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ω -Petri nets G. Geeraerts1 A. Heußner2 M. Praveen3 J.-F. Raskin1 3 1 0 1Universite´LibredeBruxelles(ULB),Belgium 2 2Otto-FriedrichUniversita¨tBamberg,Germany n 3LaboratoireSpećificationetVe´rification,ENSCachan,France a J 8 Abstract 2 We introduce ω-Petri nets (ωPN), an extension of plain Petri nets with ω- ] labeled input and output arcs, that is well-suited to analyse parametric concur- O rentsystemswithdynamicthreadcreation.Mosttechniques(suchastheKarpand L MillertreeortheRackofftechnique)thathavebeenproposedinthesettingofplain . PetrinetsdonotapplydirectlytoωPNbecauseωPNdefinetransitionsystemsthat s haveinfinitebranching. Thismotivatesathoroughanalysisofthecomputational c [ aspectsofωPN.Weshow thatanωPNcanbeturnedintoanplainPetrinetthat allowstorecoverthereachabilitysetof the ωPN,butthatdoesnot preserveter- 1 mination.Thisyieldscomplexityboundsforthereachability,(place)boundedness v andcoverabilityproblemsonωPN.Weprovideapracticalalgorithmtocompute 2 acoverabilitysetoftheωPNandtodecideterminationbyadaptingtheclassical 7 5 KarpandMillertreeconstruction. WealsoadapttheRackofftechniquetoωPN, 6 toobtain theexact complexity of theterminationproblem. Finally, weconsider . the extension of ωPN withreset and transferarcs, and show how thisextension 1 impactsthedecidabilityandcomplexityoftheaforementionedproblems. 0 3 1 1 Introduction : v i X In this paper, we introduce ω-Petrinets (ωPN), an extensionof plain Petri nets (PN) that allows input and output arcs to be labeled by the symbol ω, instead of a natu- r a ral number. An ω-labeled input arc consumes, non-deterministically, any numberof tokensinitsinputplacewhileanω-labeledoutputarcproducesnon-deterministically anynumberoftokensinitsoutputplace.WeclaimthatωPNareparticularlywellsuited formodelingparametricconcurrentsystems (see forinstanceourrecentworkon the GrandCentralDispatchtechnology[12]),andtoperformparametricverification[14] onthosesystems,asweillustratenowbymeansoftheexampleinFig1. Theexample presentaskeletonofadistributedprogram,inwhichamainfunctionforksP parallel threads(whereP isaparameteroftheprogram),eachexecutingtheone taskfunc- tion. Many distributed programsfollow this abstract skeleton that allows to perform calculationsinparallel,andbeingabletomodelpreciselysuchconcurrentbehaviorsis animportantissue. Inparticular,wewouldlikethatthemodelcapturesthefactthatP is a parameter, so thatwe can, forinstance, checkthatthe executionof the program 1 p p p 1 one_task(int k) { (a) 1 (b) 1 (c) 1 2 // some work... fork K fork • fork • 3 } ω 4 main() { p p p 5 for i := 1 to P step 1 2 2 2 6 fork(one_task(i)) 7 } one task one task one task Figure1:Anexampleofaparametricsystemwiththreepossiblemodels always terminates (assuming each individual execution of one task does), for all possiblevaluesofP. Clearly,thePetrinet(a)inFig.1doesnotcapturetheparametric natureoftheexample,asplacep containsafixednumberK oftokens. ThePN(b), 1 ontheotherhandcapturesthefactthattheprogramcan forkanunboundednumber ofthreads,butdoesnotpreservetermination: (post)ω isaninfiniteexecutionofPN (b), while the programme terminates (assuming each one task thread terminates) forallvaluesofP,becausetheforloopinline5executesexactlyP times. Finally, observethattheωPN(c)hasthedesiredproperties:firingtransitionforkcreatesnon- deterministically an unbounded albeitfinite numberof tokens in p (to modelall the 2 possibleexecutionsoftheforloopinline5), andallpossibleexecutionsofthisωPN terminate,becausethenumberoftokensproducedin p remainsfiniteandnofurther 2 tokencreationinp isallowedafterthefiringoftheforktransition. 