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Decidability Resultsfor Soundness Criteria ofResource-Constrained Workflow Nets

Ferucio Laurentiu Tiplea and Corina Bocaneala

Abstract—This paper focuses on the decidability status ofvarious forms of behavioral correctness criteria for resourceconstrained workflow nets(Petri net models of resource con-strained workflow systems). These behavioral correctness crite-ria, usually called soundness criteria, are natural extensions ofsimilar correctness criteria for workflow nets (Petri net modelsof workflow systems). While all forms of soundness are knownto be decidable for workflow nets, only soundness for resourceconstrained workflow nets with just one resource type is known tobe decidable. In this paper we show that, if we limit the numberof cases, then soundness for resource constrained workflow netswith arbitrarily many resource types is decidable. Moreover, weshow that some “intermediate” forms of soundness, as well as arestrictive form of structural soundness for resource constrainedworkflow nets, are decidable too. The proof technique is basedon instantiation nets as a general tool for dealing with arbitrarilymany cases and arbitrarily large resources in workflow netsand resource constrained workflow nets. It is also shown whythis technique cannot be extended to the most general form ofsoundness.

Index Terms—Workflow system, Petri net, workflow net, re-source constrained workflow net, soundness, decidability.

CONTENTS

I Introduction 1

II Preliminaries on Petri Nets and Workflow Nets 2

III Soundness of Resource-constrained WorkflowNets 3

III-A Resource-constrained Workflow Nets . . 3III-B Soundness of Resource-constrained

Workflow Nets . . . . . . . . . . . . . . 4III-C Closure Nets and Instantiation Nets . . 6

IV Deciding Soundness of Resource-constrainedWorkflow Nets 7

IV-A Deciding Soundness of RCWF Nets Un-der Specified Resources . . . . . . . . . 7

IV-B Deciding Soundness of RCWF Nets Un-der Unspecified Resources . . . . . . . 8

IV-C Deciding Structural Soundness ofRCWF Nets . . . . . . . . . . . . . . . 11

V Related Work and Conclusions 12

Ferucio Laurentiu Tiplea is with the Department of ComputerScience, “Al.I.Cuza” University of Iasi, Iasi, Romania (e-mail:fltiplea@info.uaic.ro).

Corina Bocaneala is with the Department of Mathematics and Com-puter Science, “Dunarea de Jos” University, Galati, Romania (e-mail:Corina.Bocaneala@ugal.ro).

References 12

I. I NTRODUCTION

The termworkflow is used to specify the way of execution(and automation) of a collection oftasks of a well-definedcomplex process, such as a business process in an enterprise.Workflows can be classified in various ways. For instance,based on the repetitiveness and predictability of tasks andon the functionality of the workflow, the trade press clas-sifies workflows intoad-hoc, production, and administrativeworkflows. The classification in [1] distinguishes betweenhuman-orientedandsystem-orientedworkflows, while [2] di-vides workflow into three categories:mail-centric, document-centric, andprocess-centricworkflows.

A specification of a workflow should incorporate executiondependencies between tasks, information flow between tasks,temporal constraints, resource constraints, exception handlingand so on. Traditionally, workflow modeling has only focusedon thecontrol flow aspect in the specification [3], [4]. Morerecently, workflow models which take into consideration timeconstraints [5], [6], [7], [8], [9], [10], [11], [12], [13] and/orresource constraints [14], [15], [9], [16], [17], [18], [19], [12],[20], [21], have been proposed.

As our paper deals with soundness properties of resourceconstrained workflows modeled by Petri nets, we will focusin the following on resource constrained workflows modeledby Petri nets and related soundness properties.

a) Petri net based models of resource constraint work-flows: The way in which Petri nets model workflows startsfrom the remark that tasks in a workflow usually havepre-conditions and post-conditions. Pre-conditions should holdbefore the task is executed, and post-conditions should holdafter the task is executed. Thus, a Petri net model of aworkflow models tasks by transitions and conditions by places.Cases, which are enactments of processes, are modeled bytokens. Petri net models of workflows are calledworkflow nets(WF nets). As with respect to resources, these are modeledin workflow nets by resource places. A resource place isa plain place associated with some resource type, which isconnected to transitions according to the system specifications.The number of tokens in a resource place gives the numberof resources of the same type that are available at a givenmoment in the workflow.

Following this idea, [14] (see also [19]) has proposed themodel of Workflow nets with resources(WFR nets). Closelyrelated to WFR nets areresource-constrained workflow nets(RCWF nets) [16]. The main difference between WFR nets

IEEE Transactions on SMC (Part A), vol. 42(1), 2012, 238-249.

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and RCWF nets is that a WFR net is an RCWF net whereeach resource place has associated a place invariant. This sup-plementary requirement ensures resource preservation alongexecutions in WFR nets, while resource preservation in RCWFnets is obtained by imposing soundness criteria. The approachin [9] has proposed the model ofmultidimension workflownetsas an extension of workflow nets with time, resource andorganization information, while the one in [12] has proposedworkflow nets with time and resource information wheretasks are modeled by places. Colored Petri nets [18], generalstochastic Petri nets [20], and nested Petri nets [21] havealso been advocated as suitable models for the modeling andanalysis of resource constrained workflows. The approachesin[15], [17] propose graph models for the analysis of resource-constrained workflow specifications.

b) Behavioral correctness criteria for workflow nets:Anecessary prerequisite for the deployment of a workflow modelis the verification of the model against some structural orbehavioral correctness criteria. One of the most appealingbe-havioral correctness criterion issoundness[22], which meansthat proper termination of an workflow execution is ensuredand no anomaly has occurred, such as deadlock or livelock.Soundness was originally formulated for unconstrained WFnets [22] and it gained much attention from researchers [14],[23], [24], [25], [19], [26]. Later, soundness was extendedtoWF nets with time constrains [10], [11], [27], [13] and toWF nets with resource constraints [14], [16], [19], [26], [28],[21]. Thestructural soundnesscriterion for WFR nets in [14]is a natural extension of the structural soundness criterion fromWF nets [14], [24]. This criterion can be formulated for RCWFnets too. Another three behavioral correctness criteria for WFRnets and RCWF nets, called(k,R)-soundness, k-soundness,and soundness, were proposed in [19] and [16], respectively.These behavioral correctness criteria are obtained by extendingthek-soundness and soundness criteria from WF nets in orderto cope with resource allocation. Resources should be durable,that is, they can be claimed and released but not created ordestroyed. Thus, no reachable marking on resource placesshould exceed the initial amount of resources, and, once allcases are completed all resources are released. Moreover, fork-soundness and soundness, the net should behave properlynot only with some fixed amount of resources but also withany greater amount.

Undoubtedly, it is of a crucial importance to know whethera workflow net with resource constraints satisfies a behavioralcorrectness criterion as the ones mentioned above. Thus, [14]proposes a characterization of structural soundness for circuit-free WFR nets, while [19] provides a characterization resultfor (k,R)-soundness of WFR nets, based onclosure nets.According to it, an WFR net is(k,R)-sound if and only if itsclosure is bounded and live. As the boundedness and livenessare decidable properties for Petri nets,(k,R)-soundness forWFR nets is decidable too. Moreover, [19] provides someother interesting characterization results fork-soundness andsoundness of particular subclasses of WFR nets, and all theseresults lead to decidability ofk-soundness and soundness forparticular subclasses of WFR nets. As with respect to RCWFnets, the only result known so far is the one announced in [28]

according to which soundness for RCWF nets with just oneresource type is decidable.

