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An EfficientAlgorithmforFinding StructuralDeadlocksin ColoredPetriNets * K.BARKAOUI C.DUTHEILLET * Laboratoire CEDRIC Conservatoire NationaldesArtsetMétiers 292rue Saint-Martin 75003Paris FRANCE e-mail [email protected] :

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S.HADDAD #

# IBP Laboratoire MASI Université P.& M.Curie Place 4, Jussieu 75252 ParisCedex0FRANCE 5e-mail [email protected] :

AbstractInthispaper,wepresentanalgorithmtocomputestructural deadlocksincolorednetsunderspecifiedconditions.Insteadofappl yingthe ordinary algorithm ontheunfoldedPetrinet,ouralgorithmtake asdvantageof thestructureothe f colorfunctions.Itisobtainedbyiterat iveoptimizationsof theordinaryalgorithm.Eachoptimizationispecifiedbyam eta-rule,whose applicationisdetectedduringthecomputationofthealgorithm. The applicationosuch f meta-rulesspeedsupastepothe f algorithm withafactor proportionaltothesizeoacfolordomain.Weillustrate theefficiencyotfhis algorithmcomparedtotheclassicalapproachonacolorednet modellingthe dining philosophersproblem.

1

Introduction

AnalysistechniquesoP f lace-Transitionnets[1]canb classes.Thefirstapproachconsistsinbuildingtherea fullinformationbutisusually very expensive [2].

egroupedintwobroad chabilitygraph,whichgives

Anotherapproach- thestructuralanalysis -consistsingettinginformation aboutthe behaviorof the modeldirectly fromthe struc tureoits funderlyingbipartite andvaluateddigraph,theinitialmarkingbeingconsideredas aparameter[3].Two kindsof structuralanalysiscanbdeistinguished: • Thealgebraicanalysis where , thestructureotfhenetisrepresentedbythe incidencematrixassociatedwithitsunderlyingdigraph.It providesresultssuchas conditionsforlivenessandboundednessof the net,orl inearinvariants[4][5]. • Thegraphtheoreticalanalysis where , thebehaviorofthenetisrelatedtothe flow relationosubnets f generatedbryemarkable subsets ofplaces,suchastructural deadlocksandtraps[6].Withsuchtechniques,livenesscan bedecidedin polynomialtime fordifferentclassesof nets[7][8][ 9]. However,modellingrealcomplexsystemsyieldssolargem difficulttohandlethecomplexityofboththestructural

odelsthatitis andthereachability

analyses.ThusHigh-Levelnets Predicate/Transition have beenintroducedtcooncisely modellarge systems.I thishigh-leveldescription,theanalysisotfhemodel referingtoitsequivalentunfoldednet,i.e.,aplace-tra behavior.Thus,alotofresearchisbeingdonetodire analysistechniquessimilarto those existingforPlace

nets[10]orColorednets[11] nordertotakeadvantageof mustbeperformedwithout nsitionnetwiththesame ctlyapplytoHigh-Levelnets -Transitionnets.

Thepresentpaperiscontribution a tographtheoretical analysisforcolorednets. OnPlace-Transitionnets,thisapproachusestheflow relationofsubnetsgenerated by structuraldeadlocksandtraps,whichareremarkablesubs etsoplaces. f Forthese nets,theproblemoffindingstructuraldeadlockswasattac kedfromdifferentpoints ofview.Thetechniqueocf omputingstronglyconnecteddeadl ocksviaapositive flowcalculusonanexpandednet[8]wasoptimizedandextended totheclassof Unary Predicate-Transitionnetswithoutguards[12].Thea pplicationothis f method to lessrestrictive classesrequiresgeneralization a ofpositiveflowcomputation[13], whichseemstobeaverydifficulttask[14].Inanycas e,theproblemofobtaining the setof minimaldeadlocksby meansof flow computatio niNP-complete. s In[15],theproblemisexpressedasalogicprogrammingprobl identifystructuraldeadlocks(traps)withthesetofsolut satisfiabilityproblem.Incolorednets,theflowrel underlyingbipartitedigraphandthefunctionslabellingthe ofacolorednetmaynotcorrespondtoadeadlockothe f of[16],whiletheconverseistrue.Yet,forarestri problemoffindingdeadlockscanbreeducedtotheproblemof theskeleton[17],andthetechniqueproposedin[15]couldbee efficientextensionothe f methodtgoeneralcoloredne

em,leadingto ionsofaHorn-clause ationisdefinedbyboththe arcs.Hence,adeadlock net’sskeletoninthesense ctedclassocf olorednets,the findingdeadlocksin xploited.Butan tsisstillanopenproblem.

Inthispaper,wedevelopamethodforsolvingefficientl Pb by reasoningdirectly onthe colorfunctions,i.e.,

ythefollowingproblem withoutaneffective unfolding.

Pb:

Forany pairof disjointssubsetsof placesP find structural a deadlockDsuchthatD or decide thatno suchdeadlockexists.

exc andP inc, ∩P exc = ∅ andP

inc ⊂D,

Thepaperios rganizedafsollows.Section2introducesa propertyodf eadlocks whichyieldspropositionaldeductionrules.These rulesare usedinanalgorithmthat solvesproblem PbforPlace-Transitionnets.Wethenextendthisapproac hto coloredPetrinets,rewritingthealgorithmintermso deduction f rulesofirst f order logic. InSection3we , optimize the applicationsof the previo rules.Thesemeta-rulestransformasetofrulessoth transformedrule correspondsto the applicationosevera f InSection4we , presentthe optimizedalgorithmthatexpl meta-rules,andweapplyitonamodelofthediningphilos compareitsexecutiononacolorednetwiththeexecution onthe equivalentunfoldednet.Section5containsthe per

usrulesby definingmetaatoneapplicationofa initial l rules. oitstheeffectsothese f ophers.Thenwe oftheordinaryalgorithm spectivesof thiswork.

2

Structural DeadlocksinColoredNets

Inthefirstpartofthissection,werecallthebas togetherwiththedefinitionoadfeadlock.Wethenexten Petrinets.Finally,we show onanexample how theco be exploitedtiomprove the efficiency odeadlock f charact 2.1 StructuralDeadlocksinOrdinaryPetriNets

icnotationsofPetrinets, dthisdefinitiontocolored lorfunctionsom af odelcould erizationalgorithms.

-W Definition2.1 APetrinetNisa4-tuplewhere P isafinitesetofplaces, T isafinitesetoftransitions, ×T → istheinput(resp.output)function. W (resp.W +P:)

Wealsodefinetheinputsetandoutputsetofasubsetof respectively the inputandoutputtransitionsof the places Definition2.2 (resp.p•) isdefinedas:

places,whichcontain underconsideration.

