Cross-Departmental Collaboration Approach for Earthquake Emergency Response Based on Synchronous Intersection between Traditional and Logical Petri Nets

Abstract

In order to reduce the harm of earthquakes to human society, all governments actively promote the construction and development of earthquake emergency rescue work. The earthquake emergency response involves many departments, multiple personnel and large rescue forces, which presents a great challenge to the ability to carry out cross-departmental rescue work in a collaborative and joint manner. A novel collaboration approach based on traditional and logical Petri nets is proposed to improve the cross-departmental collaboration in earthquake emergency response. The approach extends the synchronization of transitions in traditional Petri nets to that of traditional and logical transitions in traditional and logical Petri nets, and defines the intersection of related logical functions. The approach builds a model for the earthquake emergency response plans of various departments with the help of traditional and logical Petri net models, and then performs synchronous intersection operations on the two kinds of Petri nets and merges the two kinds of Petri nets into new logical Petri nets. Meanwhile, through realizing the collaboration ability of cross-departmental work, the approach improves the rescue efficiency and reduces the damage of the earthquake emergency.

1. Introduction

The threat of earthquakes is enormous to the safety of human society. In 2021, 13 earthquakes of magnitude 7.0 or higher occurred worldwide, which caused 2274 deaths and 13,261 injuries. In 2022, nine earthquakes of magnitude 7.0 or higher occurred worldwide, which caused 29 deaths and 761 injuries. In order to reduce the human casualties and economic losses caused by earthquakes and to lessen the impact on human society, earthquake emergency rescue work has become one of the three major work systems of governments for earthquake prevention and mitigation, which is one of the important means for human beings to respond directly to earthquake disasters.

The institutional reform of the Chinese State Council in 2018 integrated the earthquake emergency rescue responsibilities of the China Earthquake Administration with other industry emergency management functions to establish the Ministry of Emergency Management. It further standardized the rescue process. The China Earthquake Administration was changed from a component department of the State Council to a subordinate department of the Ministry of Emergency Management. This change will test its ability to coordinate across departments. In addition, the damage caused by the earthquake is enormous. After the earthquake, many rescue forces and departments need to get involved in the rescue. It also puts forward higher requirements for earthquake emergency departments to carry out cross-departmental collaboration in rescue work.

Therefore, this study argues that an efficient earthquake emergency management system should not only include earthquake prediction and emergency planning but also have efficient post-earthquake emergency rescue with cross-departmental coordination. Only in this way can it help the earthquake emergency departments carry out rescue work efficiently and minimize casualties and property damage.

How to improve cross-organizational and cross-departmental collaboration capabilities will be the focus of this study. This study introduces business process management, process mining and other related knowledge into the field of earthquake emergency management. It uses traditional and logical Petri nets to model and analyze the relevant earthquake emergency management processes. It can further improve cross-departmental collaboration capabilities of earthquake emergency departments.

The main work of this paper has the following significance:

(1)

Traditional and logical Petri nets with better representation capability are applied to the field of earthquake emergency management;

(2)

The synchronization composition of traditional Petri nets is extended from the synchronization of transitions to the synchronization between logical and traditional transitions in traditional and logical Petri nets, and the intersection operation of logical functions of corresponding transitions is presented. For the first time, the logical function is extracted for traditional transitions, and the definition of synchronous transition unit and the calculation method of its logical functions are proposed;

(3)

The synchronous intersection operation between traditional and logical Petri nets fills the research gap. It also solves the coordination problem of the earthquake emergency departments;

(4)

For the superior department, a new logical Petri net can be obtained by modeling the workflow of its subordinate units between traditional and logical Petri nets and performing synchronous intersection operations. Then, we can more directly observe the workflow of each department and judge which work needs the cooperation of the two departments.

The rest of this paper is organized as follows: Related work is introduced in Section 2. In Section 3, some basic concepts are proposed. Section 4 presents a synchronous intersection approach based on traditional and logical Petri nets. Simulation experiments for cross-departmental collaboration in earthquake response are given in Section 5. We conclude in Section 6.

2. Related Work

2.1. Business Process Management and Process Mining

Earthquake emergency management is essentially Business Process Management (BPM) for the rescue work and rescue process after an earthquake. BPM is a specific way of management science [1]. With the help of BPM, the rescue work can be efficiently promoted and the rescue process can become smooth. After the earthquake, the rescue work will involve many departments, such as public security, health, transportation, and hospitals. In the whole rescue work, multiple departments need to coordinate to complete the cross-departmental coordination emergency. Therefore, the high cross-departmental coordination systems and methods become particularly important. BPM can play an important role in solving this problem.

BPM focuses on the integration of information technology and management technology. With the help of the event log, the entire business process is simulated and monitored for better management and correction [2]. As a component of BPM, Process Mining (PM) plays an increasingly prominent role in the field of BPM, which mainly extracts data information about processes from event data. The role of BPM and PM in earthquake emergency management is mainly to abstract the process model according to the relevant earthquake emergency plan or previous rescue workflow. After the earthquake occurs again, the relevant earthquake emergency departments can quickly promote the rescue work according to the guidance of the process model. It can realize the collaboration between different departments, as well as analyze the established process model to achieve a more effective rescue process. PM relies on process models. Currently, there are many methods to establish process models, such as BPMN [3], WFN [4,5] and Petri nets [6,7], etc.