None of the Petri net based models of resource constrainedworkflow nets, except for WFR nets and RCWF nets, usessoundness or variations of it as a behavioral correctnesscriterion.

c) Paper contribution and organization:As far as weare concerned, the decidability status of thek-soundness andsoundness problems for general WF nets with (arbitrarilymany) resources is still open. In this paper we make a stepfurther and show that thek-soundness for general RCWF netsis decidable. For this, we gradually refine the soundness crite-ria for RCWF nets by taking into consideration the number ofcases a RCWF net should be able to correctly process and theamount of available resources. These soundness criteria can begrouped into three classes. The first class is that of soundnesscriteria under specified resources, where some marking onresource places is given and the main question is to decidewhether the RCWF net is sound with respect to that marking.These soundness criteria are studied in an uniform way byusing instantiation nets, and it is shown that all of them aredecidable. The second class of soundness criteria is that underunspecified resources. In this case, the main question is todecide whether there is a marking on resource places thatmakes sound the RCWF net. It is shown thatk-soundness forRCWF nets is equivalent to thek-soundness with respect tosome minimal marking on resources. Moreover, it is decidableif this minimal marking exists and, it can be effectivelycomputed when exists. Thus,k-soundness for general RCWFnets is shown decidable. The characterization above cannotbe extended to soundness due to the fact that in this case nominimal markings as those obtained fork-soundness mightexist. The last class of soundness criteria consists of two formsof structural soundness for RCWF nets. One of these, namelystructural R-soundness, is shown decidable, while the otherone “seems” as hard as the soundness problem.

Although k-soundness can be thought as a sufficient be-havioral correctness criterion for RCWF nets in practice, thedecidability of soundness for RCWF nets remains a challeng-ing open problem in the theory of workflow nets. To this,we add one more open problem, namely, the problem of de-ciding structural soundness for RCWF nets under unspecifiedresources.

The paper is organized into five sections. The second oneintroduces basic concepts and notations on Petri nets andworkflow nets. In Section III we recall the concept of anRCWF net as well as various forms of soundness criteria forRCWF nets. The decidability status of these soundness criteriais studied in Section IV. The last section discusses relatedwork to the topic of our paper and presents final conclusionsand a few open problems.

II. PRELIMINARIES ON PETRI NETS AND WORKFLOW

NETS

In this section we review the basic terminology, concepts,notations, and results concerning Petri nets and workflow nets.

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The set of integers is denoted byZ, andN stands for the setof non-negative integers (natural numbers). The free monoidgenerated by a setA under the concatenation operation isdenoted byA∗, andλ is its unity (theempty word); A+ standsfor A∗ − {λ}. If f : A → B is a function andC ⊆ A, thenf |C denotes the restriction off to C (i.e., f |C : C → B andf |C(a) = f(a), for anya ∈ C).

A Petri net [29] is a tupleΣ = (S, T, F,W ), whereS andT are two finite sets (ofplacesand transitions, respectively),S ∩ T = ∅, F ⊆ (S × T ) ∪ (T × S) is the flow relation,andW : (S × T ) ∪ (T × S) → N is the weight functionofΣ verifying W (x, y) = 0 iff (x, y) /∈ F . Given x ∈ S ∪ T ,denote•x = {y|(y, x) ∈ F} andx• = {y|(x, y) ∈ F}.

A marking of Σ is any functionM ∈ NS from S into

N, usually denoted as anS-indexed vector. Thetransitionrelation of a Petri netΣ states that a transitiont is enabledat a markingM , denotedM [t〉Σ, if M(s) ≥ W (s, t) for alls ∈ S. If t is enabled atM , then it can fire yielding a newmarkingM ′ given byM ′(s) = M(s)−W (s, t)+W (t, s) forall s ∈ S; we denote this byM [t〉ΣM

′. The transition relationis usually extended to sequences of transitions. When there isa sequencew ∈ T ∗ such thatM [w〉ΣM

′ we say thatM ′ isreachable(from M in Σ). We denote by[M〉Σ the set of allreachable markings (fromM ) in Σ. When no confusion mayarise we simplify the notation[·〉Σ to [·〉.

Given a Petri netΣ and a non-empty sequencex ∈ T+,define the functions∆x : S → Z and x− : S → N asfollows:

• if x = t ∈ T , then ∆t(s) = W (t, s) − W (s, t) andt−(s) = W (s, t), for any s ∈ S;

• if x = yt, wherey ∈ T+ andt ∈ T , then∆x = ∆y+∆tand

x−(s) =

{

y−(s), if t−(s) ≤ y−(s) + ∆y(s)t−(s)−∆y(s), otherwise,

for any s ∈ S.x− gives the minimal marking at whichx can fire at inΣ.Therefore,x is enabled at a markingM if and only if x− ≤M . Moreover, ifx is enabled atM andM [x〉M ′, thenM ′ =M +∆x.

Let M0 be a marking of a Petri netΣ. We say that:• Σ is bounded with respect toM0 if [M0〉 is finite (this

is equivalent to saying that there existsn ≥ 1 such thatM(s) ≤ n for all reachable markingsM and all placess);

• Σ is M ′-bounded onS′ with respect toM0, whereS′ isa subset of places andM ′ is a marking onS′, if M |S′ ≤M ′, for all M ∈ [M0〉;

• a transitiont of Σ is live with respect toM0 if for anyM ∈ [M0〉 there existsM ′ ∈ [M〉 such thatM ′[t〉;

• Σ is live with respect toM0 if all its transitions are livewith respect toM0;

• a markingM is ahome marking ofΣ with respect toM0

if M ∈ [M ′〉 for all M ′ ∈ [M0〉.All the properties above (boundedness, liveness, home

marking) are decidable for Petri nets [30], [31].A workflow net [3] (WF net) is a Petri netΣ with the

following two properties:

1) Σ has two special placesi and o called theinput and,respectively, theoutput placeof Σ. They satisfy•i = ∅ando• = ∅;

2) Any nodex ∈ S∪T in the graph ofΣ is on a path fromi to o.

Given a WF netΣ, a places of it, and an integerk ≥ 1,we denote byMks the marking given byMks(s) = k andMks(s

′) = 0, for all s′ 6= s. When k = 1 the notation issimplified toMs.

A WF net should satisfy some “behavioral correctnesscriteria”. The first such criterion (for WF nets) was formulatedin [3] and it was calledsoundness. It corresponds to one casehandling by a WF net and consists of:

• for any case, the procedure (represented by the WF net)eventually terminate;

• when the procedure terminates the placeo should bemarked by exactly one token and the other places shouldbe unmarked;

• there are no dead tasks (that is, it should be possibleto execute each task by following the appropriate routethrough the WF net).

Later, soundness was generalized tok-soundness[14] whichcorresponds to multiple case handling, and togeneralizedsoundness[23] which corresponds to arbitrarily many casehandling by WF nets.Structural soundness, which meansk-soundness for somek, was also considered [14], [24].k-soundness, generalized soundness, and structural soundnessare all decidable for WF nets [14], [23], [24].

In this paper we will use the soundness criteria in asimplified form as they are defined in [16]. Thus, we say thata workflow netΣ is k-sound, wherek ≥ 1, if Mko ∈ [M〉for all M ∈ [Mki〉. Σ is calledsoundif it is k-sound for allk ≥ 1.

III. SOUNDNESS OFRESOURCE-CONSTRAINED

WORKFLOW NETS

In this section we recall the workflow model we are goingto use throughout this paper (Section III-A), the soundnessproperties we are going to study (Section III-B), and twobasic constructions,closure netsand instantiation nets, thatwill intensively be used in order to develop the basic resultsof the paper (Section III-C).