2 WhileclosetoPetrinets,ωPNaresufficientlydifferentthatathoroughandcareful studyoftheircomputationalpropertiesisrequired.Thisisthemaincontributionofthe paper.AfirstexampleofdiscrepancyisthatthesemanticsofωPNisaninfinitetransi- tionsystemwhichisinfinitelybranching. ThisisnotthecaseforplainPN:theirtran- sitionsystemscanbeinfinitebuttheyarefinitelybranching. Asaconsequence,some oftheclassicaltechniquesfortheanalysisofPetrinetscannotbeapplied.Considerfor example the finite unfolding of the transition system [10] that stops the development ofabranchofthereachabilitytreewheneveranodewithasmallerancestorisfound. Thistreeisfinite(andeffectivelyconstructible)foranyplainPetrinetandanyinitial markingbecausethesetofmarkingsNk iswell-quasiordered,andfinitebranchingof plainPetrinetsallowsfortheuseofKo¨nig’slemma1. However,thistechniquecannot beappliedtoωPN,astheyareinfinitelybranching.SuchpeculiaritiesofωPNmotivate ourstudyofthreedifferenttoolsforanalysingthem. First,weconsider,inSection3, a variantof the Karp and Miller tree [15] that applies to ωPN. In orderto cope with theinfinitebranchingofthesemanticsofωPN,weneedtointroduceintheKarpand Miller tree ω’s that are notthe resultof accelerationsbut the resultof ω-outputarcs. OurvariantoftheKarpandMillerconstructionisrecursive,thisallowsustotamethe technicalityoftheproof,andasaconsequence,ourproofwhenappliedtoplainPetri nets, provides a simplification of the original proof by Karp and Miller. Second, in Section4, weshowhowtoconstruct,froman ωPN,aplainPetrinetthatpreserveits reachabilityset. Thisreductionallowsto provethatmanyboundsonthe algorithmic 1Infact,thisconstructionisapplicabletoanywell-structuredtransitionsystemwhichisfinitelybranching andallowstodecidetheterminationproblemforexample. 2 Table 1: Complexityresults on ωPN (with the section numberswhere the results are proved). ωIPN+R (ωOPN+R) and ωIPN+T (ωOPN+T) denote resp. Petri nets with reset(R)andtransfer(T)arcswithωoninput(output)arcsonly. Problem ωPN ωPN+T ωPN+R Reachability Decidable and EX- Undecidable(6) PSPACE-hard(4) Undecidable(6) Place-boundedness Boundedness EXPSPACE-c(4) Decidable(6) Coverability DecidableandAckerman-hard(6) Problem ωPN ωOPN+T,ωOPN+R ωIPN+T,ωIPN+R Termination EXPSPACE-c(5) Undecidable(6) Decidable and Ackerman-hard(6) complexityof(plain)PN problemsapplyto ωPN too. However,it doesnotpreserve termination. Thus,westudy,inSection5, asathirdcontribution,anextensionofthe self-coveringpath techniqueduetoRackoff[19]. Thistechniqueallowsto providea directproofofEXPSPACEupperboundsforseveralclassicaldecisionproblems,andin particular,thisallowstoproveEXPSPACEcompletenessoftheterminationproblem. Finally, in Section 6, as a additional contribution, and to get a complete picture, weconsiderextensionsofωPNwithresetandtransferarcs[7]. Forthoseextensions, the decidability results for reset and transfer nets (without ω