Business Process Modeling Notation (BPMN) is a process model modeling language, which has been widely used in the field of BPM. Many BPM software vendors support its use. WorkFlow net(WFN) is considered as a subclass of Petri nets. WFN contains a source place and a sink place. All nodes of WFN are on the path from the source place to the sink place. With the help of modeling languages such as BPMN, WFN and PN, the process model can be more intuitive and readable.

2.2. Traditional and Logical Petri Nets

In fact, a Petri net is a traditional Petri net, also known as a classical Petri net. The reason why Petri nets are called traditional Petri nets is to distinguish them from a series of Petri nets derived later.

Traditional Petri nets have the ability to handle concurrency, asynchronous and contain uncertain information. High-level Petri nets, time Petri nets [8,9,10], Petri nets with inhibitor arcs, and logical Petri nets are further developed on the basis of Petri nets and have been widely recognized [11,12,13,14,15,16,17].

Logical Petri nets are developed on the basis of Petri nets with inhibitor arcs and high-level Petri nets. Traditional Petri nets and logical Petri nets are two different types of Petri nets with different applications. Logical Petri nets can describe the batch processing functions and the passing value indeterminacy of the system with the help of logical functions. The passing value indeterminacy is divided into input and output indeterminacy, which can be described by logical input functions and logical output functions, respectively. Therefore, compared with traditional Petri nets, logical Petri nets have stronger representation ability. At the same time, the application of logical Petri nets has greatly reduced the number of places in the model, making the constructed model more concise and readable. Logical Petri nets have been widely used in e-commerce and medical fields [18]. However, before this paper, neither logical Petri nets nor traditional Petri nets were applied to the field of earthquake emergency.

2.3. Sharing Composition and Synchronous Composition

Although both traditional and logical Petri nets have strong representation capabilities, there are still difficulties in modeling on larger scale systems [19]. Therefore, when facing complex systems and larger organizations for modeling and analysis, the design idea of combination is usually used to compose and decompose the models, which greatly reduces the complexity of modeling and analysis of large systems.

The study belongs to the composition section. Composition is usually divided into synchronous composition and sharing composition. Synchronous composition targets transitions’ synchronization and can be used to achieve synchronous operation of traditional Petri nets. The collaboration problems of earthquake emergency departments studied in this paper belong to the synchronous composition content.

Many studies have been carried out for sharing composition and synchronous composition, but more for traditional Petri nets. In synchronous composition, a lot of research has been done on the process of synchronous composition and complete behavior invariant of traditional Petri nets, and some examples are given to illustrate the correctness and effectiveness of this method in large-scale system modeling and analysis. The synchronous composition of two Petri nets has been extended to the synchronization of multiple Petri nets.

In the context of sharing composition, the preservation of liveness problem, siphon computation and the confluence property of Petri nets systems have been studied [20,21]. In addition, the methods of hierarchical process mining and deadlock detection have greatly improved the quality of mining models and the accuracy of deadlock detection [22].

Logical Petri nets have stronger modeling capability, which also leads to more complex research on their synchronous composition and sharing composition, and less relevant research results and data. Luan Wenjing et al. [23,24] proposed interactive logical Petri nets, analyzed their liveness and boundedness, and proposed a logical Petri net composition method of cooperative systems. In addition, a vector computational method is proposed to verify the properties of the sharing composition of logical Petri nets such as liveness and boundedness, which filled the gap between the study of sharing composition of logical Petri nets [25].

2.4. Emergency Management and Cross-Organizational Collaboration

In response to emergencies, an efficient emergency management system not only serves to maintain social stability and reduce the loss of life and property, but also enables various departments to respond quickly and advance in an integrated manner. Currently, great progress has been made in using traditional and colored Petri nets to simulate and model the urban emergency linkage systems and focusing on cross-organizational task collaboration. Some scholars and researchers have studied the way of cross-organizational [26,27] and cross-modal [28,29] collaborative emergency response processes and the privacy problem [30,31] involved. A method of automatically extracting the BPMN model of the emergency response process is also applied. These research results not only greatly promote the development of cross-departmental collaborative emergency response systems, but also point out the direction for the development of the industrial emergency response system in a specific field.

Due to the complexity of earthquake emergency management research, the modeling and cross-organizational collaboration of earthquake emergency departments are still in a blank state. Compared with the general cross-departmental emergency response collaboration, the cross-departmental collaboration in emergency response is more complex. In general, the cross-departmental emergency response is relatively simple and can be realized by traditional Petri nets. Moreover, this part has been studied and is relatively mature. However, the situation is more complicated in the face of cross-departmental collaboration in an earthquake emergency, which is determined by many factors, such as the large number of departments and personnel involved in the earthquake rescue work, and the need for collaboration among multiple departments in various rescue work. Traditional Petri nets can not meet the basic modeling requirements in some cases. Therefore, this paper uses traditional and logical Petri nets for modeling to meet the needs of different situations.

In summary, it can be concluded that there are several problems with the current study as follows.

(1)

At present, the main model of emergency management is constructed using traditional Petri nets, which leads to many limitations in modeling complex situations.

(2)

The application of logical Petri nets in earthquake emergency management is still a gap, and its value has not been fully utilized.

(3)

The research on cross-departmental collaboration is still more dependent on traditional Petri nets than on logical Petri nets.