A. Resource-constrained Workflow Nets

As we have already mentioned in Section I, the goal ofthis paper is to advance the research on the decidability statusof the soundness properties of resource constrained workflownets (RCWF nets) as introduced in [16]. Therefore, we shallrecall here the concept of an RCWF net. For this, we will useplace-extensions of workflow nets, i.e., Petri nets obtained byadding new places to a given workflow net and connectingthe transitions of the workflow net with the new places in anarbitrary but fixed manner.

Definition 3.1: A place-extensionof a Petri netΣ is anyPetri netΣ′ such thatS ⊆ S′, T ′ = T , F ′|S×T∪T×S = F ,andW ′|S×T∪T×S = W .

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A place-extensionΣ′ of Σ is called empty if S′ = S.Otherwise, it is callednon-empty.

Definition 3.2: Let Σ be a WF net. Aresource constrainedworkflow net(RCWF net, for short) associated toΣ, is anynon-empty place-extensionΣr of Σ.

If Σr is an RCWF net associated toΣ, thenΣ will be calledthe underlying WF netor theproduction netof Σr. Places inΣr which are not inΣ are calledresource places.

In what follows we will simply say “Σr is an RCWF net”instead of “Σr is an RCWF net associated to the WF netΣ”,understanding implicitly that the production net ofΣr is Σ.Moreover, we will writeΣr as a 4-tuple

Σr = (S ∪ Sr, T, F ∪ F r,W ∪W r),

where S ∩ Sr = ∅, F r ⊆ Sr × T ∪ T × Sr, and W r :Sr ×T ∪T ×Sr → N verifiesW r(x, y) = 0 iff (x, y) 6∈ F r.The underlying WF net ofΣr is Σ = (S, T, F,W ) and theresource places are those inSr. In order to avoid any confusionwhen talking about an RCWF netΣr, either Σ or Sr willexplicitly be specified together withΣr (one can easily see thatΣ can be obtained fromΣr andSr, andSr can be obtainedfrom Σr andΣ).

Example 3.1:In Figure 1 an RCWF netΣr with just oneresource placer is graphically represented. Its underlying WFnet Σ is obtained fromΣr by removing the placer and thecorresponding arcs.

i

s1

o

s2

r

t1 t2

t3

2

t4

3

Fig. 1. An RCWF net

This RCWF net models a production system where (indus-trial) products are processed in two steps. The first step ismodeled byt1 and the second one byt2. After the first step,on a random basis, a verification process can be initiated byt3. This transition needs two resources (measuring and controlinstruments) fromr. After this step is completed, a transitiont4 which needs one more resource, completes the verificationprocess. Finally,t4 releases the three resources needed in theverification process and also the product for being furtherprocessed byt2.

A marking of an RCWF netΣr will be written as a pair(M,R), whereM is a marking overS andR is a markingover Sr (i.e., M is a function fromS into N, whereasR isa function fromSr into N). R will also be called aresourcemarking.

B. Soundness of Resource-constrained Workflow Nets

Any RCWF net should satisfy some behavioral correctnesscriteria. For instance, a behavioral correctness criterion for theRCWF net in Example 3.1 could be the following one. Ifkproducts are to be processed by this RCWF net (that is, theplace i is initially marked byk) then all thesek productsshould eventually be processed (that is, the placeo shouldeventually be marked byk). Moreover, if m resources areinitially available in placer, then the number of resourcesshould not increase more thanm during the production processand, finally, when all products are processed, the number ofresources inr should bem.

This correctness criterion is not satisfied by the RCWF netin Example 3.1 ifm = 2k. This is because the transitiont1 canbe appliedk times in a row, thent3 can be appliedk timesin a row and, in such a way, the net behavior is blocked.Consequently, no token will reach the placeo. However, ifm = 2k + 1, the RCWF net in Example 3.1 satisfies thiscorrectness criterion.

Our discussion above shows that the behavioral correct-ness criteria an RCWF net should satisfy may depend onthe number of cases to be processed and/or on the numberof available resources. This led researchers to define case-and/or resource-dependent correctness criteria (usuallycalledsoundness criteriaor soundness properties). Following [14],[16], [28], [19], consider the next variants of soundness forRCWF nets.

Definition 3.3: Let Σr be an RCWF net,k ≥ 1 an integer,andR a marking onSr.

1) Σr is called (k,R)-sound if, for any (M,R′) ∈[Mki, R〉Σr , the following properties hold:

a) R′ ≤ R;b) (Mko, R) ∈ [M,R′〉Σr .

2) Σr is called(≥k,R)-soundif Σr is (m,R)-sound, forall m ≥ k.

3) Σr is called(k,≥R)-soundif Σr is (k,R′)-sound, forall R′ ≥ R.

4) Σr is called (≥k,≥R)-sound if Σr is (m,R′)-sound,for all m ≥ k andR′ ≥ R.

5) Σr is calledk-sound if there existsR such thatΣr is(k,≥R)-sound.

6) Σr is called(≥k)-soundif there existsR such thatΣr

is (≥k,≥R)-sound.7) Σr is called(≤k)-soundif there existsR such thatΣr

is (m,≥R)-sound, for any1 ≤ m ≤ k.8) Σr is called sound if there existsR such thatΣr is

(≥1,≥R)-sound.

A few words about the soundness concepts in Definition3.3 are in order. A sound RCWF net model of a workflowguarantees that the model is capable to process correctlyarbitrarily many cases with unbounded resources. However,it turns out to be quite difficult (if not impossible) to alwaysmodel workflows by sound RCWF nets. Moreover, in practiceit may suffice to have models capable to process only alimited number of cases under unbounded, or even bounded,

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resources. Definition 3.3(1-7) lists a number of intermediateversions of soundness, weaker than soundness, that may sufficein practice.

The soundness concepts from Definition 3.3(1-4) might becalled soundness criteria under specified resourcesbecause,for each of them, the resource markingR is given and thesoundness should be checked with respect toR or all R′ ≥R. On the other side, the soundness concepts from Definition3.3(5-8) might be calledsoundness criteria under unspecifiedresourcesbecause, for each of them, the resource markingRwhich makes the RCWF net sound should be found.(k,R)-soundness,k-soundness, and soundness are precisely

the correctness criteria considered in [16]. They were con-sidered in [19] too, but there they have one more constraintnamely, there is no dead transitions. The other soundnesscriteria in Definition 3.3 were considered just for technicalreasons in order to pass from one to another in an incrementalway.

Example 3.2:The RCWF netΣr in Example 3.1 is(k,≥(2k + 1))-sound for anyk ≥ 1 ((2k + 1) is the markingon the resourcer). However,Σr is not (≥ k,≥ (2k + 1))-sound. Indeed, the transition sequencetk+1

1 tk+13 applied to

(M(k+1)i, (2k + 2)) yields (M ′, R′) with M ′(s1) = R′(r) =0 and M ′(s2) = k + 1. As no transition is enabled at thismarking, we deduce that(M(k+1)o, (2k+2)) is not reachablefrom (M ′, R′). Therefore,Σr is not ((k+1), (2k+2))-sound.

The following remarks are meant to clarify the relationshipbetween soundness of RCWF nets and soundness of theirunderlying WF nets.

Remark 3.1: 1) If an RCWF net Σr is (k,≥ R)-sound for someR, then its underlying WF netΣis k-sound. Indeed, for any transition sequenceuwhich leads Σ from Mki to some markingM ,there exists a resource markingR′ ≥ R such that(Mki, R

′)[u〉Σr (M,R′′), for some R′′. The (k,R′)-soundness ofΣr assures the existence of a transi-tion sequencev such that(M,R′′)[v〉Σr (Mko, R

′) and,therefore,M [v〉ΣMko which shows thatΣ is k-sound.

