Modelling Reaction Systems with Petri Nets

Modelling Reation Systems with Petri Nets

(Extended Abstrat)

JettyKleijn

1

,MaiejKoutny

2

,andGrzegorzRozenberg

1,3

1

LIACS,LeidenUniversity,2300RA,TheNetherlands

2

ShoolofComputingSiene,NewastleUniversity,NE17RU,UK

3

DepartmentofComputerSiene,UniversityofColoradoatBoulder

430UCBBoulder,CO80309-0430,U.S.A.

Abstrat. We investigate how Petri netsould be used to provide a

faithfulsemantisofreationsystems,aformalframeworkfortheinves-

tigationofproessesarried by biohemial reations. Wepropose and

disuss possible approahes to this problem usingsome existing Petri

netlassesandonurrenyonepts,suhasmaximalparallelism.After

thatweintrodueanewlassofPetrinets,alledset-nets,whihprovide

aomputationalmodelmathingveryloselythatexhibitedbyreation

systems.The keydierene betweenstandard Petri netsand set-nets

isthattheformersupportmultiset-basedtokenarithmeti,whereasthe

lattersupportset-basedoperationsontokens.

Keywords: reation system, Petrinet, living ell,natural omputing,

set-net,modeltranslation

1 Introdution

Theinvestigationof theomputationalnature of biohemialreationsisare-

searhtopi ofNatural Computing.Oneofthe goalsof thisresearh isto on-

tributetoaomputationalunderstanding ofthefuntioning oftheliving ell.

Reationsystems[2,3,710℄areaformalframeworkfortheinvestigationof

proesses arried outby biohemial reations in living ells.The entral idea

ofthisframeworkisthatthefuntioningofalivingellisbasedoninterations

between (a largenumber of) individual reations,and moreoverthese intera-

tionsareregulatedbytwomainmehanisms:failitation/aelerationandinhi-

bition/retardation. These interationsdetermine thedynami proessestaking

plaeinlivingells,andreationsystemsformaformalframeworkfordeveloping

anabstrattheoryoftheseproesses.

The model of reation systems is based on priniples remarkably dierent

from those underlying other existing models of omputation. The aim of this

paper is to develop a faithful Petri net model of reation systems. The main

motivationbehindthisistoestablishwhetherPetrinetbasedonepts(suhas

ausalproesses)andmethods (suhassynthesisofnetsfromaspeiationof

theirbehaviour)ouldbeused toprovideanalytialtoolsforreationsystems.

It is not the intention of this paper to provide diret feedbak to the area of

online: http://ceur-ws.org/Vol-724 pp.36-52

biologialappliations, butto establishbridgesbetweenbiologyand Petri nets

throughtheonnetionprovidedbyreationsystems.

Asarststep,weproposeanddisussfourdierentapproahestothemod-

eling of reation systems by using existing Petri net lasses and onurreny

onepts.However,asitturnsout,inordertoobtainagoodmathbetweenre-

ationsystemsandPetrinets,itisneessarytore-evaluateoneofthebasinet

priniples,namely, token ounting. This leadsus to theintrodution of anew

lass of Petri nets, alled set-nets, whih provide a net based omputational

model mathing verylosely the omputations exhibited by reationsystems.

Themaindierenebetweenset-netsandstandardPetrinetsisthatthelatter

support multiset-basedtokenarithmeti, whereastheformer supportset-based

(boolean)operationsontokens.Thus,theomputational`intuition'originating

from reationsystemsprovidestheinspirationto introdue anew lassofnets

withintriguingandyettodisoverproperties.Consequently,themainontribu-

tionofthispaperismorethanjustprovidingabridgebetweenreationsystems

andtheworldofPetrinets.Inthefuture,afterfullyunderstandingandmaster-

ing the properties of thenew set-nets, onewould hope to provide also anew

set oftoolsandanalysesforbiologialappliations.

Thepaperisorganisedin thefollowingway.Inthenextsetion,wedesribe

basinotionsofreationsystems.Setion3desribestwomethodsofmodelling

reationsystemusinglow-levelPetrinets,andthenextonedoesthesameusing

high-levelPetrinets.Thenewlassof set-netsis introdued inSetion 5,and

in Setion 6weexplainwhythis new lassof netsanfaithfully and elegantly

modelreationsystems.ComparisonwithrelatedworkispresentedinSetion7.

Proofsoftheresultspresentedin thispaperanbefoundin [16℄.

Notation Weusethestandardmathematialnotionsandnotation.Amultiset

overaset

X

^{is afuntion}

µ : X → N = {0, 1, 2, . . .}

^{, and itssupport is}

||µ|| = {x ∈ X | µ(x) > 0}

^{.Theemptymultiset}

∅

satises

|| ∅ || = ∅

.Amultisetmaybe represented,somewhatinformally,by listingitselementswithrepetitions, e.g.,

µ = {y, y, z}

^issuhthat

µ(y) = 2

^,

µ(z) = 1

^,and

µ(x) = 0

^{otherwise.Wetreat}

setsas multisetswithoutrepetitions.

2 Reation systems

In thissetion, we explainsomenotionsrelevant toreationsystems. It isour

intention to introdue enough onepts to allow one to follow the subsequent

disussion on the relationship between reation systemsand Petri nets. Fora

omprehensivedesriptionof reationsystems,inluding motivations, applia-

tionsandexamples,thereaderisreferredto[79℄.

Denition1 (reation system [79℄). A reation system is a pair:

A = (S, A)

^,where

S

^{isanite bakgroundset omprising the entities of}

A

^,and

A

isthesetofreationsof

A

^{.Eah reationisatripletofthe form:}

a = (R, I, P )

^,

wherethe threeomponents arenite non-emptysets:

R ⊆ S

^{istheset of rea-}

tants,

I ⊆ S

^{isthe setof} inhibitors,and

P ⊆ S

^{isthe setof produts.}

The omponents of a reation

a = (R, I, P )

^{are denoted by}

R a

^,

I a

^and

P a

^,

respetively.Denition1desribesthestati strutureofareationsystem.To

apturethedynami behaviourofreationsystems,weneedadditionalnotions.