arcs) also apply to our extensionwiththe notableexceptionof theterminationproblemthatbecomes,aswe showhere,undecidable.ThesummaryofourresultsaregiveninTable1. Relatedworks ωPN arewell-structuredtransitionsystems[10]. The set saturation technique[1]andsosymbolicbackwardanalysiscanbeappliedtothemwhilethefinite treeunfoldingisnotapplicablebecauseoftheinfinitebranchingpropertyofωPN.For thesamereason,ωPNarenotwell-structurednets[11]. In[3],Bradziletal. extendstheRackofftechniquetoVASSgameswithω output arcs. While this extension of the Rackoff technique is technically close to ours, we cannotdirectlyusetheirresultstosolvetheterminationproblemofωPN. Severalworks(see forinstance [4, 5] rely onPetri netsto modelparametricsys- temsandperformparametrisedverification. However,inalltheseworks,thedynamic creation of threads uses the same pattern as in Fig. 1 (b), and does not preserve termination. ωPNallowtomodelmorefaithfullythedynamiccreationofanunbounded numberof threads, and are thus better suited to model new programmingparadigms (such as those use in GCD [12]) that have been recently proposed to better support multi-coreplatforms. Remark: duetolackofspace,mostproofscanbefoundintheappendix. 3 2 ω-Petri nets Letus define the syntax andsemantics of ourPetri netextension, called ω Petri nets (ωPNforshort).Letωbeasymbolthatdenotes‘anypositiveintegervalue’.Weextend thearithmeticandthe≤orderingonZtoZ∪{ω}asfollows: ω+ω = ω−ω = ω; andforall c ∈ Z: c+ω = ω +c = ω −c = ω; c−ω = c; and c ≤ ω. The fact that c−ω = c might sound surprising but will be justified later when we introduce ωPN. Anω-multiset(orsimplymultiset)ofelementsfromS isafunctionm : S 7→ N∪{ω}.WedenotemultisetsmofS ={s ,s ,...,s }byextensionusingthesyntax 1 2 n {m(s )⊗s ,m(s )⊗s ,...,m(s )⊗s }(whenm(s) = 1,we write sinsteadof 1 1 2 2 n n m(s)⊗s,andweomitelementsm(s)⊗swhenm(s)= 0). Giventwomultisetsm 1 andm ,andanintegervaluecweletm +m bethemultisets.t. (m +m )(p) = 2 1 2 1 2 m (p)+m (p);m −m bethemultisets.t. (m −m )(p)=m (p)−m (p);and 1 2 1 2 1 2 1 2 c·m bethemultisets.t. (c·m )(p)=c×m (p)forallp∈P. 1 1 1 Syntax Syntactically, ωPN extend plain Petri nets [18, 20] by allowing (input and output)arcstobelabeledbyω. Intuitively,ifatransitionthasωasoutput(resp.input) effect on place p, the firing of t non-deterministically creates (consumes) a positive numberoftokensinp. Definition1 APetrinetwithω-arcs(ωPN)isatupleN =hP,Tiwhere: P isafinite setof places;T afinitesetof transitions. Eachtransitionisapairt=(I,O),where: I : P → N∪{ω}andO : P → N∪{ω},giverespectivelytheinput(output)effect I(p)(O(p))oftonplacep. Byabuseofnotation,wedenotebyI(t)(resp.O(t))thefunctionss.t.t=(I(t),O(t)). Whenconvenient,wesometimesregardI(t)orO(t)asω-multisetsofplaces. When- ever there is p s.t. O(t)(p) = ω (resp. I(t)(p) = ω), we say that t is an ω-output- transition (ω-input-transition). A transition t is an ω-transition iff it is an ω-output- transition or an ω-input-transition. Otherwise, t is a plain transition. Remark that a (plain) Petri net is an ωPN with plain transitions only. Moreover, when an ωPN containsnoω-output-transitions(resp. no ω-inputtransitions),wesaythatitisan ω- input-PN(ω-output-PN),orωIPN(ωOPN)forshort.Foralltransitionst,wedenoteby