research article confusion control in generalized petri...

Research ArticleConfusion Control in Generalized Petri Nets UsingSynchronized Events

Xiaoliang Chen12 Gaiyun Liu2 Naiqi Wu3 Abdulrahman M Al-Ahmari4

Abdulaziz Mohammed El-Tamimi4 and Emad S Abouel Nasr45

1School of Computer and Software Engineering Xihua University Chengdu 610039 China2School of Electro-Mechanical Engineering Xidian University Xirsquoan 710071 China3Institute of Systems Engineering Macau University of Science and Technology Taipa Macau4Industrial Engineering Department College of Engineering King Saud University PO Box 800 Riyadh 11421 Saudi Arabia5Mechanical Engineering Department Faculty of Engineering Helwan University Cairo 11732 Egypt

Correspondence should be addressed to Gaiyun Liu gaiyunliugmailcom

Received 5 March 2015 Accepted 26 July 2015

Academic Editor Georg Frey

Copyright copy 2015 Xiaoliang Chen et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The loss of conflicting information in a Petri net (PN) usually called confusions leads to incomplete and faulty system behaviorConfusions as an unfortunate phenomenon in discrete event systems modeled with Petri nets are caused by the frequentinterlacement of conflicting and concurrent transitions In this paper confusions are defined and investigated in boundedgeneralized PNs A reasonable control strategy for conflicts and confusions in a PN is formulated by proposing elementary conflictresolution sequences (ECRSs) and a class of local synchronized Petri nets (LSPNs) Two control algorithms are reported to controlthe appeared confusions by generating a series of external events Finally an example of confusion analysis and control in anautomated manufacturing system is presented

1 Introduction

Fault detection and avoidance through external control areimportant problems in discrete event systems [1ndash5] A faultwith the loss of conflicting information in a system shouldbe avoided and controlled which may lead to incompletedecision-making processes and confused implementationThe features of a fault are called confusions that occurfrequently in a systemwith shared resources when the systemis modeled by Petri nets Petri nets (PNs) are an effectivetool to model and analyze discrete event systems Conflictsin a PN are tied to the competition for limited resourcesbetween two or more events which usually is considered asthe information interfaces between the DESs and the humandecisions for example a switch controlled by humans or astatic conflict resolution policy designed by humans [6 7]However not all conflicts can be used for decision-makingsince frequent interlacements of concurrent and conflictingtransitions may lead to the information loss of conflicts Inother words confusions generate difficulties for resolution

of conflicts It is awkward to determine whether there areconflicts with integrated decision-making information fromthe reachability graph of a net if it contains confusions

We illustrate the significance of this study on confusionsby an example shown in Figure 1 The function of a systemis concerned with ldquoRecoveryrdquo and ldquoUpdaterdquo In Figure 1(a)a confusion arises from the interlacement between twoconcurrent events ldquoFault alarmrdquo and ldquoUpdaterdquo and twoconflicting events ldquoRecoveryrdquo and ldquoUpdaterdquo It leads to thefact that ldquoRecoveryrdquo cannot fire although it is expected tofire when ldquoFault alarmrdquo is enabled which is caused by theconcurrent behavior between ldquoFault alarmrdquo and ldquoUpdaterdquo Ifthe behavior is always sequentially involved that is ldquoUpdaterdquoldquoFault alarmrdquo the conflict between ldquoUpdaterdquo and ldquoRecoveryrdquomay never happen and the ldquoRecoveryrdquo is missing Further-more the information of the conflict containing ldquoRecoveryrdquois missing Figure 1(b) illustrates another confusion in thesystemThe interlacement between the two concurrent eventsldquoUpdaterdquo and ldquoIgnorerdquo and two conflicting events ldquoUpdaterdquo

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 636959 23 pageshttpdxdoiorg1011552015636959

2 Mathematical Problems in Engineering

Update

Setting recovery model

Fault alarm

Recovery Ignore

p3

p5

p4

p1

p6

p2

(a)

Update


Fault alarm

Recovery Ignore

p3

p5

p4

p1

p6

p2

(b)

Figure 1 A simplified model of fault recovery (a) the model at marking119872 = 1199011+ 1199012and (b) the model at marking1198721015840 = 119901

1+ 1199014

and ldquoRecoveryrdquo also leads to the loss of ldquoRecoveryrdquo sinceldquoRecoveryrdquo cannot fire when the system concurrently firesldquoIgnorerdquo and ldquoUpdaterdquo Obviously a system with confusionscan cause ambiguous behavior and difficulties in systemanalysis Solving the problem of confusion diagnosis meansthat each detected confusion should be controlled by areasonable strategy Hence this study focuses on detectionand control of confusions in PNs

Confusion problems in PNs are first investigated in [89] Several PN applications in a number of areas such asworkflow nets (WF-nets) [10ndash13] unfolding nets [14 15] safemarked ordinary nets [16ndash19] generalized (unsafe) nets [17]and generalized stochastic Petri nets (GSPNs) [20 21] involveconfusions In this section a brief survey of the existingstudies on confusions is presented

van der Aalst et al take into account the existence ofconfusions inWF-nets and show the defects ofWF-nets withconfusions [10ndash12] First the existence of confusions in aWF-net leads to a nondeterministic workflow process such that itscorrectness cannot be ensured AWF-net is said to be correctif it satisfies the two properties ldquosoundnessrdquo and being ldquowell-structuredrdquo which is reported by van der Aalst [12] Everyreachable marking of a WF-net can terminate properly if it issound AWF-net that is well-structured has a number of niceproperties For example the soundness of a WF-net can beverified in polynomial time and a sound well-structuredWF-net is safe However the correctness depends on the existenceof confusions in a WF-net If an acyclic WF-net (no directedcycle in the structure of a WF-net) contains a confusion itis certainly not sound and well-structured If a WF-net withcycles contains a confusion it may not be sound and well-structured Hence we cannot determine the correctness of aWF-net if it contains confusions

The study on occurrence nets lies in that the behaviorof PNs can be interpreted by branching unfolding semantics[15] In PNs the behavior such as sequences concurrencyconflicts trails choices and alternatives can be described andanalyzed by decomposing an occurrence net into substruc-tures given by the node relations associatedwith the behavior

However confusions cannot be described by the existingbranching semantics Hence Smith and Haar consider theindependence of events in occurrence nets and the indirectinfluences among concurrent events in [14 15] Furthermoreinterference structural conflict clusters are developed in [15]in order to describe confusions which belong to a kind ofthe substructures of occurrence nets In the work of Smithand Haar confusion detection and control problems are notconsidered

The work of Bolton [16 17] involves the detection ofconfusions in both safe and unsafe nets Truly concurrentsemantics and interleaving semantics for concurrent systemsare discussed which are useful to formalize and illustrateconfusions The main difference between the two kinds ofsemantics is that the synchronized firing of transitions andthe nondeterministic choice among the possible orders oftheir firing can be distinguished by using truly concurrentsemantics and not the interleaving semantics

For example if two transitions 1199051and 1199052are in a PN con-

currently fire three transition firing sequences are possiblewhich are denoted by 120590

1= [11990511199052] 1205902= 11990511199052 and 120590

3= 11990521199051

respectivelyThe physical interpretation of 1205901in a real system

is that the two events (represented by transitions 1199051and 1199052)

fire simultaneously In this case the firing of 1199051and 119905

2is

said to be synchronized On the other hand firing transitionsequences 120590

2= 11990511199052and 120590

3= 11990521199051means that 119905

1and 1199052

fire sequentially In [17] the definition of confusions in safePNs is extended to unsafe generalized PNs which dependson the inclusion relations of transition sets of conflicts at twomarkings However confusions in generalized PNs cannotbe completely described by the definition in [17] since thetransition sets of conflicts may remain unchanged althougha confusion occurs in a generalized PN

The approaches for confusion detection in [17] are basedon trace theory Furthermore Communicating SequentialProcess (CSP) model-checker is used to detect confusionsThe WF-net of a business procedure is constructed to depictthe logical relations among business tasks and to detectpotential faults in the procedure Confusionsmay occurwhen

Mathematical Problems in Engineering 3

the implementation of theWF-net contains the operations ofboth conflict and concurrency If a WF-net contains a confu-sion its procedure may not be completed or the procedureperforming a desired work cannot be constructed In [13]confusion detection algorithms based on the integer linearprogramming (ILP) are reported ILP methods are availableowing to the Wf-net features represented in [12] This workalso obtains a conclusion that is if an acyclic WF-systemcontains a confusion it is certainly not well-structured

ILPmethods presented in [13] netherwork in safemarkedordinary nets since WF-nets are a subclass of safe markedordinary nets In [18] a confusion detection method isdeveloped to capture confusion subnets in a safe markedordinary net Furthermore a PN system must avoid anyimproper firing order of concurrent transitions by a controlmethod such that the system can be immune to detectconfusions Two confusion online control algorithms areproposed to generate a set of control policies which cancompletely avoid the occurrences of confusions [18]

Confusion avoidance in a safe marked ordinary net canbe considered as a problem to forbid the firing of transitionsaccording to a control specification The work [19] extendsgeneralized mutual exclusion constraints (GMECs) [22 23]and proposes a class of constraints with respect to places andenabling degrees (119875119864 constraints) 119875119864 constraints can beused to design confusion avoidance supervisors by a linearalgebra method However 119875119864 constraints are unavailable ingeneralized PNs

Confusions in GSPNs are discussed in [20 21] Themarking graph of a GSPN is not a stochastic process if theGSPN contains confusions which implies that continuous-time Markov chains (CTMCs) cannot be used to analyzeconfused GSPNs Generally a classical analytical approachfor GSPNs is to assume that the subnets of immediatetransitions are confusion-free in order that the analysis canproceedHowever the assumption does not intrinsically solvethe problem of confusions since they really exist in GSPNs

In the literature confusion analysis in generalized PNsis considered in [17] only Conflict-increasing and conflict-decreasing confusions are defined in unsafe PNs whichare classified according to the change of transition sets ofconflicts However the definition of confusions in [17] isincomplete and cannot describe all confusions in generalizedPNs On the other hand the confusion detection methodin [18] is also available in generalized PNs owing to thesame confusion structures in a safe or a generalized PNHowever the confusion control policies presented in [18]cannot be appropriately enforced since conflict resolution ina generalized PN is related to transition enabling degrees thatis not appearing in safe marked ordinary PNs

This study extends the work on confusions in [17 18] byintroducing the concept of information content of conflictsThe behavior of a confusion can be clearly observed bycomparing the proposed information content of its conflictsin the confusion subnet

Confusions should be prevented by a control methodafter captured confusions in a PN This paper proposes aconfusion control strategy in generalized PNs which is based

on external event control in synchronized PNs (synchro-nized PNs are proposed in [24] and developed in [25ndash27])and based on the proposed elementary conflict resolutionsequences (ECRSs) A class of synchronized PNs with thecontrol information that is called allowable enabling degreesis introduced in which external events are featured withdesired enabling degrees of transitions and can perform avariety of dependency relations of their execution timesAny external event can be converted into a feedback controlstructure taking the form of PNs The control policy refers toa group of static rules imposing restrictions on the enablingdegrees and priorities of concurrent transitions in confusionsThe controlmethods in this paper are considered for pure andbounded PNs only

This paper is organized as follows Section 2 introducesPNs conflicts concurrency information content and con-fusions in generalized PNs Section 3 deals with the controlproblems of confusions A class of synchronized PNs withallowable enabling degrees and ECRSs is proposed Conclu-sions and future work are presented in Section 4

2 Basics of PNs and Confusions

Section 21 provides the basics of Petri nets involved inthe paper More details can be found in [28] Section 22introduces conflicts [2] and formalizes concurrency andconfusions in generalized PNs

21 Basics of PNs

Definition 1 A generalized PN119873 is a four-tuple (119875 119879 119865119882)where 119875 and 119879 are finite nonempty and disjoint sets 119875 is theset of places and 119879 is the set of transitions 119865 sube (119875 times 119879) cup

(119879times119875) is called a flow relation of the net represented by arcswith arrows from places to transitions or from transitions toplaces119882 (119875 times 119879) cup (119879 times 119875) rarr N is a mapping that assignsa weight to an arc119882(119909 119910) gt 0 if (119909 119910) isin 119865 and119882(119909 119910) = 0

otherwise where 119909 119910 isin 119875 cup 119879 and N = 0 1 2 is a set ofnonnegative integers A net is self-loop-free (pure) if ∄119909 119910 isin119875 cup 119879 119891(119909 119910) isin 119865 and 119891(119910 119909) isin 119865

The structure of a self-loop-free PN can be represented byits incidencematrix [119873] that is a |119875|times|119879| integermatrix with[119873](119901 119905) = 119882(119905 119901) minus 119882(119901 119905) [119873] can be divided into twoparts [119873]+ and [119873]minus according to the token flow by defining[119873] = [119873]

+minus [119873]

minus where [119873]+ with [119873]+(119901 119905) = 119882(119905 119901)

and [119873]minus with [119873]

minus(119901 119905) = 119882(119901 119905) are called input and

output matrices respectively

Definition 2 A marking 119872 of 119873 is a mapping from 119875 to N119872(119901) denotes the number of tokens contained in place 119901 119901 ismarked atmarking119872 if119872(119901) gt 0 (119873119872

0) is called a net sys-

tem or marked net and1198720is called an initial marking of119873

Multisets or formal sum notations are usually used todenote markings and vectors As a result a marking 119872 canbe denoted bysum

119901isin119875119872(119901)119901 For example a marking that has

five tokens in place 1199012and three tokens in place 119901

4in a net


p3p1 p2

t1 t2 t3 t4

(a)

p3p1 p2

t1 t2 t3 t4

(b)

Figure 2 (a) A structural conflict 119870 = ⟨1199012 1199051 1199052 1199053 1199054⟩ and (b) an effective conflict 119870119872 = ⟨119901

2 1199051 1199052 1199053 21199011+ 21199012⟩

with four places 1199011 1199012 1199013 and 119901

4is denoted by 5119901

2+ 31199014

instead of (0 3 0 2)119879

Definition 3 Let119909 isin 119875cup119879 be a node in a net119873 = (119875 119879 119865119882)The preset of 119909 is defined as ∙119909 = 119910 isin 119875 cup 119879 | (119910 119909) isin 119865while the postset of 119909 is defined as 119909∙ = 119910 isin 119875 cup 119879 | (119909 119910) isin

119865 Let 119883 be a set of nodes with 119883 sube 119875 cup 119879 We have ∙119883 =

cup119909isin119883

∙119909 and119883∙ = cup

119909isin119883119909∙

Definition 4 A transition 119905 isin 119879 is enabled at marking 119872if forall119901isin∙119905 119872(119901) ge 119882(119901 119905) which is denoted as 119872[119905⟩ If119905 is enabled it can fire Its firing yields another marking1198721015840 such that forall119901 isin 119875 1198721015840(119901) = 119872(119901) minus 119882(119901 119905) +

119882(119905 119901) which is denoted by 119872[119905⟩1198721015840 Marking 1198721015840 is said

to be reachable from 119872 if there exists a transition sequence120590 = 119905

11199052 119905119899and markings 119872

11198722 and 119872

119899minus1such

that 119872[1199051⟩1198721[1199052⟩1198722 119872

119899minus1[119905119899⟩1198721015840 This is denoted by

119872[120590⟩1198721015840 The set of markings reachable from 119872 in 119873

defines the reachability set of (119873119872) denoted as 119877(119873119872)A transition 119905 isin 119879 is live at 119872

0if forall119872 isin 119877(119873119872

0) exist1198721015840 isin

119877(119873119872)1198721015840[119905⟩ holds

Let119873 = (119875119879119865119882) be a net and let120590 be a finite sequenceof transitions The Parikh vector of 120590 is 119879 rarr N whichmaps 119905 in 119879 to the number of occurrences of 119905 in 120590 Let 120590 =

119905111990521199055119905211990511199051be a transition sequence of a net (119873119872) with |119879| =

5 Its Parikh vector is = (3 2 0 0 1)119879

Definition 5 A generalized PN is bounded if exist119896 isin N+ forall119872 isin

119877(1198731198720) forall119901 isin 119875119872(119901) le 119896 where N+ = 1 2

A net which is bounded implies that its reachability sethas a finite number of elements This paper deals with pureand bounded PNs

22 Basic Definitions of Conflicts Concurrencyand Confusions

Definition 6 A structural conflict in a net 119873 = (119875 119879 119865119882)

is a pair 119870 = ⟨119901 119879(119870)⟩ where 119901 isin 119875 is called the structuralconflicting place and119879(119870) = 119905

1 1199052 119905

119899 is the set of output

transitions of 119901 with 119899 = |119901∙| ge 2 The elements in 119879(119870) are

called structural conflicting transitions

Definition 7 An effective conflict in a marked net (1198731198720)

denoted by 119870119872

= ⟨119901 119879(119870119872)119872⟩ is associated with a

structural conflict 119870 = ⟨119901 119879(119870)⟩ and a marking 119872 isin

119877(1198731198720) such that forall119905

119894isin 119879(119870

119872) 119872[119905

119894⟩ is true and

119872(119901) lt sum119905119894isin119879(119870

119872)119882(119901 119905

119894) where119879(119870119872) sube 119879(119870) is the set of

enabled transitions in 119879(119870) at marking119872 with |119879(119870119872)| ge 2

Figures 2(a) and 2(b) show a structural conflict 119870 =

⟨1199012 1199051 1199052 1199053 1199054⟩ and an effective conflict 119870119872 = ⟨119901