∈P.Theinput(resp.output)setopf d, enotedby•p

Letp

T|∈ •p={t W (resp. p•={t ∈W T| ThisdefinitioncanbextendedtoasubsetDofplaces: •D = ∪ •p p ∈D

+(p, t) -(p, t)

≠0} ≠0})

D (resp.

• =



p ∈D

p•)

Asimilardefinitionexistsfortransitions. Definition2.3 t•) isdefineda:s ∈ W P| •t={p

Lett ∈ T.Theinput(resp.output)setoft,denotedb•y(resp. t -(p, t)

≠0}

(resp.

t•={p

∈ W P|

+(p, t)

≠0})

We now give the definitionosaftructuraldeadlock: Definition2.4 deadlock •D iff

LetDbeanon-emptysubsetofplaces.Disastructural ⊂D•.

Thefollowingpropertystatesthatifnoinputplaceoaf deadlock,thentheoutputplacesotfhistransitionneithe Suchapropertyisintroducedsinceitleadstotheconstr deadlocks. Property2.1 t ∈ T, [∀

transitionbelongstoa rbelongtothedeadlock. uctionofmaximal

LetDbeanon-emptysubsetofplaces.Wehave: •t ⊂(P D) \ t• ⊂ (P D) \] Disas⇔ tructuraldeadlock

Proof: Ifweconsiderthenegationoftheleft-handpartotfh canwrite: [∀ t ∈ T, ∃p ∈D,p ∈ t•  ∃p' ∈ D,p' ∈ •t] ⇔ Dis daeadlock. butwe also have ∃p ∈ D, p ∈ t• ⇔ t ∈ •D and ∃p' ∈ D, p' ∈ •t ⇔t ∈ D• Hence,the left-handpartof the property iequivalent s t o: T, t ∈ •D  t ∈ D• ∀ t ∈ whichithe s definitionosaftructuraldeadlock.

eproperty,we

Actually,Property2.1definesremovalrulesaccordingtow hichplacesthatdo notbelongtoadeadlockcanbeeliminatedfromthenet. Weknowthataplace cannotbelongtdeadlock oa iitis fanoutputplaceotaf ransitionsuchthatnoinput place othis f transitionbelongsto the deadlock. Definition2.5 • Lett ∈ T.ThenR(t) istheremovalruleassociatedwith and t iwritte s nas: R(t) : •t t•  • LetEbeasubsetofP.ThenR(t)isapplicableonEiff•t ⊂E.The ∪ t•. applicationoon tfEisgiven E:= bEy Wewillcallhypothesistheleft-handpartotfherule, whereastheright-hand partwillbecalledconclusion.UsingDefinition2.5,we canwriteageneric algorithmforfinding structural a deadlocksatisfyingtwo c onstraints: • placescontainedinset a calledP mustnotbelongttohe deadlock, exc • placescontainedinset a calledP mustbelongttohe deadlock. inc The principle othe f algorithmissimple: saetR isin itializedwithallthe rulesof thenet,asetRemovedisinitializedwiththesetP excandwetrytoapplyon RemovedthedifferentrulesoR, fi.e.,toremovethe placesthatdonotbelongtothe deadlock.Whennomoreruleisapplicable,weverifythat P incisincludedinthe complementaryofRemoved.Ifso,thealgorithmhasproduc edthemaximal deadlocksatisfyingtheconstraints.Else,thereisno deadlocksatisfyingthese constraints. Abstract Algorithm 1 Removed := Pexc While ∃ t such that R(t) is applicable on Removed do Apply R(t) on Removed Delete R(t) in R done; Deadlock := P \ Removed If Pinc ⊂ Deadlock and Deadlock ≠ ∅ then (success,Deadlock) else return(failure)

Anoptimizedimplementationofthisalgorithm[15]require proportionaltothesizeothe f net(expressedatshesum arcs).Inthenextsectionweextendthedefinitionsan nets.

return

asnexecutiontime ofthenumberonodes f and dthealgorithmtocolored

2.2 StructuralDeadlocksinColoredNets Beforegivingformal a definitionosaftructuraldeadlock the notationsassociatedwiththismodel.

inacolorednet,werecall

-W Definition2.6 AcoloredPetrinetNisa5-tuplewhere P isafinitesetofplaces, T isafinitesetoftransitions, C isthecolorfunction,mappingP ∪Tonto Ωwhere , Ωissomefinite setoffiniteandnon-emptysets.C(s) iscalledthecolorsetof s. W(resp.W +isthe ) input(resp.output)functiondefinedonP ×T,where +(p, W-(p,t) (resp.W t)) isafunctionfromC(t) toC(p) MS. MS

Inthisdefinition,E MSdenotesthesetofmultisetsoverasetEI. nformall elementof E is saubsetof Ewhere the same elementcanappearse MS

y,an veraltimes.

Notation: Inthe restof thissection,we willuse PCto denote the setcontainingcouplesof placesandassociatedcolor

∪p ∈ P{p}

theset[

Definitions2.7and2.8are the extensiontcoolorednets Definition2.7 Letpbeaplace,andc input(resp.output) setof(p,c) isdefinedbtyheset•(p,c) (r •(p,c) ={(t, W|c') (resp. (p,c)•={(t, W|c') ThisdefinitioncanbextendedtoasubsetD associatedcolorinstances: •D = ∪ •(p,c ) (p,c ) ∈D

Definition2.8 Let be t atransition,andc input(resp.output) setof(t,c) isdefinedbtyheset•(t,c) •(t,c) ={(p, W|c') (resp. (t,c)•={(p, W|c') Nowtheformalexpressionothe f definitionoasftruct to the definitionodafeadlockinanordinary Petrinet

•D

Definition2.9 iff ⊂D•.

×C(p)],i.e.,

instances. of Definitions2.2and2.3. ∈C(p)beacolorinstanceopf The . esp.(p,c)•) +(p, t)(c')(c) >0} -(p, t)(c')(c) >0}) ⊂PC,i.e.,asubsetDofplaceswith

D (resp.

• =



(p,c ) ∈D

(p,c )• )

∈C(t)beacolorinstanceot.The f (resp.(t,c)•) -(p, t)(c)(c') >0} +(p, t)(c)(c') >0}) uraldeadlockiqs uitesimilar .

LetDbeanon-emptysubsetofPC.Disastructuraldeadlock

Inthesamewayaw s edidforordinaryPetrinets,we equivalentto the definitionodafeadlock.

nowgiveapropertythatis

Property2.2 LetDbeanon-emptysubsetofPC.Wehave: [∀ (t,c)

∈ [∪t ∈ T{t} ×C •(t, (t)] c) , ⊂(PC D) \ Disas⇔ tructuraldeadlock

The proof of thisproperty iquite s similarto the proof

⊂ (PC D) \]

(t,c)• 

of Property 2.1.