effect(t)thefunctionO(t)−I(t). Remarkthateffect(t)(p)couldbeωforsomep(in particularwhenO(t)(p)=I(t)(p)=ω). Intuitively,effect(t)(p)=ωmodelsthefact thatfiring t canincrease the markingof p byan arbitrarynumberof tokens. Finally, observethat O(t)(p) = c 6= ω and I(t)(p) = ω implies effect(t)(p) = c−ω = c. This models the fact that firing t can at most increase the marking of p by c tokens. Thus,intuitively,the valueeffect(t)(p)modelsthemaximalpossibleeffectoft onp. Weextendthedefinitionofeffect tosequencesoftransitionsσ =t t ···t byletting 1 2 n effect(σ)= n effect(t ). i=1 i A marking is a function P 7→ N. An ω-marking is a function P 7→ N ∪ {ω}, P i.e. an ω-multiseton P. Remarkthat any markingis an ω-marking,and that, forall transitions t = (I,O), I and O are both ω-markings. We denote by 0 the marking s.t. 0(p) = 0forall p ∈ P. Forall ω-markingsm, weletω(m)bethesetofplaces {p | m(p) = ω}, and let nbω(m) = |ω(m)|. We define the concretisation of m 4 p p p 1 t 2 t 3 1 2 ω 2 • t t 3 4 Figure 2: An example ωPN N . The ωPN N′ is obtained by removingtransition t 1 1 4 (red). as the set of all markingsthat coincide with m on all places p 6∈ ω(m), and take an arbitrary value in any place from ω(m). Formally: γ(m) = {m′ | ∀p 6∈ ω(m) : m′(p) = m(p)}. We furtherdefine a family of orderingson ω-markingsas follows. For any P′ ⊆ P, we let m1 (cid:22)P′ m2 iff (i) for all p ∈ P′: m1(p) ≤ m2(p), and (ii) for all p ∈ P \ P′: m (p) = m (p). We abbreviate (cid:22) by (cid:22) (where P is 1 2 P thesetofplacesoftheωPN).Itiswell-knownthat(cid:22)isawell-quasiordering(wqo), that is, we can extract, from any infinite sequence m ,m ,...,m ,... of markings, 1 2 i an infinite subsequence m ,m ,...,m ,... s.t. m (cid:22) m for all i ≥ 1. For all 1 2 i i i+1 ω-markingsm,welet↓(m)bethedownward-closureofm,definedas↓(m) = {m′ | m′isamarkingandm′ (cid:22) m}. We extend↓tosetsofω-markings: ↓(S) = ∪ ↓ m∈S (m). A setD ofmarkingsis downward-closed iff ↓(D) = D. Itis well-knownthat (possiblyinfinite)downward-closedsets of markingscan alwaysbe representedby a finitesetofω-markings,becausethesetofω-markingsformsan adequatedomainof limits[13]: foralldownward-closedsetsDofmarkings,thereexistsafinitesetM of ω-markingss.t. ↓(M) = D. Weassociate,toeachωPN,anintialmarkingm . From 0 nowon,weconsidermostlyinitialisedωPNhP,T,m i. 0 Example1 An example of an ωPN (actually anωOPN) N = hP,T,m i is shown 1 0 in Fig. 2. In this example, P = {p ,p ,p }, T = {t ,t ,t ,t }, m (p ) = 1 and 1 2 3 1 2 3 4 0 1 m (p ) = m (p ) = 0. t is the onlyω-transition,with O(t )(p ) = ω. ThisωPN 0 2 0 3 1 1 2 willserveasarunningexamplethroughoutthesection. Semantics Let m be an ω-marking. A transition t = (I,O) is firable from m iff: m(p) (cid:23) I(p) for all p s.t. I(p) 6= ω. We consider two kinds of possible effects for t. The first is the concrete semantics and applies only when m is a marking. In thiscase,firingtyieldsanewmarkingm′ s.t. forallp ∈ P: m′(p) = m(p)−i+o where: i = I(t)(p) ifI(t)(p) 6= ω, i ∈ {0,...,m(p)} ifI(t)(p) = ω, o = O(t)(p) if O(t)(p) 6= ω and o ≥ 0 if O(t)(p) = ω. This is denoted by m −→t m′. Thus, intuitively, I(t)(p) = ω (resp. O(t)(p) = ω) means that t consumes (produces) an arbitrarynumberoftokensinpwhenfired. Remarkthat,intheconcretesemantics,ω- transitionsarenon-deterministic: whentisanω-transitionsthatisfirablein m,there are infinitelymanym′ s.t. m −→t m′. Thelattersemanticsisthe ω-semantics. Inthis case, firing t = (I,O) yields the (unique) ω-marking m′ = m − I + O (denoted m−→t m′). Remarkthatm−→t m′iffm−→t m′whenmandm′aremarkings. ω ω 5 We extendthe→and→ relationstofiniteorinfinitesequencesoftransitionsin ω σ theusualway. Alsowewritem−→iffσisfirablefromm. Moreprecisely,forafinite σ sequenceoftransitionsσ = t ···t ,wewritem −→ifftherearem ,...,m s.t. for 1 n 1 n all1 ≤ i ≤ n: m −t→i m . Foraninfinitesequenceoftransitionsσ = t ···t ···, i−1 i 1 j wewritem −→σ ifftherearem ,...,m ,...s.t. foralli≥1: m −t→i m . 0 1 j i−1 i Given an ωPN N = hP,T,m i, an execution of N is eithera finite sequence of 0 the form m ,t ,m ,t ,...,t ,m s.t. m −t→1 m −t→2 ··· −t→n m , or an infinite 0 1 1 2 n n 0 1 n sequenceoftheformm ,t ,m ,t ,...,t ,m ,...s.t. forallj ≥ 1: m −t→j m . 0 1 1 2 j j j−1 j σ WedenotebyReach(N)thesetofmarkings{m|∃σs.t. m −→m}thatarereachable 0 fromm inN. Finally,afinitesetofω-markingsCS isacoverabilityset ofN (with 0 initial marking m ) iff ↓(CS) =↓(Reach(N)). That is, any coverabilityset CS is a 0 finiterepresentationofthedownward-closureofN’sreachablemarkings. Example2 The sequence t tK is firable for all K ≥ 0 in N (Fig. 2). Indeed, for 1 2 1 each K ≥ 0, one possible execution correspondingto t tK is given by h1,0,0i −t→1 1 2 h0,3K,0i−t→2 h0,3K−1,2i−t→2 h0,3K−2,4i−t→2 ···−t→2 h0,2K,2Ki.Remarkthat thereareotherpossibleexecutionscorrespondingtothesamesequenceoftransitions, becausethenumberoftokenscreatedbyt inp ischosennon-deterministically.Also, 1 2 t t tω is an infinite firable sequence of transitions. Finally, observe that the set of 1 2 4 reachable markings in N is Reach(N) = {h1,0,0i}∪{h0,i,2×ji | i,j ∈ N}. 1 ThesetofωmarkingsCS = {h1,0,0i,h0,ω,ωi}isacoverabilitysetofN. Notethat ↓(CS))Reach(N): forinstance,h0,1,1i∈↓(CS),buth0,1,1iisnotreachable. LetusnowobservetwopropertiesofthesemanticsofωPN,thatwillbeusefulfor theproofsofSection3. Thefirstsaysthat,whenfiringasequenceoftransitionsσthat have non ω-labeledarcs on to and from some place p, the effect of σ on p is as in a plainPN: Lemma1 Let m and m′ be two markings and let σ = t ···t be a sequence of 1 n transitions of an ωPN s.t. m −→σ m′. Let p be a place s.t. for all 1 ≤ i ≤ n: O(t )(p)6=ω 6=I(t )(p). Then,m′(p)=m(p)+effect(σ)(p). i i Thelatterpropertysaysthatthesetofmarkingsthatarereachablebyagivensequence of transitions σ is upward-closedw.r.t. (cid:22)P′, where P′ is the set of places where the effectofσisω. Lemma2 Letm , m andm bethreemarkings,andletσ beasequenceoftransi- 1 2 3 tionss.t. (i)m1 −→σ m2, (ii)m3 (cid:23)P′ m2 withP′ = {p | effect(σ)(p) = ω}. Then, σ m −→m holdstoo. 1 3 Problems Weconsiderthefollowingproblems.LetN =(P,T,m )beanωPN: 0 1. Thereachabilityproblemasks,givenamarkingm,whetherm∈Reach(N). 2. Theplaceboundednessproblemasks,givenaplacepofN,whetherthereexists K ∈Ns.t. forallm∈Reach(N): m(p)≤K. Iftheanswerispositive,wesay thatpisbounded(fromm ). 0 6 3. TheboundednessproblemaskswhetherallplacesofN arebounded(fromm ). 