2) From the above remark we obtain that if an RCWF netΣr is k-sound or sound, thenΣ is k-sound or sound,respectively.

3) It is worth to mention that only(k,R)-soundness of anRCWF netΣr might not lead to thek-soundness ofits underlying WF netΣ. Let us consider, for instance,the RCWF net in Figure 2. It is(1, (1))-sound but itsunderlying WF net is not1-sound ((1) is the markingon the resourcer). Moreover, its underlying WF net isnot even bounded.

As we can see from Remark 3.1(2), soundness of RCWFnets under a fixed numberk of cases and unbounded resources(namely,k-soundness) implies soundness of their underlyingWF nets under the same numberk of cases. Moreover, therequirement to have “unbounded resources” is crucial for this(Remark 3.1(3)).

i

s1

s2s3

o

r

t1

t2 t3

t4

t5

Fig. 2. A (1, (1))-sound RCWF net whose underlying WF net is not1-sound

It is easy (but interesting) to see that(≥k,R)-soundness isequivalent to(≥1, R)-soundness.

Proposition 3.1:Let k ≥ 1. An RCWF netΣr is (≥k,R)-sound if and only ifΣr is (≥1, R)-sound.

Proof: Clearly, if Σr is (≥1, R)-sound, thenΣr is (≥k,R)-sound, for anyk ≥ 1.

Conversely, assume thatΣr is (≥k,R)-sound for somek >1 and let1 ≤ m < k. If Σr is not (≥m,R)-sound, then thereexists a transition sequenceu and a computation

(Mmi, R)[u〉Σr (M,R′)

such thatR′ 6≤ R or (Mmo, R) 6∈ [M,R′〉Σr .As Σr is (k,R)-sound, there exists a transition sequencew

such that(Mki, R)[w〉Σr (Mko, R). But then,

(M(k+m)i, R)[w〉Σr (Mko +Mmi, R)[u〉Σr (Mko +M,R′)

which shows thatR′ 6≤ R or (M(k+m)o, R) 6∈ [Mko +M,R′〉Σr , contradicting the(k + m,R)-soundness ofΣr.Therefore,Σr is (m,R)-sound for any1 ≤ m < k, showingthatΣr is (≥1, R)-sound.

Directly from Proposition 3.1 we obtain:

Corollary 3.1: Let Σr be an RCWF net andk ≥ 1 aninteger. Then, the following properties hold:

1) Σr is (≥k,≥R)-sound if and only ifΣr is (≥1,≥R)-sound.

2) Σr is (≥k)-sound if and only ifΣr is sound.

As it was remarked in [14], in many practical applicationsof WF nets it is important to consider the number of cases asparameters, and to monitor the number of cases which can beprocessed simultaneously such that the soundness propertyissatisfied. This led the authors of [14] to introduce the conceptof structural soundnessof WF nets and systems of WFR nets.We recall here this concept with slight variations and adaptedto RCWF nets.

Definition 3.4: Let Σr be an RCWF net.

1) Σr is calledstructurallyR-sound, whereR is a markingonSr, if there existsk ≥ 1 such thatΣr is (k,R)-sound.

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2) Σr is calledstructurally soundif there existk ≥ 1 anda markingR on Sr such thatΣr is (k,R)-sound.

Clearly, if an RCWF net is sound then it is structurallysound.

C. Closure Nets and Instantiation Nets

Some of the characterization results of the soundness prop-erty for WF nets are based onclosure nets[3], [14] andinstantiation nets[25]. Recall here these concepts and extendthem to RCWF nets.

Given a WF netΣ and an integerk ≥ 1, define thek-closureof Σ [3], [14] as being the Petri netΣ∗ obtained byadding toΣ a new transitiont∗ and two new arcs(o, t∗) and(t∗, i) with weight k. This construction can be generalized toRCWF nets as follows. Given an RCWF netΣr, an integerk ≥ 1, andR a marking onSr, define the(k,R)-closureofΣ as being the Petri net(Σr)∗ obtained by adding toΣr anew transitiont∗, two new arcs(o, t∗) and(t∗, i) with weightk, and, for eachr ∈ R, two new arcs(r, t∗) and (t∗, r) withweightR(r).

Example 3.3:A (k, (2k + 1))-closure net associated to theRCWF net in Example 3.1 is pictorially represented in Figure3 ((2k + 1) is the marking on the resourcer).

i

s1

o

s2

r

t∗

2k + 12k + 1 kk

t1 t2

t3

2

t4

3

Fig. 3. A (k, (2k+1))-closure net associated to the RCWF net in Example3.1

Given an RCWF netΣr, define thecase-resource instantia-tion netassociated toΣr as being the Petri netΣ′ graphicallyrepresented in Figure 4.Σ′ is obtained by adding toΣr thefollowing transitions and places:

(1) transitionst1, t2, and t3, and placesp1, p2, and p3. t1pumps arbitrarily many tokens ini and inp2 (p2 countsthe number of tokens pumped byt1 into i). The transitiont2 disablest1 at some time, and enablest3 to comparethe number of tokens ino against those inp2;

(2) transitionstr,1, tr,2, and tr,3, and placespr,1, pr,2, andpr,3, for any resource placer ∈ Sr. The meaning of thesetransitions and places is similar to those above.

i o

Srr

p1

p2

p3

t1

t2

t3

pr,1

pr,2

pr,3

tr,1

tr,2

tr,3

Σr

for eachr

Fig. 4. A case-resource instantiation net associated toΣr

If we remove, in the construction in Figure 4, all transitionsand places from (1) above, then the Petri net such obtained willbe called theresource instantiation netassociated toΣr, andif we remove all transitions and places associated to resourceplaces (those from (2) above) then the Petri net such obtainedwill be called thecase instantiation netassociated toΣr. Thelater one is exactly the instantiation net associated to a WFnet [25].

Example 3.4:A resource instantiation net associated to theRCWF net in Example 3.1 is pictorially represented in Figure5.

i

s1

o

s2

r

pr,1

pr,2

pr,3

tr,1

tr,2

tr,3

t1 t2

t3

2

t4

3

Fig. 5. A resource instantiation net associated to the RCWF net in Example3.1

If Σ′ is a case-resource instantiation net associated toΣr,

7

k ≥ 1 is an integer, andR is a marking onSr, denote byMki,R andMko,R the markings ofΣ′ given by:

• Mki,R marks the placei by k tokens,Sr by R, and theplacesp1 andpr,1 by one token each, for allr ∈ Sr (allthe other places are unmarked);

• Mko,R marks the placeo by k tokens,Sr by R, and theplacesp3 andpr,3 by one token each, for allr ∈ Sr (allthe other places are unmarked).

In a similar way are definedMki,R andMko,R for a case orresource instantiation net. These notations are simplifiedtoMi,R andMo,R whenk = 1.

The following lemma establishes the relationships betweenRCWF nets and their instantiation nets.

Lemma 3.1:Let Σr be an RCWF net,k ≥ 1 an integer,andR a marking onSr. Then:

1) Mko,R is a home marking of the case instantiation netof Σr w.r.t. Mki,R if and only if, for anym ≥ k, if(M,R′) ∈ [Mmi, R〉Σr , then(Mmo, R) ∈ [M,R′〉Σr .

2) Mko,R is a home marking of the resource instantiationnet ofΣr w.r.t. Mki,R if and only if, for anyR′ ≥ R, if(M,R′′) ∈ [Mki, R

′〉Σr , then(Mko, R′) ∈ [M,R′′〉Σr .