2 1199051 1199052

1199053 21199011+ 21199012⟩ that is associated with the structural conflict

119870 respectively Their transition sets can be represented as119879(119870) = 119905

1 1199052 1199053 1199054 and 119879(119870119872) = 119905

1 1199052 1199053

Definition 8 The enabling degree of a transition 119905119894in a pure

net (1198731198720) at a reachable marking119872 isin 119877(119873119872

0) denoted

by 119902119872119894 is an integer such that

119902119872

119894le min

119872(119901)

119882(119901 119905119894)| 119901isin∙119905119894 lt 119902

119872

119894+ 1 (1)

A transition 119905119894is said to be 119902119872

119894-enabled denoted by119872[(119905

119894)119902119872

119894 ⟩if 119905119894has enabling degree 119902119872

119894 Note that if 119905

119894is 0-enabled it

means that 119905119894is disabled

The enabling degree 119902119872119894

of a transition 119905119894at marking 119872

can be considered as the possible maximal number of thefiring times of the transition 119905

119894at marking 119872 Let 119872

119894be

a reachable marking of a PN and 119876119894denote the vector of

enabling degrees of all transitions at119872119894 For example let 119905

1

1199052 and 119905

3be the transitions in a PN (119873119872

0) with |119879| = 3

If they are 2- 0- and 3-enabled at the initial marking 1198720

respectively we have 1198760= [2 0 1]

119879

Definition 9 A general conflict in a marked net (1198731198720)

denoted by K119872 = ⟨119901 119879(K119872)119872⟩ is associated with astructural conflict 119870 = ⟨119901 119879(119870)⟩ and a marking 119872 isin

119877(1198731198720) such that forall119905

119894isin 119879(K119872)119872[119905

119894⟩ is true and119872(119901) lt

sum119905119894isin119879(K

119872)119902119872

119894sdot 119882(119901 119905

119894) where 119879(K119872) sube 119879(119870) is the set of

enabled transitions in 119879(119870) at marking119872 with |119879(K119872)| ge 2and 119902119872119894is the enabling degree of transition 119905

119894at marking119872

Figure 3 illustrates a general conflict K119872 = ⟨1199012 1199051 1199052


119870 = ⟨1199012 1199051 1199052 1199053 1199054⟩ at marking119872 = 2119901

1+31199012 According

to Definition 8 we have 1199021198721

= 2 1199021198722

= 3 and 1199021198723

= 3 Inother words transitions 119905

1 1199052 and 119905

3are 2- 3- and 3-enabled

respectively Hence we have

119872(1199012) = 3

lt 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) + 119902119872

3

sdot 119882 (1199012 1199053) = 8

(2)


p3p1 p2

t1 t2 t3 t4

Figure 3 A general conflictK119872 = ⟨1199012 1199051 1199052 1199053 21199011+ 31199012⟩

The number of tokens in 1199012does not suffice to fire all the out-

put transitions of 1199012according to their enabling degrees An

effective conflict in (1198731198720) is also a general conflict accord-

ing toDefinitions 7 and 9However the reverse is not true Forexample the general conflictK119872 = ⟨119901

2 1199051 1199052 1199053 21199011+31199012⟩

shown in Figure 3 is not effective since

119872(1199012) = 119882(119901

2 1199051) + 119882(119901

2 1199052) + 119882(119901

2 1199053) = 3 (3)

In this paper general conflicts are considered in confusionsLet 119879119888sube 119879 be a transition set in a net system (119873119872

0) and

let119872 isin 119877(1198731198720) be a reachable marking Notation 119864119872(119879

119888)

is used to denote the number of enabled transitions in 119879119888at

marking119872 A transition sequence 120590 is said to be a resolutionsequence of the general conflict K119872 = ⟨119901 119879(K119872)119872⟩ atmarking 119872 if each transition in 119879(K119872) becomes disabledafter firing the sequence 120590 that is 119872[120590⟩119872

1015840 where themarking1198721015840 is called the output marking ofK119872 by firing 120590

A structural conflict in a net system can producemultiplegeneral conflicts owing to different reachable markings For astructural conflict119870 = ⟨119901 119879(119870)⟩ the information content ofthe general conflictK119872 denoted by120595(K119872) associated with119870 at marking119872 is defined as follows

120595 (K119872) = 120572 sdot sum

119905119894isin119879(119870)

119902119872

119894sdot 119882 (119901 119905

119894) (4)

where

120572 =

110038161003816100381610038161003816119864119872(119879 (119870))

10038161003816100381610038161003816ge 2

0 otherwise(5)

and forall119905119894isin 119879(119870) 119902119872

119894is the enabling degree of transition 119905

119894at

marking119872The general conflicts associated with a structural conflict

119870 can be compared by their information content at markingswhich reflects the number of resolution sequences for thegeneral conflicts Note that120572 = 0 can lead to zero informationcontent that is120595(K119872) = 0 which implies that119870 at marking119872 does not produce any general conflict A general conflictK119872 associated with the structural conflict119870 in a PN (119873119872

0)

is said to have integrated decision-making information ifthere does not exist another general conflictK119872

1015840

associatedwith 119870 in (119873119872

0) such that 120595(K119872

1015840

) gt 120595(K119872)Let 120590 = (119905)

119899 denote a transition sequence in which thetransition 119905 consecutively fires 119899 times and 120590 = [119905

1 1199052 119905

119899]

denotes a synchronized transition sequence in which tran-sitions 119905

1 1199052 and 119905

119899fire synchronously that is the tran-

sitions fire simultaneously Figures 4(a) and 4(b) show two

general conflicts K119872 = ⟨1199012 1199051 1199052119872⟩ and K119872

1015840

= ⟨1199012

1199051 11990521198721015840⟩with the same structure atmarkings119872 = 119901

1+21199012

and1198721015840 = 21199011+31199012 respectively For the former that isK119872

we have

120595 (K119872) = 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) = 3 (6)

Furthermore four feasible resolution sequences 1205901= 11990511199052

1205902= 11990521199051 1205903= (1199052)2 and 120590

4= [11990511199052] can be observed

Similarly for the latter that isK1198721015840

we have

120595(K1198721015840

) = 1199021198721015840

1sdot 119882 (119901

2 1199051) + 1199021198721015840

2sdot 119882 (119901

2 1199052) = 5 (7)

The 120595(K1198721015840

) = 5 implies that the general conflict K1198721015840

hasmore resolution sequences thanK119872 which are 120590

1= (1199051)21199052

1205902

= 119905111990521199051 1205903

= 1199052(1199051)2 1205904

= 1199051(1199052)2 1205905

= 119905211990511199052

1205906= (1199052)21199051 1205907= (1199052)3 1205908= [1199051(1199052)2] and 120590

9= [(1199051)21199052]

Suppose that K119872 and K1198721015840

are only two general conflictsassociated with 119870 in a PN Then the general conflict K119872

1015840

has integrated decision-making information since120595(K1198721015840

) =

5 gt 120595(K119872) = 3General conflicts provide information for decision-

making in generalized PNs Confusions can cause the infor-mation loss The proposed control strategy of confusions isbased on general conflicts and the control objective is toensure the integrity of decision-making information of thegeneral conflicts

Definition 10 (1) Transitions 1199051and 1199052are said to be concur-

rent at marking119872 in a marked net (119873119872) if119872[1199051⟩119872[119905

2⟩

and (∙1199051cup 119905∙

1) cap (∙1199052cup 119905∙

2) = 0

(2) Let119863119872 = 1199051 1199052 119905

119899 be a transition set It is called

a set of concurrent transitions at marking119872 if forall119905119894 119905119896isin 119863119872

119894 = 119896 transitions 119905119894and 119905119896are concurrent at marking119872

(3) The concurrent degree of a set 119863119872 of concurrenttransitions denoted by 120582 is defined as 120582 = |119863119872|

(4) Let119863119872 be a set of concurrent transitions in a markednet (119873119872) with 120582 = |119863

119872| It is said to be 120582-max-concurrent

in (119873119872) if there does not exist a set of concurrent transitions119863119872

= 119863119872 with = |119863

119872| in (119873119872) such that gt 120582 A

transition in 119863119872 is said to be 120582-max-concurrent if 119863119872 is 120582-max-concurrent in (119873119872)

(5) LetD119872 = 119863119872

1 119863119872

2 119863

119872

120588 be the set of concurrent

transition sets in (119873119872) In other words forall119863119872119895

isin D119872 119863119872119895

is a set of concurrent transitions in (119873119872) D119872 is said to be120582-max-concurrent if forall119863119872

119895isin D119872 119863119872

119895is 120582-max-concurrent

in (119873119872)

The net shown in Figure 5 contains totally seven sets ofconcurrent transitions at marking 119872 = 119901

1+ 1199014+ 21199015 that

is 1198631198721= 1199051 1199055 1198631198722= 1199051 1199056 1198631198723= 1199052 1199055 1198631198724= 1199052 1199056

119863119872

5= 1199055 11990561198631198726= 1199051 1199055 1199056 and119863119872

7= 1199052 1199055 1199056 Accord-

ing to Definition 10 1198631198726

and 1198631198727

with 1205826= 1205827= 3 are 3-

max-concurrent since there does not exist a set of concurrenttransitions from119863

119872

1to1198631198727 whose cardinality is greater than

three Let D1198721

= 119863119872

6 119863119872

7 and D119872

2= 119863

119872

1 119863119872

2 119863119872

6 be


p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)

Figure 4 (a) A general conflictK119872 = ⟨1199012 1199051 1199052 1199011+ 21199012⟩ with 120595(K119872) = 3 and (b) a general conflictK119872

1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9

Figure 5 Illustration of concurrency

two sets of concurrent transition sets Then D1198721

is 3-max-concurrent andD119872

2is not 3-max-concurrent since there exist

two sets1198631198721and119863119872

2 whose concurrent degrees are less than

threeIt is denoted by 119872[(119905)

119899⟩1198721015840 that a transition 119905 isin 119879

consecutively fires 119899 times at marking119872 where 119899 le 119902119872

1and

1198721015840 is the reachable marking from119872 by firing 119905119905 sdot sdot sdot 119905⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899

Let 119879119888sube 119863119872 be a subset of concurrent transitions in

net119873 at marking119872 Let1198721015840 be the reachable marking from119872 by firing all transitions in 119879

119888according to their enabling

degrees The fact is denoted by 119872[119879119888⟩1198721015840 Note that the

firing orders of the transitions in 119879119888are nondeterministic

since 119879119888is a subset of concurrent transitions For example

if 119879119888= 1199051 1199052 and two transitions 119905

1and 1199052are 1- and 2-

enabled respectively at marking119872 fire their possible firingsequences according to enabling degrees are 120590

1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are

sequential firing sequences that is transitions fire consecu-tively and 120590

4is a synchronized firing sequence that is transi-

tions fire simultaneously Correspondingly119872[119879119888⟩1198721015840 implies

119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840

is the reachable marking from 119872 by firing any of the fourpossible firing sequences of concurrent transitions in 119879

119888

A subnet (structure) of a PN 119873 = (119875 119879 119865119882) is a four-tuple119873 = (119875 119879 119865119882) where119875 sube 119875119879 sube 119879119865 = 119865cap[(119875times119879)cup

(119879times119875)] andforall119891 isin 119865119882(119891) = 119882(119891) Let119898 and1198981015840 be positiveintegers with1198981015840 le 119898 Given a marked net (119873119872) with placeset 119875 = 119901

1 1199012 119901

119898 and its subnet 119873 with place set 119875 =

1199011198921 1199011198922 119901

1198921198981015840 sube 119875 we denote by119872 the projection of a

marking119872 isin N119898 on119873

119872 is called the submarking of119872 in119873 which is denotedby 119872 ≺ 119872 It is said to be valid if exist119895 isin 1198921 1198922 119892119898

1015840

119872(119901119895) gt 0 It is invalid if forall119895 isin 1198921 1198922 1198921198981015840119872(119901

119895) = 0

Figure 6(a) represents a net system (119873119872) with four places1199011ndash1199014and four transitions 119905

1ndash1199054at marking119872 = (1 4 1 0)

119879A subnet 119873 of 119873 with two places 119901

1and 119901

2and two

transitions 1199053and 119905

4is shown in Figure 6(b) at the valid

submarking119872 = (1 4)119879

For 119873 at another marking 1198721015840

= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)

119879 for 119873 isobtained which is invalid since any element in1198721015840 is zero

Definition 11 A confusion denoted by (119873 119879(119870)119872D119872) isa marked subnet (119873119872) in a net system (119873119872

0) such that

(1) there exists a structural conflict 119870 = ⟨119901 119879(119870)⟩ with119901 isin 119875 and 119879(119870) sub 119879

(2) there exists a nonempty setD119872 that is composed of all2-max-concurrent transition sets at the submarking119872 in (119873119872)

(3) forall119863119872119895isin D119872 exist119905

119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840

) are information contentof general conflicts associated with119870 at submarkings119872 and1198721015840 respectively

(4) there exist two reachable markings 119872 and 1198721015840 in

119877(1198731198720) such that119872 ≺ 119872 and1198721015840 ≺ 1198721015840 are valid

A confusion is said to be a conflict-increasing confusion(CIC) denoted as 119862in(119873 119879(119870)119872D119872) if 120595(K119872) lt

120595(K1198721015840

) It is said to be a conflict-decreasing confusion(CDC) denoted as 119862de(119873 119879(119870)119872D119872) if 120595(K119872) gt

120595(K1198721015840

)

Definition 11 defines confusions as a class of markedsubnets in a net system and classifies them into CICs andCDCs Condition (1) in Definition 11 imposes a requirementon marked subnets If a marked subnet is a confusion itnecessarily contains a structural conflict Condition (2) inDefinition 11 limits the size of a marked subnet (119873119872) byrestricting that all sets of concurrent transitions in (119873119872) are2-max-concurrent at submarking119872 The concurrent degree


2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)

Figure 6 (a) A generalized PN119873 at marking119872 = (1 4 1 0)119879 and (b) a subnet119873 of119873 at submarking119872 = (1 4)

119879

120582 = 2 in this condition means that there does not exist aset of concurrent transitions which contains more than twotransitions

Condition (3) in Definition 11 gives the behavior charac-teristics of confusions For any set 119863119872

119895of max-concurrent

transitions in a confusion the change of information contentof general conflicts is caused by the firing of a transition 119905 isin119863119872

119895that is not a structural conflicting transition Condition

(4) in Definition 11 ensures that the behavior described incondition (3) can actually occur in the original net systemIf condition (4) is not satisfied confusions cannot occurin the original net system although it may contain thestructures of confusions Confusions are classified into CICsand CDCs according to the change of information content inDefinition 11

Example 1 Figure 7(a) shows a CIC 119862in(1198731 119879(1198701)1198721

D1198721) where 119879(1198701) = 119905

1 1199052 is derived from the existing

structural conflict 1198701= ⟨1199012 1199051 1199052⟩ in119873

11198721= 1199011+ 21199012+

41199014is an initial submarking D1198721 = 119863

1198721

1 is the set of 2-

max-concurrent transitions sets and11986311987211

= 1199051 1199053

Figure 7(b) shows the partial reachability graph of theCIC which is generated by concurrently firing transitions 119905

1

and 1199053only In Figure 7(b) any submarking119872

119895is considered

with the information content 120595(K1198721198951) of the general conflict

K119872119895

1 The confusion is conflict-increasing since transition 119905

3

can fire twice or four times which can produce an incrementof the information content We cannot determine whetherthe general conflicts associated with119870

1occur from the initial

submarking 1198721to the submarking 119872

10according to the

concurrent firing of transitions 1199051and 1199053

As depicted in Figure 7(b) three general conflicts occurat submarkings 119872

3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725

1) = 3 respectively However the

other submarkings in Figure 7(b) do not produce any generalconflict On the other hand the general conflict K1198725

1=

⟨1199012 1199051 11990521198725⟩ at 119872

5is expected to occur since it has the

maximal information content 120595(K11987251) = 3 in Figure 7(b)

The submarking1198725can be reached by only one path that is

1198721[1199053⟩1198722[1199053⟩1198723[1199053⟩1198724[1199053⟩1198725 which implies that the tran-

sition sequence 120590 = (1199053)4 fires However transitions 119905

1and

1199053in Figure 7(a) are concurrent which implies that the firing

orders of 1199051and 1199053are nondeterministicThenondeterminism

causes that the transition sequence120590 = (1199053)4 is not guaranteed

to be fired In this case we can say that the information loss ofthe general conflictK1198725

1occurs if a transition sequence fires

which does not pass through the submarking1198725


D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055

According to condition (3) inDefinition 11 both sets11986311987211

and1198631198721

2of 2-max-concurrent transitions need to be considered

If transitions 1199051and 1199055in 11986311987211

concurrently fire the partialreachability graph shown in Figure 8(b) is obtained Byobserving the reachability graph the general conflictK1198723

2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013

has the maximal information content 120595(K11987232) = 6 The

information loss of the general conflict K11987232

occurs if thetransition sequence 120590

1= 119905511990511199055or 1205902= 119905111990551199055fires

The partial reachability graph shown in Figure 8(c) isobtained by analyzing the concurrent firing of transitions1199052and 1199055in 1198631198721

2 The same general conflict K1198723

2with the

maximal information content 120595(K11987232) = 6 is obtained in

Figure 8(c) There exists only one path 1198721[1199055⟩1198722[1199055⟩1198723in

Figure 8(c) which reaches the submarking 1198723 However

more paths exist in Figure 8(c) than Figure 8(b) from theinitial submarking 119872