Inorderto extendthe algorithmcomputingthe maximalstruc turaldeadlockthat verifiessomeconditions,weextendthedefinitionofr emovalrulestocolorednets. Considering transition a and t anassociatedcolorc,w kenowfromProperty 2.2that if no inputplace o(t, fc)inthe unfoldednetbelongsto da eadlock,thenalsononeof the outputplacesbelongsto thisdeadlock.Hence,weintro ducefunction a thatgives foreachplacepthesetofcolorsopfthatareinput( resp.output)o(ft,c).These functionsmapthecolordomainoonto tf thepowerseto the f colordomainop, fi.e., the setof subsetsof C(p),thatwe denote by P[C(p)] . Definition2.10 Letpbeaplaceand be t atransition.Thefunction C(t) →P[C(p)]definesforeverycolorinstancecoftthecorrespondinginput colorsinp: -(p, W | t)(c)(c') >0} Γ-(p,t)(c)={c' + Theoutputcolorsaredefinedby C(t) t) : Γ (p, →P[C(p)] +(p, W | t)(c)(c') >0} Γ+(p,t)(c)={c'

Γ-(p,t):

However,wealready knowthatifthereins oarcbetw eenpand (resp. t and t p), + the set Γ (p,t)(c)(resp. Γ (p,t)(c))willbeempty forany valueoc. fHence,we need todeterminewhichplacesandtransitionsareconnecte d,disregardingthecolor functions.Thesets•and t tthat • wedefinenowaret hesetsthatwouldbcealculated onthe ordinary Petrinetobtainedwhenignoringthe co lorfunctionsonthe arcs. Definition2.11 Letbe t atransition.Theinput(resp.output)placesoare tf definedbtyheset•t(resp.t•): •t={p |∈ P ∃c ∈C(t), Γ-(p,t)(c) ≠ ∅} |∈ P ∃c ∈C(t), Γ+(p,t)(c) ≠ ∅}) (resp. t•={p Thefollowingdefinitionintroducesnewnotationsthat expressionoset f of placesandassociatedcolors. Definition2.12 Letpandqbetwoplaces,EandFbesubsetsoC(p) f andC(q) respectively.Wedefine: • [p, {(p, E] =c) c| ∈E} [p, F] =E] ∪[q,F] • [p,E] ∧[q, Nowwehavealltheelementsthatallowustodefine willdetermine ipaflace belongsto deadlock a onot. r

allowamorecompact

therules,whoseapplication

Definition2.13 T.ThenR(t) istheremovalruleassociatedtand ot iwritten s as • Lett ∈ -(p, t )]  R(t) : [p, Γ ∧ p ∈ •t



p ∈ t•

: [p, Γ +(p, t )]

• •

ThecolordomainC[R(t)]othe f ruleithe s colordomainothe f transit ∈C[R(t)].R(t)isapplicablefor ocnEiff LetEbeasubsetofPC,andc -(p, t )(c)] ⊆ E [p, Γ ∧



TheapplicationoR(t) f for ocnEisgivenby E := E ∪

ion.

p ∈ •t



p ∈ t•

[p, Γ +(p, t )(c)]

Weuseremovalrulestowriteanalgorithmthatcomputes structuraldeadlocks. WefirstdefineasetP (resp. P o ) p f laces a nd associated color i nstances that exc inc areexcludedfromthedeadlock(resp.thatmustbelongtothe deadlock).We initializeasetRwiththerulesotfhenet,aset RemovedwithP excandwetryto applytheremovalrules.IfaruleR(t)isapplicablefor acolorcwe , addtheoutput placeso(t, fc)toRemoved,i.e.,weremovethesepla cesbecausetheycannotbelong tothedeadlock,andwaelsoremove from c thecolordom ainothe f rule.Whenthis colordomainisempty,weremovetherulefromR.When nomoreruleis applicable,placesthathavenotbeenincludedinRemoved formthemaximal deadlockundertheconstraintsoPf .P inc isincludedinthecomplementaryof excIf Removed,thenthe researchhasbeensuccessful. Abstract Algorithm 2 Removed := Pexc; While ∃ t and c such that R(t) is applicable for c on Removed do Apply R(t) for c on Removed ; C[R(t)] := C[R(t)] \ {c}; If C[R(t)] = ∅ then Delete R(t) in R; done; Deadlock := PC \ Removed; If Pinc ⊂ Deadlock and Deadlock ≠ ∅ then return (success, Deadlock) else return (failure)

Clearly,the complexity othis f algorithmdependsonthe eveniwe f extendtheoptimizedversionthatexistsfor analgorithmdoesnotexploitatallthestructureofthe correspondstoparticularstructuresothe f graphothe f unf functionsoftenhavearegularstructurethatcanbue sed of removalrules.We presenttwo examplesof thisoptimi 2.3 Two IntroductoryExamples

sizeothe f unfoldednet, ordinaryPetrinets.Butsuch colorfunctions,which oldednet.Actually,these tooptimizetheapplication zationinthe nextsection.

Inthissection,we presenttwo structuresof ordinary usualcolorfunctions,andweshowhowthesecolorfunct improve the efficiency oalgorithms f forfindingstructur

Petrinetsthatcorrespondto ionscanbeexploitedto aldeadlocks.

Thefirstcasewherewecanbenefitfromthecolor severalplacesothe f unfoldednetcanbeliminatedat inputtransitionshavethesameinputplaces.Thisiswh deduction".Weshowonanexamplehowitworks.Letuscon netinFigure 1The . removalrule associatedwiththe tr [P1, ε]



functionsisthecasewhere thesametime,becausetheir atwecall"parallel siderthecoloredPetri ansitionothe f colorednetis:

[P2, X]

Butwhenconsideringtheunfoldednet,weimmediatelyrem P2bandP2chavethesamesetofinputplaces,namelyplac tryingthepossibleassignmentsovariable f Xif,wea belongtdeadlock, ao we couldeliminate allof the insta

arkthatplacesP2a, eP1.Hence,insteadof lreadyknowthatP1doesnot ncesof P2athe t same time.

To dsoo,we shouldrewrite the formerremovalrule as: [P1, ε]

where function



[P2, C1.all]

C1.all isdefined by:

C1.all(x)

=



{x}

x ∈ C1

P1

P1

P1

transformation of the rule

unfolding

X

P2 P2a

P2b

P2a

P2c

P2b

P2c

classC1is[a,b,c];

Fig.1:Paralleldeduction

Theapplicabilityothis f newruledoesnotdependonany originalrulerequiredtheassignmentofX.Hence,thenum rule idivided s btyhe cardinality othe f classonwhich

variable,whereasthe berofapplicationsothe f Xisdefined.