(4)

In the composition of logical Petri nets, due to its complexity, the current research focus is mainly on the sharing composition. Therefore, the synchronous composition of logical Petri nets, which has the significance of cross-organizational, cross-departmental collaboration research, has not achieved research results.

2.5. Solution

In order to solve the above problems, this paper introduces the application of traditional and logical Petri nets to the field of earthquake emergency response. In this paper, the emergency plans or workflows of each earthquake emergency department are transformed into traditional or logical Petri nets models. Since there is no research on synchronous composition of logical Petri nets, there is no way to directly apply traditional and logical Petri nets across departments. In order to solve the problem of cross-departmental collaboration of traditional and logical Petri nets, this study researches the logical Petri nets and proposes the synchronous intersection operation for the two kinds of Petri nets to realize the synchronous composition. Then, it can realize the cross-departmental collaboration for traditional and logical Petri nets in order to realize the collaboration of earthquake emergency departments.

3. Preliminaries

Before studying the synchronous intersection operations of traditional and logical Petri nets to achieve cross-departmental collaboration, it is necessary to acquire knowledge about traditional and logical Petri nets. For example, the definitions of traditional Petri nets, logical Petri nets and the related meanings of transition and place are as follows:

Definition 1.

(traditional Petri nets) Let$A$be a set of activities, and let PN be a five-tuple traditional Petri nets over the set of activities$A$,$PN=\left(P,T;F,M,\alpha \right)$. P is a finite set of places on PN; T is a finite set of transitions on PN;$F\subseteq \left(P\times T\right)\cup \left(T\times P\right)$is a finite set of directed arcs on PN;$M:P\to {{\rm N}}^{+}$is a marking function on PN,$M\left(p\right)$represents the number of token in p;$\alpha :T\to A$is a mapping from a set of transitions to a set of activities. The transition firing rules for traditional Petri net PN are as follows:

(1)

For$t_i\in T$, if$\forall p_i\in •t_i$,$M\left(p_i\right)\ge 1$, t_i is enabled at M, then t_i can be fired, it can be denoted by M[t_i>.

(2)

When t_i is fired under the marking M, a new marking M′ is created, and it can be denoted by M[t_i>M′. If$p_i\in •t_i$, and$p_i\notin t_i•$, then$M\left(p_i\right)-1=M^{\prime }\left(p_i\right)$; if$p_i\in t_i•$, and$p_i\notin •t_i$, then$M\left(p_i\right)+1=M^{\prime }\left(p_i\right)$; else,$M\left(p_i\right)=M^{\prime }\left(p_i\right)$.

For$t_i\in T$,$•t_i=\left\{p_i\right|\left(p_i,t_i\right)\in F\}$,$•t_i$is called a preset of$t_i$. For$t_i\in T$,$t_i•=\left\{p_i\right|\left(t_i,p_i\right)\in F\}$,$t_i•$is called a postset of$t_i$.

The main difference between traditional and logical Petri nets is the constraint of the logical function of the transitions to the places. It enhances the representation capability of logical Petri nets.

Definition 2.

(Logical Petri nets) Let$A$be a set of activities, logical Petri net is a seven-tuple on the set of activities$A$, can be denoted by LPN,$LPN=\left(P,T;F,I,O,M,\alpha \right)$, where

(1)

P is a finite set of places on LPN;

(2)

T is a finite set of transitions on LPN,$T=T_D\cup T_I\cup T_O$, where

(a)

$T_D$denotes the set of traditional transitions of LPN, whose firing rules are consistent with those of the transitions in traditional Petri nets, with specific reference to the contents of Definition 1;

(b)

$T_I$denotes the set of logical input transitions of LPN; for$\forall t_i\in T_I$, the firing of t_i is constrained by the logical input function$f_I\left(t_i\right)$of t_i for its place$p_i$, and$p_i\in •t_i$. For$\forall t_i\in T_I$, if$f_I\left(t_i\right){|}_M=.T.$, t_i is enabled at M, then t_i can be fired, it can be denoted by M[t_i>; When t_i is fired under the marking M, a new marking M′ is created, and it can be denoted by M[t_i>M′. For$\forall p\in t_i•$,$M\left(p\right)=0$and$M^{\prime }\left(p\right)=1$; for$\forall p\notin •t_i\cup t_i•$,$M\left(p\right)=M^{\prime }\left(p\right)$;

(c)

$T_O$denotes the set of logical output transitions of LPN; for$\forall t_i\in T_O$, the firing result is constrained by the logical output function$f_O\left(t_i\right)$of t_i for its place$p_i$, and$p_i\in t_i•$. For$\forall t_i\in T_O$, if$\forall p_i\in •t_i$and$M\left(p_i\right)\ge 1$,$t_i$is enabled at M, then$t_i$can be fired, it can be denoted by M[t_i>; when t_i is fired under the marking M a new marking M′ is created, and it can be denoted by M[t_i>M′. For$\forall p\in •t_i$,$M\left(p\right)=1$and$M^{\prime }\left(p\right)=0$; for$\forall p\notin •t_i\cup t_i•$,$M\left(p\right)=M^{\prime }\left(p\right)$;

(3)

$F\subseteq \left(P\times T\right)\cup \left(T\times P\right)$is a finite set of directed arcs on LPN;

(4)

I is a mapping from a set of logical input transitions to a logical input function, for$\forall t_i\in T_I$,$I\left(t_i\right)=f_I\left(t_i\right)$;

(5)

O is a mapping from a set of logical output transitions to a logical output function, for$\forall t_i\in T_O$,$O\left(t_i\right)=f_O\left(t_i\right)$;

(6)

$\alpha :T\to A$is a mapping from a set of transitions to a set of activities;

(7)

$M:P\to {{\rm N}}^{+}$is a marking function on LPN,$M\left(p\right)$represents the number of tokens in p.