1to the submarking 119872

6such that the

information loss of the general conflictK11987232

occurs In otherwords K1198723

2does not occur if a path in Figure 8(c) does not

pass through the submarking1198723from119872

1to1198726

Example 3 Figure 9(a) shows a CDC that consists of twostructural conflicts 119870

3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩

Only one structural conflict in the confusion is considered


2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)

Figure 7 (a) A CIC 119862in(1198731 119879(1198701)1198721D1198721 ) with |D1198721 | = 1 and (b) its partial reachability graph obtained by concurrently firing 119905

1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)

Figure 8 (a) A CIC 119862in(1198732 119879(1198702)1198721D1198721 ) with |119863119872| = 2 (b) the partial reachability graph of (a) obtained by concurrently firing 119905

1and

1199055 and (c) the partial reachability graph of (a) obtained by concurrently firing 119905

2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)

Figure 9 (a) A CDC 119862de(1198733 119879(1198703)1198721D1198721 ) with |119863119872| = 1 and (b) its partial reachability graph obtained by concurrently firing 119905

1and 1199053

due to symmetry In this example the structural conflict1198703is focused that is the CDC 119862de(1198733 119879(1198703)1198721D

1198721) isanalyzed From Figure 9(a) we have 119879(119870

3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053

Figure 9(b) depicts the partial reachability graph of theCDC which can be obtained by concurrently firing transi-tions 119905

1and 1199053 The maximal information content of general

conflicts associated with the structural conflict 1198703can be

found at the initial submarking 1198721= 2119901

1+ 31199012 that is

120595(K1198721

3) = 3 Hence the decisions on conflicting transitions

1199051and 1199052should be made for the general conflict K1198721

3=

⟨1199011 1199051 11990521198721⟩ However a transition 119905

3notin 119879(119870

3) can

concurrently fire such that the information content is reducedfrom 120595(K

1198721

3) = 3 to 120595(K

1198724

3) = 0 Consequently the

information loss ofK11987213

occursOn the other hand firing 119905

1also reduces the information

content which is not concerned with the information losssince 119905

1is a structural conflict transition for decisions The

submarking1198722= 1199011+ 31199012+1199013with 120595(K1198722

3) = 2 is reached

after firing 1199051at 1198721 which is also prone to the information

loss of the general conflictK11987223

= ⟨1199011 1199051 11990521198722⟩ by firing

1199053

Example 4 Figure 10(a) depicts a CDC with two struc-tural conflicts 119870

5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2

1199053 1199054 1199055⟩ Similarly we consider 119870

5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=

1199052 1199055 In this case the partial reachability graph shown in

Figure 10(b) (resp Figure 10(c)) is obtained by concurrentlyfiring two transitions 119905

1and 1199055(resp 119905

2and 1199055)The confusion

is prone to the information loss of the general conflictK11987215

=

⟨1199011 1199051 1199052 1199053 11990541198721⟩ since firing transition 119905

5reduces the

information content from 120595(K1198721

5) = 8 to 120595(K1198723

5) = 4 and

1199055does not have a structural conflict transition in 119879(119870

5)

Confusion control in a generalized PN means thata reasonable strategy should be constructed to ensure

the appearance of the general conflicts with maximal infor-mation content and if an output marking of a general conflictis preassigned the control policy can guarantee the markingto be reachable

3 Control of Conflicts and ConfusionsUsing Synchronized PNs

Confusions may occur in a PN since a class of specialstructures in the PN is closely related to the appearances ofconfusions and the improper firing orders of concurrent tran-sitions exist in its subnets at somemarkingsThe informationcontent of general conflicts becomes nondeterministic if aconfusion occurs A PN system must avoid any improperfiring order of concurrent transitions such that the system canbe immune to the information loss of general conflicts

Seeking a control mechanism carried out off-line toprevent the appearance of confusions is expected In thissection we achieve this purpose by using synchronized PNsSynchronized PNs are established to reinforce the firingconditions of each transition by introducing a reasonableexternal event In PNs a transition can fire if it is enabledHowever when it will be fired is unknown In contrast ina synchronized PN a transition fires if it is enabled and theassociated external event is satisfied

External events usually have practical meanings Forexample a synchronized PN in which external events areassociated with time or probability is called a 119879-timed PN ora stochastic PN which is introduced by Ramchandani [29]and Florin and Natkin [30] respectively The synchronizedfiring of transitions is allowed in a synchronized PN withexternal events In other words two enabled transitions canfire simultaneously by the control of their external events

In this section a transition firing mechanism is devel-oped which applies external events to each transition withcontrol information that is called allowable enabling degreesThey are featured with desired enabling degrees of transitionsand can perform a variety of external event dependencyrelations by assigning their dependency orders The depen-dency relations of two external events can be described asthe fact that the execution times of an external event dependon that of another external event Then an approach to


22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)

Figure 10 (a) A CDC 119862de(1198734 119879(1198705)1198721D1198721 ) with |119863119872| = 2 (b) the partial reachability graph of (a) obtained by concurrently firing 119905

1

and 1199055 and (c) the partial reachability graph of (a) obtained by concurrently firing 119905

2and 1199055

confusion control is proposed which refers to a group ofstatic rules imposing restrictions on the enabling degreesand the dependency relations of external events controllingconcurrent transitions in confusions A major advantageis that any assigned external event can be converted intoa feedback structure taking the form of PNs such thatconfusions can be controlled in system design stages

Let 120572 isin N be a nonnegative integer It is said to be anallowable enabling degree of a transition 119905 if 119905 is 1205721015840-enabledand 1205721015840 ge 120572

Definition 12 A synchronized PN119873119904with allowable enabling

degrees is a three-tuple (119873 119864 119878) where119873 = (119875 119879 119865119882) is aPN forall119905

119894isin 119879 120572

119894is the allowable enabling degree of 119905

119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] with 119899 le |119879| being a set of external

events and 119878 is a function from the transition set 119879 of 119873 to119864 (1198731199041198720) is called a synchronized net system where119872

0is

an initial marking

In a synchronized net system (119873119904119872) if a transition 119905

119894isin

119879 is controlled by an external event 119890119896[120572119892119896]

with 120572119892119896= 120572119894and is

enabled at119872 it does not become fireable unless the enablingdegree of 119905

119894is greater than or equal to the assigned allowable

enabling degree 120572119894 If 119905119894obtains a proper enabling degree at

119872 then it can immediately fire 120572119894times

A given external event can control more than onetransition such that they can synchronously fire accordingto their allowable enabling degrees Let 119905

119894and 119905

119895be two

transitions in a synchronized PN with 119894 = 119895 Suppose that

119905119894and 119905119895are controlled by an external event 119890119896

[120572119892119896]that can

impose different allowable enabling degrees 120572119894and 120572

119895to

119905119894and 119905119895 respectively Transitions 119905

119894and 119905

119895are said to be

synchronized with 119890119896

[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572

119895= 3 are assigned and

119905119894and 119905119895are 2- and 4-enabled at marking 119872 respectively

they can synchronously fire twice and thrice respectivelyConsequently the transition sequence 120590 = [(119905

119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]

be two external events in 119864 with 119894 = 119895

The fact that the execution times of 119890119895[120572119892119895]

rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895

[120572119892119895] In this case we can say that

the external events 119890119894[120572119892119894]

and 119890119895

[120572119892119895]satisfy the dependency

relation 119890119894[120572119892119894]

≫ 119890119895

[120572119892119895] For instance let transitions 119905

1and 1199055

be controlled by an external event 1198901[1205721198921]

with the functions119878(1199051) = 1198901

[2]and 119878(119905

5) = 1198901

[3]and let transition 119905

2be controlled

by another external event 1198902[1205721198922]

with the function 119878(1199052) = 1198902

[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872

5= 3 be their enabling degrees at marking

119872 respectively Then they are fireable since forall119896 isin 1 2 5119902119872

119896ge 120572119896is true Suppose that the external events satisfy

the dependency relation 1198902[1205721198922]

≫ 1198901

[1205721198921] Then the transition

sequence [(1199051)2(1199055)3] can fire 120588 times if the sequence [(119905

2)2]

had fired 120588 times where 120588 isin N


p1

p2

t1

t2 p3

t3

(a)

0

3

0lfloorlceil rfloorrceil

1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]

0[2]lfloorlceil rfloorrceil )

0

1

2

0[1]

1[1]

2[2]lfloorlceil rfloorrceil )1

1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]

0[1]lfloorlceil rfloorrceil )M3 Q3

1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)

Figure 11 (a) A net system (1198731198720) (b) the reachability graph of (119873119872

0) (c) a synchronized net system (119873

1199041198720) (d) the reachability graph

of (1198731199041198720) (e) a synchronized net system (119873

1015840

1199041198720) with an event dependency relation and (f) the reachability graph of (1198731015840

1199041198720)

Let119873119904= (119873 119864 119878) be a synchronized net with 119899 external

events where 119899 le |119879| The transition set 119879 of 119873119904can be

partitioned into 119899 subsets 1198791 1198792 and 119879

119899according to the

control of the 119899 external events In other words an externalevent 119890119896

[120572119892119896]controls all the transitions in the subset 119879

119896 where

119896 isin 1 2 119899 119873119904at a marking 119872 satisfies the following

transition firing rules

(1) Event dependency relations are not given for externalevents a transition 119905

119895isin 119879119896sube 119879 is fireable and

immediately fires 120572119895times at the marking119872 if forall119905

119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true

(2) Event dependency relations are given for 119899 externalevents for example 1198901

[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫

sdot sdot sdot ≫ 119890119899

[120572119892119899] a transition 119905


immediately fires 120588sdot120572119895times at themarking119872 ifforall119905

119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true and exist1198721015840 exist120590 such that

1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894

is the enabling degree of the transition 119905119894

at marking 119872 120572119894is the allowable enabling degree of the

transition 119905119894controlled by external events and 120588 isin N is an

integer Any synchronized transition (a transition controlledby an external event) in synchronized PNs is presumed asan immediate transition [26] The transition firing rules areillustrated in Examples 5 and 6 respectively

Figures 11(a) and 11(b) show a PN (1198731198720) and its reacha-

bility graphwith 10markings respectively Two synchronizedPNs (119873

1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are

illustrated in Figures 11(c) and 11(e) respectivelyTheir evolu-tions are depicted as reachability graphs and shown in Figures11(d) and 11(f) respectively The restrictions generated by thecontrol of external events can be observed by comparing theirreachability graphs As shown in the examples a node 119899

119896

in the reachability graph of a synchronized PN contains thefollowing information

(1) The marking119872119896

(2) The vector 119876119896of enabling degrees at marking119872

119896

(3) A list of allowable enabling degrees for transitionswhich is shownon the right side of119876

119896in a reachability

graph

The weight on an arc is denoted by the form 120590119890119894

[120572119892119894]that

illustrates the marking change due to the firing of transitionsequence 120590 by the control of external event 119890119894

[120572119892119894]isin 119864


Example 5 Figure 11(c) shows a synchronized PN (1198731199041198720)

that has the same structure and the initial marking of(119873119872

0) where 119905

1and 1199052are synchronized with 119890

1

[1205721198921]and

both 1199051and 119905

2are controlled by 119890

1

[1]with the allowable

enabling degrees 1205721= 1205722= 1 They can synchronously

fire if 1199021198721

ge 1205721and 119902

119872

2ge 1205722are satisfied at a reachable

marking 119872 Otherwise they cannot fire Transition 1199053is

controlled by event 1198902[2]

with 1205723= 2 Figure 11(d) depicts the

reachability graph of (1198731199041198720) where the initial marking

1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this

marking the vector of enabling degrees of 1199051 1199052 and 119905

3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]

119879According to 119876

0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720

3lt 1205723 The synchronized transitions 119905

1and 1199052can fire

The marking1198724= [0 1 2]

119879 can be obtained from the initialmarking 119872

0by firing the transition sequence 120590 = [119905

11199052]

Then the vector of enabling degree1198760= [1 1 1]

119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4

are considered we have 11990211987241

lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723

However transition 1199052is not allowed to fire owing to the con-

trol of the external event 1198901[1205721198921]

It can fire if its synchronizedtransition 119905

1is also fireable On the other hand transition

1199053is not synchronized with other transitions Hence it can

fire After the transition sequence 120590 = (1199053)2 fires marking

1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]

119879 are obtained Thetransition sequence 120590 = [119905

11199052] can immediately fire since the

allowable enabling degrees of 1199051and 1199052are satisfied atmarking

1198723 Finally the state returns to the initial marking119872

0

Example 6 Figure 11(e) depicts a synchronized net system(11987310158401199041198720) where transitions 119905

1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If

the dependency relations are set for the external events 1198901[1205721198921]

and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901

[1205721198921] the reachability graph shown

in Figure 11(f) is obtained By considering the initial marking1198720= [1 1 1]

119879 we obtain the vector 1198760= [1 1 1]

119879 Thethree transitions 119905

1 1199052 and 119905

3are fireable since their allowable

enabling degrees are satisfied In this case 1199051and 1199052can fire

if 1199053has fired which is ensured by the assigned event depen-

dency relation Hence 1199051and 1199052cannot fire at119872

0until 119905

3fires

once Marking1198723= [2 1 0]

119879 is reached by firing 1199053from the

initial marking1198720 Then the transition sequence 120590 = [119905

11199052]

fires once and the state returns to the initial marking1198720

The control of external events generates restrictionson the reachability set of (119873

1199041198720) that is 119877(119873

1199041198720) sube

119877(1198731198720) On the other hand the set of feasible transition

firing sequences of (1198731199041198720) is also included in that of

(1198731198720) [26]

To illustrate the feasibility of the control of external eventsin synchronized PNs we provide formal definitions andillustrations by converting the control into feedback controlstructures taking the form of PNs

Let 1198791015840 = 1199051 1199052 119905

120578 sube 119879 be the set of the transitions

controlled by an external event 119890119896[120572119892119896]

isin 119864 with 119896 isin 1 2

|119864| in a synchronized PN 119873119904and forall119905

119894isin 1198791015840 let 120572

119894be the

allowable enabling degree of the transition 119905119894 The feedback

control structure of 119890119896[120572119892119896]

can be constructed according to thefollowing steps

(1) Let 119905119903119896and 119905119888119896be transitions and let 119901

119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896

performs reading operation that canread the token information of the preset places ofthe controlled transitions according to their allowableenabling degrees If the tokens in the preset placessatisfy the firing rules of the controlled transitionsaccording to their allowable enabling degrees thetransition 119905

119903119896fires and outputs a token to the place119901

119903119896

The construction of this step can be formally definedas follows define the arc (119905

119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =

1 forall119905119894isin 1198791015840 forall119901isin∙119905

119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896

restores the tokens in the pre-set places of the controlled transitions and outputscontrol tokens to the places 119901

1198881ndash119901119888120578

such that thetokens in them can control the firing of the controlledtransitions The construction of this step can beformally defined as follows define the arc (119901

119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1

The constructed feedback control structures of externalevents are illustrated with examples First in a synchronizednet system an external event can perform control on theallowable enabling degree of a transition For exampletransition 119905

1in Figure 12(a) is considered which has two

input places 1199011and 119901

2 An external event 1198901

[1205721198921]with 119878(119905

1) =

1198901

[2](1205721= 2) is designated to control 119905

1and the allowable

enabling degree 1205721= 2 is used to control that the transition 119905

1

cannot fire unless its enabling degree is greater than or equalto twoThe feedback structure shown in Figure 12(b) achievesthe control purposes

In Figure 12(b) two tokens in place 1199011198981

imply 1205721= 2

Transition 1199051199031

performs reading operations for the enablingdegree of 119905

1at amarking If the enabling degree of 119905

1is greater

than or equal to two transition 1199051199031fires Then transition 119905

1198881

outputs control tokens such that 1199051fires twice

Second synchronized firing of transitions can be per-formed by external events Transitions 119905

2and 1199053shown in

Figure 12(c) are controlled by an external event 1198902[1205721198922]

with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both

enabling degrees of 1199051and 1199052satisfy their allowable enabling

degrees they become fireable and fire otherwise none ofthem is allowed to fire

Figure 12(d) depicts the feedback control structure thatachieves the same control purposes of 1198902

[1205721198922]for transitions


(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1

Figure 12 Illustration of the control of external events (a) a transition with an external event (b) the feedback structure of (a) (c) twosynchronized transitions (d) the feedback structure of (c) (e) two synchronized transitions with a common preset place (f) the feedbackstructure of (e) (g) setting dependency relation 1198901

[1205721198921]≫ 1198902

[1205721198922]for two external events and (h) the feedback structure of (g)


p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)

Figure 13 Illustration of the control of resolution sequences in a general conflict

1199052and 1199053 Both enabling degrees of 119905

2and 1199053are detected

by transition 1199051199032 Furthermore transition 119905

1198882outputs control

tokens for the two transitions simultaneously after 1199051199032fires

Suppose that 1199052and 1199053in Figure 12(c) have commonpreset

places For example the two places 1199013and 119901

4in Figure 12(c)

are merged into the place 1199013shown in Figure 12(e) The

PN construction of Figure 12(e) can be obtained by merging1199013 1199014 and their arcs in Figure 12(d) which is shown in

Figure 12(f)Let 1198791015840 and 119879

10158401015840 be subsets of the transition set 119879 in asynchronized PN 119873

119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895

[120572119892119895]be two external events where the transitions in1198791015840 (resp