3) Mko,R is a home marking of the case-resource instan-tiation net ofΣr w.r.t. Mki,R if and only if, for anym ≥ k andR′ ≥ R, if (M,R′′) ∈ [Mmi, R

′〉Σr , then(Mmo, R

′) ∈ [M,R′′〉Σr .

Proof: It follows exactly the same line as the proof ofTheorem 2 in [25] and, therefore, it is omitted.

IV. D ECIDING SOUNDNESS OFRESOURCE-CONSTRAINED

WORKFLOW NETS

Each of the soundness concepts in Definition 3.3 andDefinition 3.4 leads to a corresponding decision problem.For instance, the(k,≥R)-soundness decision problemis theproblem to decide, given an RCWF netΣr, an integerk ≥ 1,and a resource markingR, whetherΣr is (k,≥R)-sound.

We will prove that all these decision problems, but(≥k)-soundness (and, therefore, soundness) and structural sound-ness, are decidable. The main idea is to combine the instan-tiation net technique in [25] with some special properties ak-sound RCWF net should satisfy.

A. Deciding Soundness of RCWF Nets Under Specified Re-sources

In this section we will focus on the decidability status of thesoundness criteria for RCWF nets under specified resources. Acharacterization result fork-soundness of WF nets, based ontheir k-closure, was proposed in [14]. Later [19], this resultwas extended to WFR nets. As RCWF nets are not exactlyWFR nets, and as the soundness concepts we use are notexactly those from [19], the characterization result in [19] doesnot work for RCWF nets. However, we can adapt it as follows.

Proposition 4.1:Let Σ be a WF net,Σr an RCWF netassociated toΣ, k ≥ 1 an integer, andR a marking onSr.Then, the following properties hold:

1) Σ is k-sound if and only if itsk-closureΣ∗ is boundedw.r.t. Mki and t∗ is live w.r.t. Mki.

2) Σr is (k,R)-sound if and only ifΣr is R-bounded onSr w.r.t. (Mki, R), its (k,R)-closure(Σr)∗ is boundedw.r.t. (Mki, R), andt∗ is live w.r.t. (Mki, R).

3) Σr is (k,R)-sound if and only ifΣr is R-bounded onSr w.r.t. (Mki, R), and(Mko, R) is a home marking ofΣr w.r.t. (Mki, R).

Proof: (1) If Σ is k-sound then itsk-closureΣ∗ isbounded and live [14] w.r.t.Mki and, therefore,Σ∗ is boundedw.r.t. Mki and t∗ is live w.r.t. Mki.

Conversely, assume that thek-closureΣ∗ of Σ is boundedw.r.t. Mki and t∗ is live w.r.t. Mki, and letM ∈ [Mki〉Σ.Then, there existsM ′ reachable fromM such thatM ′[t∗〉Σ(by the liveness oft∗). Therefore,M ′(o) ≥ k. One can easilysee thatΣ∗ would be unbounded if we assumeM ′ > Mko.Therefore,Σ is k-sound.

(2) Assume first thatΣr is (k,R)-sound. If we assume that(Σr)∗ is not bounded w.r.t.(Mki, R), then there are two reach-able markings(M1, R1) and (M2, R2) from (Mki, R) suchthat (M1, R1) < (M2, R2). The marking(Mko, R) should bereachable from(M1, R1) by some transition sequencew. It isclear thatw is enabled at(M2, R2) and the marking yieldedby applyingw at (M2, R2) is strictly greater than(Mko, R),contradicting the(k,R)-soundness ofΣr.

The R-boundedness onSr w.r.t. (Mki, R) and liveness oft∗ w.r.t. (Mki, R) are direct consequences of(k,R)-soundnessof Σr.

Conversely, let(M1, R1) be a reachable marking from(Mki, R). By R-boundedness onSr w.r.t. (Mki, R) we haveR1 ≤ R, and by liveness oft∗ w.r.t. (Mki, R) we havethat there exists(M2, R2) reachable from(M1, R1) such thatM2 ≥ Mko and R2 ≥ R. Using again theR-boundednessproperty on Sr we obtain R2 = R. If we assume thatM2 > Mko, then(Σr)∗ would be unbounded w.r.t.(Mki, R).Therefore,(M2, R2) = (Mko, R), which shows thatΣr is(k,R)-sound.

(3) Definition 3.3(1a) is equivalent to theR-boundednesson Sr w.r.t. (Mki, R), and Definition 3.3(1b) is equivalentto the property “(Mko, R) is a home marking ofΣr w.r.t.(Mki, R)”.

The items (1) and (2) of Proposition 4.1 show how thek-soundness characterization based onk-closures extends froma WF netΣ to an RCWF net whose underlying WF net isΣ.

Corollary 4.1: The (k,R)-soundness problem for RCWFnets is decidable.

Proof: From Proposition 4.1(2) and from the fact thatR-boundedness, boundedness, and transition liveness are alldecidable for Petri nets.

Proposition 4.2:Let Σr be an RCWF net,k ≥ 1 an integer,andR a marking onSr. Then,Σr is (≥k,R)-sound if andonly if the following two properties hold:

1) Mko,R is a home marking of the case instantiation netof Σr w.r.t. Mki,R;

2) Σr is R-bounded onSr w.r.t. (Mmi, R), for all m ≥ k.

8

Proof: From definitions and Lemma 3.1(1).

Corollary 4.2: The (≥k,R)-soundness problem for RCWFnets is decidable.

Proof: The property in Proposition 4.2(2) is equivalentto the R-boundedness of the case instantiation net ofΣr

w.r.t.Mki,R. Therefore, the corollary follows from Proposition4.2 and from the fact that the home marking and theR-boundedness problems are both decidable.

Proposition 4.3:Let Σr be an RCWF net,k ≥ 1 an integer,andR a marking onSr. Then,Σr is (k,≥R)-sound if andonly if the following two properties hold:

1) Mko,R is a home marking of the resource instantiationnet ofΣr w.r.t. Mki,R;

2) Σr isR′-bounded onSr w.r.t. (Mki, R′), for all R′ ≥ R.

Proof: From definitions and Lemma 3.1(2).

In a similar way to Proposition 4.3 we obtain the followingresult.

Proposition 4.4:Let Σr be an RCWF net,k ≥ 1 an integer,andR a marking onSr. Then,Σr is (≥k,≥R)-sound if andonly if the following two properties hold:

1) Mko,R is a home marking of the case-resource instanti-ation net ofΣr w.r.t. Mki,R;

2) Σr is R′-bounded onSr w.r.t. (Mmi, R′), for all m ≥ k

andR′ ≥ R.

Corollary 4.3: The (k,≥ R)- and (≥ k,≥ R)-soundnessproblems for RCWF nets are decidable.

Proof: We will only focus on (≥ k,≥ R)-soundnessproblem (the other case being similar to this). According toProposition 4.4, we only need to show that the property (2) inthis proposition is decidable. For this, we use the constructionin Figure 6. Assume thatΣr, starting withm ≥ k tokensinto i and a resource markingR′ ≥ R on Sr, reaches amarking(M1, R1) with R1 6≤ R′ by some transition sequenceu. Moreover, assume thatu is minimal with this property inthe sense that if we remove the last transition ofu then themarking onSr is less than or equal toR′. In the Petri netΣ′ in Figure 6 there exists a computation which proceeds asfollows:

• first, m− k tokens are pumped byt1 into i andR′(r)−R(r) tokens are pumped bytr,1 into r, for any r ∈ Sr.We remark thatR′(r)−R(r) tokens are also inserted intor′;

• then,u is applied. The marking obtained isM1, on S,andR1 on Sr;

• if we assume thatR1(r) > R′(r), then by applyingtr for R′(r) − R(r) times, a markingM2 is reachedwith the propertyM2(r

′) = 0 and M2(r) = R1(r) −R′(r) + R(r) > R(r). Moreover, M2(r) − R(r) ≤max{∆t(r)|t ∈ T}.