This paper uses some symbols $\otimes$, $\oplus$, $. \mathrm{T} .$ and the reachable marking functions $M^{\prime }=M+\left(R^T\oplus W^T\right)\otimes X$. Please refer to [11] for the use of these symbols and functions. In this paper, "*" represents the number 0, 1, or −1. Please refer to [11] for its use and calculation method.

Figure 1 gives an example of logical Petri nets, denoted by LPN. For LPN, $P=\left\{p_1,p_2,p_3,p_4,p_5,p_6,p_7\right\}$, $T=T_D\cup T_I\cup T_O$, $T_D=\left\{t_1\right\}$, $T_I=\left\{t_2\right\}$, $T_O=\left\{t_3\right\}$, $M={\left[1,1,0,0,0,0,0\right]}^T$. $t_1$ is a traditional transition; $t_2$ is a logical input transition, and $I\left(t_2\right)=f_I\left(t_2\right)=\left(p_2\otimes p_4\right)\wedge p_3$; $t_3$ is a logical output transition, and $O\left(t_3\right)=f_O\left(t_3\right)=\left(p_6\otimes p_7\right)$. Under the marking $M$, both p₁ and p₂ contain a token; at this time, t₁ is enabled. After t₁ is fired, p₃ contains a token; when t₂ is fired, p₅ contains a token; p₂ and p₄ do not contain any tokens at the same time.

Figure 1. LPN model.

Since both p₂ and p₃ contain a token, and p₄ does not contain a token, t₂ is enabled. When t₂ is fired, p₅ contains a token. At this point, t₃ is enabled. When t₃ is fired, there will be two kinds of cases: (1) p₆ contains a token; (2) p₇ contains a token. Assuming that case (1) occurs, p₆ contains a token and p₇ does not contain a token.

Each logical function is transformed into a disjunctive normal form with the help of discrete mathematical knowledge, and that is unique. Each disjunctive clause in the disjunctive normal form is a conjunction of all the positive or negative places associated with a logical transition. Each conjunct in the disjunctive can be converted to a vector. Then, we calculate the reachable marking of the logical Petri nets; e.g., for $I\left(t_2\right)=f_I\left(t_2\right)=\left(p_2\otimes p_4\right)\wedge p_3$, it can be transformed into a disjunctive normal form: $\left(p_2\wedge \neg p_4\wedge p_3\right)\vee \left(\neg p_2\wedge p_4\wedge p_3\right)$. Calculating the reachable marking of the logical Petri nets requires a lot of other knowledge, e.g., LPN conjunct vector, logical transition vector sets, the logical incidence matrix, the logical function matrix, reachable marking functions, and so on, which can be found in the related literature [11] and will not be discussed in this paper.

Using $LPN=\left(P,T;F,I,O,M,\alpha \right)$ in Figure 1 as an example, the logical incidence matrix of LPN is constructed:

$$R=\left[\begin{array}{ccccccc}-1& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -1& 0& 0\end{array}\right]$$

The logical function matrix of the LPN is as follows:

$$W=\left[\begin{array}{ccccccc}0& 0& 0& 0& 0& 0& 0\\ {(*,*)}^T& {(-1,0)}^T& {(-1,-1)}^T& {(0,-1)}^T& {(*,*)}^T& {(*,*)}^T& {(*,*)}^T\\ {(*,*)}^T& {(*,*)}^T& {(*,*)}^T& {(*,*)}^T& {(*,*)}^T& {(1,0)}^T& {(0,1)}^T\end{array}\right]$$

According to the firing process of LPN, a final marking $M^{\prime }={\left[0,0,0,0,0,1,0\right]}^T$ and a vector $X={\left[1,\left(1,0\right),\left(0,1\right)\right]}^T$ are obtained.

Finally, the reachable marking functions of LPN is as follows: $M^{\prime }=M+\left(R^T\oplus W^T\right)\otimes X={\left[0,0,0,0,0,1,0\right]}^T$.

4. Synchronous Intersection of Traditional and Logical Petri Nets

4.1. Definition of Synchronous Intersection

The relevant rescue work of the earthquake emergency department often needs the collaboration between two or more departments, which has the characteristics of multi- department collaboration. When solving the problem of collaboration between two departments, the problem of synchronous composition between traditional and logical Petri nets often arises. In essence, it is to solve the problem of synchronous composition between traditional and logical Petri nets. Therefore, we propose a synchronous intersection operation for the two kinds of Petri nets. It is mainly an extension on the basis of transition synchronization in the synchronous composition of traditional Petri nets, so that the synchronous composition of traditional and logical Petri nets includes the synchronization between traditional and logical transitions and the intersection of logical functions. The synchronous intersection operation is actually a synchronous intersection operation of some transitions and logical functions in the two kinds of Petri nets. Therefore, on the basis of logical Petri nets, this paper further generalizes the meaning of transitions and the logical functions. Firstly, the corresponding logical functions are extracted for the traditional transitions in traditional Petri nets. Then, the definition and calculation approach of synchronous transition units and the logical functions of synchronous transition units are proposed. It needs to start with two transitions with the same meaning, i.e., the mapping meaning of these transitions on the set of activities is consistent. Finally, the logical functions of synchronous transition units are obtained, and then the synchronous intersection operations between traditional and logical Petri nets are realized.