0 4. Thecoveringproblemasks,givenamarkingmofN,whetherthereexistsm′ ∈ Reach(N)s.t. m′ (cid:23)m. 5. TheterminationproblemaskswhetherallexecutionsofN arefinite. RemarkthatacoverabilitysetoftheωPNissufficienttosolveboundedness,place boundednessandcovering,asinthecaseofPetrinets.IfCS isacoverabilitysetofN, then: (i)pisboundediffm(p)6= ωforallm ∈ CS;(ii)N isboundediffm(p) 6= ω forall p and forall m ∈ CS; and(iii), N can coverm iffthere exists m′ ∈ CS s.t. m (cid:22) m′. AsintheplainPetrinetscase, asufficientandnecessaryconditionofnon- terminationistheexistenceofa selfcoveringexecution. Aselfcoveringexecutionof anωPNN =hP,T,m iisafiniteexecutionoftheformm −t→1 m ···−t→k m −t−k−+→1 0 0 1 k ···−t→n m withm (cid:23)m : n n k Lemma3 AnωPNterminatesiffitadmitsnoself-coveringexecution. Example3 ConsideragaintheωPNN inFig.2. RecallfromExample2that,forall 1 K ≥ 0, t tK isfirableandallowstoreachh0,2K,2Ki. Allthesemarkingsarethus 1 2 reachable. These sequences of transitions also show that p and p are unbounded 2 3 (hence,N isunboundedtoo),whilep isbounded. Markingh0,1,1iisnotreachable 1 1 butcoverable,whileh2,0,0iisneitherreachablenorcoverable. Finally,N doesnot 1 terminate(becauset t tω isfirable),while N′ does. Inparticular,inN′, t canfire 1 2 4 1 1 3 onlyafinitenumberoftime,becauset willalwayscreateafinite(albeitunbounded) 1 numberoftokensinp . ThisanimportantdifferencebetweenωPNandplainPN:no 2 unboundedPNsterminates,whilethereareunboundedωPNthatterminate,e.g.N′. 1 3 A Karp and Miller procedure for ωPN Inthissection,wepresentsanextensionoftheclassicalKarp&Millerprocedure[15], adapted to ωPN. We show that the finite tree built by this algorithm (coined the KM tree),allows,asinthecaseofPNs,todecideboundedness,placeboundednes,cover- abilityandterminationonωPN. Before describing the algorithm, we discuss intuitively the KM trees of the ωPN N andN′ givenin Fig. 2. Theirrespective KM trees(forthe initialmarking m = 1 1 0 h1,0,0i) are T and T′, respectivelythe tree in Fig. 3 and its black subtree (i.e., ex- 1 1 cluding n ). As can be observed, the nodes and edges of a KM tree are labeled by 7 ω-markingsandtransitionsrespectively. Therelationship betweena KMtree andthe executions of the corresponding ωPN can be formalised using the notion of stuttering path. Intuitively, a stuttering path is a sequence of nodes n ,n ,...,n s.t. for 1 2 k all i ≥ 2: either n is a son of n , or n is an ancestor of n that has the same i i−1 i i−1 label as n . For instance, π = n ,n ,n ,n ,n ,n ,n ,n ,n ,n is a stuttering i−1 1 2 4 2 3 6 3 5 3 5 path in T′. Then, we claim (i) that every execution of the ωPN is simulated by a 1 stuttering path in its KM tree, and that (ii) every stuttering path in the KM tree cor- responds to a family of executions of the ωPN, where an arbitrary numberof tokens 7 n 1 h1,0,0i t 1 n 2 h0,ω,0i t t 2 3 n n 3 4 h0,ω,ωi h0,ω,0i t t 2 4 t 3 n n n 5 6 7 h0,ω,ωi h0,ω,ωi h0,ω,ωi Figure3:TheKMtreesT (wholetree)andT′(blacksubtree)ofresp. N andN′. 