11987910158401015840) are controlled by the event 119890119896

[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895

[120572119892119895]satisfy the event dependency relation 119890119896

[120572119892119896]≫ 119890119895

[120572119892119895] The

corresponding feedback control structure can be constructedaccording to the following steps

(1) Let 1199051199041and 1199051199042be two transitions and let 119901

1199041ndash1199011199043be

three places where forall119894 = 1 2 3119872(119901119904119894) = 0

(2) Transition 1199051199041fires if the control of the external event

119890119896

[120572119892119896]is executed The execution times of 119890119896

[120572119892119896]are

stored into the place 1199011199043by firing 119905

1199042 The tokens in 119901

3

control the execution times of another external event119890119895

[120572119892119895]such that the execution times of 119890119895

[120572119892119895]rely on

119890119896

[120572119892119896]The construction of this step is formally defined

as follows define arcs (119905119903119896 1199011199041) (1199011199041 1199051199041) (1199051199041 1199011199042)

(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)

As shown in Figure 12(g) the execution times of theevent 1198902

[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]

By considering Figure 12(h) the dependency relation can beachieved by adding two transitions 119905

1199041and 1199051199042 three places

1199011199041ndash1199011199043 and their arcs

31 Control of General Conflicts in Synchronized PNs Theresolution control of a general conflict is to choose a transi-tion sequence such that the general conflict can be resolvedaccording to the selected sequence to reach an expectedmarking

The resolution control is illustrated by an exampleK119872 =

⟨1199011 1199051 1199052 31199011+ 21199012⟩ shown in Figure 13(a) is a general

conflict at marking 119872 = 31199011+ 21199012 Figure 13(b) depicts

K119872 in the corresponding synchronized net 119873119904of 119873 with

119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901

[1205722] The resolution control of K119872

means to make decision on firing times of transitions 1199051and

1199052 In otherwords the values of120572

1and1205722are determined such

that a resolution sequence ofK119872 fires Figures 13(c) and 13(d)show two feasible control policies

If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]

are employed the general conflict K119872 will fire resolutionsequence 120590

1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901

[2]are applied the resolution

sequence 1205902= [1199051(1199052)2] is ensured to fire in (119873119872)

In a general conflict some resolution sequences can leadto the same output marking which have different transitionfiring orders In these resolution sequences the synchronizedone is unique which is used to resolve the general conflict insynchronized PNs For example 120590

1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590

4= [1199051(1199052)2] are four resolution sequences

of the general conflict shown in Figure 13(a) Firing any ofthem leads to the same output marking In this example 120590

4is

used to resolve the general conflict in the synchronized PNA strategy to find all synchronized resolution sequences

for a general conflict is proposed Let 119879(119890119894[120572119892119894]

119872) be the setof transitions that are controlled by the external event 119890119894

[120572119892119894]at

marking119872 Let 119886 and 119887 be two vectors Notation 119886 ge 119887meansfor every 119894 119886(119894) ge 119887(119894) and 119886 ⪈ 119887means for every 119894 119886(119894) ge 119887(119894)

and for at least one 119894 119886(119894) gt 119887(119894)

Definition 13 The sequence 120590119896is said to be an elementary

conflict resolution sequence (ECRS) with an external event119890119894

[120572119892119894]in (119873119904119872) if it meets the following four conditions

(1) All the transitions in 120590119896belong to 119879(119890119894

[120572119892119894]119872)

(2) All the transitions in 120590119896belong to 119879(K119872)

(3) [119873119904]minussdot997888rarr120590119896le 119872 where [119873

119904]minus with [119873

119904]minus(119901 119905) =

119882(119901 119905) is the output incidence matrix and 997888rarr120590119896is the

Parikh vector of 120590119896

(4) There is no sequence 119896such that

997888rarr119896⪈997888rarr120590119896


An ECRS of a general conflict in a net (119873119904119872) corre-

sponds to a synchronized resolution sequence Conditions(1) and (2) in Definition 13 guarantee that the fundamentalelements (transitions) in an ECRS are controlled by 119890

119894

[120572119892119894]

at marking 119872 and the transitions are in the set of generalconflicting transitions Condition (3) in Definition 13 ensuresthat the conflicting transitions in 119879(K119872) according to theECRS are feasible (the tokens in places meet the firingcondition of their output transitions in 120590

119896) Finally according

to condition (4) in Definition 13 120590119896is ensured as a resolution

sequence of the general conflictK119872 In other words all con-flicting transitions become disabled if 120590

119896fires Algorithm 14

is proposed to generate all ECRSs of a synchronized PN at amarking

Algorithm 14 is illustrated through an example shown inFigure 14 in which the weight of each arc equals one for thesake of simplicity since the weights do not affect the functionof Algorithm 14 Actually a weight means a coefficient inthe computation of enabling degrees A synchronized PN(119873119904119872) is shown in Figure 14(a) where 119872 = 3119901

2+ 21199013+

21199014+1199016 Transitions from 119905

1to 1199054are controlled by an external

event 1198901[1205721198921]

outputting the functions from 119878(1199051) = 119890

1

[1205721]to

119878(1199052) = 119890

1

[1205724] respectively Transition 119905

5is controlled by

another external event 1198902[1205721198922]

with 119878(1199055) = 119890

2

[1205725] The values

of allowable enabling degrees of conflicting transitions canbe obtained according to the firing times of the transitionsin every ECRS

Algorithm 14 (generation of all ECRSs of a synchronized PN(119873119904119872))

Input is as follows (119873119904119872) and 119890

119894

[120572119892119894] 119890119894[120572119892119894]

is anexternal event and employed in119873


Output is as follows all ECRSs

Step 1 delete all transitions not belonging to119879(119890119894

[120572119892119894]119872) and their input and output arcs

according to condition (1) in Definition 13Step 2 according to condition (2) inDefinition 13 delete all the transitions notbelonging to 119879(K119872) and their input and outputarcsStep 3 delete all the arcs from transitions toplaces since an ECRS is related to the out-put matrix only according to condition (3)in Definition 13 the reduction leads to a newsynchronized PN 119873

119904 All the transitions in 119873

119904

are conflicting transitionsStep 4 construct the reachability graph of(119873119872) Then each deadlock marking impliesan ECRS (119873119872) is the generalized PN with[119873] = [119873

119904]

Step 5 output the ECRSs

First the input of Algorithm 14 is (119873119904119872) and the

external event 1198901[1205721198921]

After Step 1 in Algorithm 14 a subnet

of 119873119904is obtained as shown in Figure 14(b) where transition

1199055and its corresponding arcs in 119873

119904are removed since 119905

5is

not controlled by 1198901

[1205721198921] Then transitions 119905

1 1199054 and their

corresponding arcs are removed by Step 2 in Algorithm 14since they are not conflicting transitions which is depictedin Figure 14(c) The subnet 119873

119904of 119873119904can be found as shown

in Figure 14(d) according to Step 3 in Algorithm 14The generalized PN (119873119872) with [119873] = [119873

119904] contains a

general conflictK119872 = ⟨1199012 1199052 1199053 31199012+ 21199013+ 21199014+ 1199016⟩ Its

reachability graph is shown in Figure 14(e) There are threedeadlock markings that correspond to three different ECRSs1205901= [(1199052)3] 1205902= [1199052(1199053)2] and 120590

3= [(1199052)21199053] respectively

Each ECRS can deduce the values of 1205721 1205722 1205723 and 120572

4on

external event 1198901[1205721198921]

For example 1205902= [1199052(1199053)2] is an ECRS

of K119872 which means that K119872 can be resolved by firingtransitions 119905

2once and 119905

3twice simultaneously Hence 120572

1=

0 1205722= 1 120572

3= 2 and 120572

4= 0 are obtained The external event

1198901

[1205721198921]with the four allowable enabling degrees can perform

control such that the transitions inK119872 fire according to theECRS 120590

2at marking119872

The validity of Algorithm 14 is clear since the reachabilitygraph of the PN that is composed of a general conflict onlycontains all feasible paths from an initial marking to all deadmarkings

32 Control of Confusions Using External Events As statedin Section 2 a confusion is prone to the information loss ofgeneral conflicts owing to nondeterministic firing orders ofconcurrent transitions In a CIC a general conflictK119872 withmaximal information content is implied in the evolution ofmarkings The marking119872 at which the general conflictK119872occurs is expected It should be ensured that the firing ofa transition sequence can reach the marking 119872 Also thetransition firing sequences that lead to the information lossof the general conflictK119872 should be avoided

If a general conflict K119872 with maximal informationcontent in a confusion occurs all the ECRSs are neededand can be obtained Furthermore an ECRS is selected asa decision to generate the allowable enabling degrees oftransitions The general conflict K119872 can always be resolvedaccording to the selected ECRS by the control of externalevents Here we assume that confusions have been foundat a marking and we aim to control them Algorithm 15 isproposed to control a given CIC

Algorithm 15 (general conflict control in CIC (119873119872))

Input is as follows (119873119872)

Output is as follows output 119878(119905119894) forall119905119894isin 119879 and the

dependency relation between 119890119895

[120572119892119895]and 119890119896[120572119892119896]

119879 is

the set of transitions in 119873 119878 is a function mappingfrom 119879 to 119864 119864 = 119890

119895

[120572119892119895] 119890119896

[120572119892119896] is a set of external

events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2

1205901 = [(t2)3]1205902 = [t2(t3)2]1205903 = [(t2)2t3]Figure 14 Application of Algorithm 14 to a PN119873

119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]

Figure 15 Illustration of the control of confusions using external events (a) a CIC (11987311198721) (b) application of Algorithm 15 to (119873

11198721) (c)

a CDC (11987331198721) and (d) application of Algorithm 16 to (119873

31198721)

Step 1 find the set 119879(119870) of the structuralconflicting transitions in119873 Let119867 = 119879 119879(119870)forall119905119894isin 119867 design a function 119878(119905

119894) = 119890

119895

[120572119894] forall119905119894isin

119879(119870) design a function 119878(119905119894) = 119890119896

[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894

is the enablingdegree of 119905

119894at the submarking119872

Step 3 design the dependency relation 119890119895[120572119892119895]

≫

119890119896

[120572119892119896]

Step 4 obtain the submarking 1198721015840 by firing

all transitions in 119867 at 119872 according to theirenabling degreesStep 5 find all ECRSs of (119873119872

1015840

) byAlgorithm 14 with the inputs (1198731198721015840) and 119890119896

[120572119892119896]

Step 6 choose an ECRS For any transition 119905119894in

119879(119870) obtain the allowable enabling degree 120572119894

by the ECRSStep 7 output 119878(119905

119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]

When the general conflict of aCIC is identified atmakingits control can be enforced by adding external events accord-ing to some of the established rules Algorithm 15 presentsthe rules In Step 1 of Algorithm 15 the transition set 119879 of aCIC is partitioned into two parts 119879(119870) and 119867 = 119879 119879(119870)According to Definition 11 a CIC occurs if the transitionsin 119879(119870) fire before those in 119867 Hence Algorithm 15 designsa dependency relation by Step 3 in Algorithm 15 betweenthe event controlling the transitions in 119867 and the event

controlling the transitions in 119879(119870) such that forall119905119894isin 119867 forall119905

119895isin

119879(119870) 119905119894fires 119902119872

119894times before firing 119905

119895 If the transitions in119867

fire according to their enabling degrees the general conflictassociated with 119870 with the maximal information content isobtainedWe can obtain all ECRSs according toAlgorithm 14Any ECRS is a control policy Then an ECRS is selectedand the allowable enabling degrees of conflicting transitionsto perform this ECRS can be obtained Finally the outputfunctions and the event dependency relation presented inStep 7 of Algorithm 15 can control the CIC such that theinformation loss of the general conflict that has the maximalinformation content does not occur

The CIC 119862in(1198731 119879(1198701)1198721D1198721) in Example 1 is also

depicted in Figure 15(a) to illustrate Algorithm 15 where119879(1198701) = 119905

1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905

1 1199053 By the analysis in Section 2 the

general conflict K11987251

= ⟨1199012 1199051 11990521198725⟩ at the submarking

1198725= 1199011+ 21199012+ 41199013has the maximal information content

120595(K1198725

1) = 3 Transitions 119905

1and 1199053are concurrent

The information loss of K11987251

in the CIC occurs if theconflicting transition 119905

1fires before firing transition sequence

(1199053)4 For example the firing of the sequence 120590

1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial

submarking1198721causes the information loss ofK1198725

1 On the

contrary the information loss of K11987251

does not occur in theCIC if the transition sequence (119905

3)4 fires before firing 119905

1 For

example the firing of the sequence1205905= (1199053)41199051does not cause

the information loss since the submarking 1198725leading to


middot middot middot middot middot middot

middot middot middot

middot middot middotmiddot middot middotmiddot middot middot

middot middot middotmiddot middot middot

Ω1 Ω2Ωlowast

p1 p2

Figure 16 Schema of a CDC (119873119872) without considering the weights of arcs where 1198701198921= ⟨1199011 119901∙

1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ωlowast and Ω

2= 119879(119870

1198922) Ωlowast

K1198725

1can be reached through firing the sequence 120590

5= (1199053)41199051

from1198721 that is119872

1[(1199053)4⟩1198725[1199051⟩11987210The control processes

for this confusion contain the following steps according toAlgorithm 15

(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905

3be controlled by 1198902

[1205721] 1198902[1205722]

and 1198901[1205723]

respectively (Step 1) Suppose that 119895 = 1 and 119896 = 2

(2) The allowable enabling degree of transition 1199053is

obtained by 1205723= 1199021198721

3= 4 (Step 2)

(3) The dependency relation 1198901

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)

(4) The submarking1198725= 1199011+ 21199012+ 41199013is reached by

firing 1199053four times (Step 4)

(5) Two ECRSs 1205901= [11990511199052] and 120590

2= [(1199052)2] of the general

conflictK11987251

are obtained by Algorithm 14 (Step 5)(6) The ECRS 120590

1is selected We have the allowable

enabling degrees 1205721= 1205722= 1 by the ECRS (Step 6)

(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901

[4]are obtained The dependency relation 119890

1

[1205721198921]≫

1198902

[1205721198922]needs to be applied (Step 7)

Finally the controlled CIC is shown in Figure 15(b) whereonly the transition sequence 120590 = (119905

3)4[11990511199052] can fire The


is preventedThe structure of a CDC is composed of two structural

conflicts However only one structural conflict can be used toanalyze the information loss of general conflicts in the CDCwhich is illustrated with the example shown in Figure 16

Figure 16 shows the schema of a CDC (119873119872) withoutconsidering the weights of arcs where 119870

1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙

2are two structural conflicts in the CDC Let K119872

1198921and

K1198721198922

be two general conflicts associated with 1198701198921

and 1198701198922

at submaking 119872 respectively All transitions in (119873119872) arepartitioned into three subsets Ω

lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω

lowast Let D119872 =

119863119872

1 119863119872

2 119863

119872

|Ω1|sdot|Ω2| denote the set of 2-max-concurrent

transition sets of (119873119872) forall119905119894isin Ω1 forall119905119896isin Ω2 transitions 119905

119894

and 119905119896are concurrent

According to Definition 11 forall119905119894isin Ω1 forall119905119896isin Ω2 exist119863119872119895=

119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)

such that the information loss of K1198721198921

occurs by firing thesequence 120590 = 119905

119896119905119894 However we also have 119905

119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872

119895119879(1198701198922) such that the firing of the same sequence

120590 = 119905119896119905119894does not cause the information loss of K119872

1198922 Hence

the contradiction occurs For a CDC only one structuralconflict can be used to analyze the information loss of generalconflicts in the CDC

Algorithm 16 (general conflict control in CDC (119873119872))


Output is as followd 119878(119905119894) forall119905119894

isin 119879 and thedependency relation between 119890

119895

[120572119892119895]and 119890119896[120572119892119896]

119879 isthe set of transitions in 119873 119878 is a function mappingfrom 119879 to 119864 119864 = 119890

119895

[120572119892119895] 119890119896


events with 119895 = 119896

Step 1 forall119905119894isin 119879 design a function 119878(119905

119894) = 119890119895

[120572119894]

Find all ECRSs of (119873119872) by Algorithm 14 withthe inputs (119873119872) and 119890119895

[120572119892119895] Choose an ECRS

For any transition 119905119894that meets 119878(119905

119894) = 119890

119895

[120572119894]

obtain the allowable enabling degree 120572119894by the

ECRSStep 2 find the two sets 119879(119870

1198921) and 119879(119870

1198922) of

the structural conflicting transitions in 119873 LetΩlowast= 119879(119870

1198921) cap 119879(119870

1198922) Ω1= 119879(119870

1198921) Ωlowast and

Ω2= 119879(119870

1198922) Ωlowast The structure of any CDC

is composed of two structural conflictsStep 3 choose a set of structural conflict transi-tions If 119879(119870

1198921) is considered go to Step 4 else

go to Step 5 Determine a structural conflictto be controlledStep 4 forall119905

119894isin Ω2 replace the function 119878(119905

119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6


Step 5forall119905119894isin Ω1 replace the function 119878(119905

119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6

Step 6 design the the dependency relation119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]

Step 7 output 119878(119905119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]

When a CDC is detected its structure is known [18]According to Figure 16 any CDC contains two general con-flicts in which one is focused to control Hence Algorithm 16designs rules to add external control events such that thefocused general conflict can be resolved before the otherAlgorithm 16 achieves this purpose First the partition of thetransitions in a CDC (119873119872) is performed Let us considerthe structural conflict 119870

