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