1 1 1 1 canbeproducedintheplacesmarkedbyωintheKMtree. Forinstance,theexecution m ,t ,h0,42,0i,t ,h0,41,0i,t ,h0,40,2i,t ,h0,39,2i,t ,h0,38,4i,t ,h0,37,6i,of 0 1 3 2 3 2 2 N′ is witnessed in T′ by the stuttering path π given above – observe that the se- 1 1 quence of edge labels in π’s equalsthe sequence of transitions of the execution, and that all markingsalong the executionare covered by the labels of the corresponding nodes in π: m ∈ γ(n ), h0,42,0i ∈ γ(n ), and so forth. On the other hand, the 0 1 2 stuttering path n ,n ,n of N summarisesall the (infinitely many) possible execu- 1 2 3 1 tions obtained by firing a sequence of the form t tn. Indeed, for all k ≥ 1, ℓ ≥ 0: 1 2 m ,t ,h0,k + ℓ,0i,t ,h0,k + ℓ − 1,2i,t ,...,t ,h0,k,2 × ℓi is an execution of 0 1 2 2 2 N , so,anarbitrarynumberoftokenscanbeobtainedinboth p andp byfiringse- 1 2 3 quencesoftheformt tn.Finally,observethataself-coveringexecutionofN ,suchas 1 2 1 m ,t ,h0,1,0i,t ,h0,0,2i,t ,h0,0,2icanbedetectedinT ,byconsideringthepath 0 1 2 4 1 n ,n ,n ,n ,andnotingthatthelabelof(n ,n )ist witheffect(t )(cid:23)0. 1 2 3 7 3 7 4 4 TheBuild-KMalgorithm LetusnowshowhowtobuildalgorithmicallytheKM of an ωPN. Recall that, in the case of plain PNs, the Karp& Miller tree [15] can be regardedasafiniteover-approximationofthe(potentiallyinfinite)reachabilitytreeof thePN.Thus,theKarp&Milleralgorithmworksbyunfoldingthetransitionrelationof thePN,andaddstwoingredientstoguaranteethatthetreeisfinite. First,anodenthat hasanancestorn′ with thesame labelis notdeveloped (ithasnochildren). Second, when a noden with label m has an ancestor n′ with label m′ ≺ m, an acceleration functionis applied to producea marking m s.t. m (p) = ω if m(p) > m′(p) and ω ω m (p) = m(p) otherwise. This acceleration is sound wrt to coverability since the ω sequenceoftransitionthathasproducedthebranch(n,n′)canbeiteratedanarbitrary numberoftimes,thusproducingarbitrarylargenumbersoftokensintheplacesmarked byω inm . Remarkthatthesetwoconstructionsarenotsufficienttoensuretermina- ω tionofthealgorithminthecase ofωPN, asωPN arenotfinitelybranching(firingan ω-output-transition can produce infinitely many different successors). To cope with 8 thisdifficulty,oursolutionunfoldstheω-semantics→ insteadoftheconcreteseman- ω tics→. Thishasanimportantconsequence:whereasthepresenceofanodelabeledby mwithm(p)=ωintheKMtreeofaPNN impliesthatN doesnotterminate,thisis nottrueanymoreinthecaseofωPN.Forinstance,allnodesbutn inT′ (Fig.3)are 1 1 markedbyω,yetthecorrespondingωPNN′ (Fig.2)doesterminate. 1 OurversionoftheKarp&MillertreeadaptedtoωPNisgiveninFig.4. Itbuildsa treeT = hN,E,λ,µ,n iwhere: N isasetofnodes;E ⊆ N ×N isasetofedges; 0 λ : N 7→ (N∪{ω})P isafunctionthatlabelsnodesbyω-markings2;µ : E 7→ T is alabelingfunctionthatlabelsarcsbytransitions; and n ∈ N istherootofthetree. 0 Foreachedgee,weleteffect(e) = effect(µ(e)). LetE+ andE∗ berespectivelythe transitiveandthetransitivereflexiveclosureofE. Astutteringpathisafinitesequence n ,n ,...,n s.t. forall1 ≤ i ≤ ℓ: either (n ,n ) ∈ E or (n ,n ) ∈ E+ and 0 1 ℓ i−1 i i i−1 λ(n )=λ(n ). Astutteringpathn ,n ,...,n isa(plain)pathiff(n ,n )∈E i i−1 0 1 ℓ i−1 i forall1 ≤ i ≤ ℓ. Giventwonodesnandn′ s.t. (n,n′) ∈ E∗,wedenotebyn n′ the(uniquepath)fromnton′. Givenastutteringpathπ = n ,n ,...,n ,wedenote 0 1 ℓ by µ(π) the sequence µ(n ,n )µ(n ,n )···µ(n ,n ) assuming µ(n ,n ) = ε 0 1 1 2 ℓ−1 ℓ i i+1 when(n ,n )6∈E;andbyeffect(π)= ℓ effect(n ,n ),lettingeffect(n ,n )= i i+1 i=1 i−1 i i−1 i 0when(n ,n )6∈E. i i+1 P Build-KM follows the intuition given above. At all times, it maintains a fron- tier U of tree nodes that are candidate for development (initially, U = {n }, with 0 λ(n )=m ).Then,Build-KMiterativelypicksupanodenfromU(seeline4),and 