1198921 The information loss of the

corresponding general conflict K1198721198921

occurs if the transitionsin Ω2(Ω2

sube 119879(1198701198921)) fire before the resolution of K119872

1198921

On the contrary the information loss of K1198721198921

does notoccur if K119872

1198921is resolved before the transitions in Ω

2fire

Hence Algorithm 16 designs a dependency relation betweenthe event controlling the transitions in 119879(119870

1198921) and that

controlling the transitions in Ω2 which also assigns the

allowable enabling degrees of the transitions in 119879(1198701198921) to

resolve the general conflictK1198721198921 Finally the output functions

and the dependency relation can control the CDC such thatthe information loss ofK119872

1198921does not occur

The 119862de-conf 119862de(1198733 119879(1198703)1198721D1198721) in Example 3 is

also illustrated in Figure 15(c) where 1198703

= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870

3is focused only By the analysis in

Section 2K11987213

= ⟨1199011 1199051 1199052 21199011+ 31199012⟩ is a general conflict

with the maximal information content 120595(K11987213) = 3 at the

initial submarking 1198721 The decisions on K

1198721

3are made

However 1199051and 1199053are concurrent The information loss of

K1198721

3does not occur unless transition 119905

3fires after the general

conflictK11987213

is resolved The control processes for this CDCcontain the following steps according to Algorithm 16

(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901

[1205723]are obtained Two ECRSs 120590

1= [11990511199052] and 120590

2=

[(1199051)21199053] are obtained by Algorithm 14 Suppose that

ECRS 1205902is selectedThree allowable enabling degrees

1205721= 2 120572

2= 0 and 120572

3= 1 are assigned (Step 1)

Suppose that 119895 = 1 and 119896 = 2(2) Five sets 119879(119870

1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ

2= 1199053 are obtained (Step 2)

(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in

Section 2 is selected (Step 3)(4) The function 119878(119905

3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]

(Step 4)(5) The dependency relation 119890

1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902


Finally the controlled CDC is obtained as shown inFigure 15(d) where only transition sequence120590 = [(119905

1)2]1199053can

fireK11987213

can be resolved before 1199053fires

The computational complexity of Algorithms 14 15 and16 is analyzed First the reachability graph of a synchronizedPN to compute ECRSs is necessary which is the main costin Algorithms 14 15 and 16 However the reachability graphfor a general conflict is only a small part of that of the wholePN which is necessary to find all synchronized resolutionsequences Second in Algorithms 15 and 16 finding thestructural conflicts in a confusion (119873119872) and partitioningthe transitions in 119879 are involved For the former structuralconflicts can be found by scanning the incidence matrix [119873]which can be completed in 119874(119898119899) time where 119898 and 119899 arethe numbers of places and transitions in119873 respectively Forthe latter the partitions follow the basic operations of setswhich can be completed in 119874(1198992) time

Third adding and outputting external events of transi-tions in Algorithms 15 and 16 can be completed in119874(1198992) timeThe computational complexity of adding external eventsto the transitions in 119873 is considered According to theconstructions in Figure 12 only 5 elements (two transitionsand three places) are needed for an external event to controla transition in119873 In the worst case we need 119899 external eventsto control totally 119899 transitions in 119873 In other words 5 sdot 119899elements in total are added by adding rows and columns inthe incidence matrix [119873] which can be completed in 119874(1198992)time since the number of rows and columns added has linearrelation with the number 119899 of transitions

Finally Algorithm 15 or Algorithm 16 outputs an eventdependency relation for the two external events in a con-fusion subnet According to the constructions in Figure 12only 5 elements (two transitions and three places) are usedto represent the event dependency relation The number ofelements added for the event dependency relation has norelation with the numbers119898 of places and 119899 transitions in119873Hence adding an event dependency relation in Algorithm 15or Algorithm 16 can be completed in 119874(1) time

The control strategies (Algorithms 14 15 and 16) ofconfusions are based on synchronized PNs in which anytransition is controlled by an external event The strategiesare applied to generalized PNs It is unnecessary to controlthe transitions not in confusions by external events Hence aclass of local synchronized PNs (LSPNs) is proposed

Definition 17 A marked LSPN is a seven-tuple 119873119897

=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904

= 0 are the sets of nonsynchronized and synchronizedtransitions resp) are finite nonempty and disjoint sets 119865 sube

(119875 times 119879) cup (119879 times 119875) is called a flow relation of the net 119882

(119875 times 119879) cup (119879 times 119875) rarr N is a mapping that assigns a weight toan arc119882(119909 119910) gt 0 if (119909 119910) isin 119865 and119882(119909 119910) = 0 otherwisewhere 119909 119910 isin 119875 cup 119879 119864 = 119890

1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)

Figure 17 (a) A flexible assembly cell and (b) its Petri net model (1198731198720)

external events 119878 is a function defined from 119879119904to 119864119872

0is an

initial marking

As stated 119873 = (119875 119879 119865119882) is a generalized PN and119873119897is

its corresponding LSPN It means [119873] = [119873119897] forall119905 isin 119879

119904 119905 is


in119873119897and forall119905 isin 119879

119899 119905 is a

nonsynchronized transition in 119873119897 In other words the tran-

sitions in 119879119899are not controlled by any external event whose

firing coincides with those in generalized PNs 119879119899

= 0 (resp119879119904

= 0) means that there necessarily exists at least a nonsyn-chronized (resp synchronized) transition in an LSPN

The method of confusion control in a PN is mapping thePN into an LSPN such that the behavior of confusions cannotoccur which contains the following two steps

(1) If a CIC (119873119872) is known in a PN Algorithm 15 isperformed with the input (119873119872) If a CDC (11987310158401198721015840)is known in a PN Algorithm 16 is performed with theinput (11987310158401198721015840) Then for any transition 119905

119894in confu-

sions we obtain a function 119878(119905119894) In other words the

external events that control confusions are obtained(2) The obtained external events are applied to the LSPN

of the PN by assigning the set119864 of external events andthe function 119878

33 Control of Confusions in LSPNs An Example An auto-mated manufacturing system (AMS) is taken as an exampleto demonstrate the proposed confusion control methods AnAMS is composed of machining tools buffers conveyersand robots Several raw and semifinished parts are manufac-tured through predefined routes The realization of resource

allocation depends on conflicts and concurrency Henceconfusions may appear in an AMS which can lead to the lossof decision-making information

A flexible assembly cell is shown in Figure 17(a) Thedescription of the operations to be performed in the cell isstated as follows Robot R1 loads raw Part 1 to machine toolM1 After being processed bymachine toolM1 a semifinishedPart 1 is produced and stored in buffer B1 The semifinishedPart 1 is loaded into machine tool M2 when the part is movedby robot R1 from conveyors C1 to C3 After this stage a fin-ished Part 1 is obtained Correspondingly machine tools M1M2 and robotR1 can be considered as shared resources whereM1 (resp M2) can process two (resp three) raw parts simul-taneously Part 2 is also produced by them First machinetool M1 loads raw Part 2The produced semifinished Part 2 isstored in buffer B2Then robot R1 moves semifinished Part 2from conveyors C2 to C4 At this moment the semifinishedPart 2 is available and is loaded bymachine toolM2 After thisstage a finished Part 2 is obtained The PNmodel (119873119872

0) of

the cell is shown in Figure 17(b) and the detailed descriptionsare shown in Table 1 The net is bounded and live

Two confusions exist in the PN which are shown inFigures 18(a) and 18(c) The marked subnet (119873

11198721) with

1198721= 31199010+1199016+21199018+211990112+311990113depicted in Figure 18(a) is a

CDC It contains two structural conflicts 1198701= ⟨11990112 1199050 1199056⟩

and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905

0)2] can be used to make

decisions Firing 1205901implies that a raw Part 1 and a raw Part

2 are processed by machine tool M1 In other words twoprocesses for the production of two different finished parts


Table 1 Descriptions of the system model shown in Figure 17(b)

Element Meanings

1199010

Raw Part 1 is available and conveyor belt C1requests robot R1

1199011 Machine tool M1 is working1199012 Semifinished Part 1 is stored in buffer B11199013 Robot R1 is working1199014 Semifinished Part 1 is available1199015 Machine tool M2 is working1199016 Raw Part 2 is available1199017 Machine tool M1 is working

1199018

Semifinished Part 2 is stored in buffer B2 andconveyor belt C2 requests robot R1

1199019 Robot R1 is working11990110 Semifinished Part 2 is available11990111 Machine tool M2 is working11990112 Machine tool M1 is available11990113 Robot R1 is available11990114 Machine tool M2 is available

1199050

Robot R1 responds to conveyor belt C1 andmachine tool M1 begins working

1199051 The working of machine tool M1 is finished

1199052

Move semifinished Part 1 with robot R1 fromconveyor belts C1 to C3

1199053 Conveyor belt C1 releases robot R11199054 Machine tool M2 begins working

1199055

The working of machine tool M2 is finished andthe finished Part 1 is available

1199056 Machine tool M1 begins working1199057 The working of machine tool M1 is finished

1199058

Robot R1 responds to conveyor belt C2 and movessemifinished Part 2 from conveyor belts C2 to C4


11990511


start simultaneouslyThefiring of1205902means that two rawParts

2 are processed by machine tool M1 Finished Parts 2 areproduced only and M1 can process two parts at a momentAs stated previously two ECRSs 120590

1= [11990501199056] and 120590

2= [(1199050)2]

can be used to make decisions However transitions 1199056and 1199058

are concurrent If 1199058fires before 119905

6 two tokens in place 119901

13are

removed Consequently 1199050becomes 1-enabled The decision

for 1205902= [(1199050)2] cannot be performed In other words the


occursFor this assembly cell suppose that both production

processes are expected to work The decision on 1205901is made

By Algorithm 16with the input (11987311198721) we can obtain three

functions that is 119878(1199050) = 1198901

[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]

and the event dependency relation 1198901[1205721198921]

≫ 1198902

[1205721198922]

Themarked subnet (11987331198723)with119872

3= 31199014+1199019+211990110+

311990114shown in Figure 18(c) is a CIC that contains a structural

conflict 1198703= ⟨11990114 1199054 11990510⟩ The general conflict K1198723

3=

⟨11990114 1199054 119905101198723⟩with the information content 120595(K1198723

3) = 4

occurs at submarking 1198723 which is not the general conflict

with themaximal information content in theCIC Transitions1199054and 1199059are concurrent at119872

3 If 1199059fires we obtain another

general conflictK11987243

= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5

at submarking1198724= 31199014+ 311990110+ 311990114 The general conflict

needs to be resolved since 120595(K11987243) = 5 is the maximal

information content in the evolution of the CICTwo ECRSs 120590

3= [119905411990510] and 120590

4= [(11990510)3] can be used to

make decisions If 1205903is selected to fire semifinished Parts 1

and 2 are processed by machine tool M2 simultaneously Thedecision coincides with 120590

1= [11990501199056] of the general conflict

K1198721

1in (119873

11198721) which makes both production processes

work If 1205904is selected to fire only semifinished Parts 2 can be

processed by machine tool M2Suppose that the ECRS 120590

3= [119905411990510] is the decision to

be made By Algorithm 15 with the input (11987331198723) we can

obtain three functions that is 119878(1199054) = 1198904

[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4

[1] and the event dependency relation 119890

3

[1205721198923]≫

1198904

[1205721198924]All the obtained external events and the event depen-

dency relations are added to the original PN to controlthe two confusions (119873

11198721) and (119873

31198723) Then an LSPN

without the behavior of confusions is finally obtained where119879119899= 1199051 1199052 1199053 1199055 1199057 11990511 119879119904= 1199050 1199054 1199056 1199058 1199059 11990510 and 119864 =

1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924

Figure 18(b) shows a subnet1198732of a CIC obtained by the

detection algorithms in the next section In the subnet theinformation loss of the general conflicts associated with119870

3=

⟨11990114 1199054 11990510⟩ does not occur Hence confusion control is not

applied to1198732

In this paper the integrity of decision-making infor-mation of a general conflict is ensured such that a systemdesigner can make a decision based on all ECRSs In areal-world system the decisions on ECRSs are made bycontrol engineers If two inappropriate ECRSs are selectedthe system may lead to deadlocks For example in theconsidered assembly cell 120590

1= [11990501199056] and 120590

4= [(119905

10)3]

are ECRSs for general conflicts K11987211

and K1198724

3 respectively

If K11987211

and K1198724

3are resolved according to ECRSs 120590

1and

1205904 respectively the system will be in a deadlock state since

transition 1199054becomes disabled by the control of 1198904

[0] Con-

secutively transition 1199050becomes disabled Then transition 119905

6

is not allowed to fire although it is enabled since 1199050and 1199056

are synchronized by the control of 1198901[1] The problem can be

solved by classifying ECRSs according to the relations amongconflicts They will be considered in our future work

On the other hand the proposed control methods canbe applied to industrial cases since any external event canbe designed as a controller according to a feedback control


2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)

Figure 18 (a) A CDC (11987311198721) and its control (b) the structure of a CIC119873

2 and (c) a CIC (119873

31198723) and its control

structure First a monitor is designed to observe whetherthe enabling degree of a transition is equal to the allowableenabling degree Then a controller is designed as a triggerthat can output control information to transitions Theevent dependency relations can be performed by designingcounters

4 Conclusions and Future Work

This paper investigates confusion issues in generalized PNsTwo problems are consideredThe first is the formalization ofconfusions in generalized PNs Confusions are well definedby proposing a novel concept of information content BothCICs and CDCs in generalized PNs can be well described bythe proposed definition

The second is the confusion control Synchronized PNswith allowable enabling degrees and LSPNs are developed tocontrol two classes of confusions The work can be regardedas an effectivemethod to control Petri nets by using Petri netsFirst the concept of ECRSs of a general conflict is proposedEach ECRS can deduce a series of control information ofexternal events Second two algorithms are proposed to auto-matically generate external events and the event dependencyrelations which can be successfully applied to the controlof confusions such that the information loss of the generalconflicts in the confusions does not occur Finally an exampleof confusion detection and control in the PN model of amanufacturing cell is given Consequently confusions in thePN can be effectively detected and controlled by using theproposed methods

Futurework includes the control of confusions in general-ized PNs by an algebraic method On the other hand we alsoaim to detect confusions in a PN without computing a reach-ability graphThemethods can involve finding markings thatare prone to the information loss of general conflicts by stateestimation and structural analysis methods

Conflict of Interests

The first author and any of the coauthors do not have anyconflict of interests regarding this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grant nos 61374068 and61304051 the Fundamental Research Funds for the CentralUniversities under Grant no JB150404 the FundamentalResearch Funds for Science and Technology Departmentof Sichuan Province under Grant no 2013JY0089 the Fun-damental Research Funds for Educational Department ofSichuan Province under Grant no 08ZA029 the Founda-tion of Sichuan Educational Committee under Grant no15ZB0134 the Key Scientific Research Fund of Xihua Uni-versity under Grant no z1412616 the Preeminent YouthFund for School of Computer and Software Engineering inXihua University the Science and Technology DevelopmentFund MSAR under Grant no 0662013A2 and the NSTIPstrategic technologies Program no 12-ELE2506-02 in thekingdom of Saudi Arabia

References

[1] Y F Chen and Z W Li Optimal Supervisory Control of Auto-mated Manufacturing Systems CRC Press Taylor amp FrancisNew York NY USA 2013

[2] Y Chen Z Li and M C Zhou ldquoOptimal supervisory controlof flexible manufacturing systems by petri nets a set classifica-tion approachrdquo IEEE Transactions on Automation Science andEngineering vol 11 no 2 pp 549ndash563 2014

[3] Z W Li G Y Liu H-M Hanisch and M C Zhou ldquoDeadlockprevention based on structure reuse of petri net supervisors forflexible manufacturing systemsrdquo IEEE Transactions on SystemsMan and Cybernetics Part A Systems vol 42 no 1 pp 178ndash1912012


[4] W-Z Jia B-H Mao T-K Ho H-D Liu and B Yang ldquoBot-tlenecks detection of track allocation schemes at rail stations byPetri netsrdquo Journal of Transportation Systems Engineering andInformation Technology vol 9 no 6 pp 136ndash141 2009

[5] V Calderaro C NHadjicostis A Piccolo and P Siano ldquoFailureidentification in smart grids based on Petri net modelingrdquo IEEETransactions on Industrial Electronics vol 58 no 10 pp 4613ndash4623 2011

[6] J F Zhang M Khalgui Z W Li G Frey O Mosbahi and HB Salah ldquoReconfigurable coordination of distributed discreteevent control systemsrdquo IEEE Transactions on Control SystemsTechnology vol 23 no 1 pp 323ndash330 2015

[7] Y F Chen Z W Li and K Barkaoui ldquoNew Petri net structureand its application to optimal supervisory control intervalinhibitor arcsrdquo IEEE Transactions on Systems Man and Cyber-netics Systems vol 44 no 10 pp 1384ndash1400 2014

[8] G Rozenberg and P Thiagarajan ldquoPetri nets basic notionsstructure behaviourrdquo in Current Trends in ConcurrencyOverviews and Tutorials J de Bakker W de Roever and GRozenberg Eds vol 224 of Lecture Notes in Computer Sciencepp 585ndash668 Springer Berlin Germany 1986

[9] G Rozenberg and J Engelfriet ldquoElementary net systemsrdquoin Lectures on Petri Nets I Basic Models W Reisig and GRozenberg Eds vol 1491 of Lecture Notes in Computer Sciencepp 12ndash121 Springer Berlin Germany 1998