WeconsidernowthecolorednetinFigure2,whosegraphc AccordingtoProperty2.2,weknowthat(P,a)cannotbel thenaccordingtD o efinition2.13,the removalrule assoc 

True

If we considernow transitionT2,the associatedremov [P, X]



ontainsaloop. ongtoadeadlock,and iatedwithT1is:

[P, a]

alrule is: [P, X++1]

where X++1 standsforthesuccessorfunction,modulothecolorclas s.Bytrying successivelythedifferentassignmentsothis f rule,we deducethatif(P,a)doesnot belongtoadeadlock,thenneither(P,b)nor(P,c)be longstothedeadlock.Infact, thesuccessiveapplicationsotfheruleareequivalentto findingadeadlockinthe unfoldednet.These successive applicationsare whatwe c all"iterative deduction".

However,ifwetrytoiteratetheapplicationofthe thenwiemmediately findthatthisrule canbreeplaced [P, X]

rulebeforeanyassignment, by the followingrule: 

[P, C.all]

forwhichonlyoneapplicationisnecessaryinordert information.Theapplicationothis f newrule,combine ruleassociatedwithT1ensuresthatnooccurrenceoPf thisrule,thenumberosteps f othe f removalalgorithm of the colordomainoT2. f

oobtainthewanted dwiththeapplicationofthe belongstoadeadlock.With isdividedbythecardinality

Infact,the transformationothe f rule ibased s ont ofthecolorednet.Onelapinthecoloredcycleprovide colorfunctions,onthecolorinstancesopf lacestha deadlock.Thisinformationiimmediately s reusedttory t

he cyclic structure othe f graph sinformation,throughthe m t ustbeexcludedfromthe obtainmore results.

Hence,byacombinationofthecolorfunctionsthat obtaindirectly the informationgivenbrepeated ay appli

appearinthecycle,we cationothe f originalrule.

Thesetwoexamplesshowthatforparticularstructuresof representedbyspecificstructuresobf oththecolorfunc net,the deadlockcomputationalgorithmcanbiemproved.

theunfoldednet, tionsandthegraphofthe

T1a

T1

T1a

a

Pa Pa P unfolding

transformation of the rule

T2a

T2a

X

X++1

Pb

Pb

T2 T2b

T2b

classCis[a,b,c];

Pc Pc

T2c T2c

Fig.2:Iterativededuction

Whatwehavepresentedontheexamplesisinfactaver namelytransformingtheremovalrulesotfheunfoldedne onthem.Inorderto obtainautomatic transformations approachinthe nextsection.

3

ygeneralapproach, bt yapplyingmeta-rules we , developandformalizethe

Removal RulesandMeta-Rules

Theaimofthemeta-rulesitsorewriteremovalrules application.Atevery stepothe f computationodafeadloc mustprovideam s uchinformationapsossible.Thecomputat

inordertoimprovetheir k,oneapplicationorafule ionodafeadlockrelies

ontheinitialcreationofonerulepertransitionof transformationthatappliestoaruleRor ,toaseto ruleR',orasetofnewrules.Ameta-rulecanmodify colordomain,i.e.,thepossibleassignmentsothe f va Theapplicationoam f eta-ruletransformsaruleinsuch ofthenewrulecorrespondstoaseriesoaf pplications unfoldednet.The efficiency othe f approachithus s linke rulesthathave beenreplacedbtyhistransformation.

thenet.Ameta-ruleisa rules, f andthatproducesanew notonlytherule,butalsoits riablesthatappearintherule. awaythatoneapplication oftheoriginalruleinthe dttohenumberofordinary

Asremovalrulescorrespondttoransitions,we caniden colorednetfromwhichtheyhavebeenwritten.Them resultsfromtheapplicationofameta-rulecanbeseen associatedPetrinet.Basedonthisidentification,we existforcolorednets.Theskeletonoarfulewillbe therule,disregardingthecolorinstancesthatareasso willalso use pto •denote the setof rulesforwhichp

tify set a ofruleswiththe odificationofarulethat asatransformationofthe extendtorulesnotionsthat thesetofplacesthatappearin ciatedwiththeseplaces.We appearsinthe hypothesis.

3.1 BasicMeta-Rules InanordinaryPetrinet,thealgorithmforcomputinga structuraldeadlock consistsinremovingtheplacesthatcannotbelongto thedeadlock.Inacolored Petrinet,weonly removecolorinstancesothe f pla ces.However,ifalltheinstances ofaplacehavebeenremoved,thenwecanremovethe place.Removingtheplace fromthenetisequivalenttoremoveifrom t therules Actually, . inbothcasesthis suppressionaccountsforthe factthatnothingmore will be provedforthisplace,and alsothatanyhypothesisrequiringthatasubsetofcolo irnstancesothis f placedoes notbelongtothedeadlockwillbetrue.Hence,ourfirst meta-ruleconsistsin removingthe placesforwhicheverythinghasbeenprove d. Definition3.1(MR1) Letp 0beaplacesuchthatnoelementof[p 0,C(p 0)] belongstoadeadlock.Thenmeta-ruleMR1consistsinreplacingruleRbyr uleR' suchthatC(R') =C(R) and: R :



[p, fp ]

R' :







p ∈ P1

[q, fq]

q ∈ P2

[p, fp ]

p ∈ P 1 \{p 0 }

OnceMR1hasbeenapplied,somerulesmaynolongerhave rulescanbreemovedbecause they willgive nfourtheri





[q, fq]

q ∈ P 2 \{p 0 }

aconclusion.These nformation.

Definition3.2(MR2) LetRbearulewhoseconclusioniesmpty.Thenmeta-rule MR2consistsinremovingR. Thetwoformermeta-rulesdonotdependonthecolorfunc tionsthatappear aroundthetransitioncorrespondingtoruleRH . ence,the ycanbeappliedonthe skeletonothe f colorednetaswell.Wepresentnowtw ometa-rulesthatimprovethe removalprocessbyusingthestructureofthecolorednet throughthecolor functions.

3.2 ParallelDeduction Thefollowingmeta-rulecorrespondstotheparalleldeducti haveshownontheexampleoSection f 2.3,wecandefine casewherethehypothesisoarule f ips artiallyindepen meaningofthismeta-rulecanbeexplainedasfollows.W functionsaroundatransitionthat,intheunfoldednet, sameinputplaces.Thesemanticsotfhemeta-ruleisth transitionsby unique a transition,whose inputplacesar andwhoseoutputplacesaretheunionoftheoutputplacesof the substitutioniperformed s athe t colorednetlevel.

onprocess.Aswe ameta-ruleexploitingthe dentoftheconclusion.The erecognizeonthecolor sometransitionshavethe entosubstituteallthese tehose commoninputplaces thetransitions.And

Inordertogiveaformalexpressionofthismeta-rule, functions: Definition3.3 LetC1andC2btewosets. Thep• rojection πofC1 ×C2onC1idefined s by: • ΣisafunctiononC1thatdefinesthesumovertheelementsofC2: Σ(x) = { ∪

weintroducetwo

π()=x }

y ∈C2

×C2,whereC1andC2canin independenceothe f hypothesis unctionsinthehypothesis, assignmentofthevariable

Considernow rauleRwithdomain a C(R)=C1 turnbC e artesianproductsof colorsets.For paartial andtheconclusion,wewantaconditiononthecolorf suchthatthehypothesisivs erifiedindependentlyothe f inC2.

