A Subclass of Petri Net with Reachability Equivalent to State Equation Satisfiability: Live Single Branch Petri Net
 Author: Gao Qian, Cho Young Im
 Organization: Gao Qian; Cho Young Im
 Publish: International Journal of Fuzzy Logic and Intelligent Systems Volume 13, Issue3, p200~207, 25 Sep 2013

ABSTRACT
Petri Nets are a system description and analysis tool. Reachability is one of the most basic properties in Petri Net research. In a sense, reachability research is the foundation study for other dynamic properties of Petri Nets through which many problems involving Petri Nets can be described. Nowadays, there are two mature analysis methodsthe matrix equation and the reachability tree. However, both methods are localized, i.e., it is difficult to find a general algorithm that can determine reachability for an arbitrary Petri Net, especially an unbounded Petri Net. This paper proposes and proves three propositions in order to present a subclass of a Petri Net, the live singlebranch Petri Net, whose reachability is equivalent to the satisfiability of the state equation.

KEYWORD
Petri Net , Reachability , Live single branch , State equation

1. Introduction
Highlevel Petri Nets provide a graphical and mathematical tool for system description and analysis that can be applied effectively to issues of parallel and asynchronous workflow [1]. Vigorous research and development on Petri Nets in recent years has led to a widening of their field of application. Petri Nets have become the preferred modeling tool for net systems that have a variety of abstraction levels [2]. They have been used widely in the areas of discrete event systems analysis, flexible manufacturing systems (FMS), automatic control, computer network performance analysis and protocol verification, parallel program design and analysis, knowledge inference, artificial neural networks, and decision models [3].
One of the main purposes of using Petri Nets to model an actual system is to analyze the properties and functions of the actual system with the assistance of a system model. We research Petri Nets for the purpose of applying them to areas of artificial intelligence, such as knowledge expression and the establishment of formalized reasoning models, expressing relationships among various activities, reasoning logically based on modeling through a known initial state and initial conditions, calculating the credibility of each inference result through fuzzy inference reasoning, and determining reasonable inference results. In order to achieve this purpose, we first study the reachability of a Petri Net.
In research on Petri Nets, reachability is probably the most basic dynamic property and the decidability of the reachability problem is probably the most important open problem in the mathematical theory of Petri Nets and related formalisms. In a sense, research on reachability is the foundation for research on other dynamic properties of Petri Nets, because many other problems can be described using reachability. The reachability problem for Petri Nets is to decide, given a Petri Net
N and a markingM , whetherM ∈R(N) . Clearly, this is a matter of marking the reachability graph defined above, until we reach the requested marking or we know it can no longer be found. This is more difficult than it might seem at first: the reachability graph is generally infinite, and it is not easy to determine when it is safe to stop. In fact, this problem was shown to be EXPSPACEhard [4] years before it was shown to be decidable at all. Papers continue to be published on how to decide the reachability efficiently [5].While reachability seems to be a good tool for finding erroneous states, the constructed graph usually has far too many states to calculate for practical problems. To alleviate this drawback, linear temporal logical (LTI) is customarily used in conjunction with the tableau method to prove that such states cannot be reached. LTL uses the semidecision technique to find if a state can be reached, by finding a set of necessary conditions for the state to be reached and then proving that those conditions cannot be satisfied.
Thus, if the decision problem of reachability can be solved, then other dynamic properties such as activity are expected to be solved. Unfortunately, it is not easy to find a general algorithm that can determine the reachability of any Petri Net, especially an unbounded Petri Nets. Since the reachability decision problem has at least exponential space complexity and no effective algorithm is found by now, so if we can find and prove some Petri Net subclasses that can use the satisfiability of the state equation to decide the reachability, it is also an important and indispensable work.
It has been proved that, for the live
T system and live weightedT system, the reachability is equivalent to the satisfiability of the state equation, based on these existing conclusions, this paper proposed a subclass of live weighted Petri NetLive Single Branch Petri Net, and proved that for this kind of Petri Net, the reachability is equivalent to the satisfiability of the state equation. That is to say, for living single branch Petri Net, we can make use of the satisfiability of state equation to determine its reachability.The rest of this paper is organized as follows. Section 2 gives a short review and comparison of related studies and then presents the contributions of our paper. Section 3 describes the basic notations and definitions of Petri Nets. Section 4 first shows two examples to illustrate why using state equations cannot decide the reachability of any kind of Petri Net, then proposed a subclass of Petri Net, that can use the satisfiability of state equation to decide the reachability, and then proved the validity. Section 5 introduces parts of existing reachability decision methods. Finally we draw conclusions and suggest future studies.