Denition2 (state of reation system). A state of a reation system is

any set

C

^{of its entities. Then an} initialised reation system is a triplet

A = (S, A, C 0 )

^,where

(S, A)

^{isareationsystemand}

C 0 ⊆ S

^{isthe initialstate.}

Inthis and in the nextsetion, we will onsider asa running examplethe

initialised reationsystem

A 0 = ({w, x, y, z}, {a, b, c}, {x, z})

^{,withbakground}

set

{w, x, y, z}

^{,initialstate}

{x, z}

^{,andthreereations:}

a = ({x}, {y}, {y, z}) b = ({y}, {x}, {x, z}) c = ({z}, {w}, {z}) .

A reationsystemwith bakgroundset

S

^hasexatly

2 ^|S|

^{potentialstates.}

To desribe possible transitions between these states, we need to say what is

meantbyanourreneofareationorasetofreations.

Denition3 (state hange). A reation

a

^{is enabled at a state}

C ⊆ S

^if

R a ⊆ C

^and

I a ∩C = ∅

;theresultofareation

a

^at

C

^isdenedby

res _a (C) = P a

if

a

^isenabledat

C

^and

res _a (C) = ∅

otherwise. The resultof

A

^on

C

^{, denoted}

by

res A (C)

^{onsistsof theprodutsofall reations from}

A

^{enabledatC, thatis}

res A (C) = [

a∈A

res _a (C) .

This statehange isdenotedby

C −→ res A (C)

^.

NotethatthestatehangesapturedbyDenition3aredeterministi.More-

over,allentitiesin

C \ S

a∈A res a (C)

^{disappear.Asaresult,andunlikeinother}

formalmodelsofdynamisystems,thereisnopersistenyinareationsystem

inthesensethatanentitypresentinastatedisappearsunlessitissustainedby

atleastonereation.

Fortheexamplereationsystem

A 0

^,wehave:

{x, z} −→ {y, z}

^and

{y, z} −→ {x, z}

^and

{w, x, y} −→ ∅ .

Onemay observethat thereis noonit between reationsin the `lassi'

sense that the ourrene of one reation might imply that another reation

whihisalsoenabledattheurrentstate,annotour.This,again,isafeature

notfound inmost otherformal modelsof dynami systems.Inpartiular,it is

worthwhiletopointexpliitlytothe`non-ounting'featuresofreationsystems:

entities are either present or not, and produed or not, and reationsan or

annot our based only on thepresene orabsene of ertain entities. There

is norepresentation ofmultiple instanesof entities ormultiple ourrenesof

reations. Thus reation systems are a qualitative rather than a quantitative

model.

Wealsonote thatthere isanalternativenotionofonit-freenessforaset

ofreations,alled onsisteny.Aset ofreations

R

^{isonsistent ifforanytwo}

reations

a, b ∈ R

^,

R a ∩ I b = R b ∩ I a = ∅

. Clearly,if aset of reationsis not onsistent,thenthereationsitomprisesannotbeexeutedsimultaneously.

Although the goal of this paper is a faithful `translation' of reation sys-

temsintoPetrinets,weonludethissetionwithanumberofommentsabout

researh on reation systems. This researh happens in the framework of re-

ation systemswhere a reationsystemonstitutes thebasi tehnial notion.

Depending onthegoalof aspei researh theme,manyother onstrutsare

introduedand studied (see,e.g., [2,9,10℄) theyform variousextensions of

thebasinotionofreationsystem.Forexample,therearemanybiologialsitu-

ationswhereoneneedstoassignquantitativeparameters(time,onentrations,

...)tostatesofabiohemialsystem.Althoughreationsystemsareaqualita-

tivemodel(theyannot`ount'),theyanbeextendedsothatsuhquantitative

parametersanbeaommodated.Thisisdonethroughtheuseofmeasurement

funtions whih lead to reation systems with measurements (see [2,3,9,10℄),

where variousnumerialparametersanbeassignedto (alulatedfor)onse-

utivestatesofdynamiproesses.

Finally, we want to point out that (beause living ells are open systems)

reationsystemshaveanenvironmentandtheyoperate/evolvewithinahanging

ontext (withentities omingfrom theenvironmentinueningthe transitions

of dynami proesses). In this paper, however, we will onsider only ontext-

independent proessesdened byareationsystemwithaninitial state,where

eahnext stateis obtainedsolelyasthe resultof reationstakingplae in the

previous state (thus assuming that the environment does not inuene state

transitions).

3 Reation systems and low-level Petri nets

Inthis setion, wedisuss twopossiblewaysof modelling ontext-independent

proessesofreationsystemsusinglow-levelPetri nets(pt-netsextended with

withinhibitorandativatorars).

Inaddition to the standardnotions of reationsystems, in order to better

explainhowtheyrelateto Petri nets,throughouttherest ofthispaperwewill

say that aset

R ⊆ A

^{is enabledat}

C

^{if eah reationof}

R

^isenabledat

C

^{. If}

R ⊆ A

^isenabledat

C

^,then

C = ^R ⇒ res R (C) = [

a∈R

P a .

denotestheeetof

R

^at

C

^.

Denition4 (pt-nets with inhibitorand ativator ars [14℄).A pt-net

withinhibitorandativatorars(or ptia-net)

N = (Pl, Tr , Flw , Inh, Act, M 0 )

isatuple suhthat

Pl

^and

Tr

^{arenite,disjoint setsof}respetively plaesand transitions, and:

Flw ⊆ (Pl × Tr) ∪ (Tr × Pl )

^,

Inh ⊆ Pl × Tr

^,

Act ⊆ Pl × Tr

are respetively the sets of ow, inhibitorand ativatorars. Moreover,

M 0

^is

amultisetofplaes, the initial markingof

N

^{;ingeneral,anymultisetof plaes}

isalleda marking.

Indiagrams,plaesaredrawnasirlesandtransitionsasretangles.Mark-

ingsare thepossibleglobal ongurations(states) of

N

^{.Wesay thataplae}

q

is marked under a marking

M

^if

M (q) > 0

^{, where}

M (q)

^{denotes thenumber}

ofourrenesof

q

ⁱⁿ

M

^{.Indiagrams,markingsareindiatedbyputting}

M (q)

tokens insidetheirlerepresenting

q

^.If

(x, y) ∈ Flw

^,then

(x, y)

^isanarlead-

ing from node

x

^{to node}

y

^{. A double headedarrowbetween}

q

^and

t

^indiates

that

(q, t), (t, q) ∈ Flw

^{. Aninhibitorarendswithasmallopenirle,whilean}

ativatorarendswithasmallblakirle.