[10] W van der Aalst and K Hee Workflow Management ModelsMethods and Systems MIT Press Cambridge UK 2004

[11] W van der Aalst ldquoWorkflow verification finding control-flowerrors using Petri-net-based techniquesrdquo in Business ProcessManagement Models Techniques and Empirical Studies Wvan der Aalst J Desel and A Oberweis Eds vol 1806 ofLecture Notes in Computer Science pp 19ndash128 Springer BerlinGermany 2000

[12] W van der Aalst ldquoVerification of workflow netsrdquo in Applicationand Theory of Petri Nets P Azma and G Balbo Eds vol 1248of Lecture Notes in Computer Science pp 407ndash426 SpringerBerlin Germany 1997

[13] X L Chen Z Y Jiang and J H Ye ldquoConfusion analysis anddetection for workflow netsrdquo Discrete Dynamics in Nature andSociety vol 2014 Article ID 825313 14 pages 2014

[14] E Smith ldquoOn the border of causality contact and confusionrdquoTheoretical Computer Science vol 153 no 1-2 pp 245ndash270 1996

[15] S Haar ldquoClusters confusion and unfoldingsrdquo FundamentaInformaticae vol 47 no 3-4 pp 259ndash270 2001

[16] C Bolton ldquoAdding conflict and confusion to CSPrdquo in Pro-ceedings of the International Conference on Formal Methods JFitzgerald J Hayes and A Tarlecki Eds vol 3582 of LectureNotes in Computer Science pp 205ndash220 Springer BerlinGermany 2005

[17] C Bolton ldquoCapturing conflict and confusion in CSPrdquo inIntegrated Formal Methods 6th International Conference IFM2007 Oxford UK July 2ndash5 2007 Proceedings J Davies and JGibbons Eds vol 4591 of Lecture Notes in Computer Sciencepp 413ndash438 Springer Berlin Germany 2007

[18] X-L Chen Z-W Li A M Al-Ahmari A M El-Tamimi andE S Abouel Nasr ldquoConfusion diagnosis and control of discreteevent systems using synchronized Petri netsrdquo Asian Journal ofControl vol 15 no 6 pp 1736ndash1751 2013

[19] X L Chen Z W Li N Q Wu A M Al-Ahmari A MEl-Tamimi and E S Abouel Nasr ldquoConfusion avoidance fordiscrete event systems by PE constraints and supervisory

controlrdquo IMA Journal of Mathematical Control and Information2014

[20] J P Katoen ldquoGSPNs revisited simple semantics and newanalysis algorithmsrdquo in Proceedings of the 2012 InternationalConference on Application of Concurrency to System Design pp6ndash11 Hamburg Germany 2012

[21] M A Marsan G Balbo G Conte S Donatelli and GFranceschinisModelling with Generalized Stochastic Petri NetsJohn Wiley amp Sons 1995

[22] Z W Li and M C Zhou ldquoElementary siphons of Petrinets and their application to deadlock prevention in flexiblemanufacturing systemsrdquo IEEE Transactions on Systems Manand Cybernetics Part ASystems and Humans vol 34 no 1 pp38ndash51 2004

[23] Z Y Ma Z W Li and A Giua ldquoDesign of optimal Petrinet controllers for disjunctive generalized mutual exclusionconstraintsrdquo IEEE Transactions on Automatic Control vol 60no 7 pp 1774ndash1785 2015

[24] M Moalla J Pulou and J Sifakis ldquoSynchronized petri nets a model for the description of non-autonomous sytemsrdquo inMathematical Foundations of Computer Science 1978 Proceed-ings 7th Symposium Zakopane Poland September 4ndash8 1978 JWinkowski Ed vol 64 of Lecture Notes in Computer Sciencepp 374ndash384 Springer Berlin Germany 1978

[25] G SerugendoDMandrioli D Buchs andNGuelfi ldquoReal-timesynchronised Petri netsrdquo in Proceedings of the 23th InternationalConference on Application and Theory of Petri Nets J Esparzaand C Lakos Eds vol 2360 of Lecture Notes in ComputerScience pp 281ndash298 Springer Berlin Germany 2002

[26] R David and H Alla Discrete Continuous and Hybrid PetriNets Springer Berlin Germany 2005

[27] R David and H AllaDu Grafcet Aux Reseaux de Petri HermesPublication Paris France 2nd edition 1992

[28] J L Peterson Petri Net Theory and the Modeling of SystemsPrentice-Hall Englewood Cliffs NJ USA 1981

[29] C Ramchandani Analysis of asynchronous concurrent systemsby timed Petri nets [PhD thesis] MIT Cambridge Mass USA1973

[30] G Florin and S Natkin ldquoLes reseaux de Petri stochastiquesrdquoTechnique et Science Informatiques vol 4 no 1 pp 143ndash1601985

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Update


Fault alarm

Recovery Ignore

p3

p5

p4

p1

p6

p2

(a)

Update


Fault alarm

Recovery Ignore

p3

p5

p4

p1

p6

p2

(b)

Figure 1 A simplified model of fault recovery (a) the model at marking119872 = 1199011+ 1199012and (b) the model at marking1198721015840 = 119901

1+ 1199014

and ldquoRecoveryrdquo also leads to the loss of ldquoRecoveryrdquo sinceldquoRecoveryrdquo cannot fire when the system concurrently firesldquoIgnorerdquo and ldquoUpdaterdquo Obviously a system with confusionscan cause ambiguous behavior and difficulties in systemanalysis Solving the problem of confusion diagnosis meansthat each detected confusion should be controlled by areasonable strategy Hence this study focuses on detectionand control of confusions in PNs

Confusion problems in PNs are first investigated in [89] Several PN applications in a number of areas such asworkflow nets (WF-nets) [10ndash13] unfolding nets [14 15] safemarked ordinary nets [16ndash19] generalized (unsafe) nets [17]and generalized stochastic Petri nets (GSPNs) [20 21] involveconfusions In this section a brief survey of the existingstudies on confusions is presented

van der Aalst et al take into account the existence ofconfusions inWF-nets and show the defects ofWF-nets withconfusions [10ndash12] First the existence of confusions in aWF-net leads to a nondeterministic workflow process such that itscorrectness cannot be ensured AWF-net is said to be correctif it satisfies the two properties ldquosoundnessrdquo and being ldquowell-structuredrdquo which is reported by van der Aalst [12] Everyreachable marking of a WF-net can terminate properly if it issound AWF-net that is well-structured has a number of niceproperties For example the soundness of a WF-net can beverified in polynomial time and a sound well-structuredWF-net is safe However the correctness depends on the existenceof confusions in a WF-net If an acyclic WF-net (no directedcycle in the structure of a WF-net) contains a confusion itis certainly not sound and well-structured If a WF-net withcycles contains a confusion it may not be sound and well-structured Hence we cannot determine the correctness of aWF-net if it contains confusions

The study on occurrence nets lies in that the behaviorof PNs can be interpreted by branching unfolding semantics[15] In PNs the behavior such as sequences concurrencyconflicts trails choices and alternatives can be described andanalyzed by decomposing an occurrence net into substruc-tures given by the node relations associatedwith the behavior

However confusions cannot be described by the existingbranching semantics Hence Smith and Haar consider theindependence of events in occurrence nets and the indirectinfluences among concurrent events in [14 15] Furthermoreinterference structural conflict clusters are developed in [15]in order to describe confusions which belong to a kind ofthe substructures of occurrence nets In the work of Smithand Haar confusion detection and control problems are notconsidered

The work of Bolton [16 17] involves the detection ofconfusions in both safe and unsafe nets Truly concurrentsemantics and interleaving semantics for concurrent systemsare discussed which are useful to formalize and illustrateconfusions The main difference between the two kinds ofsemantics is that the synchronized firing of transitions andthe nondeterministic choice among the possible orders oftheir firing can be distinguished by using truly concurrentsemantics and not the interleaving semantics

For example if two transitions 1199051and 1199052are in a PN con-

currently fire three transition firing sequences are possiblewhich are denoted by 120590

1= [11990511199052] 1205902= 11990511199052 and 120590

3= 11990521199051

respectivelyThe physical interpretation of 1205901in a real system

is that the two events (represented by transitions 1199051and 1199052)

fire simultaneously In this case the firing of 1199051and 119905

2is

said to be synchronized On the other hand firing transitionsequences 120590

2= 11990511199052and 120590

3= 11990521199051means that 119905

1and 1199052

fire sequentially In [17] the definition of confusions in safePNs is extended to unsafe generalized PNs which dependson the inclusion relations of transition sets of conflicts at twomarkings However confusions in generalized PNs cannotbe completely described by the definition in [17] since thetransition sets of conflicts may remain unchanged althougha confusion occurs in a generalized PN

The approaches for confusion detection in [17] are basedon trace theory Furthermore Communicating SequentialProcess (CSP) model-checker is used to detect confusionsThe WF-net of a business procedure is constructed to depictthe logical relations among business tasks and to detectpotential faults in the procedure Confusionsmay occurwhen













21 Basics of PNs





+minus [119873]

minus where [119873]+ with [119873]+(119901 119905) = 119882(119905 119901)

and [119873]minus with [119873]









4in a net


p3p1 p2

t1 t2 t3 t4

(a)

p3p1 p2

t1 t2 t3 t4

(b)


2 1199051 1199052 1199053 21199011+ 21199012⟩



2+ 31199014

instead of (0 3 0 2)119879



cup119909isin119883

∙119909 and119883∙ = cup

119909isin119883119909∙




11199052 119905119899and markings 119872

11198722 and 119872

119899minus1such

that 119872[1199051⟩1198721[1199052⟩1198722 119872




0if forall119872 isin 119877(119873119872

0) exist1198721015840 isin

119877(119873119872)1198721015840[119905⟩ holds










1 1199052 119905





denoted by 119870119872



119877(1198731198720) such that forall119905

119894isin 119879(119870

119872) 119872[119905


119872(119901) lt sum119905119894isin119879(119870

119872)119882(119901 119905





2 1199051 1199052



1 1199052 1199053 1199054 and 119879(119870119872) = 119905

1 1199052 1199053



0) denoted


119902119872

119894le min

119872(119901)

119882(119901 119905119894)| 119901isin∙119905119894 lt 119902

119872

119894+ 1 (1)



119894)119902119872


119894 Note that if 119905






119894at marking 119872 Let 119872

119894be



1

1199052 and 119905


0) with |119879| = 3



119879



119877(1198731198720) such that forall119905

119894isin 119879(K119872)119872[119905

119894⟩ is true and119872(119901) lt

sum119905119894isin119879(K

119872)119902119872

119894sdot 119882(119901 119905






119870 = ⟨1199012 1199051 1199052 1199053 1199054⟩ at marking119872 = 2119901



= 2 1199021198722

= 3 and 1199021198723


1 1199052 and 119905



119872(1199012) = 3

lt 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) + 119902119872

3

sdot 119882 (1199012 1199053) = 8

(2)


p3p1 p2

t1 t2 t3 t4






2 1199051 1199052 1199053 21199011+31199012⟩


119872(1199012) = 119882(119901

2 1199051) + 119882(119901

2 1199052) + 119882(119901

2 1199053) = 3 (3)


0) and


119888)





120595 (K119872) = 120572 sdot sum

119905119894isin119879(119870)

119902119872

119894sdot 119882 (119901 119905

119894) (4)

where

120572 =

110038161003816100381610038161003816119864119872(119879 (119870))

10038161003816100381610038161003816ge 2

0 otherwise(5)

and forall119905119894isin 119879(119870) 119902119872


119894at



0)


1015840


0) such that 120595(K119872

1015840

) gt 120595(K119872)Let 120590 = (119905)


1 1199052 119905

119899]


1 1199052 and 119905




1015840

= ⟨1199012


1+21199012


we have

120595 (K119872) = 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) = 3 (6)


1205902= 11990521199051 1205903= (1199052)2 and 120590

4= [11990511199052] can be observed


we have

120595(K1198721015840

) = 1199021198721015840

1sdot 119882 (119901

2 1199051) + 1199021198721015840

2sdot 119882 (119901

2 1199052) = 5 (7)

The 120595(K1198721015840



1= (1199051)21199052

1205902

= 119905111990521199051 1205903

= 1199052(1199051)2 1205904

= 1199051(1199052)2 1205905

= 119905211990511199052

1205906= (1199052)21199051 1205907= (1199052)3 1205908= [1199051(1199052)2] and 120590

9= [(1199051)21199052]



1015840


) =





2⟩

and (∙1199051cup 119905∙

1) cap (∙1199052cup 119905∙

2) = 0

(2) Let119863119872 = 1199051 1199052 119905








= 119863119872 with = |119863

119872| in (119873119872) such that gt 120582 A


(5) LetD119872 = 119863119872

1 119863119872

2 119863

119872



isin D119872 119863119872119895


119895isin D119872 119863119872


in (119873119872)


1+ 1199014+ 21199015 that

is 1198631198721= 1199051 1199055 1198631198722= 1199051 1199056 1198631198723= 1199052 1199055 1198631198724= 1199052 1199056

119863119872

5= 1199055 11990561198631198726= 1199051 1199055 1199056 and119863119872

7= 1199052 1199055 1199056 Accord-


and 1198631198727

with 1205826= 1205827= 3 are 3-


119872


three Let D1198721

= 119863119872

6 119863119872

7 and D119872

2= 119863

119872

1 119863119872

2 119863119872

6 be


p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)


1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9





two sets1198631198721and119863119872





1and


119899








1and 1199052are 1- and 2-


1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are




119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840


119888



1 1199012 119901


1199011198921 1199011198922 119901




1015840


119895) = 0


1ndash1199054at marking119872 = (1 4 1 0)


1and 119901

2and two



submarking119872 = (1 4)119879


= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)



0) such that




119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840





120595(K1198721015840


120595(K1198721015840

)



2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















21 Basics of PNs





+minus [119873]

minus where [119873]+ with [119873]+(119901 119905) = 119882(119905 119901)

and [119873]minus with [119873]









4in a net


p3p1 p2

t1 t2 t3 t4

(a)

p3p1 p2

t1 t2 t3 t4

(b)


2 1199051 1199052 1199053 21199011+ 21199012⟩



2+ 31199014

instead of (0 3 0 2)119879



cup119909isin119883

∙119909 and119883∙ = cup

119909isin119883119909∙




11199052 119905119899and markings 119872

11198722 and 119872

119899minus1such

that 119872[1199051⟩1198721[1199052⟩1198722 119872




0if forall119872 isin 119877(119873119872

0) exist1198721015840 isin

119877(119873119872)1198721015840[119905⟩ holds










1 1199052 119905





denoted by 119870119872



119877(1198731198720) such that forall119905

119894isin 119879(119870

119872) 119872[119905


119872(119901) lt sum119905119894isin119879(119870

119872)119882(119901 119905





2 1199051 1199052



1 1199052 1199053 1199054 and 119879(119870119872) = 119905

1 1199052 1199053



0) denoted


119902119872

119894le min

119872(119901)

119882(119901 119905119894)| 119901isin∙119905119894 lt 119902

119872

119894+ 1 (1)



119894)119902119872


119894 Note that if 119905






119894at marking 119872 Let 119872

119894be



1

1199052 and 119905


0) with |119879| = 3



119879



119877(1198731198720) such that forall119905

119894isin 119879(K119872)119872[119905

119894⟩ is true and119872(119901) lt

sum119905119894isin119879(K

119872)119902119872

119894sdot 119882(119901 119905






119870 = ⟨1199012 1199051 1199052 1199053 1199054⟩ at marking119872 = 2119901



= 2 1199021198722

= 3 and 1199021198723


1 1199052 and 119905



119872(1199012) = 3

lt 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) + 119902119872

3

sdot 119882 (1199012 1199053) = 8

(2)


p3p1 p2

t1 t2 t3 t4






2 1199051 1199052 1199053 21199011+31199012⟩


119872(1199012) = 119882(119901

2 1199051) + 119882(119901

2 1199052) + 119882(119901

2 1199053) = 3 (3)


0) and


119888)





120595 (K119872) = 120572 sdot sum

119905119894isin119879(119870)

119902119872

119894sdot 119882 (119901 119905

119894) (4)

where

120572 =

110038161003816100381610038161003816119864119872(119879 (119870))

10038161003816100381610038161003816ge 2

0 otherwise(5)

and forall119905119894isin 119879(119870) 119902119872


119894at



0)


1015840


0) such that 120595(K119872

1015840

) gt 120595(K119872)Let 120590 = (119905)


1 1199052 119905

119899]


1 1199052 and 119905




1015840

= ⟨1199012


1+21199012


we have

120595 (K119872) = 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) = 3 (6)


1205902= 11990521199051 1205903= (1199052)2 and 120590

4= [11990511199052] can be observed


we have

120595(K1198721015840

) = 1199021198721015840

1sdot 119882 (119901

2 1199051) + 1199021198721015840

2sdot 119882 (119901

2 1199052) = 5 (7)

The 120595(K1198721015840



1= (1199051)21199052

1205902

= 119905111990521199051 1205903

= 1199052(1199051)2 1205904

= 1199051(1199052)2 1205905

= 119905211990511199052

1205906= (1199052)21199051 1205907= (1199052)3 1205908= [1199051(1199052)2] and 120590

9= [(1199051)21199052]



1015840


) =





2⟩

and (∙1199051cup 119905∙

1) cap (∙1199052cup 119905∙

2) = 0

(2) Let119863119872 = 1199051 1199052 119905








= 119863119872 with = |119863

119872| in (119873119872) such that gt 120582 A


(5) LetD119872 = 119863119872

1 119863119872

2 119863

119872



isin D119872 119863119872119895


119895isin D119872 119863119872


in (119873119872)