2. Related Work
Our research relates to a number of areas, especially Petri Netbased formalisms, dynamic behavior, and synchronized concurrent systems research. Since for some subclasses of Petri Net, the state equation has nonnegative integer solution is not only the necessary condition but also the sufficient condition of
M can reach fromM _{0}, so for these kind of Petri Net, the reachability is equivalent to the satisfiability of the state equation. Therefore, it can significant simplify the decision of reachability (get rid of the judgment of the existence of legitimate sequence). So to find and prove which kinds of Petri Nets subclasses can have this property is an item of work with extraordinary significance. Many researchers have devoted themselves to find the Petri Net subclasses with this property. Here we highlight some relevant works in these fields.Liu and Chen [6] have found a subnet of Petri Net: ∑ = (
S, T; F;W;M _{0}), if it satisfied the condition ∀_{s} ∈S → │？s │ ≤ 1, then for this kind of Petri Net subclass, the reachability is equivalent to the satisfiability of the state equation. Murata [7] has found that for the liveT system, the reachability is equivalent to the satisfiability of the state equation. Xu and Wu [8] further found and proved that for the live weightedT system, the reachability is also equivalent to the satisfiability of the state equation. Bao et al. [9] have defined a minimaltrapcircuit Petri Net and backwardcircuit Petri Net, and proved that the reachability problem of this two kinds of Petri Net subclasses is equivalent to the satisfiability of the state equation if no tokenfree minimal circuit exits under the initial marking.Despite these previous researches, some other researchers devote themselves to use Petri Net as a tool for context inference, modeling and analyzing of discrete event and so on. Lee et al. [10] propose a context inference model which is based on the socalled fuzzy colored timed Petri Net. The model represents and handles the sequential occurrence of some events along with involving timing constraints, deals with the multiple entities using the colored Petri Net model, and employs the concept of fuzzy tokens to manage the fuzzy concepts [10]. Mo et al. [11] proposed a fuzzy transition timed Petri Net with fuzzy theory to determine the optimal transition time. Kim et al. [12] models traffic system with fuzzy discrete event system’s character in fuzzy transition timed Petri Net, and based on it to design a traffic signal controller.
Based on the abovementioned studies, in this paper, we put forward a subnet of a Petri Neta live single branch Petri Net whose reachability is equivalent to the satisfiability of the state equation.
3. Basic Notations and Definitions of Petri Net
3.1 Features and Definition of Petri Net
Petri Net has many good features that can be used to analyze and describe the system. It can simulate the system from the perspective control and management, accurately describe the dependency and dependency relationship of the event of in the system, describe the system structure and behavior with unified language and describe the synchronized concurrent systems, provide a new way to solve the different kinds of problems, refer to the literature [13].
Petri Net is a model used to describe distributed systems. It can not only describe the structure of the system, but also simulate the operation of the system. The part that describes the system structure is referred to as a network. In form, a net is a directed bipartite graph without isolated nodes.