Given anode

x

^{, we denote by}

^• x

^{the set of input nodes of}

x

^{, i.e., those}

y

for whih

(y, x) ∈ Flw

^{, and by}

x ^•

^{the set of output nodes of}

x

^{, i.e., those}

y

for whih

(x, y) ∈ Flw

^{. For atransition}

t

^weuse:

^◦ t = {q | (q, t) ∈ Inh}

^and

t = {q | (q, t) ∈ Act}

^{todenotetheinhibitorandativatorplaesof}

t

^{.All four}

notations extendin the usual way to sets of nodes. As in the ase of reation

systems,wenowformalisethenotionofmarking(state) hange.

Denition5 (marking hange). A multiset of transitions

U

^{(also alled a}

step) is enabled ata marking

M

^if

^◦ U ∩ ||M || = ∅

,

U ⊆ ||M ||

^{and, for every}

plae

q

^,

M (q) ≥ P

t∈q ^• U (t)

^{(reall that}

||M ||

^{isthe set of}

q

^{whih our in}

M

^,

and

U (t)

^{isthe numberof ourrenesof}

t

ⁱⁿ

U

^).

In suh a ase,

U

^{an be red with its eet on}

M

^{being given by the result-}

ing marking

M ^′

^{suh that, for every plae}

q

^:

M ^′ (q) = M (q) − P

t∈q ^• U (t) + P

t∈ ^• q U (t)

^{. We denote this by}

M [UiM ^′

^{.Moreover, if}

U

^{is a maximal (w.r.t.}

multiset inlusion) step of transitions enabled at

M

^{, then we may denote this}

markinghangealso by

M [U i max M ^′

^.

Notethat wheneverastep

U

^{is enabledat marking}

M

^{itmust bethease}

thatallativatorplaesoftransitionsin

||U ||

^{aremarked(arein}

||M ||

^)andnone

oftheinhibitorplaesoftransitions in

||U ||

^aremarked.

We now make some general observations and assumptions about the rela-

tionship betweenreationsystemsandnets.

Entitiesanberepresentedbyplaes,andreationsbynettransitions.

Sine there are no onits between reations,ativator ars an be used

totestfor thepreseneofreatants(ratherthan laimingresouresforthe

exlusiveuseaswithordinaryarsandinputplaes).

Allreationsthatanourinareationsystemdoour,andtheonlyen-

titiesleftafterastatehangearethenewlygeneratedproduts.InthePetri

netframework,these featuresorrespondto maximalparallelism desribed

attheendofDenition5,andplaeresetting [6℄desribedlateron.

(a)

N I (A ⁰ )

x

y

z

w

q a

q b

q c

r x

r y

r z

r w

a

b

c

(b)

N ^II (A ⁰ )

x

y

z

w r

a

b

c

⋆

⋆

⋆

⋆

Fig.1.MethodIandIIrepresentationsofthereationsystem

A 0

^.

MethodI. TherstattemptisillustratedinFigure1(

a

^{)fortheexamplerea-}

tionsystem

A 0

^{.Method Iproduesaptia-net}

N I (A 0 )

^suhthat:

Transitions

a

^,

b

^and

c

^{useativatorarsandinhibitorarstotest}respetively forthepreseneandabseneoftokensin theplaes

w

^,

x

^,

y

^and

z

^.

Plaes

q a

^,

q b

^and

q c

^{ensure that thethree} transitions modelling reations, i.e.,

a

^,

b

^and

c

^{, reatmostonein anystep.Thisorrespondsto the`non-}

ounting'ofourreneinstanesofthesamereationinareationsystem.

Transitions

r w

^,

r x

^,

r y

^and

r z

^{(in a maximal step) empty the four plaes}

modelling entities

w

^,

x

^,

y

^and

z

^{. This doesnot have any inuene on the}

ringofthetransitions

a

^,

b

^and

c

^.

Inasinglemaximalstep,

M [U i max M ^′

^{,the netresamaximal multisetof}

transitions

U

^{enabledatmarking}

M

^{andthenproduesanewmarking}

M ^′

^.

ForthenetinFigure1(

a

^{),suharingrulegives:}

{x, z, q a , q b , q c } [{r x , r z , a, c}i max {y, z, z, q a , q b , q c } {x, x, x, z, q a , q b , q c } [{r x , r x , r x , r z , a, c}i max {y, z, z, q a , q b , q c } .

Formally,given aninitialised reation system

A = (S, A)

^{, Method I yields}

aptia-net

N I (A)

^{suhthat theplaes,}transitionsand theinitialmarkingare, respetively:

Pl = {q a | a ∈ A} ∪ S

^,

Tr = {r s | s ∈ S} ∪ A

^and

M 0 = {q a | a ∈ A} +C 0

^{.Moreover,thesetsofow,inhibitorandativatorarsare,}respetively:

Flw = {(s, r s ) | s ∈ S} ∪ {(a, q a ), (q a , a) | a ∈ A} ∪ {(a, s) | a ∈ A ∧ s ∈ P a } Inh = {(s, a) | a ∈ A ∧ s ∈ I a } Act = {(s, a) | a ∈ A ∧ s ∈ R a } .

Notethatthiskindofmodellinginombinationwiththe`resetting'ofplaes

w

^,

x

^,

y

^and

z

^{in eah red step,} implemented by the auxiliarytransitions

r w

^,

r x

^,

r y

^and

r z

^{,meansthattheresultingPetri netisbounded(ineveryreahable}

markingthemultipliityofeahplaeisnevermorethanthenumberofreations

of

A

^if

A

^{hasat leastonereation).}

In order to relate the behaviour of the original reationsystem

A

^{and its}

ptia-netrepresentation

N I (A)

^{justintrodued,weneedtwomappings.Therst}

one takesamarking

M

^of

N I (A)

^{andreturnsastateof}

A

^{,and theother takes}

astep

U

^oftransitions of

N I (A)

^{andreturnsaset ofreationsof}

A

^,asfollows

ν I (M ) = S ∩ ||M ||

^and

ϕ I (U ) = A ∩ ||U||

^{. It isthen possibleto showanumber}

of results, where amarking

M

^{of the ptia-net}

N I (A)

^{is alled} well-formed if

M (q a ) = 1

^,forevery

a ∈ A

^.