1+ 1199014+ 21199015 that

is 1198631198721= 1199051 1199055 1198631198722= 1199051 1199056 1198631198723= 1199052 1199055 1198631198724= 1199052 1199056

119863119872

5= 1199055 11990561198631198726= 1199051 1199055 1199056 and119863119872

7= 1199052 1199055 1199056 Accord-


and 1198631198727

with 1205826= 1205827= 3 are 3-


119872


three Let D1198721

= 119863119872

6 119863119872

7 and D119872

2= 119863

119872

1 119863119872

2 119863119872

6 be


p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)


1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9





two sets1198631198721and119863119872





1and


119899








1and 1199052are 1- and 2-


1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are




119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840


119888



1 1199012 119901


1199011198921 1199011198922 119901




1015840


119895) = 0


1ndash1199054at marking119872 = (1 4 1 0)


1and 119901

2and two



submarking119872 = (1 4)119879


= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)



0) such that




119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840





120595(K1198721015840


120595(K1198721015840

)



2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p3p1 p2

t1 t2 t3 t4

(a)

p3p1 p2

t1 t2 t3 t4

(b)


2 1199051 1199052 1199053 21199011+ 21199012⟩



2+ 31199014

instead of (0 3 0 2)119879



cup119909isin119883

∙119909 and119883∙ = cup

119909isin119883119909∙




11199052 119905119899and markings 119872

11198722 and 119872

119899minus1such

that 119872[1199051⟩1198721[1199052⟩1198722 119872




0if forall119872 isin 119877(119873119872

0) exist1198721015840 isin

119877(119873119872)1198721015840[119905⟩ holds










1 1199052 119905





denoted by 119870119872



119877(1198731198720) such that forall119905

119894isin 119879(119870

119872) 119872[119905


119872(119901) lt sum119905119894isin119879(119870

119872)119882(119901 119905





2 1199051 1199052



1 1199052 1199053 1199054 and 119879(119870119872) = 119905

1 1199052 1199053



0) denoted


119902119872

119894le min

119872(119901)

119882(119901 119905119894)| 119901isin∙119905119894 lt 119902

119872

119894+ 1 (1)



119894)119902119872


119894 Note that if 119905






119894at marking 119872 Let 119872

119894be



1

1199052 and 119905


0) with |119879| = 3



119879



119877(1198731198720) such that forall119905

119894isin 119879(K119872)119872[119905

119894⟩ is true and119872(119901) lt

sum119905119894isin119879(K

119872)119902119872

119894sdot 119882(119901 119905






119870 = ⟨1199012 1199051 1199052 1199053 1199054⟩ at marking119872 = 2119901



= 2 1199021198722

= 3 and 1199021198723


1 1199052 and 119905



119872(1199012) = 3

lt 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) + 119902119872

3

sdot 119882 (1199012 1199053) = 8

(2)


p3p1 p2

t1 t2 t3 t4






2 1199051 1199052 1199053 21199011+31199012⟩


119872(1199012) = 119882(119901

2 1199051) + 119882(119901

2 1199052) + 119882(119901

2 1199053) = 3 (3)


0) and


119888)





120595 (K119872) = 120572 sdot sum

119905119894isin119879(119870)

119902119872

119894sdot 119882 (119901 119905

119894) (4)

where

120572 =

110038161003816100381610038161003816119864119872(119879 (119870))

10038161003816100381610038161003816ge 2

0 otherwise(5)

and forall119905119894isin 119879(119870) 119902119872


119894at



0)


1015840


0) such that 120595(K119872

1015840

) gt 120595(K119872)Let 120590 = (119905)


1 1199052 119905

119899]


1 1199052 and 119905




1015840

= ⟨1199012


1+21199012


we have

120595 (K119872) = 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) = 3 (6)


1205902= 11990521199051 1205903= (1199052)2 and 120590

4= [11990511199052] can be observed


we have

120595(K1198721015840

) = 1199021198721015840

1sdot 119882 (119901

2 1199051) + 1199021198721015840

2sdot 119882 (119901

2 1199052) = 5 (7)

The 120595(K1198721015840



1= (1199051)21199052

1205902

= 119905111990521199051 1205903

= 1199052(1199051)2 1205904

= 1199051(1199052)2 1205905

= 119905211990511199052

1205906= (1199052)21199051 1205907= (1199052)3 1205908= [1199051(1199052)2] and 120590

9= [(1199051)21199052]



1015840


) =





2⟩

and (∙1199051cup 119905∙

1) cap (∙1199052cup 119905∙

2) = 0

(2) Let119863119872 = 1199051 1199052 119905








= 119863119872 with = |119863

119872| in (119873119872) such that gt 120582 A


(5) LetD119872 = 119863119872

1 119863119872

2 119863

119872



isin D119872 119863119872119895


119895isin D119872 119863119872


in (119873119872)


1+ 1199014+ 21199015 that

is 1198631198721= 1199051 1199055 1198631198722= 1199051 1199056 1198631198723= 1199052 1199055 1198631198724= 1199052 1199056

119863119872

5= 1199055 11990561198631198726= 1199051 1199055 1199056 and119863119872

7= 1199052 1199055 1199056 Accord-


and 1198631198727

with 1205826= 1205827= 3 are 3-


119872


three Let D1198721

= 119863119872

6 119863119872

7 and D119872

2= 119863

119872

1 119863119872

2 119863119872

6 be


p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)


1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9





two sets1198631198721and119863119872





1and


119899








1and 1199052are 1- and 2-


1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are




119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840


119888



1 1199012 119901


1199011198921 1199011198922 119901




1015840


119895) = 0


1ndash1199054at marking119872 = (1 4 1 0)


1and 119901

2and two



submarking119872 = (1 4)119879


= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)



0) such that




119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840





120595(K1198721015840


120595(K1198721015840

)



2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p3p1 p2

t1 t2 t3 t4






2 1199051 1199052 1199053 21199011+31199012⟩


119872(1199012) = 119882(119901

2 1199051) + 119882(119901

2 1199052) + 119882(119901

2 1199053) = 3 (3)


0) and


119888)





120595 (K119872) = 120572 sdot sum

119905119894isin119879(119870)

119902119872

119894sdot 119882 (119901 119905

119894) (4)

where

120572 =

110038161003816100381610038161003816119864119872(119879 (119870))

10038161003816100381610038161003816ge 2

0 otherwise(5)

and forall119905119894isin 119879(119870) 119902119872


119894at



0)


1015840


0) such that 120595(K119872

1015840

) gt 120595(K119872)Let 120590 = (119905)


1 1199052 119905

119899]


1 1199052 and 119905




1015840

= ⟨1199012


1+21199012


we have

120595 (K119872) = 119902119872

1sdot 119882 (119901

2 1199051) + 119902119872

2sdot 119882 (119901

2 1199052) = 3 (6)


1205902= 11990521199051 1205903= (1199052)2 and 120590

4= [11990511199052] can be observed


we have

120595(K1198721015840

) = 1199021198721015840

1sdot 119882 (119901

2 1199051) + 1199021198721015840

2sdot 119882 (119901

2 1199052) = 5 (7)

The 120595(K1198721015840



1= (1199051)21199052

1205902

= 119905111990521199051 1205903

= 1199052(1199051)2 1205904

= 1199051(1199052)2 1205905

= 119905211990511199052

1205906= (1199052)21199051 1205907= (1199052)3 1205908= [1199051(1199052)2] and 120590

9= [(1199051)21199052]



1015840


) =





2⟩

and (∙1199051cup 119905∙

1) cap (∙1199052cup 119905∙

2) = 0

(2) Let119863119872 = 1199051 1199052 119905








= 119863119872 with = |119863

119872| in (119873119872) such that gt 120582 A


(5) LetD119872 = 119863119872

1 119863119872

2 119863

119872



isin D119872 119863119872119895


119895isin D119872 119863119872


in (119873119872)


1+ 1199014+ 21199015 that

is 1198631198721= 1199051 1199055 1198631198722= 1199051 1199056 1198631198723= 1199052 1199055 1198631198724= 1199052 1199056

119863119872

5= 1199055 11990561198631198726= 1199051 1199055 1199056 and119863119872

7= 1199052 1199055 1199056 Accord-


and 1198631198727

with 1205826= 1205827= 3 are 3-


119872


three Let D1198721

= 119863119872

6 119863119872

7 and D119872

2= 119863

119872

1 119863119872

2 119863119872

6 be


p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)


1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9





two sets1198631198721and119863119872





1and


119899








1and 1199052are 1- and 2-


1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are




119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840


119888



1 1199012 119901


1199011198921 1199011198922 119901




1015840


119895) = 0


1ndash1199054at marking119872 = (1 4 1 0)


1and 119901

2and two



submarking119872 = (1 4)119879


= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)



0) such that




119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840





120595(K1198721015840


120595(K1198721015840

)



2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p1 p2

t1 t2

(a)

p1 p2

t1 t2

(b)


1015840

= ⟨1199012 1199051 1199052 21199011+ 31199012⟩ with

120595(K1198721015840

) = 5

p1 p2

t1 t2

p3

p4

t3 t4

p5

p6

t5 t6

p7 p8 p9





two sets1198631198721and119863119872





1and


119899








1and 1199052are 1- and 2-


1= 1199051(1199052)2

1205902= (1199052)21199051 1205903= 119905211990511199052 and 120590

4= [1199051(1199052)2] where 120590

1ndash1205903are




119872[1205901⟩1198721015840 119872[120590

2⟩1198721015840 119872[120590

3⟩1198721015840 and 119872[120590

4⟩1198721015840 where 1198721015840


119888



1 1199012 119901


1199011198921 1199011198922 119901




1015840


119895) = 0


1ndash1199054at marking119872 = (1 4 1 0)


1and 119901

2and two



submarking119872 = (1 4)119879


= (0 0 0 2)119879 with

119872[1199051⟩1198721[(1199054)2⟩1198721015840 the submarking 1198721015840 = (0 0)



0) such that




119894isin 119863119872

119895 119879(119870) and exist119896 isin 1 2 119902119872

119894

such that 119872[(119905119894)119896⟩1198721015840 and 120595(K119872) = 120595(K119872

1015840

)where 120595(K119872) and 120595(K119872

1015840





120595(K1198721015840


120595(K1198721015840

)



2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

2

p3

t1 t2

p2p1

p4

t3 t4

(a)

2

p2p1

t3 t4

(b)


119879








D1198721) where 119879(1198701) = 119905



11198721= 1199011+ 21199012+


1198721

1 is the set of 2-


= 1199051 1199053


1


119895is considered


K119872119895


3




10according to the



3 1198724 and 119872

5due to 120595(K

1198723

1) = 2

120595(K1198724

1) = 2 and 120595(K1198725



1=

⟨1199012 1199051 11990521198725⟩ at 119872






1and








D1198721) where 119879(1198702) = 119905

1 1199052 1199053 1199054 1198721

= 21199011+ 21199014

D1198721 = 1198631198721

1 1198631198721

2 11986311987211

= 1199051 1199055 and 119863

1198721

2= 1199052 1199055


and1198631198721




2=

⟨1199011 1199051 1199052 1199053 11990541198723⟩ at submarking119872

3= 21199011+ 21199012+ 21199013




1= 119905511990511199055or 1205902= 119905111990551199055fires



2with the






6such that the





1to1198726


3= ⟨1199011 1199051 1199052⟩ and 119870

4= ⟨1199012 1199052 1199053⟩



2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2t1 t2

p2 p3p1

p4

p5 p6

t3

(a)

M1 = p1 + 2p2 + 4p4

M2 = p1 + 2p2 + p3 + 3p4 M6 = p2 + 4p4 + p5

M3 = p1 + 2p2 + 2p3 + 2p4 M7 = p2 + p3 + 3p4 + p5

M4 = p1 + 2p2 + 3p3 + p4 M8 = p2 + 2p3 + 2p4 + p5

M5 = p1 + 2p2 + 4p3 M9 = p2 + 3p3 + p4 + p5

M10 = p2 + 4p3 + p5 120595(119974M101 ) = 0

120595(119974M11 ) = 0

120595(119974M91 ) = 0120595(119974M5

1 ) = 3

120595(119974M81 ) = 0120595(119974M4

1 ) = 2

120595(119974M71 ) = 0120595(119974M3

1 ) = 2

120595(119974M61 ) = 0120595(119974M2

1 ) = 0

t1

t1

t1

t3

t3

t3 t3

t1t3 t3

t1t3 t3

(b)


1and 1199053

22 2

p5 p6p8p7

p3

t1

t5

t2

p2p1

p4

t3 t4

(a)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

120595(119974M22 ) = 4 120595(119974M4

2 ) = 0

120595(119974M32 ) = 6 120595((119974M5

2 ) = 0

120595(119974M62 ) = 0

t1t5

t1 t5

t1t5 t5

M2 = 2p1 + p2 + p3 + p4 M4 = 2p4 + p5

M3 = 2p1 + 2p2 + 2p3 M5 = p2 + p3 + p4 + p5

M6 = 2p2 + 2p3 + p5

(b)

M1 = 2p1 + 2p4 120595(119974M12 ) = 4

t2

t5

t5

t2 t5

t2

t2

t2

t2

t5

t5

t5

M2 = 2p1 + p2 + p3 + p4 120595(119974M22 ) = 4

120595(119974M32 ) = 6

120595(119974M62 ) = 0

120595(119974M102 ) = 3 120595(119974M11

2 ) = 0

120595(119974M72 ) = 0

120595(119974M82 ) = 0

120595(119974M92 ) = 0

M7 = p1 + 2p4 + p6

M8 = p1 + p2 + p3 + p4 + p6

M3 = 2p1 + 2p2 + 2p3

M10 = p1 + 2p2 + 2p3 + p6 M11 = p2 + p3 + p4 + 2p6

M6 = 2p2 + 2p3 + p5

M9 = 2p4 + 2p6

(c)


1and


2and 1199055


2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2 2

p3

t1 t2

p2p1

p4 p5

t3

(a)

t1 t3

t1 t1t3

t1t3

120595(119974M13 ) = 3

120595(119974M23 ) = 2 120595(119974M4

3 ) = 0

120595(119974M33 ) = 0 120595(119974M5

3 ) = 0

120595(119974M63 ) = 0

M1 = 2p1 + 3p2

M2 = p1 + 3p2 + p3 M4 = 2p1 + p2 + p5

M3 = 3p2 + 2p3 M5 = p1 + p2 + p3 + p5

M6 = p2 + 2p3 + p5

(b)


1and 1199053



3) = 119905

1 1199052 1198721=

21199011+ 31199012 D1198721 = 1198631198721

1 and1198631198721

1= 1199051 1199053





1+ 31199012 that is

120595(K1198721



3=


3notin 119879(119870

3) can


1198721

3) = 3 to 120595(K

1198724








3) = 2 is reached



= ⟨1199011 1199051 11990521198722⟩ by firing

1199053


5= ⟨119901

1 1199051 1199052 1199053 1199054⟩ and 119870

6= ⟨119901

2


5in the confusion

119862de(1198734 119879(1198705)1198721D1198721) only where 119879(119870

5) = 119905

1 1199052 1199053 1199054

1198721= 21199011+1199012D1198721 = 1198631198721

1 1198631198721

211986311987211

= 1199051 1199055 and1198631198721

2=



1and 1199055(resp 119905



=


5reduces the


5) = 8 to 120595(K1198723

5) = 4 and


5)









22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










22 2

p1 p2

t1 t2

p3 p4

t3 t4

p5 p6

t5

p7

(a)

t1

t1

t5

t5

120595(119974M15 ) = 8

120595(119974M25 ) = 0 120595(119974M3

5 ) = 4

120595(119974M45 ) = 0

M1 = 2p1 + p2

M2 = p2 + p3 M3 = 2p1 + p7

M4 = p3 + p7

(b)

t2

t2 t2

t2

t5

t5

t5

120595(119974M15 ) = 8

120595(119974M55 ) = 0 120595(119974M3

5 ) = 4

120595(119974M65 ) = 0 120595(119974M7

5 ) = 0

120595(119974M85 ) = 0

M3 = 2p1 + p7

M1 = 2p1 + p2

M5 = p1 + p2 + p4

M6 = p2 + 2p4 M7 = p1 + p4 + p7

M8 = 2p4 + p7

(c)


1


2and 1199055





119894isin 119879 120572


119894 119864 =

1198901

[1205721198921] 1198902

[1205721198922] 119890

119899



0is

an initial marking


119894isin


with 120572119892119896= 120572119894and is






119894and 119905

119895be two



[120572119892119896]that can


119895to


119894and 119905



[120572119892119896]if two functions 119878(119905

119894) = 119890

119896

[120572119894]and

119878(119905119895) = 119890

119896

[120572119895]exist If 120572

119894= 2 and 120572




119894)2(119905119895)3] fires

Let 119890119894[120572119892119894]

and 119890119895[120572119892119895]



rely on that of 119890119894[120572119892119894]

is denoted by 119890119894[120572119892119894]

≫ 119890119895



and 119890119895


relation 119890119894[120572119892119894]

≫ 119890119895


1and 1199055



[2]and 119878(119905

5) = 1198901


2be controlled



[2]

implying that 1205721= 2 120572

2= 2 and 120572

5= 3 Let 119902119872

1= 2

119902119872

2= 3 and 119902119872





≫ 1198901



2)2]



p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p1

p2

t1

t2 p3

t3

(a)