0 0 developsit(line6onwards)if n hasnoancestorn′ with thesamelabel(line5). De- velopinganodenamountstocomputingallthemarkingms.t. λ(n)→ m(line17), ω performingaccelerations(line19)ifneedbe,andinsertingtheresultingchildreninthe tree. RemarkthatBuild-KMisrecursive(seeline9): everytimeamarkingmwith an extra ω is created, it performsa recursivecall to Build-KM(N,m), usingm as initialmarking3. Therestofthesectionisdevotedtoprovingthatthisalgorithmiscorrect. Westart by establishing termination, then soundness (every stuttering path in the tree corre- spondstoanexecutionoftheωOPN)andfinallycompleteness(everyexecutionofthe ωOPNcorrespondstoastutteringpathinthetree). Tothisend,werelyonthefollow- ing notions. Symmetrically to self-covering executions we define the notion of self- covering(stuttering)pathinatree: a(stuttering)pathπ isself-coveringiffπ = π π 1 2 witheffect(π )≥0. Aself-coveringstutteringpathπ =π π isω-maximaliffforall 2 1 2 nodesn,n′alongπ : nbω(n)=nbω(n′). 2 Termination LetusshowthatBuild-KMalwaysterminates.Firstobservethatthe depth of recursivecalls is at most by |P|+1, as the numberof places marked by ω alonga branchdoesnotdecrease, andsince we performa recursivecallonlywhena placegetsmarkedbyωandwasnotbefore.Moreover,thebranchingdegreeofthetree isboundedbythenumber|T|oftransitions. Thus,byKo¨nig’slemma,aninfinitetree wouldcontainaninfinitebranch. Weruleoutthispossibilitybyaclassicalwqoargu- 2WeextendλtosetofnodesSintheusualway:λ(S)={λ(n)|n∈S}. 3Althoughthisdiffersfromclassicalpresentations oftheKarp&Millertechnique, wehaveretainedit becauseitsimplifiestheproofsofcorrectness. 9 Input anωOPNN =hP,Tiandanω-markingm 0 Output theKMofN,startingfromm 0 Build-KM(N,m ): 0 1 T := hN,E,λ,µ,n0i where N ={n0} with λ(n0)=m0 2 U := {n0} 3 while U 6=∅: 4 select and remove n from U 5 if ∄n st (n,n)∈E+ and λ(n)=λ(n): 6 forall t in T s.t. ∀p∈P: I(t)(p)6=ω implies λ(n)(p)≥I(t)(p): 7 m′ := Post(N,λ(n), t) 8 if nbω(m′)>nbω(λ(n)): 9 T′ := Build-KM(N,m′) 10 add all edge and nodes of T′ to T 11 let n′ be the root of T′ 12 else 13 n′ := new node with λ(n′)=m′ 14 U := U ∪ {n′} 15 E := E∪(n,n′) s.t. µ(n,n′)=t. 16 return T Post(N,n,t): 17 m′ := λ(n)−I(t)+O(t) 18 if ∃n: n,n)∈E+∧λ(n)≺λ(n) : m′(p) ifeffect(n n·t)(p)≤0 19 mw(p):(cid:0)= (cid:1) (ω otherwise 20 return mw 21 else: 22 return m′ Figure4:ThealgorithmtobuildtheKMofanωPN. ment: if there were an infinite branchin the tree computedby Build-KM(N,m ), 0 then there would be two nodes n along the branch n (where n is an ancestor of 1 2 1 n )s.t. λ(n ) (cid:22) λ(n )andeffect(n n ) (cid:23) 0. Sincethedepthofrecursivecalls 2 1 2 1 2 is bounded, we can assume, wlog, that n and n have been built during the same 1 2 recursivecall,henceλ(n ) ≺ λ(n )isnotpossible,becausethiswouldtriggeranac- 1 2 celeration,createanextraω andstartanewrecursivecall. Thus, λ(n ) = λ(n ),but 1 2 inthiscasethealgorithmstopsdevelopingthebranch(line5). Seetheappendixfora fullproof. Proposition1 For all ωPN N and for all marking m , Build-KM(N,m ) termi- 0 0 nates. Then,followingtheintuitionthatwehavesketchedatthebeginningofthesection, we show that KM is sound (Lemma 4) and complete (Lemma 6). Note that we first establishtheseresultsassumingthattheωPNN givenasparameterisanωOPN,then 10