First,

M 0

^{is a} well-formed marking satisfying

ν(M 0 ) = C 0

^{, and if}

M

^{is a}

well-formedmarkingand

M [U iM ^′

^,then

M ^′

^isalsowell-formed.Seond,if

M

^is

awell-formedmarking,thenforeveryreation

a ∈ A

^,

a

^isenabledat

M

ⁱ

{a}

isenabledatstate

ν I (M )

^{.Wethenanshowthatthe}translationissound.

Theorem1. If

M

^isawell-formedmarking then:

1.

M [U iM ^′

^implies

ν I (M ) ^ϕ = ^{I (U)} ⇒ ν I (M ^′ )

^.Moreover,if

M [U i max M ^′

^,then

ϕ I (U )

omprisesallreationsenabledat

ν I (M )

^.

2.

ν I (M ) = ^R ⇒ C

^implies

M [U iM ^′

^{for some}

U

^and

M ^′

satisfying:

ϕ I (U) = R

and

ν I (M ^′ ) = C

^{. Moreover, if}

R

^{omprises all reationsenabledat}

ν I (M )

^,

then

M [U i max M ^′

^.

Thus, eah maximal omputational step in the Petri net orresponds to a

unique exeution of the reation system, and eah exeution in the reation

systemorrespondstoatleast onemaximalstepinthePetrinet.Forexample,

thetwoexeutionsgivenaboveforthePetri netinFigure1(

a

^{)bothorrespond}

to

{x, z} ^{a,c} = ⇒ {y, z}

^{in thereationsystem}

A 0

^.

NotethatinFigure1(

a

^{)oneannotsimplydeletetheauxiliaryplaesofthe}

form

q r

^{astheneahofthe}transitionsrepresentingreationsouldbeunbound- edlyenabled.Toaddressthisproblemoneouldhangetheativatorarsfrom

plaesrepresentingentitiesintoowars.Then,however,itwouldbeneessary

to add weights

|R|

^{to thears}orrespondingto theprodution of newentities in ordertoavoidonits ontheplaesrepresentingthereatants.

MethodII. Therstattemptto modelontext-independentreationsystems

providesasound translation, but itis notsimple asitemploysfeatures whih

an makeformal analysisand veriationfarfrom easy.Onewayofimproving

Method Iouldbetoreplaemultisetsofredtransitionsbysetsofredtran-

sitionsleadingto amaximal set-semantis.Thisanbeahievedbyusingreset

ars [6℄,onneting plaesto transitionsand indiatedby

⋆

^{'s in the diagrams,}

whihalwaysemptytheirsoureplae.Formally,resetars

Reset ⊆ Pl × Tr

^do

nothaveanyinueneontheenablednessofastep

U

^{,butthealulationofthe}

markingofaplae

q

^{aftertheringof}

U

^{(nowaset)atmarking}

M

^hangesto:

M ^′ (q) =

M (q) − |q ^• ∩ U | + | ^• q ∩ U |

^if

({q} × U ) ∩ Reset = ∅

| ^• q ∩ U |

^otherwise

.

Theresultingptia-netwithresetars

N II (A 0 )

^{isshowninFigure1(}

b

^).Tran-

sition

r

^{is always enabled and, when red, removes all the tokens from the}

plaesmodellingtheentities.ForthenetinFigure1(

b

^{),thenewringrulegives}

{x, z} [{r, a, c}i max {y, z, z}

^and

{x, x, x, z} [{r, a, c}i max {y, z, z}

^.Oneanthen

showthataounterpartofTheorem1holdsalsointhisase,with

ν II

^denedas

ν I

^beforeand

ϕ II (U ) = U \ {r}

^{.Astransition}

r

^{isalwaysenabled,wenowhavea}

one-to-oneorrespondenebetweengroupsofexeutedreationsandtransitions,

attheprieofintroduingnon-standardresetars.

Toremove theneed to have resetars or,equivalently, to obtainaone-to-

oneorrespondenebetweenstatesandmarkings,oneouldhangetherulesfor

insertingtokensintoplaes,bybasiallyapplyinganOR-treatmentforarriving

tokens. This would, of ourse, be aradial departure from the standard Petri

netapproah,butoneworthinvestigating.Theresultingmodelof set-nets will

bedesribedinSetion5.

4 Reation systems and high-level Petri nets

Thetwotranslationsdesribedintheprevioussetionuselow-levelpt-netsex-

tended with reset ars in addition to inhibitor and ativator ars as well as

maximalparallelism.Resetarsareanon-standardmehanismand,inpartiu-

lar,theydonotasyetsupportaausalproesssemantis.Moreover,theeetof

aresetardependsontheurrentmarkingratherthanonaxedinput/output

relationwithitsneighbourhood.Toopewiththisproblem,wewillnowoutline

twotranslationsfrom ontext-independent reationsystemsto high-level Petri

nets. We assume familiarity with the basi oneptsof high-level nets [13℄,in

partiular, ar insriptions, ativatorand inhibitor ars, and simple transition

guards.

MethodIII. Thersttranslationisillustratedbythehigh-levelnet

N III (A 0 )

shownin Figure2(

a

^{).Inthisase,tokensarepositiveintegersatingasthough}

theyweretime-stamps.Intuitively,atoken

n

^{isativeonlyinthe}

n

^-thexeution

yle ofthereationsystem.Beause thesametoken annotbeaessedmore

thanoneinastepsequeneevolution,resetarsarenotneededanymore.Sine

the

X

transition res in eah maximal step, the yle number

n

^{held in the}

`lok'plae

clk

^isknowntoalltransitionsrepresentingreations.Intheplaes representingentities,theyhekonlyfortokens

n

^{,ignoringalltheothertokens}

produedinpreviousyles,andthenproduetokenswithvalue

n +1

^tobeused

inthenextyle.Theinitialmarking

M 0

^{isformedbyinsertingasingletoken1}

intoplae

clk

^{andalltheplaes}

s

^suhthat

s ∈ C 0

^{.Notethat theresultingnet}

maybeunboundedasthetokensinplaesrepresentingentitiesarenot`garbage

olleted'.Forthehigh-levelnet

N III (A 0 )

^{in Figure2(}

b

^),wehave:

{x 7→ {1}, y 7→ ∅ , z 7→ {1}, w 7→ ∅ , clk 7→ {1}}

[{a n7→1 , c n7→1 , X _n7→1 }i max

{x 7→ {1}, y 7→ {2}, z 7→ {1, 2, 2}, w 7→ ∅ , clk 7→ {2}}

[{b n7→2 , c n7→2 , X _n7→2 }i max

{x 7→ {1, 3}, y 7→ {2}, z 7→ {1, 2, 2, 3, 3}, w 7→ ∅ , clk 7→ {3}} .