Definition 3.

(Synchronous transition unit) Let$A_1$and$A_2$be two sets of activities, and$A_1\cap A_2\ne \phi$.$P{N}_1=\left(P_1,T_1;F_1,M_1,{\alpha }_1\right)$is a traditional Petri net based on the activities set$A_1$, and$LP{N}_2=\left(P_2,T_2;F_2,I_2,O_2,M_2,{\alpha }_2\right)$is a logical Petri net based on the activities set$A_2$. For$\exists a_i\in A_1$, and$t_i\in T_1$,${\alpha }_1\left(t_i\right)=a_i$; for$\exists a_j\in A_2$, and$t_j\in T_2$,${\alpha }_2\left(t_j\right)=a$; if$a_i=a_j$, i.e.,${\alpha }_1\left(t_i\right)={\alpha }_2\left(t_j\right)$, synchronous transition unit can be denoted by$<t_i,t_j>$.

We define the meaning of the synchronous transition unit $<t_i,t_j>$ as a transition, which is approximately similar to what is represented by transitions t_i, t_j and has all the characteristics of transitions t_i, t_j. In fact, $<t_i,t_j>$ is obtained by performing synchronous mergence and synchronous intersection operations on the transitions t_i, t_j. Therefore, the synchronous transition unit is essentially a transition. Its nature, characteristics and meaning are the same as those of a transition, but the denotation approach is different. Its purpose is mainly to replace the original transition in the appropriate scene, to make a unified symbolic representation and to distinguish it from the traditional and logical transitions.

In this paper, there are three cases considered for the synchronous intersection operation between traditional and logical Petri nets: (1) the synchronous intersection operation between a pair of traditional transitions, and the synchronous transition unit is regarded as a traditional transition; (2) the synchronous intersection operation between a traditional transition and a logical input transition, and the synchronous transition unit is regarded as a logical input transition; (3) the synchronous intersection operation between a traditional transition and a logical output transition, and the synchronous transition unit is regarded as a logical output transition.

Both synchronous transition unit, traditional transition and logical transition all have input and output. The logical input transition and the logical output transition have related logical function to constrain their firing. The relevant logical functions can also be extracted for the traditional transition, and the logical function of the traditional transition is often used in performing synchronous intersection operations.

Definition 4.

(Extraction of traditional transition logical functions) Let$LPN=\left(P,T;F,I,O,M,\alpha \right)$be a logical Petri net. For$\forall t_i\in T$, a logical input function$f_I\left(t_i\right)$and a logical output function$f_O\left(t_i\right)$can be extracted, where

• For$\forall t_i\in T_D\cup T_O$, the logical input function is$f_I\left(t_i\right)={\wedge }_{p_i\in •t_i}{p}_i$;
• For$\forall t_i\in T_D\cup T_I$, the logical output function is$f_O\left(t_i\right)={\wedge }_{p_i\in t_i•}{p}_i$;
• For$\forall t_i\in T$in traditional Petri nets, its logical functions are extracted in the same way as the way for$\forall t_i\in T_D$in logical Petri nets. In addition, obtaining functions for the synchronous transition unit can be done by intersecting the logical functions of the two transitions that have not been merged before.

Definition 5.

(Synchronous transition unit logical functions) Let$P{N}_1=\left(P_1,T_1;F_1,M_1,{\alpha }_1\right)$be a traditional Petri net based on$A_1$, let$LP{N}_2=\left(P_2,T_2;F_2,I_2,O_2,M_2,{\alpha }_2\right)$be a logical Petri net based on$A_2$, and$A_1\cap A_2\ne \phi$, there exists a synchronous transition unit$<t_i,t_j>$in$P{N}_1$and$LP{N}_2$.$f_I\left(<t_i,t_j>\right)$is called a logical input function of$<t_i,t_j>$, and$f_O\left(<t_i,t_j>\right)$is called a logical output function of$<t_i,t_j>$, where (1)$f_I\left(<t_i,t_j>\right)=f_I\left(t_i\right)\wedge f_I\left(t_j\right)$; (2)$f_O\left(<t_i,t_j>\right)=f_O\left(t_i\right)\wedge f_O\left(t_j\right)$.

4.2. Computation of Synchronous Intersection

The synchronous intersection of traditional and logical Petri nets is actually the synchronous intersection of some transitions, which enables some transitions to be synchronized and merged in order to realize the collaboration between two departments in some work. In the two kinds of Petri nets, the transitions are divided into the traditional transitions, the logical input transitions, and the logical output transitions. Therefore, there are many cases included in the synchronous intersection operation of traditional and logical Petri nets. In this paper, we mainly consider three cases of synchronous intersection operations: (1) The traditional transitions and the traditional transitions; (2) The traditional transitions and the logical input transitions; (3) The traditional transitions and the logical output transitions.

