . . , c

p

andif

t

isasin Fig-ure7(e),where

γ

isa ommuni ationareaoforder

1

andwhere

s

isasimplenet whi hsatises

P

I

a,b

1

,...,b

n

,c

1

,...,c

p

s

. 6

Thereisasimpleinterpretationofofsolodiagramsintodierentialintera tionnets, whi husesonlyourtoolbox withoutpromotion sothat solodiagrams anbe seen as anintermediategraphi al languagewhi h anbe implemented inthe low level dierentialsyntax.Ourtranslationofthe

π

- al ulusresultsfromananalysisanda simpli ationofthe omposedtranslation

π

- al ulus

→

solodiagrams

→

dierential nets.Thesimpli ationresultsfromsomerewiringandfrom theuseoftheboxed identity ellswhi hiseasilyrepli able.Thetranslationofsolosintodierentialnets leadsto y les(whi happearwhenanameisidentiedwithitself)whi hareavoided inthepresentdire ttranslation.Wellbehaved onditionsonsolosforavoidingsu h

t

a

1

a

n

. . .

c

(a)Notation

b

n

. . .

b

1

?

∗

c

(b)Emptypro ess

s

a

b

1

b

n

. . .

c

( )Restri tion

b

1

b

n

. . .

b

1

b

n

. . .

. . .

c

c

c

t

1

t

2

γ

1

γ

n

?

∗

(d)Parallel omposition

!

I

!

•

?

•

s

. . .

a

c

1

. . .

c

p

c

c

1

c

p

b

1

. . .

. . .

b

n

a

c

l

γ

?

∗

(e)Inputprex

•

?

?

•

I

!

. . .

b

n

b

1

. . .

. . .

b

1

b

n

. . .

1

n

q

0

c

f

. . .

c

. . .

s

l

γ

n

δ

!

∗

_γ

1

(f) Outputprex

I

!

I

!

e

. . .

. . .

. . .

. . .

. . .

1

1

p

n

c

c

. . .

s

1

. . .

δ

s

N

(g)State

Fig.7.Pro essandstatetranslation

4.2.5 Outputprex. Let

l

∈ L

.Let

b

1

, . . . , b

n

bealistofpairwisedistin t namesandlet

Q

= [l]b

f(0)

hb

f(1)

. . . b

f(q)

i · P

,where

f

: {0, 1, . . . , q} → {1, . . . , n}

is afun tion. Onehas

Q

I

b

1

,...,b

n

t

ifall thefree names of

P

are ontainedin

b

1

, . . . , b

n

andif

t

isasinFigure7(f),where

γ

1

, . . . , γ

n

are ommuni ationareas of order

1

,

δ

is an

f

-identi ation stru tureand where

s

is asimplenetwhi h satises

P

I

b

1

,...,b

n

s

.

4.2.6 States. Let

S

= (P

1

, e

1

) . . . (P

N

, e

N

)

be a soup and

b

1

, . . . , b

n

be a repetition-free list of names ontainingall the odomains of the environments

e

1

, . . . , e

N

.Onehas

S

I

b

1

,...,b

n

t

if,forsomesimplenets

s

i

(

i

= 1, . . . , N

)onehas

P

i

I

b

i

1

,...,b

i

ni

s

i

where

b

i

1

, . . . , b

i

n

i

isarepetition-freeenumerationofthedomainof

e

i

,and

t

isobtainedby onne tingthepairoffreeportsof

s

i

asso iatedtoea h

b

i

k

tothe orrespondingpairoffreeportofanidenti ationstru tureasso iated tothefun tion

e

denedby

e(b

i

k

) = e

i

(b

i

k

)

,see Figure7(g).

Last, if we are moreover given

L

⊆ N

and a repetition-free list of names

b

1

, . . . , b

n

ontainingallthefreenamesofthestate

(S, L)

,onehas

(S, L) I

b

1

,...,b

n

u

ifonehas

S

I

b

1

,...,b

n

,c

1

,...,c

p

t

forsomerepetition-freeenumeration

c

1

, . . . , c

p

of

L

(assumedof oursetobedisjointfrom

b

1

, . . . , b

n

)and

u

isobtainedbyplugging

ommuni ationareasof order

−1

onthe pairsof freeports of

t

orresponding tothe

c

j

s.

5 Comparing the transition systems

Wearenowready tostateabisimulation 7

theorem. Givenarepetition-freelist

b

1

, . . . , b

n

ofnames,wedenearelation

I

e

b

1

,...,b

n

betweenstatesandsimplenets by:

(S, L) e

I

b

1

,...,b

n

s

if there exists asimplenet

s

0

su h that

(S, L) I

b

1

,...,b

n

s

0

and

s

0

∼

d

s

.

Theorem2. The relation

I

e

b

1

,...,b

n

is astrongbisimulation between the labeled transitionsystems

S

L

and

D

L

.

Con lusion. Themaingoalofthisworkwasnottodeneonemoretranslation ofthe

π

- al ulusinto yetanotherexoti formalism.Wewantedto illustrateby ourbisimulationresultthat dierentialintera tionnets aresu iently expres-sive for simulating on urren y and mobility, as formalized in the

π

- al ulus. We believe that dierential intera tion nets have their own interest and nd a strong mathemati aland logi aljusti ation in their onne tion with linear logi , in the existen e of various denotational models and in the analogy

be-tween its basi onstru ts and fundamental mathemati al operations su h as

dierentiation and onvolution produ t. The fa t that dierential intera tion nets support on urren yand mobilitysuggeststhat theymightprovidemore onvenientmathemati alandlogi alfoundationsto on urrent omputing.

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