(a) N ^III (A 0 )

1 x

y

1 z

w

1

clk X

a

b

c

n

n + 1

(b) N ^IV (A 0 )

1 x

y

1 z

w

1 clk a

1 clk b

1 clk c

n n + 1

n n + 1

a y

a x hm ≥ ni a

b

c

n n + 1

n n + 1

m

n n + 1

Fig.2. Method III and IV representations of reation system

A 0

^{. Note that}

n

^and

m

^{are net variables, and that to avoid lutter not all ars have been annotated: all}

the ow(thiker)arsto plaes

x

^,

y

^,

z

^{are infatannotatedwith}

n + 1

^{,and all the}

unannotated inhibitor and ativator arsare annotatedwith

n

^{. In(}

b

^{), the auxiliary}

plaesfortransitions

b

^and

c

^{areomitted.Notethat}

hm ≥ ni

^{istheguardoftransition}

a x

^,andallothertransitionshavethetrivial

true

^guard.

AsintheaseofMethodI,noteverymarking

M

^of

N III (A)

^anrepresenta

validstateofthereationsystem

A

^.Wesaythat

M

^islok-onsistent ifthere isasingletoken

k

^inplae

clk

^{,andallthetokens}

l

^{inotherplaessatisfy}

l ≤ k

^.

Relatingtheresultingnetandtheoriginalreationsystemanbedoneusing

the following two mappings:

ν III (M ) = {s ∈ S | ||M (clk )|| ∩ ||M (s)|| 6= ∅ }

and

ϕ III (U ) = U \ { X }

^{. One anshowthat}

M 0

^{is a}lok-onsistentmarking satisfying

ν (M 0 ) = C 0

^,andif

M

^isalok-onsistentmarkingand

M [U i max M ^′

then

M ^′

^isalso lok-onsistent.

Theorem2. If

M

^isalok-onsistentmarkingthen:

1.

M [U i max M ^′

^implies

ν III (M ) ^ϕ = ^III ⇒ ^(U) ν III (M ^′ )

^.

2.

ν III (M ) = ^R ⇒ C

^implies

M [U i max M ^′

^forsome

U

^and

M ^′

satisfying:

ϕ III (U ) = R

^and

ν III (M ^′ ) = C

^.

MethodIV. Intheseond high-levelnetonstrutiontheaimis toeliminate

theneedformaximalparallelismusinginformationpresentinthetime-stamped

tokens.Wereplaetheglobal

clk

^{plaebyindividual}

clk a

^{plaes,whihareinre-}

mentedbytransitions

a

representingreations.Moreover,whenever

a

^isbloked

from ring in aertain yle oneof the auxiliarytransitions orresponding to

thepossible`reasons'forthebloking

a

^{isredto inrementthetoken in}

clk a

^.

This results in an inrement of the yle number for this transition (in ase

there is more than one reason for bloking, an auxiliary transition is hosen

non-deterministially).

Therearetwopossiblereasonswhy

a

^{mightbeblokedinyle}

n

^.Oneisthe

preseneofatoken

n

^intheplaerepresentinganinhibitorof

a

^,andtohekfor

thisweuseatransitionwithanativatorar,e.g.,

a y

^inFigure2(

b

^).Theotheris

moreompliatedasitisalakoftoken

n

^intheplaerepresentingareatant

s

for

a

^{,andtohekforthisweuseatransitionwithaninhibitorar.However,we}

alsoneedtoensurethatalltransitionswhihfeedtokensto

s

^{havealreadyhad}

ahane todoso,andwehekthisusing extraativatorarstogetherwitha

transitionguardwhihevaluatestotrueifallsuhfeedingtransitionshavetheir

loalylesuientlyhigh,e.g.,transition

a x

^inFigure2(

b

^{).Theoverallresult}

forthereationsystem

A 0

^{isahigh-levelnet}

N IV (A 0 )

^{showninFigure2(}

b

^).

Theresultinghigh-levelnetisexeutedaordingtothestandardsequential

(interleaving)ringruleand itsbehaviourloselysimulatesthat ofthenetob-

tainedbyMethodIII,andsoalsothebehaviouroftheoriginalreationsystem.

We skip the full desription of the relationship between these twonets. Intu-

itively,amarking

M

^oftheseondtranslationorrespondsdiretlytoamarking of therst oneifall theplaes of theform

clk _a

^{ontainthe samesingletoken}

k

^{,andallthetokens}

l

^{in otherplaessatisfy}

l ≤ k

^{.(Notethatfromeahreah-}

ablemarkingoftheseondtranslationoneanexeuteasequeneoftransitions

leadingtoamarkingwiththisproperty.)

5 Set-nets

Inourattemptstoobtainadiretandeleganttranslationfromreationsystems

into Petrinets,amajorand asfaraswe antellinsurmountableproblem was

the fat that several transitions may insert tokens into a plae representing

the preseneof asingle entity. Inthis setion, we introdue set-nets,amodel

thatresultedfromloserinvestigationsintothepossibilitiesofanOR-treatment

of arriving tokens representing the prodution of entities by reations. Note

that OR-treatmentofausalityhasbeenonsidered in[20℄,butthe underlying

prinipletherewasompletely dierentfromwhatwearegoingtopropose.

Themain idea is that in aset-netthere is noonept of ounting. Plaes

aremarkedornotmarkedandarshavenoweights.Set-nets resembleelemen-

tarynetsystems(en-systems)[19℄whihisafundamentalmodeltostudybasi

features of onurrentsystems, inluding onit, ausalityand independene.

However,theirexeutionsemantisis dierent.Inset-nets,amarkedplaein-

diatesthepreseneofaresourewithoutanyquantiation.Heneanynumber

of transitions that take input from this plae an be red at the same time.

Moreover,ringatransitionemptiesallitsinputplaes.Thustherearenoon-

itsovertokensinset-nets,unlikein en-systemsorpt-nets.Similarly,plaes

donotountthetokens,andthering ofatransitionsimplymarkseah ofits

outputplaes(whetherornottheywerealreadymarked).Wewillbuild upthe

newmodelin twostages,introduingrstset-netswithonlyowars.