Let $P{N}_1=\left(P_1,T_1;F_1,M_1,{\alpha }_1\right)$ be a traditional Petri net based on $A_1$, let $LP{N}_2=\left(P_2,T_2;F_2,I_2,O_2,M_2,{\alpha }_2\right)$ be a logical Petri net based on $A_2$, and $A_1\cap A_2\ne \phi$. For LPN₂, $T_2=T_{2,D}\cup T_{2,I}\cup T_{2,O}$. The three cases of synchronous intersection operations for traditional and logical Petri nets are discussed separately.

4.2.1. Synchronous Intersection Operation of Traditional Transitions

For $\exists t_i\in T_1$ and $\exists t_j\in T_{2,D}$, ${\alpha }_1\left(t_i\right)=a_i={\alpha }_2\left(t_j\right)=a_j$, $a_i\in A_1$, $a_j\in A_2$, at this time, $t_i$ and $t_j$ can perform synchronous intersection operations to make $t_i$ and $t_j$ merge together; it can be recorded as $<t_i,t_j>$, and replace $t_i$ and $t_j$. A new logical Petri net $LP{N}_3=\left(P_3,T_3;F_3,I_3,O_3,M_3,{\alpha }_3\right)$ is obtained by PN₁ and LPN₂ after the synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, mapping I composition, mapping O composition, marking composition, and mapping $\alpha$ composition.

(1)

The set of activities composition

$A_3=A_1\cup \left(A_2-a_j\right)$. Composing the set of activities $A_1$, $A_2$ of PN₁ and LPN₂. Since the activities mapped in the set of activities $A_1$ and $A_2$ are the same for transitions t_i and t_j, i.e., $a_i=a_j$, it needs to delete one of the activities $a_i$ or $a_j$. Then preforming the union operation on $A_1$ and $A_2$, $A_3$ can be obtained;

(2)

The set of places composition

$P_3=P_1\cup P_2$. Since the place deletion is not included in the synchronous intersection operation, preforming the union operation on $P_1$ and $P_2$, $P_3$ can be obtained;

(3)

The set of transitions composition

$T_{3,D}=\left(T_1-t_i\right)\cup \left(T_{2,D}-t_j\right)\cup <t_i,t_j>$. Since t_i, t_j and $<t_i,t_j>$ can be regarded as a traditional transition and belong to the set of traditional transitions, hence, it is necessary to delete t_i, t_j from $T_1$ and $T_{2,D}$, respectively. Then preforming the union operation on $T_1$, $T_{2,D}$ and $<t_i,t_j>$, $T_{3,D}$ can be obtained;

$T_{3,\mathrm{I}}=T_{2,\mathrm{I}}$, $T_{3,\mathrm{O}}=T_{2,\mathrm{O}}$. Since this part performs the synchronous intersection operation of two traditional transitions, and PN₁ is a traditional Petri net and LPN₂ is a logical Petri net, so assign $T_{2,\mathrm{I}}$ to $T_{3,\mathrm{I}}$ and $T_{2,\mathrm{O}}$ to $T_{3,\mathrm{O}}$ to obtain the set of logical input transitions and the set of logical output transitions of the new logical Petri net LPN₃;

(4)

The set of directed arcs composition

$F_3=F_1-(t_i,p_{j_{p_j\in t_i•}})-(p_{i_{p_i\in •t_i}},t_i)$. Firstly, delete all arcs from F₁ that contain t_i, F₃ can be obtained: $F_3=F_3\cup (F_2-(t_j,p_{j_{p_j\in t_j•}})-(p_{i_{p_i\in •t_j}},t_j))$; delete all arcs from F₂ that contain t_j, and then preform the union operation on F₃ and F₂;

$F_3=F_3\cup (<t_i,t_j>,p_{j_{p_j\in t_i•}})\cup (p_{i_{p_i\in •t_i}},<t_i,t_j)$; add the new directed arcs to F₃ by replacing t_i in the previously deleted arcs with $<t_i,t_j>$; $F_3=F_3\cup (<t_i,t_j>,p_{j_{p_j\in t_j•}})\cup (p_{i_{p_i\in •t_j}},<t_i,t_j>)$; add the new directed arcs to F₃ by replacing t_j in the previously deleted arcs with $<t_i,t_j>$;

(5)

Mapping I composition

I₃ = I₂. Because this part is the synchronous intersection of two traditional transitions, it does not include mapping I composition. Assign I₂ to I₃ in order to get the mapping I of LPN₃;

(6)

Mapping O composition

O₃ = O₂. Assign O₂ to O₃ in order to get the mapping O of LPN₃;

(7)

Marking composition

$M_3=M_1\cup M_2$. Since the place deletion is not included in the synchronous intersection operation, marking composition is the same as the set of places composition approach. Preforming the union operation on M₁ and M₂, M₃ can be obtained;

(8)

Mapping composition

${\alpha }_3=\left({\alpha }_1-{\alpha }_1\left(t_i\right)\right)\cup \left({\alpha }_2-{\alpha }_2\left(t_j\right)\right)$. Delete ${\alpha }_1\left(t_i\right)$, ${\alpha }_2\left(t_j\right)$ from ${\alpha }_1$, ${\alpha }_2$. Then, preforming the union operation on ${\alpha }_1$ and ${\alpha }_2$, ${\alpha }_3$ can be obtained;

${\alpha }_3={\alpha }_3\cup \left({\alpha }_3\left(<t_i,t_j>\right)=a_i\right)$; add the mapping ${\alpha }_3\left(<t_i,t_j>\right)=a_i$ of $<t_i,t_j>$ for ${\alpha }_3$.