Anecessary andsufficientconditionithat s the functi onsof the hypothesiscanbe written gas = fp p o π,where f is fpaunctiondefinedonC1.Aftertheapplicationof themeta-rule,thecompositionoffunctionf pwithaprojectionisnolonger necessary,asthe domainothe f rule hasbeenreducedto C1. Intheconclusionorule f Rfa,unctionf that toC1 ×C2iasssociatedto q applies eachplaceqBut . theaimofthemeta-ruleitsobtain anewrule,whosedomainis onlyC1andwhoseconclusionistheunionoftheconclus ionsobtainedforallthe assignmentsothe f variableinC2.Hence,onceanass ignmenthasbeendoneforthe variableinC1,wefirstcomputeallthecouplesthatas sociatethisassignmenttoan elementofC2.Thisids onebyapplyingfunction Σ.Thenweapplyf qtoallthese couples. Definition3.4(MR3) LetRbearulewhosedomaincanbw e rittenaC s (R)=C1 C2,andwhoseexpressionis: R :



[p, fp o π]





p ∈ P1

[q, fq]

q ∈ P2

Thenthemeta-ruleMR3producesanewruleRwhose ' domainiC s (R’) = whoseexpressioni:s R' :



p ∈ P1

[p, fp ]





q ∈ P2

C1,and [q, fq o Σ]

×

However,theproblemistodetectinwhichcasesthis meta-rulecanbe applied.The easiesttodetectandalsothemostfrequent caseiw s hentheexpression of raule containsinthe conclusionvariable a that doesnotappearinthehypothesis. Theapplicationofthemeta-rulethenconsistsinrepla cingthisvariablebythe constantrepresentingallthe elementsof the domaino the f variable. Wenowpresentthelastmeta-rulethatcorrespondsto process.

theiterativededuction

3.3 IterativeDeduction Thefollowingmeta-rulecorrespondstotheiterativededuc tionprocess.Aswe haveshownontheexampleoSection f 2.3,wecandefine ameta-ruleexploitingthe casewherethereexistsacircuitsuchthateachofit tsransitionhasonlyoneinput place inthe graphothe f Petrinet.Wecallexternalpl acesw.r.t.caircuitplacesthat areconnectedtoatransitionofthecircuitbutdonot belongtothecircuit.Weare thusinterestedincircuitswithoutexternalinputplaces. Moreover,if the functionsvaluatingthe inputarcsof t areallidentityfunctions,thenallthetransitions place inthe unfoldednet.Hence,if palace othe f unfol thedeadlock,thennodescendantofthisplaceobtainedby transitionsof the circuitcanbelongttohe deadlock.

he transitionsof thecircuit ofthecircuitonlyhaveoneinput dedcircuitdoesnotbelongto apathmeetingonly

Themeaningothe f meta-rulecanbeexplainedafsollows outputplacesoatfransitionthesetofdescendantplaces ofplacescanbeobtainedbycomputingthetransitivecl consideration.Analgorithmforthiscomputationcanbe transitive closure ocafolorfunctionthatwe recall

We . substitutetothe ofthistransition.Thisset osureotfhesubnetunder foundin[18].Itusesthe

Definition3.5 transitiveclosuref

P(E) →P(F).The

LetEandFbetwosets,be f afunction *is of defined f by:

f *(c) ⇔ ∃n c' ∈ nf wheref =

now.

0 suchthatc'

∈f n(c)

o(n … times). fo

Forthesakeoclarity, f wefirstpresentasimplified version,we only consider caircuitwithoutexternalout

versionothe f meta-rule.Inthis putplaces.

Letp 0…, ,p be the placesbelongingttohe circuit,andletf be colorfunction n-1 ithe valuatingthe inputarc oplace f p . fromaninstance of cp , canreach i+1Starting iwe the descendantinstancesc'of p that are s uch t hat: j c' ∈ fj-1 o... ofi (c) Hence,asthegraphiscyclic,fromaninstancecof pi,wemayreachallthe instancesc'of p that are s uch t hat i c' ∈ fi-1 o ...

ofi (c)

Butthese instancesmay inturnreachinstancesc"suc

hthat

c" ∈ fi-1 o ...

of i(c'), i.e., c" ∈ fi-1 o ...

Byrepeatingtheprocess,fromaninstancecofp instancesc'of p that suchthat i are c' ∈ (f i - 1 o ...

of i o fi-1 o ...

, ecanreachthedescendant iw *

of i ) (c)

Asknowhowtocomputethecolorsthatcanbereachedf any placep belonging tothecircuit we , cannowgivethegeneral j meta-rule for caircuitwithoutexternaloutputplaces.

romthesecolorsopf i for expressionothe f

Definition3.6(simplifiedMR4) Letp 0…, ,p n-1 benplacesbelongingtoacycleotfhecolorednet.Lett outputtransitionopf in be ithecycle,andR i theruleassociatedwitht thefollowingexpression: Ri :

[pi , id]



[pi, id]

be i the .R has iIf i

[pi+1 , fi]

theapplicationothe f meta-ruletransformsR R'i :

of i (c)

in iR'

such i that:

n-1





j = 0

*

[pj , fj-1 o ... ofi o(fi-1 o ... ofi ) ]

wheretheoperationsontheindicesareperformedmodulon. Theintroductionoexternal f outputplacesdoesnotchanget colorfunctions,buttheseplacesmustnowappearinall becomes:

heexpressionothe f therules.Themeta-rule

Definition3.7(MR4) Letp 0…, ,p n-1 benplacesbelongingtoacycleotfhecolorednet.Lett outputtransitionopf in be ithecycle,andP i thesetofoutputplacesotf notbelongtothecycle.LetR be t he r ule a ssociated tot .R i iIf expression: Ri :

[pi , id]



[pi +1 , fi ]

be i the that i do has t he f ollowing i

[q, gq]

q ∈ Pi

theapplicationothe f meta-ruletransformsR R'i :

[pi, id]

in iR'

such i that:

n-1





j = 0

q

*

[pj, fj-1 o... ofi o(fi-1 o... ofi ) ]

∧ Pj [q,

*

gq ofj-1 o... ofi o(fi-1 o... ofi) ]



wheretheoperationsontheindicesareperformedmodulon. Wehaveconsideredonlyidentityfunctionsoninputarcs. easilyextendedtothecasewherethefunctionsoninput the same kindotransformation f afor s reductionsinco

Theresultcanbe arcsarebijective,byusing lorednets[19].