As a conclusion, if a marking(M1, R1) with R1 6≤ R′ isreachable inΣr from (Mmi, R

′), for some m ≥ k andR′ ≥ R, then a markingM2 is reachable inΣ′ which has

i o

Srr

r′

p1

p2

p3

t1

t2

t3

pr,1

pr,2

pr,3

tr,1

tr,2

tr,3

Σr

for eachr

tr

Fig. 6. Testing property (2) in Proposition 4.4

the propertyM2(r′) = 0 and R(r) < M2(r) ≤ R(r) +

max{∆t(r)|t ∈ T}, for somer ∈ Sr. It is easy to see thatthe converse of this holds true as well.

Therefore, the negation of the property (2) in Proposition4.4 can be reduced to a few number of instances of the sub-marking reachability problem for Petri nets. As this problem isdecidable [30], we deduce that the property (2) in Proposition4.4 is decidable.

Example 4.1:The proof of Corollary 4.3 highlights analgorithm to decide the(k,≥R)- or (≥k,≥R)-soundness ofRCWF nets. The two main steps of this algorithm, applied tothe RCWF netΣr in Example 3.1 in order to prove that it is(k,≥(2k + 1))-sound (see also Example 3.2), are:

– show thatMko,(2k+1) is a home marking of the resourceinstantiation net associated toΣr from Mki,(2k+1) (seeFigure 5);

– show that no markingM with M(r′) = 0 andM(r) =2k + 2, or M(r′) = 0 andM(r) = 2k + 3, is reachablein the Petri net in Figure 7 (the numbers inside placesiandr represents the initial marking of the net).

B. Deciding Soundness of RCWF Nets Under UnspecifiedResources

Deciding soundness of RCWF nets under unspecified re-sources is much harder than deciding soundness under spec-ified resources. The main difficulty occurs due to the factthat in this case a lower bound for the resource markings forwhich soundness is fulfilled should be estimated first, and thensoundness should be checked with respect to this lower bound.We will show that such a lower bound can be computed in thecase ofk-soundness. The technique used to compute it doesnot scale to the case of soundness.

9

ki

s1

o

s2

2k + 1

r

pr,1

pr,2

pr,3

r′tr,1

tr,2

tr,3

tr

t1 t2

t3

2

t4

3

Fig. 7. Testing property (2) in Proposition 4.3 forΣr in Example 3.1 w.r.t.(Mki, (2k + 1))

Definition 4.1: Let Σ be a WF net andk ≥ 1 an integer.A loop sequenceof Σ from Mki is any non-empty transitionsequencew of Σ such thatM [w〉ΣM for someM ∈ [Mki〉Σ.

Denote byLl(Σ, k) the set of all loop sequences ofΣ fromMki. It is clear that ifLl(Σ, k) 6= ∅, thenLl(Σ, k) is infinite.

Example 4.2:For anyk ≥ 1, the transition sequencet3t4is a loop sequence fromMki of the underlying WF net of theRCWF net in Figure 1. Moreover,(t3t4)n is a loop sequencetoo, for anyn ≥ 1.

Loop sequences of the underlying WF net of ak-soundRCWF net do not change the resource markings. This is be-cause, otherwise, arbitrarily large changes on resource placescan be generated by iterating loop sequences, contradictingk-soundness. Formally, we have the following result.

Lemma 4.1:Let Σr be an RCWF net andk ≥ 1 an integer.If Σr is k-sound then∆x|Sr = 0, for anyx ∈ Ll(Σ, k).

Proof: Let Σr be ak-sound RCWF net. Then, there existsa resource markingR such thatΣr is (k,R′)-sound for anyR′ ≥ R. Givenx ∈ Ll(Σ, k) there exist a transition sequenceu and a markingM such thatMki[u〉ΣM [x〉ΣM . Moreover,one can easily see that there exists a resource markingR′ ≥ Rsuch that

(Mki, R′)[u〉Σr (M,R′

1)[x〉Σr (M,R′2)

for some resource markingsR′1 and R′

2. By the (k,R′)-soundness ofΣr we deduce thatR′

1 ≤ R′ andR′2 ≤ R′.

Now, assume by contradiction that∆x|Sr 6= 0. Then, twocases are to be considered:Case 1. ∆x(r) ≥ 0 for all r ∈ Sr and ∆x(r0) > 0 forsomer0 ∈ Sr. Then,R′

2 > R′1 and, therefore,x can fire at

(M,R′2). As a matter of fact we have

(M,R′1)[x

m〉Σr (M,R′1 +m∆x|Sr ),

for any m ≥ 1. As there existsm ≥ 1 such thatR′1 +

m∆x|Sr 6≤ R′, Σr cannot be(k,R′)-sound; a contradiction.Case 2: ∆x(r0) < 0 for somer0 ∈ Sr. By the (k,R′)-soundness ofΣr, there exists a transition sequencev suchthat

(Mki, R′)[u〉Σr (M,R′

1)[x〉Σr (M,R′2)[v〉Σr (Mko, R

′).

From this it follows that∆u|Sr +∆v|Sr +∆x|Sr = 0.Consider now the least markingR′′ on Sr such thatR′′ ≥

R′ and

(Mki, R′′)[u〉Σr (M,R′′

1 )[x〉Σr (M,R′′2 )[x〉Σr (M,R′′

3 )

[v〉Σr (Mko, R′′4 )

for someR′′1 , R′′

2 , R′′3 , andR′′

4 . From this it follows

R′′4 = R′′ +∆u|Sr +∆v|Sr + 2∆x|Sr = R′′ +∆x|Sr

which shows thatR′′4 6= R′′, contradicting the(k,R′′)-

soundness ofΣr.

The next lemma specifies a very simple but importantproperty of firebility in Petri nets.

Lemma 4.2:Let Σ be a Petri net,x, y, z ∈ T ∗, and Mand M1 markings ofΣ such thatM [x〉ΣM1[y〉ΣM1. Then,M [xyz〉Σ if and only if M [xz〉Σ.

Proof: If we assumeM [xyz〉Σ, then there exists a mark-ing M2 such that

M [x〉ΣM1[y〉ΣM1[z〉ΣM2.

But then,M [x〉ΣM1[z〉ΣM2 which shows thatM [xz〉Σ.Conversely, ifM [xz〉Σ, then there exists a markingM2 such

thatM [x〉ΣM1[z〉ΣM2.

As M1[y〉ΣM1, it follows

M [x〉ΣM1[y〉ΣM1[z〉ΣM2,

which shows thatM [xyz〉Σ.

Definition 4.2: Let Σ be a WF net andk ≥ 1 an integer.A simple transition sequenceof Σ from Mki is any non-empty transition sequencew = t1 · · · tm of Σ such that, if wedenoteMki[t1〉ΣM1 · · · [tm〉ΣMm, then one of the followingtwo properties holds:

1) Mm = Mko andMi 6= Mj for any 1 ≤ i, j ≤ m withi 6= j;

2) Mm = Mp for somep < m and Mi 6= Mj for any1 ≤ i, j < m with i 6= j.

A simple transition sequence ofΣ from Mki starts fromMki and either reachesMko without repeating any markingor, if it reaches a marking already encountered, then it stopsthere.

We denote byLs(Σ, k) the set of all simple transitionsequences ofΣ from Mki.