0

3


1

2


2

1


1

1


0

1


0

0


3

0


2

0


1

0


0

2


t1

t1

t1

t1

t1

t1 t2

t2

t2t2t2

t2

t3 t3 t3

t3

t3

t3

M1

M2

M3 M4

M5

M6

M0

M7

M8

M9

(b)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[2]

(c)

2

1

0

2[1]

1[1]


0

1

2

0[1]

1[1]


1

1

1[1]

1[1]


M3

M0 M4

Q3

Q4Q0

[t1t2]e1[1205721198921] (t3)2e2[1205721198922]

[t1t2]e1[1205721198921] ((

(

(d)

p1

p2

t1

t2 p3

t3

e1[1]

e1[1]

e2[1]

e2[1205721198922] ≫ e1[1205721198921]

(e)

2

1

0

2[1]

1[1]


1

1

1

1[1]

1[1]


[t1t2]e1[1205721198921]

t3e2[1205721198922]

( ((f)





1015840


1199041198720)







119896 where






119894isin

119879119896119872[119905119894⟩ and 119902119872

119894ge 120572119894are true


[1205721198921]≫ sdot sdot sdot ≫ 119890

119896minus1

[120572119892119896minus1]≫ 119890119896

[120572119892119896]≫


[120572119892119899] a transition 119905



119894isin

119879119896119872[119905119894⟩ and 119902119872


1198721015840[120590⟩119872 and forall119905

119895isin 119879119896minus1

119905119895fires 120588 sdot 120572

119895times at the

marking1198721015840

In the rules 119902119872119894







1199041198720) and (1198731015840

1199041198720) with [119873] = [119873

119904] = [119873

1015840

119904] are


119896


(1) The marking119872119896


119896



graph


[120572119892119894]that


[120572119892119894]isin 119864




0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












0) where 119905


1

[1205721198921]and

both 1199051and 119905


1



fire if 1199021198721

ge 1205721and 119902

119872






1198720

= [1198720(1199011)1198720(1199012)1198720(1199013)]119879

= [1 1 1]119879 At this


3is

1198760= [1199021198720

1 1199021198720

2 1199021198720

3]119879= [1 1 1]


0 we have 1199021198720

1= 1205721 11990211987202

= 1205722 and

1199021198720



The marking1198724= [0 1 2]



11199052]


119879 is updatedto 1198764= [0 1 2]

119879 at 1198724 When marking 119872

4and vector 119876

4


lt 1205721 11990211987242

= 1205722 and 1199021198724

3= 1205723







1198723= [2 1 0]

119879 and vector 1198763= [2 1 0]





0


1 1199052 and 119905

3are controlled by

1198901

[1] 1198901[1] and 119890

2

[1]with 120572

1= 1205722= 1205723= 1 respectively If


and 1198902[1205721198922]

with 1198902[1205721198922]

≫ 1198901





1 1199052 and 119905





0until 119905

3fires

once Marking1198723= [2 1 0]



11199052]



1199041198720) that is 119877(119873

1199041198720) sube



(1198731198720) [26]


Let 1198791015840 = 1199051 1199052 119905



isin 119864 with 119896 isin 1 2


119894isin 1198791015840 let 120572

119894be the





119903119896 1199011198881ndash119901119888120578 and

1199011198981ndash119901119898120578

be places Define119872(119901119903119896) = 0119872(119901

119898119894) = 120572119894

and forall119894 = 1 2 120578119872(119901119888119894) = 0

(2) Transition 119905119888119896



119903119896


119903119896 119901119903119896) with119882(119905

119903119896 119901119903119896) =


119894 define arcs (119901 119905

119903119896) (119905119894 119901119898119894) and

(119901119898119894 119905119903119896) where119882(119901 119905

119903119896) = 120572119894sdot119882(119901 119905

119894)119882(119905

119894 119901119898119894) =

1 and119882(119901119898119894 119905119903119896) = 120572119894

(3) Then transition 119905119888119896


1198881ndash119901119888120578


119903119896 119905119888119896)

with 119882(119901119903119896 119905119888119896) = 1 forall119905

119894isin 1198791015840 forall119901isin∙119905

119894 define arcs

(119905119888119896 119901) (119905

119888119896 119901119888119894) and (119901

119888119894 119905119894) with 119882(119905

119888119896 119901) = 120572

119894sdot

119882(119901 119905119894)119882(119905

119888119896 119901119888119894) = 120572119894 and119882(119901

119888119894 119905119894) = 1





[1205721198921]with 119878(119905

1) =

1198901


1and the allowable


1



imply 1205721= 2




1is greater


1198881





with119878(1199052) = 119890

2

[3](1205722= 3) and 119878(119905

3) = 119890

2

[2](1205723= 2) If both






(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a)

2

2

2

2

2 2

44

2

2

22

2

2

3

5

5

3

3

3

3

3

(c)

(b)

(d)

2

(f)

(g)

(h)

(e)

2

2

22

33

3

3

222

22

2

2

44

p1p1

p3

p3

p3

p3

p2p2

p4p4

p3

p3

p4

p4

pm1

pm2 pm3

pm2 pm3

t1 t1

t3

t3

t3

t3

t3t2

t2

t2

t2

t2

e1[2]

e2[3]

e2[3]

e2[2]

t3t2 e2[3] e2[2]

e2[2]

e1[1205721198921]

e1[1205721198921]

p1

p1

p2

p2

t1

t1

e1[2]

e2[1205721198922]

e2[1205721198922]

tr1

tr2

tr2

tc1

tc2

tc2

pr1

pr2

pr2

pc1

pc2 pc3

pc3

pc3

pc2

tr2

ts1

ts2

tc1 tc2

pr2pr1

pc1 ps1ps2

ps3

pc2

e1[1205721198921] ≫ e2[1205721198922]

pm2pm1 pm3

e2[1205721198922]

tr1


[1205721198921]≫ 1198902



p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










p1 p2

t1 t2

(a)

p1 p2

t1 t2

e1[1205721] e1[1205722]

(b)

p1 p2

t1 t2

e1[2] e1[1]

1205721 = 2 1205722 = 1

(c)

p1 p2

t1 t2

e1[2]e1[1]

1205721 = 1 1205722 = 2

(d)









4in Figure 12(c)



Figure 12(f)Let 1198791015840 and 119879


119904with 119879

1015840cap 11987910158401015840= 0 and let 119890119896

[120572119892119896]and

119890119895



[120572119892119896](resp 119890119895

[120572119892119895]) If 119890119896[120572119892119896]

and

119890119895


[120572119892119896]≫ 119890119895

[120572119892119895] The



1199041ndash1199011199043be



119890119896


[120572119892119896]are


1199042 The tokens in 119901

3



[120572119892119895]rely on

119890119896



(1199011199042 1199051199042) (1199051199042 1199011199043) and (119901

1199043 119905119903119895) where119882(119905

119903119896 1199011199041) =

1 119882(1199011199041 1199051199041) = 1 119882(119905

1199041 1199011199042) = 1 119882(119901

1199042 1199051199042) = 1

119882(1199051199042 1199011199043) = 1 and119882(119901

1199043 119905119903119895) = 1 forall119905

119894isin 1198791015840 define

the arcs (119901119898119894 1199051199041) and (119905

1199042 119901119898119894) where 119882(119901

119898119894 1199051199041) =

119882(1199051199042 119901119898119894) = 119872(119901

119898119894)


[1205721198922]rely on that of 1198901

[1205721198921] that is 1198901

[1205721198921]≫ 1198902

[1205721198922]


1199041and 1199051199042 three places







119878(1199051) = 1198901

[1205721]and 119878(119905

2) = 1198901






If functions 119878(1199051) = 1198901

[1205721]= 1198901

[2]and 119878(119905

2) = 1198901

[1205722]= 1198901

[1]


1= [(1199051)21199052] in a net system (119873119872) If 119878(119905

1) =

1198901

[1205721]= 1198901

[1]and 119878(119905

2) = 1198901

[1205722]= 1198901




1= 1199051(1199052)2 1205902= 119905211990511199052

1205903= (1199052)21199051 and 120590



4is




[120572119892119894]at







[120572119892119894]119872)



119904]minus with [119873

119904]minus(119901 119905) =




997888rarr119896⪈997888rarr120590119896




119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894

[120572119892119894]








2+ 21199013+



event 1198901[1205721198921]


1

[1205721]to

119878(1199052) = 119890

1


5is controlled by


with 119878(1199055) = 119890

2





119894

[120572119892119894] 119890119894[120572119892119894]








119904


119904]



external event 1198901[1205721198921]





5is



1 1199054 and their




119904] contains a



3= [(1199052)21199053] respectively


4on

external event 1198901[1205721198921]



2once and 119905


1=

0 1205722= 1 120572

3= 2 and 120572


1198901



2at marking119872








[120572119892119895]and 119890119896[120572119892119896]

119879 is


119895

[120572119892119895] 119890119896


events with 119895 = 119896


Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Step 5

(a) (b)

Step 1

Step 2

Step 4

(c)

Step 3

(d)

(e)

(f)

ECRSs

p1 p3p2

p1 p3p2

p1 p3p2

p4 p5 p6

p4 p5 p6

p4 p5 p6

t1

t4 t4t5

t3t2

t3

t3

t3

t3

t2

t2t2

t2

t1 t3t2e1[1205721] e1[1205722]

t2e1[1205722]

e1[1205724] e1[1205724]e2[1205725]

e1[1205723]

t3e1[1205723]

p1 p3p2

p4 p5 p6

t2e1[1205722] t3e1[1205723]

e1[1205721] e1[1205722] e1[1205723]

M0 = 3p2 + 2p3 + 2p4 + p6

M1 = 2p2 + 2p3 + 2p4 + p6

M2 = p2 + 2p3 + 2p4 + p6

M3 = 2p3 + 2p4 + p6

M4 = 2p2 + p3 + 2p4 + p6

M5 = p2 + 2p4 + p6

M6 = 2p4 + p6

M7 = p2 + p3 + 2p4 + p6

M8 = p3 + 2p4 + p6

1205721 = 0 1205722 = 3 1205723 = 0 1205724 = 0

1205721 = 0 1205722 = 1 1205723 = 2 1205724 = 0

1205721 = 0 1205722 = 2 1205723 = 1 1205724 = 0

e1[120572]

t3t2


119904


2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

(a)

(c)

2 2

2

(b)

2 2

(d)

p1

p1

p3

p3

p2

p2 p1 p2

p5

p5

p6

p4

p4 p3 p5p4

p4

t1

t1

t3 t3

t2

t2 t3 t1 t2 t3

p1 p3p2

p5 p6

t1 t2

e1[2] e2[1]

e1[4]

e1[0]

e2[1] e2[1]

e1[1205721198921] ≫ e2[1205721198922]

e1[1205721198921] ≫ e2[1205721198922]


11198721) (c)


31198721)


119894) = 119890

119895

[120572119894] forall119905119894isin


[120572119894]

Step 2 forall119905119894isin 119867 120572

119894= 119902119872

119894 119902119872119894




≫

119890119896

[120572119892119896]



1015840


[120572119892119896]




119894) forall119905119894isin 119879 and 119890119895

[120572119892119895]≫ 119890119896

[120572119892119896]



119895isin

119879(119870) 119905119894fires 119902119872






1 1199052 1198721= 1199011+ 21199012+ 41199014 D1198721 = 119863

1198721

1

and 1198631198721

1= 119905





120595(K1198725







1= 1199051(1199053)4

1205902= 11990531199051(1199053)3 1205903= (1199053)21199051(1199053)2 or 120590

4= (1199053)311990511199053at the initial


1 On the




1 For








Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Ω1 Ω2Ωlowast

p1 p2


1⟩ 1198701198922= ⟨1199012 119901∙

2⟩ Ωlowast = 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870


2= 119879(119870

1198922) Ωlowast

K1198725


5= (1199053)41199051




(1) Two sets 119879(119870) = 1199051 1199052 and 119867 = 119905

3 are obtained

Let 1199051 1199052 and 119905


[1205721] 1198902[1205722]

and 1198901[1205723]



obtained by 1205723= 1199021198721

3= 4 (Step 2)


[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 3)



(5) Two ECRSs 1205901= [11990511199052] and 120590


conflictK11987251




(7) Three functions 119878(1199051) = 1198902

[1] 119878(1199052) = 1198902

[1] and 119878(119905

3) =

1198901


1

[1205721198921]≫

1198902



3)4[11990511199052] can fire The





1198921= ⟨1199011 119879(1198701198921)⟩

with 119879(1198701198921) = 119901

∙

1and 119870

1198922= ⟨1199012 119879(1198701198922)⟩ with 119879(119870

1198922) =

119901∙


1198921and

K1198721198922


and 1198701198922


lowast= 119879(119870

1198921) cap 119879(119870

1198922)

Ω1= 119879(119870

1198921) Ω

lowast and Ω2= 119879(119870

1198922) Ω


119863119872

1 119863119872

2 119863

119872



119894



119905119894 119905119896 isin D119872 we have 119905

119894isin 119879(119870

1198921)cap119863119872

119895and 119905119896isin 119863119872

119895119879(1198701198921)




119896isin 119879(119870

1198922) cap 119863119872

119895

and 119905119894isin 119863119872



1198922 Hence






119895

[120572119892119895]and 119890119896[120572119892119896]


119895

[120572119892119895] 119890119896


events with 119895 = 119896


119894) = 119890119895

[120572119894]


[120572119892119895] Choose an ECRS


119894) = 119890

119895

[120572119894]



1198921) and 119879(119870

1198922) of


1198921) cap 119879(119870

1198922) Ω1= 119879(119870


Ω2= 119879(119870






119894) =

119890119895

[120572119894]with 119878(119905

119894) = 119890119896

[120572119894] Go to Step 6



119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894) = 119890119895

[120572119894]

with 119878(119905119894) = 119890119896

[120572119894] Go to Step 6


[120572119892119895]≫ 119890119896

[120572119892119896]


[120572119892119895]≫ 119890119896

[120572119892119896]






1198921




2fire


1198921) and that








= ⟨1199011 119879(1198703)⟩

119879(1198703) = 119905

1 11990521198721= 21199011+ 31199012 D1198721 = 119863

1198721

1 and 1198631198721

1=

1199051 1199053 In this case 119870


Section 2K11987213




1198721

3are made


K1198721



conflictK11987213


(1) Three functions 119878(1199051) = 1198901

[1205721] 119878(1199052) = 1198901

[1205722] and 119878(119905

3) =

1198901


1= [11990511199052] and 120590

2=



1205721= 2 120572

2= 0 and 120572



1198921) = 119905

1 1199052 119879(119870

1198922) = 119905

2 1199053 Ωlowast =

1199052Ω1= 1199051 andΩ


(3) 119879(1198701198921) = 119905

1 1199052 that is 119879(119870

3) in Example 3 in


3) = 119890

1

[1]is replaced by 119878(119905

3) = 119890

2

[1]


1

[1205721198921]≫ 1198902

[1205721198922]is assigned

(Step 6)

(6) Three functions 119878(1199051) = 1198901

[2] 119878(1199052) = 1198901

[0] and 119878(119905

3) =

1198902


1

[1205721198921]≫

1198902



1)2]1199053can

fireK11987213







=

(119875 119879 119865119882 119864 1198781198720) where 119875 and 119879 = 119879

119899cup 119879119904(119879119899

= 0 and119879119904




1

[1205721198921] 1198902

[1205721198922] 119890

119899

[120572119892119899] is a set of


C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










C1R1

B1 B2

C2

C4

M1

M2

C3

(a)

2

2

2

2

p1

p2

p3

p4

p5

p6

p7

p8

p9

p10

p11

t1

p0

p12

p13

p14

t0

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

(b)



0is an

initial marking



119904 119905 is



119899 119905 is a




= 0 (resp119879119904




119894in confu-







0) of



11198721) with



and 1198702

= ⟨11990113 1199050 1199058⟩ 1198701at marking 119872

1produces a


= 11990112 1199050 11990561198721 where two ECRSs

1205901

= [11990501199056] and 120590

2= [(119905






Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Element Meanings

1199010



1199018



1199050



1199052



1199055



1199058



11990511




1= [11990501199056] and 120590

2= [(1199050)2]




13are








[1] 119878(1199056) = 1198901

[1] and 119878(119905

8) = 1198902

[1]


≫ 1198902

[1205721198922]


3= 31199014+1199019+211990110+



3=


3) = 4





= ⟨11990114 1199054 119905101198724⟩ with 120595(K1198724

3) = 5




3= [119905411990510] and 120590

4= [(11990510)3] can be used to




K1198721

1in (119873







[1] 119878(1199059) = 1198903

[1] and

119878(11990510) = 119890

4


3

[1205721198923]≫

1198904



11198721) and (119873



1198901

1205721198921 1198902

1205721198922 1198903

1205721198923 1198904

1205721198924



3=


applied to1198732


1= [11990501199056] and 120590

4= [(119905

10)3]


and K1198724

3 respectively

If K11987211

and K1198724


1and



[0] Con-


6






2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2

e2[1]e1[1] e1[1]

e1[1205721198921] ≫ e2[1205721198922]

p1

p6

p7

p8p13p0p12

p9

t6 t0 t8

(a)

2

p3

p4

p5

p10

p11

p14

t3

t4 t10

(b)

2

e4[1] e4[1]

e3[1]

e3[1205721198923] ≫ e4[1205721198924]

p4

p5

p14

p9

p10

p11

t4

t9

t10

(c)


2 and (c) a CIC (119873









Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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