Note: Since the synchronous transition unit obtained by the synchronous intersection operation of two traditional transitions is still a traditional transition, the extraction of logical functions and the union operation are not involved. In addition, there has been much research on the synchronous intersection operation of traditional transitions. For specific examples, please refer to relevant literature, and this paper will not give any examples.

4.2.2. Synchronous Intersection Operation of Traditional and Logical-Input Transitions

For $\exists t_i\in T_1$ and $\exists t_j\in T_{2,I}$, ${\alpha }_1\left(t_i\right)=a_i={\alpha }_2\left(t_j\right)=a_j$, $a_i\in A_1$, $a_j\in A_2$; at this time, $t_i$ and $t_j$ can perform synchronous intersection operations to make $t_i$ and $t_j$ merge together; it can be recorded as $<t_i,t_j>$, and replace $t_i$ and $t_j$. A new logical Petri net $LP{N}_4=\left(P_4,T_4;F_4,I_4,O_4,M_4,{\alpha }_4\right)$ is obtained by PN₁ and LPN₂ after a synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, the logical input functions composition, mapping I composition, mapping O composition, marking composition, and mapping $\alpha$ composition.

Some of the steps and approaches in the synchronous intersection operation of traditional transitions and logical input transitions are consistent with the synchronous intersection operation of traditional transitions. The same aspects will not be introduced here; only the differences will be introduced. The same aspects mainly include the set of activities composition, the set of places composition, the set of directed arcs composition, marking composition, and mapping $\alpha$ composition. The different aspects are as follows:

(1)

The set of transitions composition

$T_{4,D}=\left(T_1-t_i\right)\cup \left(T_{2,D}\right)$, $T_{4,I}=\left(T_{2,I}-t_j\right)\cup ⟨t_i,t_j⟩$. Since $<t_i,t_j>$, obtained after synchronous intersection operation of $t_i$ and $t_j$, can be regarded as a logical input transition and belongs to the set of logical input transitions, therefore it is necessary to delete t_i from $T_1$. Then, preforming the union operation on $T_1$ and $T_{2,D}$, $T_{4,D}$ can be obtained. Finally, deleting t_j from $T_{2,I}$ and preforming the union operation on $T_{2,I}$ and $<t_i,t_j>$, $T_{4,I}$ can be obtained.

$T_{4,O}=T_{2,O}$. Because this part is the synchronous intersection of a traditional transition and a logical input transition, it does not include the logical output transitions composition. Assign $T_{2,O}$ to $T_{4,O}$, and then get the set of logical output transitions of LPN₄.

(2)

The logical input functions composition

$f_I\left(<t_i,t_j>\right)=f_I\left(t_i\right)\wedge f_I\left(t_j\right)$, obtain the logical input function $f_I\left(t\right)={\wedge }_{p_i\in •t_i}{p}_i$ for t_i; then, intersect $f_I\left(t_i\right)$ with $f_I\left(t_j\right)$ to obtain the logical input function $f_I\left(<t_i,t_j>\right)$ of $<t_i,t_j>$.

(3)

Mapping I composition

$I_4=I_2-I_2\left(t_j\right)$; since PN₁ is a traditional Petri net, it does not involve mapping I. After deleting mapping $I_2\left(t_j\right)=f_I\left(t_j\right)$, I₂ is assigned to I₄ in order to get the mapping I₄ of LPN₄.

$I_4=I_4\cup \left(I_4\left(<t_i,t_j>\right)=f_I\left(<t_i,t_j>\right)\right)$; since $<t_i,t_j>$ can be regarded as a logical input transition, the mapping from $<t_i,t_j>$ to $f_I\left(<t_i,t_j>\right)$ needs to be added to I₄ in order to finally obtain the mapping I₄ in LPN₄.

(4)

Mapping O composition

O₄=O₂, because this part is the synchronous intersection of a traditional transition and a logical input transition, it does not include mapping O composition. Assign O₂ to O₄ in order to get the mapping O of LPN₄.

An example of a traditional Petri net is given in Figure 2a, denoted by PN₅. An example of a logical Petri net is given in Figure 2b, denoted by LPN₆. $P{N}_5=\left(P_5,T_5;F_5,M_5,{\alpha }_5\right)$ is based on A₅, $LP{N}_6=\left(P_6,T_6;F_6,I_6,O_6,M_6,{\alpha }_6\right)$ is based on A₆, and $A_5\cap A_6\ne \phi$. For $P{N}_5$, $T_5=\left\{t_1\right\}$, $P_5=\left\{p_1,p_2\right\}$, $M_5={\left[1,0\right]}^T$. For $LP{N}_6$, $T_{6,D}=\phi$, $T_{6,I}=\left\{{t_1}^{\prime }\right\}$, $T_{6,O}=\phi$, $P_6=\left\{{p_1}^{\prime },{p_2}^{\prime },{p_3}^{\prime },{p_4}^{\prime }\right\}$, $I_6\left({t_1}^{\prime }\right)=f_I\left({t_1}^{\prime }\right)=\left({p_1}^{\prime }\otimes {p_3}^{\prime }\right)\wedge {p_2}^{\prime }$, $O=\phi$, $M_6={\left[1,1,0,0\right]}^T$.