(a) r

q s r ↓

q ↓

s ↓

a

b

(b) q

s

r q

r b q

q ↓

s ↓

r ↓

a b q ↓

Fig.3. A set-netrepresenting reation system

A 1 (a)

^{; and an ourrene net on-}

strutedforitsstepsequene

{b, q ↓ , s ↓ }{a, b, r ↓ , q ↓ } (b)

^.

Denition6 (basi set-net). A tuple

SN = (Pl , Tr, Flw , M 0 )

^{is a (basi)}

set-netif therstthreeomponentsareas inDenition4, and

M 0 ⊆ Pl

^isthe

initial marking(ingeneral,any setof plaesisa marking).

Thegraphialrepresentationof set-netsisthesameasinthe aseof Petri

nets.Wenowformalisetheringruleforset-nets.

Denition7 (marking hange). A setof transitions

U

^{(also alleda step)}

is enabledata marking

M

^if

^• U ⊆ M

^{.In suha ase,}

U

^{an be redwith its}

eeton

M

^{beinggivenbytheresultingmarking}

M ^′ = (M \ ^• U )∪U ^•

^.Wedenote

this by

M [UiM ^′

^{.Moreover, if}

U

^{isthe setof all}transitions enabledat

M

^(i.e.,

all transitions

t

^satisfying

^• t ⊆ M

^{),thenwe maywrite}

M [Ui max M ^′

^.

Heneastep

U

^{enabledatamarking}

M

^{mayontaintwodistint}transitions

t

^and

u

^forwhih

^• t ∩ ^• u 6= ∅

or

t ^• ∩ u ^• 6= ∅

and yet the ommonplaes will neverontainmorethanonetoken.Sinetokensaremanipulatedusingset-based

arithmeti wehavehosenthename`set-nets'forthenewlassofPetri nets.

We have introdued rst basi set-nets (without inhibitor and ativator

ars), as it seems that one anattempt to develop for them aounterpart of

`struturetheory'ofpt-nets.Toillustrateourpoint,letusonsiderabasiset-

net

SN = (Pl, Tr, Flw , M 0 )

^{with at least one} transition. A non-empty set of plaes

Sphn ⊆ Pl

^{is alled asiphon if}

^• Sphn ⊆ Sphn ^•

^{. Similarly, anon-empty}

set ofplaes

Trap ⊆ Pl

^{isalled atrap if}

Trap ^• ⊆ ^• Trap

^{.It anbeeasilyseen}

that an empty siphon annot aquire a token by ring any transition, and a

markedtrap annot beome empty by ring anytransition. Both type of sets

of plaes anbe used to provide asuient onditionfor deadlok-freenessin

pt-netswhihwasamajormotivationbehindthedevelopmentoftheirstruture

theory.Asitturnsout,thesameanbedonein aseof set-nets.

Theorem3. If in the initial marking, every siphon ontains a marked trap,

thenthe set-netisdeadlok free.

Wenextintrodueset-netswithinhibitorandativatorars.

Denition8 (set-net). Atuple

SNIA = (Pl , Tr, Flw , Inh, Act , M 0 )

^{isa set-}

net if the rst ve omponents are as in Denition 4, and the last one as in

Denition 6.

Thedenitionsandnotationsonerningthemarkinghangein

SNIA

^arethe

sameasfor

SN

^{inDenition7withoneexeption,namelyasetof}transitions

U

isenabledatamarking

M

^if

^• U ∪ U ⊆ M

^and

^◦ U ∩ M = ∅

.Itisinterestingto observethatanenabledstep

U

^{isalwaysonsistentinthesensethat}

( ^• U ∪ U )∩

◦ U = ∅

. Suh aproperty hasa naturaland diret (as wewill see) onnetion withthenotionofonsistenyintroduedforreationsystems.

Asbefore, givenatransition

t

representingareation,thesets

• t

^,

^◦ t

^and

t

orrespond to thereatants,inhibitors andprodutsof this reation.However,

wedo not require that these sets be non-empty in a set-net (at least at this

point)assuhanassumptionisnotneessary.

6 Reation systems and set-nets

Reationsystems and set-netst together well in thesense that both donot

ount tokens and both hange states on the basis of the presene/absene of

resoures,representedbysets.Moreover,undertheset-netsemantis,ordinary

ars(transitions)anbeusedtoemptyplaes.Inthissemantis,resetarswith

theireetdependingontheurrentnumberoftokensinaplaearemeaningless.

Finally,followingtheassumptionthat allreationsthat antakeplaedo take

plae,themaximalset-semantisanbeemployed.

Figure 3(

a

^{) depits a set-net} orresponding to a ontext-independent ini- tialisedreationsystem

A 1 = ({r, q, s}, {a, b}, {q, s})

^,where

a = ({r, q}, ∅ , {r})

and

b = ({q}, ∅ , {r, q})

^{.(Forreasonsoflarity,weallowinthissetionreations}

withoutanyinhibitors.) Asbefore, plaes represententities.Transitions

r ↓

^,

q ↓

and

s ↓

^{ensurethat one theset-net isativeonlytokens produedin thelast}

maximalsteparepresentintheurrentmarking.Forexample,wehave:

{q, s} [{b, q ↓ , s ↓ }i max {r, q} [{a, b, r ↓ , q ↓ }i max {r, q} ,

andso

σ = {b, q ↓ , s ↓ }{a, b, r ↓ , q ↓ }

^{isamax-stepsequene.Relatingthebehaviour}

of theset-netmodeland theoriginal reationsystemis easyand weobtaina

ounterpartofTheorem1with

ν(M ) = M

^and

ν(U ) = U \ {s ↓ | s ∈ S}

^.

For a set-net without inhibitor and ativator ars as in Figure 3(

a

^{), one}

an investigate the ausality semantis of reation systems based on the un-

foldings of the orresponding set-nets. Figure 3(

b

^{) showshow suh an our-}

renenetouldbederivedfortheset-netinFigure 3(

a

^{)anditsstepsequene}

{b, q ↓ , s ↓ }{a, b, r ↓ , q ↓ }

^{whih orrespondsof the state sequene}

{b}{a, b}

^{of the}

original reation system.It is worthobserving that the proess hasbranhing

plaes whih isnotpossible,in theaseof proessesof en-systemsorpt-nets.