> Definition 1:
A triple
N = (S, T; F ) which satisfies the following conditions is called a net:1)
S ∩T ≠？ 2)
S ∩T =？ 3)
F ⊆ (S ×T ) ∪ (T ×S ))4)
dom(F) ∪cod(F) =S ∪T Where:
dom(F) = {x ∈ S ∪ T│∃y ∈ S ∪ T : (x, y) ∈ F}
cod(F) = {x ∈ S ∪ T│∃y ∈ S ∪ T : (y, x) ∈ F}
S andT are two disjoint sets (In general they are assumed to be the finite set), that are the basic elements sets of netN. S is called place set andT is called transition set. The elements ofS are known as theS place, which can also be called location and the elements ofT are known as theT transition.F is the flow relation of netN [14].> Definition 2:
Suppose
N=(S, T; F) is a net, then for all thex ∈S ∪T , record？x = {y│y ∈ S ∪ T ∧ (y, x) ∈ F}
x？ = {y│y ∈ S ∪ T ∧ (x, y) ∈ F}
Here we name ^{？}
x as the preset or input set ofx ,x ^{？} as the postset or output set ofx , name ^{？}x ∪x ^{？}as the extension of element [14].An extension of a place is the subset of transition set
T , and an extension of a transition is the subset of place setS .> Definition 3:
Suppose
N = (S, T; F;M ) is a Petri Net, if there exists at ∈T , which satisfiesM[t >M’ , thenM’ is directly reachable formM . If there exists transition sequencet _{1},t _{2}, ...,t_{k} and identity sequenceM _{1},M _{2},...,M _{k}, which satisfies:M[t _{1} >M_{1}[t _{2} >M _{2}[t _{3} ... >M_{k？1}[t_{k} >M_{k} , Then we say thatM_{k} is reachable formM . All the identifications that are reachable fromM are named asR(M) , andM ∈R(M) . If the transition sett _{1},t _{2}, . . . ,t _{k} is labeled as σ, then theM[t _{1}>M_{1}[t _{2}>M _{2}[t_{3} . . .>M_{k？1}[t_{k} >M_{k} can be labeled asM[σ >M_{k} [14].When we use a Petri Net to simulate an actual system, the net ∑ = (
S, T; F ) describes the structure of the system. Initial markingM_{0} is marked of the initial states of the system, andR (M_{0} ) is the totally probable state set of the system that occurs in the operation process.Based on this, we can define the net system ∑ as follows: here
M_{0} is the initial marking of ∑, and the reachability marking set of ∑ is the Minimum Set that satisfies the following two conditions:1)
M_{0} ∈R (M_{0} )2) If
M ∈R (M_{0} ), and there existst ∈T that satisfiesM[t >M , thenM ∈R (M_{0} ) ][14].For the bounded Petri Net, since its reachability marking set
R (M_{0} ) is a finite set, we can takeR (M_{0} ) as a vertex set, using the directed reachable relations between the markings as arc set to make up a directed graph. This kind of directed graph is called a reachable marking graph of the Petri Net.> Definition 4:
Suppose ∑ = (
S, T; F;W;M_{0} ) is a Petri Net,M_{0} is the initial identifier,t ∈T . If for ∀M ∈R(M_{0} ) always existM’ ∈R(M) , makeM’[t >, then the transitiont is live. If ∀t ∈T, t always lives, then the ∑ is a live Petri Net [14].4. The Subclass of Petri Net with Reachability Equivalent to State Equation Satisfiability
4.1 The Limitations of Using State Equations to Decide Reachability for Any Kinds of Petri Net
For a general Petri Net, ∑ = (
S, T; F;W;M_{0} ), since exiting an n (n=│T│) dimensional nonnegative integer vector satisfied the state equationM =M_{0} +A^{T}X is just a necessary condition but not the sufficient condition ofM is reachable fromM_{0} , so we cannot using state equation to decide the reachable marking of ∑.In the following, we’ll use two examples to show why we cannot use the state equation to decide the reachability for any kinds of Petri Nets.
First, considering the matrix method for analyzing Petri Nets, a matrix A cannot reflect the structure of a Petri Net accurately, because if the transition section uses one position to input and output simultaneously, there will be a selfloop, i.e., the same position of A^{+} and A^{？} will occur at the same number, which will be mutually offset by A^{？}  A^{+}.
Second, there will be missing sequence information in the triggered vector.
For example, suppose we want to confirm whether the identification (0, 0, 0, 0, 1) can be reached from (1, 0, 0, 0, 0) in Figure 1. If we use the state equation to decide reachability, we get Equation (1):
(For the matrix, the column indicate the place
p _{1},p _{2},p _{3},p _{4},p _{5}, the row indicate the transitiont _{1},t _{2},t _{3},t _{4},t _{5},t _{6})Obviously, the solution of this equation is not unique. From this general form, we can know only the relationships among the numbers of the initiation of the transition node, but cannot confirm the sequence of the initiation.
Third, the solution calculated using the state equation might be a false solution.