4

AnAlgorithm toComputeStructural Deadlocks

4.1 PresentationandDefinitionothe f Algorithm ThisalgorithmdiffersfromAlgorithm2onthe two foll • The occurrence oplaces f inrulesisusedonly athe t "s • Thestructuresobf oththecolorfunctionsandtheskel meta-rulesMR3andMR4. Moreprecisely,thealgorithmworksafsollows.Thefi initializesthecolorinstancesopf lacesthataree rulestobexamined those - withahypothesiscontaini one colorisexcludedfromthe deadlock and - triesto apply Themainloop(instructions8-20)thenappliesoneruleat possibleassignments,updatingtheexcludedcolorinstanceso tobexamined.Themainoptimization(instructions15-17) applicationothe f meta-rules;duetothetestin(15),th alwaysappliesatleastonemeta-rule(seebelow).In instructionsworkathe t skeletonlevelandcolorsonly anordinary rule.The endothe f algorithm(instructions Algorithm2.The algorithmisdevelopedbelow. NotationGivenE,somesubsetoPf C,weneedtoknowthesubset place includedinESo . wientroduce the followingnotatio Abstract Algorithm 3 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22)

owingpoints: keleton"level. etonareexploitedby rststep(instructions1-7) xcludedfromthedeadlock,the ngaplaceforwhichaleast t the meta-rulesonce. atimewithallthe places f andtherules isthedetectionofthe executionoinstruction f 16 thisloop,allthecontrol appearintheapplicationof 21-22)isidenticaltothatof

n:E.p={c (p, | c)

ofcolorsoaf ∈ E}

Compute the elementary circuits of the skeleton Removed := Pexc To_Examine := ∅ For all place p such that Removed.p ≠ ∅ do Insert p• in To_Examine done Apply the Meta-rules on Removed While To_Examine ≠ ∅ Do Extract R from To_Examine A := {c | R is applicable for c on Removed} For all c in A do Apply R for c on Removed ; For all p such that Removed.p has increased do Insert p• in To_Examine; If Removed.p = C(p) then Apply the Meta-rules endif done done done Deadlock := PC \ Removed If Pinc ⊂ Deadlock and Deadlock ≠ ∅ then return (success, Deadlock) else return (failure).

Explanationsanddetailsofimplementation (1)Inordertoapplymeta-ruleMR4,wecomputealltheele the skeleton.We choose tcoompute the circuitsbefore becausetheresultotfhiscomputationcanbeusedfordif samenet,suchastheresearchofdeadlocksandtraps,or withdifferentconditions,namely differentsetsP The computationothe f circuitscanbdeoneintime a pr ofthesizeotfheskeletonandthenumberocf ircuits independentofthesizeofthecolordomains,whichis complexityforcolorednets.Totestefficientlyifa MR4,weassociatetoeachcircuitthenumberoefxterna eachplacetothecircuitsowhich f iis tanexternal MR1,we update these numbersandsee inew f applicationsof

mentarycircuitsof eliminatingthe placesinP exc ferentproblemsonthe theresearchofdeadlocks exc.

oportionaltotheproduct [18].Anyway,thistimeis therelevantcriterionof circuitfulfilstheconditionof input l placesandwelink inputplace.Eachtimewaepply MR4are possible.

(3),(5),(8),(9),(14) To_Examineisaset-typevariablewhichprovidesa terminationtesttothealgorithm.Againtheupdatingof thisvariableisrelatedto the skeletonothe f setof rules.Eachtime color a o paflace iadded s tR o emoved,the ruleswiththisplaceappearinginthehypothesisaresel ectedforexamination.With alinkbetweenaplaceandeachrulewheretheplaceappea rsinthehypothesis,the updatingof To_Examineisquick. (10),(12)Thesestepsarethemosttime-consumingonesand theydependonthe colordomains.Manyheuristics,usedinsimulationtechn iques,(andalready applicabletoAlgorithm2)optimizethisstepbutthecompl exityremainscolor domaindependent. ButtheaimofAlgorithm3itsoreducethecolordomainothe f rulesandtotransformtheconclusionsotfherulesinsuchawaythatthec ostof testingisminimalandtheinformationbroughtbytheapplicationoftheruleis maximal. (13),(15)TheevolutionofRemoved.pcaneasilybememoris edwithtwo counters,oneforthecurrentcardinalityandonefort heprecedingone.Hencethese two testsare comparison a ointegers. f Ifthetesto instruction f (15)issuccessful,we already know thatwe canapply the deletionmeta-rule MR1 onplacepand , possibly any othe f three othermeta-rules: • MR2becomesapplicable ipw f asthe lastelementof th ceonclusionorafule • MR3becomesapplicableifpappearedinthehypothesisof aruleand determinedpartotfheassignment.Thiscanbedetectedb yaverysimple syntactic analysisothe f colorfunctions,suchafso instance r thevanishingof avariableinthehypothesisofarule.Thenextsecti onwillgivean illustrationothis. f • MR4becomesapplicable ipw f asthe lastexternalinputpl ace ocafircuit. Onecanseethattheoverheadtimeaddedbythetestof minimalandonce againindependentof the colordomain. 4.2 Application:TheDiningPhilosophers

themeta-rulesis

ThenetinFigure3modelsthediningphilosophers,inthe casewheretheydo nottake bothforksatthe same time.Initially allt he philosophersare Thinkingand all Forksarefree.Whenaphilosopher Xwantstoeat,hetakeshisrightfork (X++1and ) Waitfsorhisleftfork( X).Ifhisleftforkifsree,thenhe Eats.When he stopseating,he releasesthe two forksandstartst hinkingagain.WewilluseCto denote the setof philosophers(andoforks). f Wearelookingforpossiblydeficient,i.e.,insufficien the philosophersnet,such deadlock a may appearwithwaiti

tlymarkeddeadlocks.For ngphilosophers.