10

Remark 4.1:Let Σ be a WF net. IfΣ is k-sound, thenLs(Σ, k) is finite and can effectively be computed. This is adirect consequence of the fact thatΣ is bounded with respectto Mki (Proposition 4.1(1)).

Given an RCWF netΣr and an integerk ≥ 1 such thatΣ isk-sound, one can effectively computeLs(Σ, k) and the leastmarking RΣr,k with the propertyRΣr,k ≥ w−|Sr , for anyw ∈ Ls(Σ, k). RΣr,k gives the minimal amount of resourcesneeded to all simple transition sequences ofΣ from Mki inorder to fire inΣr from (Mki, RΣr,k).

Example 4.3:If Σ is the underlying WF net of the RCWFnetΣr in Figure 1, thenLs(Σ, 1) = {t1t2, t1t3t4}. Moreover,by a simple computation we obtainRΣr,k = (2k+1) for anyk ≥ 1.

If Σ is the underlying WF net of the RCWF netΣr inFigure 2, thenLs(Σ, 1) is infinite. It containst1tm2 t4t

m3 t5, for

all m ≥ 1. In this case,RΣr,1 does not exist.

We will prove now an important result which shows thatRΣr,k is a sufficient resource marking to all transition se-quences inΣ from Mki in order to fire inΣr, in case thatΣr

is k-sound.

Lemma 4.3:Let Σr be an RCWF net andk ≥ 1 an integer.If Σr is k-sound, thenRΣr,k exists and(Mki, R)[w〉Σr , foranyR ≥ RΣr,k and anyw ∈ T+ with Mki[w〉ΣMko.

Proof: Assume thatΣr is k-sound. Then,Σ is k-soundand, therefore,RΣr,k can effectively be computed. LetR ≥RΣr,k andw ∈ T+ such thatMki[w〉ΣMko.

If w ∈ Ls(Σ, k) then, directly from the definition ofRΣr,k

we have(Mki, R)[w〉Σr .If w is not a simple transition sequence ofΣ from Mki

then, by inspecting from left to right the markings generatedby w, we can decomposew into w = xyz such thatxy is asimple transition sequence ofΣ from Mki andy ∈ Ll(Σ, k).Then, from the definition ofRΣr,k we have

(Mki, R)[x〉Σr (M,R1)[y〉Σr (M,R′1),

for someR1 and R′1. Moreover, by Lemma 4.1,R1 = R′

1.But now, from Lemma 4.2 we have that(Mki, R)[xyz〉Σr ifand only if (Mki, R)[xz〉Σr . Therefore, we may apply theprocedure above toxz and, after finitely many steps weeventually obtain a simple transition sequenceu ∈ Ls(Σ, k)such that(Mki, R)[w〉Σr if and only if (Mki, R)[u〉Σr . Now,lemma follows from the definition ofRΣr,k.

We are now ready to prove our main result.

Theorem 4.1:LetΣr be an RCWF net andk ≥ 1 an integer.Then,Σr is k-sound if and only ifRΣr,k exists andΣr is(k,≥RΣr,k)-sound.

Proof: Clearly, if RΣr,k exists andΣr is (k,≥RΣr,k)-sound thenΣr is k-sound.

Conversely, assume thatΣr is k-sound. By Lemma 4.3,RΣr,k exists. Let R ≥ RΣr,k and u ∈ T+ such that(Mki, R)[u〉Σr (M,R1) for someR1. It is straightforward to

see thatR1 should satisfyR1 ≤ R (otherwise, for anyR′ ≥ R there existsR′

1 such that(Mki, R′)[u〉Σr (M,R′

1) andR′

1 6≤ R′, which shows thatΣr is not k-sound).By thek-soundness ofΣ it follows that there existsv ∈ T ∗

such thatMki[u〉ΣM [v〉ΣMko. Then, Lemma 4.3 corroboratedwith the definition ofRΣr,k shows that

(Mki, R)[u〉Σr (M,R1)[v〉Σr (Mko, R′),

for someR′. By a similar argument as the one above, one caneasily show thatR′ should beR. Hence,Σr is (k,≥RΣr,k)-sound.

Corollary 4.4: k-soundness problem for RCWF nets is de-cidable.

Proof: Let Σr be an RCWF net andk ≥ 1. If Σ is notk-sound, thenΣr is notk-sound (recall thatk-soundness of WFnets is decidable). IfΣ is k-sound, thenRΣr,k can effectivelybe computed. Then, the corollary follows from Theorem 4.1and Corollary 4.3.

The proof of Corollary 4.4 highlights the following algo-rithm for the decidability of thek-soundness problem forRCWF nets.

Algorithm 1 : k-soundness of RCWF nets

input : RCWF netΣr and integerk ≥ 1;output: “Σr is k-sound” if Σr is k-sound, and “Σr is

not k-sound”, otherwise;

beginif Σ is not k-soundthen

“Σr is not k-sound”else

computeRΣr,k;if Σr is not (k,≥RΣr,k)-soundthen

“Σr is not k-sound”else

“Σr is k-sound”

end

Corollary 4.5: Let Σr be an RCWF net andk ≥ 1 aninteger. Then,Σr is (≤k)-sound if and only ifRΣr,k existsandΣr is (≤k,≥RΣr,k)-sound.

Proof: If RΣr,k exists andΣr is (≤k,≥RΣr,k)-sound,thenΣr is (≤k)-sound.

Conversely, ifΣr is (≤k)-sound thenΣ is m-sound for any1 ≤ m ≤ k. Moreover,RΣr,m can effectively be computedfor all 1 ≤ m ≤ k and,RΣr,m ≤ RΣr,k, for any 1 ≤ m ≤ k.Then, the corollary follows from Theorem 4.1.

Corollary 4.6: (≤k)-soundness problem for RCWF nets isdecidable.

Proof: Let Σr be an RCWF net andk ≥ 1. If Σ is notm-sound for all1 ≤ m ≤ k, thenΣr is not (≤k)-sound. IfΣis m-sound for all1 ≤ m ≤ k, thenRΣr,k can effectively becomputed and(m,≥RΣr,k)-soundness ofΣr can be decidedfor any 1 ≤ m ≤ k.

11

Given an RCWF netΣr and an integerk ≥ 1, denoteby RΣr,≥k the minimal marking onSr, if it exists, with thepropertyRΣr,≥k ≥ w−|Sr , for anyw ∈ Ls(Σ,m) andm ≥ k.

Corollary 4.7: Let Σr be an RCWF net andk ≥ 1 aninteger. If RΣr,≥k exists, thenΣr is (≥k,≥RΣr,≥k)-soundif and only if Σr is (≥k)-sound.

Proof: Similar to the proof of Corollary 4.5 but with thedifference thatRΣr,m ≤ RΣr,≥k, for all m ≥ k.

As with respect to the characterization in Corollary 4.7, twoimportant questions arise:

• Does there exist any(≥k)-sound RCWF netΣr for whichRΣr,≥k does not exist?

• Given an RCWF netΣr and k ≥ 1, is it decidablewhetherRΣr,≥k exists?

The first question has a positive answer.

Example 4.4:Consider the RCWF netΣr in Figure 8. Aswe can see, it is a simple variation of the RCWF in Figure 1:it includes a transitionst5 which undoes the effect oft3. Thus,whenΣr starts with(M(k+1)i, (2k+2)) and reaches a marking(M ′, R′) by tk+1

1 tk+13 , whereR′(r) = 0, the transitiont5

undoes the lastt3 and thent4 can firek times in a row. Then,by applyingtk+1

2 , the marking(M(k+1)o, (2k+2)) is reached.Corroborating this with Example 3.2 we can easily obtain thatΣr is (≥k,≥(2k + 1))-sound, for anyk ≥ 1.

i

s1

o

s2

r

t1 t2

t3

2

t4

3

t5

2

Fig. 8. A (≥k,≥(2k + 1))-sound RCWF net for whichRΣr,≥k does notexist

However,RΣr,≥k does not exist because, for anyk ≥ 1,tk1t

k3 is a simple transition sequence ofΣ and it needs at least

(2k + 1) tokens intor in order to fire at(Mki, (2k + 1)).