For example, suppose we want to confirm whether the identification (0, 0, 0, 1) can be reached from (1, 0, 0, 0) in Figure 2. If we use state equation to decide reachability, we can get Equation (2):
This equation has a solution of (1, 1). The equation corresponds to the two initiating sequences
t _{1}t _{2} ort _{2}t _{1}, but neither can be reached, becauset _{1} andt _{2} cannot be enabled in (1, 0, 0, 0). Therefore, we can see from this example that the solution calculated using the state equation might be a false solution.However, there exits some special Petri Net, the reachability problem is easily to be decided, so it has great significant to find such Petri Net subclass. In the following, we will define
a Petri Net subclass and prove that, for this kind of Petri Net subclass, the reachability is also equivalent to the satisfiability of the state equation.
4.2 The Proposed Petri Net Subclass and Demonstration
For a Petri Net∑ = (
S, T; F;W;M_{0} ), if identifierM is reachable fromM_{0} (mark as:M ∈R (M_{0} ), then there must exist an ndimensional nonnegative integer vector X, satisfyinghere
A is the incidence matrix of ∑, which is ann ×m integer matrix ( │T │=n , │s │=m ). Equation (3) is the state equation of Petri Net ∑. But for a general Petri Net, Equation (3) is just the necessary condition ofM ∈R (M _{0}), not the sufficient condition, i.e., for any Petri Net, we cannot determine the initiating sequence caused by the transition node by judging whether the Petri Net satisfies the state equation.However, for some subclass of Petri Net, Equation (3) is the sufficient and necessary condition of
M ∈R (M _{0}). In the following, we’ll first give the definition of live single branch Petri Net, and then prove for this kind of Petri Net, the reachability is equivalent to the satisfiability of the state equation.> Definition:
Suppose ∑ = (
S, T; F;W;M_{0} ) is a live Petri Net, if ∀_{s} ∈S , it must have │^{？}s │ > 1 ⇒ (s, s ) ？F ^{+}. Then the Petri Net is called a live single branch Petri Net. HereTX F ^{+} is the transitive closure of flow relationF . It means that the Live Single Branch Petri Net cannot have multi preset, but can have multi postset. An example of live single branch Petri Net is shown as Figure 3, here │s_{7}│=2.Obviously the live single branch Petri Net includes the live weighted
T system Petri Net which has only one input and one output (∀_{s} ∈S : │^{？}s │ = │s ^{？}│ = 1), but the live single branch Petri Net which we proposed in our paper has one input but multi outputs, it is more general than the live weightedT system.Xu and Wu [8] have proved that for the live weighted
T system Petri Net, the reachability is equivalent to the satisfiability of the state equation. In the following, we’ll prove that for the wider one, we can also depend on the state equation satisfiability to decide the reachability.Since it is depicted in extensive literature that the equation
M =M _{0} +A^{T}X is the necessary condition of the reachability of a Petri Net, we will prove here only that for a live singlebranch Petri Net, equationM =M _{0} +A^{T}X is also the sufficient condition for reachability.Before we prove this, we first prove two propositions.
Proposition 1:
Suppose ∑ = (
S, T; F;W;M_{0} ) is a live weighted Petri Net. If for anyM ∈R (M _{0}) for which the ndimensional nonnegative integer vectorX satisfiesM +A^{T}X ≥ 0, we can find at ∈TX that satisfiesM[t >, then the sufficient and necessary condition of identifierM_{d} being reachable fromM _{0} is that there exists an ndimensional nonnegative integer vector X satisfying the state equationM_{d} =M _{0} +A^{T}X . HereTX ={t_{i} │t_{i} ∈T and X(i) >0}(X(i) is the ith component of vectorX ) .Proof:
(1) The necessity of this theorem follows obviously from the basic properties of Petri Nets.
(2) Now we prove the sufficiency
Suppose
Since
M_{d} =M _{0} +A^{T}X ≥ 0, there existst_{il} ∈TX , 1 ≤i _{1} ≤n , satisfyingM _{0}[ti_{l} >M_{l} . In addition, becauseM _{1} =M _{0} +A^{T} E _{i1} (here Eil is an ndimensional nonnegative vector whosei1 th component is1 , and whose other components are0 ), supposeX_{l} =XE _{i}_{1}, thenM _{1} +A^{T}X _{1} =M _{1} +A^{T}X (X ？E _{i1}) =M _{0} +A^{T}X ≥ 0. Also, becauseM _{l} ∈R (M _{0}), there existsti _{2} ∈TX _{l}, that satisfiesM_{1} [ti _{2}>M _{2}. SupposeX _{2}=X _{1}E _{i2}, thenM _{2} +A^{T}X _{2} ≥ 0 andM _{2} ∈R (M _{0}), so there existsti _{3} ∈TX _{2}, satisfyingM _{2}[ti _{3}>M _{3}. Continuing the recursion, we can getM3[ti4 > M4; ... ;MK？1[tiK > MK = Md.