SowelookfordeadlocksthatdonotcontainplaceWait, algorithmwithP exc=[Wait,C(Wait)]andP inc= transitionsT1,T2andT3respectively are: R1 R2 R3

andwestartour ∅.Therulesassociatedwith

[Think, X] ∧ [Forks, X++1]  [Wait, X] [Wait, X] ∧ [Forks, X]  [Eat, X] [Eat, X]  [Forks, X + X++1] ∧ [Think, X]

We first(instruction1compute ) the elementary circuits

andwfeindthree ones:

(Think,Wait,Eat),(Forks,Wait,Eat), Forks, ( Eat) Thenwienitialize(instruction2Removed ) with[Wait, weupdateTo_ExaminewithruleR2.Wenowapplythemeta-rule MR1iapplicable s onWait,andthe rulesbecome: R1 R2 R3

C]and(instructions3-6) (sinstruction7):

[Think, X] ∧ [Forks, X++1]  True [Forks, X]  [Eat, X] [Eat, X]  [Forks, X + X++1] ∧ [Think, X] Think

X X++1

X Forks Wait X

X

X

X+X++1

Eat

X X

Fig.3:Thediningphilosophers withforkstakenseparately.

MR2iapplicable s onR1andthe rulesbecome: R2 R3

[Forks, X]  [Eat, X] [Eat, X]  [Forks, X + X++1] ∧ [Think, X]

X*is

MR4isapplicableonthecircuit(Forks,Eat).As C.Allthe , rulesbecome: R2 R3

Xand(

X + X++1)*is

[Forks,X]  [Forks,C.all]∧ [Eat,C.all] ∧ [Think,C.all] [Eat,X]  [Eat,C.all]∧[Forks,C.all] ∧ [Think,C.all]

ThenetinFigure4graphicallydescribesthetworules.A To_examine containsR2,so wextractitandtry taopply

tinstruction7, it.

However,asRemoved.forksiesmpty,noassignmentispo (instruction10).Thusweexitfromthemainloopandretur followingdeadlock:

ssibleandAisempty nsuccesswiththe

[Eat, C.all] ∧ [Forks, C.all] ∧ [Think, C.all]

Thisdeadlockcontainsalltheforks.Isthereadeadlock thatcontainsastrict subsetofforks?Toanswerthisquestion,weinitializ eP exc=[Wait,C(Wait)] ∪ [Forks,c]andP canarbitrarycolor.Allthestepsuntili nstruction inc= ∅whereis 10are identicalto the formerones. We findnow thatrule R2iapplicable s for acnditsapplic toPC.Theninstructions15-16deleteallplacesandrules. withanempty deadlock,hence returningfailure.

ationupdatesRemoved Thealgorithmfinishes

ItmustbenotedthatthemostefficientimplementationofAlgorithm2 haverequired2.n(nitshenumberophilosophers) f applicationsorules, f wher Algorithm3requiresonlyoneapplication! For laast

illustrationoAlgorithm f 3,we testif ourfirstdeadloc

kcontainstraps.

Think

X

X++1

Forks

C.all X

C.all

C.all C.all

Eat

X

C.all

C.all

Fig.4:Transformednetafter oneapplicationoMR1, f MR2andMR4

would eas

Sowestartouralgorithmwiththereversednetandwit andP inc = ∅.The modelispresentedinFigure 5The . rulesbecome: R1 R2 R3

[Wait, X]  [Think, X] ∧ [Forks, X++1] [Eat, X]  [Wait, X] ∧ [Forks, X] [Forks, X + X++1] ∧ [Think, X]  [Eat, X]

Thecircuitsarethereverseothe f precedingcircuits. withR1.Letuslookathe t applicationothe f meta-rules rulesbecome: R1 R2 R3

hP exc=[Wait,C(Wait)]

To_Examineiisnitialized MR1 : deletesWaitandthe

True  [Think, X] ∧ [Forks, X++1] [Eat, X]  [Forks, X] [Forks, X + X++1] ∧ [Think, X]  [Eat, X]

NowMR3isapplicableonruleR1,thedecompositionofthe particularcaseC={ ε}xCandids etectedavsariableXappearsintheconclus anddoesnotappearinthe hypothesis.Thusthe rulesbeco me: R1 R2 R3

True  [Think, C.all] ∧ [Forks, C.all] [Forks, X] [Eat, X]  [Forks, X + X++1] ∧ [Think, X]  [Eat, X]

TheloopiesxecutedoncewithanapplicationofR1;Remo [Think,C]and[Forks,C],sowteesttheapplicationof placesThinkandForks;MR2deletesrulesR1andR2;MR3trans R3

domainisa

True 

vedisupdatedwith themeta-rules.MR1deletes formsrule R3in:

[Eat, C.all]

Thesecondexecutionothe f loopappliesruleR3whichupdates Theapplicationofmeta-rulesisdetectedagainandplaceE deleted.Thenthe algorithmexitsfromthe loopandreturn empty.Hence,thedeadlockexcludingplaceWaitwehadobtain tarap. Think

X X++1

Forks

X

X

X+X++1

Eat

X X

Fig.5:Searchfortraps,Waitexcluded

RemovedtoPC. atandruleR3are failure s sinceDeadlockis eddoesnotcontain

ion

ItmustbenotedthatthemostefficientimplementationofAlgorithm2 haverequired2.n(nbeingthenumberofphilosophers)applicationsofrules, whereasAlgorithm3requiresonlytwoapplications!

would

4.3 ComparisonBetweenAlgorithms2and3 TheefficiencyofAlgorithm3comparedwithAlgorithm2de numberoaf pplicationsom f eta-rulesMR3andMR4alongits difficulttogiveanytheoreticalmeasureotfhecomplexi overheadtimeaddedtoAlgorithm3ins egligiblecomparedwit the rules,andthusthe complexity iin sthe same order. However,toestimatetheaveragecomplexity,wecanob applicableasoonaplace as conditioningtheassignment andthishappensfrequently whenthealgorithmisappliedto realsystems.MoreoverMR4isbasedontheexistenceo numerousinthe skeletonocafolorednet,especially wh arerequired.Theadditionalconstraintsareusuallynotf Neverthelessthedeletionofplacesincreasestheposs structuralcondition(noexternalinputplaces)andthetra domainsbymeta-ruleMR3yieldstheoccurrenceofthefun (existence oidentities). f

pendsonthe execution.Soitis ty.Intheworstcase,the htheassignmentof servethatMR3becomes oftaransitiondisappears, colorednetsmodelling fcircuits,whichare enlivenessandboundedness ulfilledinitially. ibilityofsatisfyingthe nsformationofthecolor ctionalcondition

OneapplicationofMR3followedbytheapplicationofthe transformedrule correspondstonapplicationsothe f initialrulewhere nitshesizeothe f vanishing colordomain.OneapplicationofMR4followedbytheappli cationofthe transformedrule correspondsto aleast t one application of allthe rulesothe f circuit. Infactassoonastheoutputfunctionsothe f circuita redifferentfromidentity,the reductionfactorisproportionaltotheproductofcaolor domainbtyhelengthothe f circuit. Wepointoutthatinmanycasesthecomputationispara validforafamilyomodels f whereonlythesizeotf thedeadlockcharacterizationweperformonthemodelof independentofthenumberofphilosophers.Suchacharacte beenimpossiblewithAlgorithm2.Wenowplantodevelop of Algorithm3forwell-formednets[20].