For the second question we do not have yet any answer, butwe conjecture the following.

Conjecture 4.1:It is decidable, given an RCWF netΣr andan integerk ≥ 1, whetherRΣr,≥k exists.

C. Deciding Structural Soundness of RCWF Nets

We will show in this section that structuralR-soundness isdecidable by using a technique similar to the one in [24].

GivenΣr and a markingR onSr, denote bykΣr,R the leastk ≥ 1, if it exists, with the property(Mko, R) ∈ [Mki, R〉Σr .Such ak ≥ 1 exists if and only ifM0o,R is reachable fromM0i,R in the case instantiation net ofΣr. Therefore, it isdecidable whetherkΣr,R exists and, when exists, it can beeffectively computed.

Proposition 4.5:An RCWF netΣr is structurallyR-sound,whereR is a marking onSr, if and only if kΣr,R exists andΣr is (kΣr,R, R)-sound.

Proof: Clearly, if kΣr,R exists andΣr is (kΣr,R, R)-sound, thenΣr is structurallyR-sound.

Conversely, assume thatΣr is structurallyR-sound. Then,there existsk ≥ 1 such thatΣr is (k,R)-sound. Therefore,kΣr,R exists and can effectively be computed. Moreover,kΣr,R ≤ k.

We show thatΣr is (kΣr,R, R)-sound. Assume, by contra-diction, thatΣr is not (kΣr,R, R)-sound. Then, there exists acomputation

(MkΣr,Ri, R)[u〉Σr (M,R′),

where u ∈ T ∗, such thatR′ 6≤ R or (MkΣr,Ro, R) 6∈[M,R′〉Σr . We will discuss these two cases separately.Case 1:R′ 6≤ R. As kΣr,R ≤ k,

(Mki, R)[u〉Σr (M ′, R′),

for someM ′, which shows thatΣr is not (k,R)-sound; acontradiction.Case 2:(MkΣr,Ro, R) 6∈ [M,R′〉Σr . Let w ∈ T ∗ such that(MkΣr,Ri, R)[w〉Σr (MkΣr,Ro, R), and letm ≥ 1 and0 ≤ n <kΣr,R such thatk = mkΣr,R + n. Two new cases are to beconsidered:

Case 2.1:n = 0. Then,

(Mki, R)[wm−1〉Σr (M(m−1)kΣr,Ro +MkΣr,Ri, R)

[u〉Σr (M(m−1)kΣr,Ro +M,R′)

which shows thatΣr is not (k,R)-sound because(MkΣr,Ro, R) is not reachable from(M,R′); a contra-diction.Case 2.2:0 < n < kΣr,R. Then,

(Mki, R)[wm〉Σr (MmkΣr,Ro +Mni, R)

which shows thatΣr is not (k,R)-sound because(Mno, R) is not reachable from(Mni, R) (otherwise,nwould contradict the choice ofkΣr,R); a contradiction.

As both possible cases led to a contradiction, our assumptionis false and, therefore,Σr is (kΣr,R, R)-sound.

Corollary 4.8: The structural R-soundness problem forRCWF nets is decidable.

Proof: As we have argued above, it is decidable whetherkΣr,R exists and, when exists, it can be effectively computed.Then, the corollary follows from Proposition 4.5 and Corollary4.1.

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By using Proposition 3.1 and Corollary 3.1, we obtainthat the problem “existsk ≥ 1 such thatΣr is (≥ k,R)-sound ((≥k,≥R)-sound)” is decidable because this problemis equivalent to the problem “Σr is (≥1, R)-sound ((≥1,≥R)-sound)”.

As with respect to the structural soundness problem forRCWF nets, we do not have any solution. It seems that thisproblem is as hard as the soundness problem.

V. RELATED WORK AND CONCLUSIONS

In this section we briefly discuss related work to the maintopic of this paper, namely, soundness properties for Petrinetbased models of resource constraint workflows.

There exists an impressive amount of work on using Petrinets to model and analyse resource management in flexiblemanufacturing systems ([32], [33] provide an excellent reviewon this topic). Most of this work deals with strategies tohandle deadlock problems in flexible manufacturing systems.In contrast to this, researches on soundness properties ofresource constraint workflow nets aim to construct robust Petrinet based models that are free of deadlocks no matter ofthe available resource amount beyond a “minimal” one. Afirst approach along this direction was the one in [22] whichproposed a behavioral correctness property for workflow nets,calledsoundness. In [14], the formalism of workflow nets wasextended in order to cope with resource constraints. Thus,[14] has proposedworkflow nets with resources(WFR nets)which are defined as RCWF nets are (Definition 3.2) butwith the difference that they have one more constraint: foreach resource placer there exists a positive place invariantwhose support contains just one resource place, namelyr. Byassociating a place invariant to each resource place, resourcepreservation along any process execution is ensured. Thebehavioral correctness property studied in [14] is that ofstructural soundness. For circuit-free WFR nets, structuralsoundness is equivalent to boundedness, quasi-liveness, andthe controlled-siphon property with respect to some initialmarking. Therefore, for this subclass of WFR nets, structuralsoundness is decidable. Later, in [19], three subclasses ofWFRnets were identified,orderedWFR nets,root WFR nets, andclosure consistentWFR nets, for which soundness can bestructurally characterized and decided effectively.

The approach in [16] to resource constrained workflowsproposesresource constrained workflow nets(RCWF nets) asdefined in Section III-A. As we have explained above, WFRnets are RCWF nets where a place invariant is associated toeach resource place. Conversely, it can be shown that eachresource place in a sound RCWF net has a place invariant [16].In [28] it is shown that soundness of RCWF nets with just oneresource type (place) is decidable. First, soundness is showndecidable for RCWF nets with just one resource type andwhose production nets are state machines (i.e., each transitionhas exactly one pre-condition and exactly one post-condition).Then, soundness of RCWF nets with just one resource isreduced to the soundness of this particular subclass of RCWFnets.

Our paper makes a step further in studying the decidabilitystatus of the soundness property for RCWF nets and shows

that thek-soundness of general RCWF nets (with arbitrarilymany resource types) is decidable. This result was obtainedby showing that an RCWF net can bek-sound if and only ifit is (k,≥R)-sound, whereR is a minimal resource markingwhich enables all simple transition sequences of the underlyingworkflow net. Moreover, it is decidable whenR exists and itcan be effectively computed in such a case. This result doesnot hold for the soundness problem as there may exist soundRCWF nets for which such minimal markings do not exist.

Another contribution of the paper is that it gradually refinessoundness of RCWF nets, establishes necessary and sufficientconditions that can be decided effectively in an uniformway for each intermediate soundness criterion. The paperalso tackles the structural soundness problem for RCWF nets[14]. When the resource markingR is a priori specified, thisproblem is shown to be decidable. Otherwise, the problem“seems” as hard as the soundness problem.

Although soundness and structural soundness remain twoimportant open problems, the(≤k)-soundness property (whichwas shown decidable in this paper) might be sufficient inpractice because, in such a case, we can always estimate themaximum numberk of cases that the workflow should be ableto process.

Acknowledgments. We would like to thank the anony-mous referees for providing us with constructive commentsand suggestions which have led to the improvement of thepaper.

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