Finally we can conclude that identifier
M_{d} is reachable fromM _{0}.This proves the sufficiency, and thus also proposition 1.
Proposition 2:
Suppose ∑=(
S, T; F;W;M_{0} ) is a live singlebranch Petri Net, then for anyn dimension nonnegative integer vectorX that satisfiesM _{0} +A^{T}X ≥ 0, there existst ∈TX , satisfyingM[t > (HereTX ={t_{i} │t_{i} ∈T and X(i) >0 }).Proof: We use reductio ad absurdum.
Suppose an
n dimensional nonnegative integer vector satisfiesM _{0} +A^{T}X ≥ 0, then for allt ∈TX , there is noM _{0}[t >. By constructing the sequence ofti_{l} ,si _{1},ti _{2},si _{2},...,ti_{r} with the following algorithm, we will find a contradiction in the sequence.Algorithm:
Step 0: ① Randomly select a t∈TX, labeled as til;
② S1 = {s│s ∈ ？ti1, and M0(s) < W(s, ti1) //W is the weighted function of ∑, because the //M0[til> is false, so S1≠0.
③ Counting variable j:=1;
Step l: ①Tj+1 = {t│t ∈ ？Sj, and t ∈ TX
② If Tj+1 ≠= ？ then
{
Randomly select a t∈Tj+1, labeled as tij+1, and label sij as the place S of ？tij∩ t？ij+1 (here, if there are many places in ？tij ∩ t？ij+1, then select one of them randomly);
}
Else // Tj+1=？
Jump to Step 2.
③ If tij+1 = tk and 1 ≤ k ≥ j then Jump to Step 2. Else j:=j+l, Jump to Step l.
Step 2: End.
It is easy to see that since the elements in T are finite, the algorithm can be terminated. We can finally get the sequence:
ti_{l} ,si _{1},ti _{2},si _{2}, . . . ,ti_{r} , whereti_{j} ∈TX ,j = l, 2,. . . ,r .We discuss the sequence
ti_{l} ,si _{1},ti _{2},si _{2}, . . . ,ti_{r} according to the termination conditions of the above algorithm with the following two cases:(1) Case 1:
When
ti_{j} =ti_{k} (1 ≤k <r ), the sequence will be a return circuitAnalyze the loop
ti_{k} ,si_{k} ,. . . ,ti _{r？1},ti_{k} . Since │^{？}S │ > 1 ⇒ (s, s ) ？F ^{+}, so│^{？}S i_{j} │ = 1(j =k ,k + 1, ...r ？ 1). Also because some placessi_{j} in the sequence do not have enough tags to initiate the transitionti_{j} (j =k,k +1 ,. . . ,r 1 ) of the sequence, and for anyS i_{j} (j =k,k +1 ,. . . ,r 1 ) in the sequence, ^{？}S i_{j} = {ti_{j} +1 }, therefore tags outside of the sequence cannot flow to places within the sequence, i.e., any transitionsti_{j} (j =k,k +1 ,. . . ,r 1 ) within the sequence will not be initiated, which is contradictory to the fact that ∑ is a live Petri Net.(2) Case 2:
When
T_{j} _{+1} ≠？ , the sequence will be:(where
ti_{j} ≠t_{ik} ,l ≤k <r )Since
M _{0}[ti > is false, there exists a place sir, that satisfies_{l} M _{0}(si_{r} )<W(si_{r},ti_{r}) , and there is no transitiont in the ^{？}si_{r} that satisfiest ∈TX . Now letM _{1} =M _{0} +A^{T}X , sinceti_{r} ∈TX , i.e.,X (i_{r}) >0 , but the number of the tags ofsi_{r} is smaller than the weight of the sideW(si_{r},ti_{r}) , so there must exist aM _{1}(si_{r} )<0 , which is contradictory withM _{1} =M _{0} +A^{T}X ≥ 0Thus proposition 2 is proved.