This,however,isfullyonsistentwiththeexeutionsemantisof set-nets.

Modelling inhibitionaspetsofreations is ratherstraightforwardusingin-

hibitorars,asillustratedbytheset-netinFigure4(

a

^),representingtheontext- independentinitialisedreationsystem

A 2 = ({r, q, s}, {a, b}, {q})

^,where:

a = ({r, q}, ∅ , {r})

^and

b = ({q}, {s}, {r, q})

^and

c = ({q}, ∅ , {s}) .

Using inhibitor ars givesa ompat translation of reationsystems whih is

in asenseminimal w.r.t.thenumberofplaes, arsandtransitions. Moreover,

relatingthebehaviouroftheresultingset-netsandtheoriginalreationsystems

an be doneas before. Formally,the plaes, transitions and initial marking of

thetranslationaregivenby:

Pl = S

^,

Tr = A ∪ {s ↓ | s ∈ S}

^and

M 0 = C 0

^.There

arenoativatorars, andtheowandinhibitorarsareasfollows:

Flw = {(s, s ↓ ) | s ∈ S } ∪ {(s, a) | a ∈ A ∧ s ∈ R a } ∪ {(a, s) | a ∈ A ∧ s ∈ P a } Inh = {(s, a) | a ∈ A ∧ s ∈ I a } .

Thedevelopmentofaausalproesssemantisof set-netswithinhibitorarsis

morediult.It isthereforeinterestingtoonsider modelsof reationsystems

usingset-netswithoutanyinhibitorars,asoutlinednext.

Figure4(

b

^{)showsaset-netwithoutinhibitorarsmodelling}

A 2

^.Thewayin

whihitdoesitisnowmoreinvolved.Morepreisely,eahexeutionstepofthe

reationsystemissimulatedintwophasesbytheset-netoperatingaordingto

themaximal parallelismexeutionsemantis.Tokeepthesetwophaseslearly

separated,theyareontrolledbyanadditionalylisubnetwithtwoplaes.The

keyaspet oftheonstrutionistheuseof a`omplement'

s ^cpl

^{ofthe`regular'}

plae

s

^{whihatthetimeofhekingwhether}

s

^{isemptybyreation}

b

^ontains

atokeni

s

^isempty.

Figure4(

c

^{)providesageneripitureofhow,intheproposed}onstrution, aset-net(without inhibitorars)handles anentity

r

^{inits roleasareatant,}

inhibitor,andprodut.Notethat

r

^isrepresentedbytwoplaes,

r

^and

r ^cpl

^,and

if

r ^cpl

^{ismarkedthentheentity}

r

^{inabsentintheurrentstate.Moreover,eah}

reation

d

^is representedbytwotransitions,

d

^and

d ^′

^{.The rstorresponds to}

theenablingstageof

d

^{,andtheseondtothegenerationofitsproduts.}

Therst phase of the simulation always startsin aonsistent marking

M

in whih there is a token in plae

phI

^{; for every}

s ∈ S

^,

s ∈ M ⇔ s ^cpl ∈ / M

^,

and otherwise all plaes are empty. Inthis phasetransitions orresponding to

reationsbeome ativeonthebasisof thepreseneandabsene oftheirrea-

tantsandinhibitors.Simultaneously,transitionsoftheform

r ↓

^and

r ↑

^takeare

thatalltheentitiespresentintheurrentstateeasetoexist(theirorrespond-

ing plaesareemptied and theomplement plaeslled). Intheseond phase,

eahenabled transition

d ^′

^{nishestheexeutionof the}orrespondingreation, andmarkstheplaes orrespondingto theentities produedbyreation

d

^and

emptiestheiromplements.

Relatingthebehaviouroftheset-netmodelandtheoriginalreationsystem

ismoreompliated,usingthefollowingtwomappings:

ν(M ) = M \({phI }∪{s ^cpl | s ∈ S}) ϕ(U ) = U \({I}∪{s ↓ | s ∈ S}∪{s ↑ | s ∈ S}) .

(a) r

q s r ↓

q ↓

s ↓

a b c

(b) phI

phII II

I

r q s s ^cpl r ↓

q ↓

s ↓

s ↑

a b c

a ^′ b ^′ c ^′

(c) r

r ^cpl r ↓

r ↑

a

c b

a ^′

c ^′ b ^′

Fig.4. Twoset-netsrepresenting

A ²

⁽

a, b

^).Generitranslationwithoutinhibitorars:

here

r

^{isareatantforreation}

a

^,produtfor

b

^{,andinhibitorfor}

c

⁽

c

^{).Notethatnot}

allplaesandarsareshown;inpartiular,eahreationhasatleastonereatantand

henetransitionslike

c

^{anonlyreintherstphase.}

Oneanthenshowthat

M 0

^{isonsistentandsatisfying}

ν(M 0 ) = C 0

^{,and if}

M

isaonsistentmarkingand

M [U i max M ^′′ [U ^′ i max M ^′

^then

M ^′

^{isalsoonsistent.}

Theorem4. If

M

^{isaonsistent markingthen:}

1.

M [U iM ^′′ [U ^′ iM ^′

^implies

ν(M ) ^ϕ(U) = ⇒ ν(M ^′ )

^.

2.

ν(M ) = ^R ⇒ C

^implies

M [U iM ^′′ [U ^′ iM ^′

^{for some}

U

^,

U ^′

^,

M ^′

^and

M ^′′

^satisfy-

ing:

ϕ(U ) = R

^and

ν (M ^′ ) = C

7 Related work and onluding remarks

Whenintroduinganewlassof Petrinets,espeiallyafundamental one,itis

neessarytoputitintheontextofexistingformalisations.Tomakeomparison

fair,wewillnowdroptheassumptionaboutmaximalparallelismintheexeution

of set-nets(whih isimplied bytheexeution mode ofreationsystems), and

onsidersemantiswhihallowsanysetofenabledtransitionstobered.