meterized:irtemains hecolorclasseschanges.Thus thephilosophersis rizationwouldhave pa arameterizedversion

Lastbutnotleast,theresultsareeasiertointerpre functionofthehigh-leveldescription,andthususesthe beengivenbtyhe designerto describe the model.Unlike providesanextensive representationodeadlocks. f

t.Theirexpressionisonlya samenotationsthathave ouralgorithm,Algorithm2

Alltheresultscanbeeasilytransposedforthedetec showninthe example othe f philosophers.

tionoftraps,aswehave

5

Conclusions

Inthispaperwehavepresentedanefficientalgorithmf orfindingdeadlocksand trapsincolorednets.Thealgorithmexploitsboththes tructureotfhenetandthe structureothe f colorfunctions.Wehaveshownonan examplehowthemeta-rules speedupthecomputationofcoloreddeadlocks.Theefficiency ofthealgorithm stronglydependsonthenumberoftimesmeta-rulescanbe applied.Thefirst experimentshaveshownthatinmostcases,theconditi onsofapplicationare fulfilled.We are now workingontheintegrationothe f algorithmintheCASEAMI [21]inorderto obtainstatisticalresultsonthe effi ciency othe f algorithm. Aforthcomingworkitshespecializationothis f algori definednets,withcomplexity a almostindependentofthe Aftercharacterizingclassesocolored f netsforwhic necessaryorsufficientlivenesscondition,thisalgor livenessforsuchnets.

thmforsyntacticallywell sizeothe f colordomains. hsomestructuralpropertyias ithmwillallowustodecide

References [1] W.Reisig: Place-TransitionSystems I.nPetriNets:Centralmodelsandtheir properties,W.Brauer,W.ReisigandG.Rozenbergeds.,LNCS n°254, Springer-Verlag,1987,pp117-141. [2] E.W.Mayr: AnAlgorithmfortheGeneralPetriNetReachabilityProblem In . SIAM.Journalof Computingn1°3,1984. [3] E.Best: StructureTheoryoPetri f Nets:theFreeChoiceHiatus In . PetriNets: Centralmodelsandtheirproperties,W.Brauer,W.Reisiga ndG.Rozenberg eds.,LNCSn2°54,Springer-Verlag,1986,pp168-205. [4] J.Martinez,M.Silva: ASimpleandFastAlgorithmtoObtainallInvariantsof aGeneralizedPetriNet .InInformatikFachberichten°52,C.Giraultand W.Reisigeds.,Springer-Verlag,1982,pp301-310. [5] K.Lautenbach: LinearAlgebraicTechniquesforPlace/TransitionNets .In PetriNets:Centralmodelsandtheirproperties,W.Braue r,W.Reisigand G.Rozenbergeds.,LNCSn2°54,Springer-Verlag,1986,pp142-167. [6] F.Commoner: DeadlocksinPetrinets In . AppliedDataRes.Inc.,Wakefield, MA,1972. [7] J.Esparza,M.Silva: APolynomial-TimeAlgorithmtoDecideLivenessof BoundedFree-ChoiceNets H . ildesheimerInformatikberichte12/90,Institut fürInformatik,Univ.Hildesheim. [8] K.Lautenbach: LinearAlgebraicCalculationofDeadlocksandTraps .In ConcurrencyandNets,K.Voss,H.GenrichandG.Rozenber geds.,Springer Verlag,1987,pp315-336. [9] K.Barkaoui,M.Minoux: APolynomial-TimeGraphAlgorithmtoDecide LivenessoSf omeBasicClassesoB f oundedPetriNets I. nApplicationand Theory oPetri f Nets92,Proc.ofthe13thConference, K.Jensened.,LNCSn° 616,Springer-Verlag,Sheffield,UK,1992,pp62-74.

[10] H.J.Genrich: Predicate Transition / Nets In .High-levelPetriNets.Theoryand Application,K.JensenandG.Rozenbergeds.,Springer-Ver lag,1991,pp343. [11] K.Jensen: ColouredPetriNets:AHighLevelLanguageforSystemDesignand Analysis.InHigh-levelPetriNets.TheoryandApplication,K. JensenandG. Rozenbergeds.,Springer-Verlag,1991,pp44-119. [12] J.Ezpeleta,J.M.Couvreur: ANewTechniqueforFindingaGeneratingFamily ofSiphons,TrapsandST-Components.ApplicationtoColoredPetriNets In . proc.ofthe12thInternationalConferenceonApplicatio nandTheoryoPetri f Nets,Gjern,Denmark,June 1991,pp145-164. [13] J.M.Couvreur,S.Haddad,J.F.Peyre: ComputationofGenerativeFamiliesof PositiveFlowsinColouredNets In .proc.ofthe12thInternationalConference onApplicationandTheory oPetri f Nets,Gjern,Denmar k,June 1991. [14] J.F.Peyre: RésolutionParamétréedeSystèmesLinéaires.Applicationau Calculd'InvariantsetàlaGénérationdeCodeParallèle .Thèsede l'Université Paris6,March1993(inFrench). [15] M.Minoux,K.Barkaoui: DeadlocksandTrapsinPetriNetsasHornSatisfiabilitySolutionsandsomeRelatedPolynomiallySolvableProblems . Discrete AppliedMathematicsn°29,1990. [16] J.Vautherin: ParallelSystemsSpecificationswithColouredPetriNetsand AlgebraicSpecifications I. nAdvancesinPetriNets1987,Springer-Verlag, 1987,pp293-308. [17] G.Findlow: ObtainingDeadlock-PreservingSkeletonsforColouredNets, in ApplicationandTheoryofPetriNets92,Proc.ofthe13th Conference,K. Jensened.,LNCSn6°16,Springer-Verlag,Sheffield,UK,1992, pp173-192. [18] D.B.Johnson: FindingallElementaryCircuitsoafDirectedGraph S , IAM J.Computer,vol.4,n°11975. , [19] S.Haddad: AReductionTheoryforColouredNets I.nHigh-levelPetriNets. TheoryandApplication,KJensen . andG.Rozenbergeds., Springer-Verlag, 1991,pp399-425. [20] G.Chiola,C.Dutheillet,G.Franceschinis,S.Hadda d: StochasticWell-Formed NetsandSymmetricModelingApplications t,oappearinIEEETransactions onComputers. [21] J.M.Bernard,J.L.Mounier,NB . eldiceanu,S.Haddad: AMIanExtensible PetriNetInteractiveWorkshop ,Proc.ofthe9thEuropeanWorkshopon ApplicationandTheory oPetri f Nets,Venice,Italy,J une 1988.

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