Proposition 3:
Based on the above conclusions (conclusions 1 and 2) we will prove, for any live singlebranch Petri Net ∑ = (
S, T; F;W;M_{0} ), the sufficient and necessary condition of identifierM_{d} being reachable fromM_{0} is that there exists an ndimensional nonnegative integer vectorX , that satisfies the state equationM_{d} =M_{0} +A^{T}X .Proof:
The necessity of this theorem follows obviously from the basic properties of Petri Nets.
Now let us prove the sufficiency:
Since the ∑ = (
S, T; F;W;M_{0} ) is a live singlebranch Petri Net, according to conclusion 2, we know that for any ndimensional nonnegative integer vectorX that satisfiesM_{0} +A^{T}X . ≥ 0, there must exist at ∈TX , making theM_{0} [t >. HereTX = {ti │ti ∈T and X(i) >0}Then according to proposition 1, since the ∑ = (
S, T; F;W;M_{0} ) is a live singlebranch Petri Net, for any ndimensional nonnegative integer vectorX that satisfiesM ∈R (M_{0} ), andM +A^{T}X ≥ 0, there must existt ∈TX , that makes theM[t >. Hence, the sufficient and necessary condition of identifierM_{d} being reachable fromM_{0} is that there exists an ndimensional nonnegative integer vectorX that satisfies the state equationM_{d} =M_{0} +A^{T}X , whereTX = {ti1ti ∈T and X(i) >0}(X(i ) is thei^{th} component of vectorX )As a result, we can draw the conclusion that: Suppose ∑ = (
S, T; F;W;M_{0} ) is a live Petri Net, if ∀_{s} ∈S , it must have │^{？}s │ > 1 ⇒ (s, s ) ？F ^{+}, then for this kind of Petri Net, if exits and only exits ann dimensional nonnegative integer vectorX , satisfies the state equationM_{d} =M_{0} +A^{T}X , then the identifierM_{d} being reachable fromM_{0} . Till now, we have found a way to decide the reachability of Live Single Branch Petri Net.5. Conclusion
Petri Nets are a tool for system description and analysis. Reachability is one of the most basic properties in Petri Net research. We research Petri Nets for the purpose of applying them to areas of artificial intelligence, such as knowledge expression and the establishment of formalized reasoning models, expressing relationships among various activities, reasoning logically based on modeling through a known initial state and initial conditions, calculating the credibility of each inference result through fuzzy inference reasoning, and determining reasonable inference results. In order to achieve this purpose, the first task is to research the reachability of Petri Nets. In a sense, reachability is the foundation of the study of other dynamic properties of Petri Nets. Many problems on Petri Nets can be described by it, so the reachability decision problem is therefore one of the most important topics in Petri Net theory.
Methods that can be used to analyze the reachability of a Petri Net include the reachable marking graph and coverability tree, the correlation matrix and state equation, and Petri Net language. Each method has its advantages and disadvantages. Sometimes we can combine them, so as to analyze the dynamic properties of a Petri Net.
Although, using state equation cannot decide the reachability for all kinds of Petri Net, bur since this method is very easy to be grasped and is convenient to be used without the judgment of the existence of legitimate sequence, so if we can prove that for some kinds of Petri Net subclasses, the state equation satisfiability is not only a necessary condition but also a sufficient condition, then it is also an important and indispensable work.
The previous researches have proved that some subclass of Petri Net, such as transitionsingleinputcircuit Petri Nets, minimumtransitionsingleinputtrapcircuit Petri Nets, minimumsiphon circuit Petri Nets, and live weighted
T system Petri Net, the reachability is equivalent to the satisfiability of the state equation.Based on these previous researches, we found a subclass that includes the live weighted
T system can also use the state equation satisfiability to decide the reachability. In this paper, we defined the Petri Net subclass, and proved that for this kind of Petri Net, if and only if exits an n dimensional nonnegative integer vector X, satisfies the state equationM_{d} =M _{0}+A^{T}X , then the identifierM_{d} is reachable fromM _{0}.However, our proposed method is suitable only for determining the reachability of a subclass of a Petri Net, so in the future, the key issue in our further study will be to determine how to combine our proposed method in this paper with an improved coverability tree proposed in reference [15] to decide reachability for a general Petri Net.
> Conflict of Interest
No potential conflict of interest relevant to this article was reported.

[Figure 1.] Petri Net ∑1.

[Figure 2.] Petri Net ∑2.

[Figure 3.] Live single branch Petri Net ∑3.