Set-netsaresosimplewhenitomestotheirdenition,thatitisreasonable

toexpetthattherewereinthepastnetlasseswithsimilarfeatures.Indeed,the

fundamentallassof en-systems[19℄extendedwithinhibitoraswellasativator

ars [12,17,18℄ basially havethe samestati struture as set-nets.However,

their treatment of onits between transitions aessing the same token, as

wellblokingatransitionwhihouldaddatokentoamarkedplae,aretotally

(a)

q

a b

c (b)

q ⁰ q ¹

a ⁰ a ¹ b ⁰

b ¹ c

Fig.5.Boolean net(set-net withsequentialsemantis) (

a

^{), and1-safept-netsimu-}

latingits(sequential)behaviour(

b

^).

dierent.Thelatterissuehasbeennotedinthepast,andtheonstraintrelaxed.

Forexample,therearevariationsofPetrinets,suhasBooleanPetrinets,where

addingatokentoanalreadymarkedplaedoesnotaddanothertoken[4,5,11℄.

Also,behaviourofthiskindwasmentionedin[1℄intheontextofnetsynthesis.

Having said that, the semantis onsidered in prior works known to us was

based on single transition rings, rather than (maximal) steps as is the ase

for set-nets.Therefore,the previousmodelswerenotonerned withmultiple

inputs of tokensto asingle plae somethingwhih is essentialif onewants to

faithfully model reation systems. Furthermore, by aiming at a set-semantis,

we had to introdue the non-onit feature on the ow ars onsuming the

tokens. Therefore,asfar as weare aware,the model of set-nets is anoriginal

ontributiontotheeld ofPetri nets.

Aswealreadymentioned,set-netswith interleavingsemantisarenothing

but Boolean nets used, for instane, in [5℄.In suh aase, the lak of onit

whenringtwotransitionssharinganinputplaeisanirrelevantissue,andthe

onlynon-standardaspetsisthatringatransitionwithamarkedoutputplae

does notinreasethetoken ountin that plae. Suh afeature, moreover,an

easilybemodelledusingordinary1-safept-nets,aordingtothefollowingidea.

First, one splits eah plae

q

^{into plaes}

q ⁰

^and

q ¹

^, respetively representing the lak and presene of a token in

q

^{. Then, eah transition}

t

^{adding tokens}

to plae

q

^{is split into}

t ⁰

^and

t ¹

^{to aount for two dierent states the plae}

q

^{an be in} represented by

q ⁰

^and

q ¹

^{. Figure 5} illustrates this onstrution.

It an be easily seenthat both nets generatethe same sequential reahability

graphsassumingthat

a ⁰

^and

a ¹

^{areinstanes of}

a

^,and

b ⁰

^and

b ¹

^areinstanes

of

a

^{. However, one we start treating the net in Figure 5(}

a

^{) as set-net, the}

situation hanges radially. The reason is that we then have three rings of

thefollowingform:

∅ [{a}i{q}

^,

∅ [{b}i{q}

^and

∅ [{a, b}i{q}

^{.Now,thestandard}

lasses ofPetri netsenjoythe so-alledsubset propertywhih means that ifa

step

U

^{isenabledatmarking}

M

^{,thenalsoanyofitssubsetsisenabledaswell.}

Suppose,then,thatthereisaPetrinet

N

^{satisfyingthispropertyandsuhthat}

its step reahability graph is the same asthat of the set-net in Figure 5(

a

^),

perhapsafter renaming

λ

^{being applied to the}transitions of theformer. Then wehavetohavetwotransitions,

t

^and

u

^,in

N

^suhthat

λ(t) = a

^,

λ(u) = b

^and

M 0 [{t, u}iM

^{. Then, by the subset losure property, we also have}

M 0 [{t}iM ^′

and

M 0 [{u}iM ^′′

^{.Hene,by the}reahabilitygraphisomorphism,wemusthave

M = M ^′ = M ^′′

^aswellas

M 0 6= M

^.Henewehave:

M 0 [{t, u}iM

^and

M 0 [{t}iM

and

M 0 [{u}iM

^and

M 0 6= M

^{. In the standard Petri nets, inluding various}

extensionsofpt-nets,

M 0 [{t, u}iM

^and

M 0 [{t}iM

^{wouldimplythat}

u

^doesnot

hangetheurrentmarking.Similarly,

M 0 [{t, u}iM

^and

M 0 [{u}iM

^wouldimply

that

t

^{doesnothangetheurrentmarking.Yetthe}simultaneousringof

t

^and

u

^{does hange the marking as}

M 0 6= M

^{. This would produe a} ontradition.

Whatwejust presentedisintuitionratherthanproof,however,weexpet that

detailed arguments an be developed for any of the standard net lasses. An

importantonsequene,however,isthatset-netsaresemantiallydierentfrom

the existing net lasses and therefore deserve to be reognised as an original

ontribution.

8 Conlusions

ThemaininitialmotivationofourinvestigationwastoseehowPetri netbased

oneptsouldbedeployedtoanalysereationsystems.Inpartiular,wewanted

todisovermethodsforhekingpropertiesofreationsystemsbyrelatingthem

tothepropertiesof theorrespondingPetrinetsandausalproesses.

We proposed modelling methods resulting both in low-level and high-level

nets.Inallfourases,weestablishedaloseorrespondenebetweenthemark-

ingsofPetrinetsandstatesoftheoriginalreationsystems.Thesamewastrue

oftheevolutionsoftwoorrespondingmodels.Infat,weestablishedthatthey

haveessentiallyisomorphistatespaes.Allthesenetmodels,however,exhibited

deienies w.r.t.simpliity and/oreleganeand/or tratabilityof thetransla-

tion.For example,bothhigh-levelnetmodelsareintrinsiallyunbounded, and

the seond of thelow-leveltranslations usesreset ars. We thereforeproposed

anewlassofPetrinets,alled set-nets,whihwefeelprovideastrongmath

withthereationsystemsandtheirsemantis.

Inthiswaywethinkwederivednewinterestingnotionsandontributionsto

Petrinettheorybasedonourexperieneswithreationsystemsinasimilarway

astheoneptsofloalitiesandloallymaximalonurrenywerederivedfrom

ourpreviousinvestigationofaPetrinetsemantisofmembranesystems[15℄.

Aknowledgement Wewouldliketothanktheanonymousreviewersfortheir

suggestions and omments. This researh was supported by the Pasal Chair

awardfrom LeidenUniversityandtheEpsrVerdadprojet.

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2. R.Brijder,A.Ehrenfeuht,M.G.MainandG.Rozenberg:Reationsystemswithdu-

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3. R.Brijder, A.Ehrenfeuht,M.G.Main andG.Rozenberg:A TourofReation Sys-

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