Figure 2. Synchronous intersection of a traditional transition and a logical input transition. (a) The traditional Petri net model PN₅. (b) The logical Petri net model LPN₆. (c) The logical Petri net model LPN₇.

(1)

Reachable marking functions about PN₅

Construct the logical incidence matrix of PN₅: $R_5=\left[\begin{array}{cc}-1& 1\end{array}\right]$, since PN₅ is a traditional Petri net, it is not necessary to construct the logical function matrix. t₁ is enabled under the marking $M_5={\left[1,0\right]}^T$, and it can be fired. Then, a vector $X_5=\left[1\right]$ is obtained.

Finally, the reachable marking functions of PN₅ is as follows: ${M_5}^{\prime }=M_5+R_5^T\otimes X_5={\left[0,1\right]}^T$, the final marking ${M_5}^{\prime }={\left[0,1\right]}^T$ is obtained. It means that p₂ contains a token, whereas p₁ does not.

(2)

Reachable marking functions about LPN₆

Construct the logical incidence matrix of LPN₆: $R_6=\left[\begin{array}{cccc}0& 0& 0& 1\end{array}\right]$; construct the logical function matrix of LPN₆: $W_6=\left[\begin{array}{cccc}{(-1,0)}^T& {(-1,-1)}^T& {(0,-1)}^T& {(*,*)}^T\end{array}\right]$. ${t_1}^{\prime }$ is enabled at $M_6={\left[1,1,0,0\right]}^T$, and it can be fired. Let p₁′∧p₂′ be fired under the marking $M_6={\left[1,1,0,0\right]}^T$, then a vector $X_6=\left[{(1,0)}^T\right]$ is obtained.

Finally, the reachable marking functions of LPN₆ is as follows: ${M_6}^{\prime }=M_6+\left(R_{6^T}\oplus W_6^T\right)\otimes X_6={\left[0,0,0,1\right]}^T$, the final marking ${M_6}^{\prime }={\left[0,0,0,1\right]}^T$ is obtained. It means that p₄′ contains a token, whereas p₁′, p₂′, and p₃′ do not.

(3)

Synchronous intersection operation of PN₅ and LPN₆

By the synchronous intersection operation of PN₅ and LPN₆, LPN₇ can be obtained. The specific structure of LPN₇ is given in Figure 2c. The steps are as follows:

Step 1 The set of places composition: $P_7=P_5\cup P_6=\left\{p_1,p_2,{p_1}^{\prime },{p_2}^{\prime },{p_3}^{\prime },{p_4}^{\prime }\right\}$;

Step 2 The set of transitions composition:

$T_{7,D}=\left(T_5-t_1\right)\cup \left(T_{6,D}\right)=\phi$, $T_{7,I}=\left(T_{6,I}-{t_1}^{\prime }\right)\cup <t_1,{t_1}^{\prime }>$, $T_{7,O}=T_{6,O}=\phi$;

Step 3 The set of directed arcs composition:

$F_7=F_5-(t_1,p_{j_{p_j\in t_1•}})-(p_{i_{p_i\in •t_1}},t_1)$,$F_7=F_7\cup (F_6-({t_1}^{\prime },p_{j_{p_j\in {t_1}^{\prime }•}}))$, $F_7=F_7\cup (F_6-(p_{i_{•{t_1}^{\prime }}},{t_1}^{\prime }))$, $F_7=F_7\cup (<t_1,{t_1}^{\prime }>,p_{j_{p_j\in t_1•.}})$, $F_7=F_7\cup (p_{i_{p_i\in •t_1}},<t_1,{t_1}^{\prime }>)$, $F_7=F_7\cup (<t_1,{t_1}^{\prime }>,p_{j_{p_j\in {t_1}^{\prime }•.}})$, $F_7=F_7\cup (p_{i_{p_i\in •{t_1}^{\prime }}},<t_1,{t_1}^{\prime }>)$;

Step 4 The logical input functions composition:

Obtain the logical input function $f_I\left(t_1\right)={\wedge }_{p_i\in •t_1}{p}_i=p_1$ for t₁, then intersect $f_I\left(t_1\right)$. with $f_I\left({t_1}^{\prime }\right)$ of ${t_1}^{\prime }$ to obtain the logical input function $f_I\left(<t_1,{t_1}^{\prime }>\right)=f_I\left(t_1\right)\wedge f_I\left({t_1}^{\prime }\right)=p_1\wedge \left({p_1}^{\prime }\otimes {p_3}^{\prime }\right)\wedge {p_2}^{\prime }$ of $<t_1,{t_1}^{\prime }>$;

Step 5 Mapping I composition:

$I_7=I_6-I_6\left({t_1}^{\prime }\right)$, since PN₅ is a traditional Petri net, it does not involve mapping I. After deleting mapping $I_6\left({t_1}^{\prime }\right)=f_I\left(t^{\prime }\right)$, I₆ is assigned to I₇ in order to get the mapping I₇ of LPN₇.

$I_7=I_7\cup \left(I_7\left(<t_1,{t_1}^{\prime }>\right)=f_I\left(<t_1,{t_1}^{\prime }>\right)\right)$, the mapping from $<t_1,{t_1}^{\prime }>$ to $f_I\left(<t_1,{t_1}^{\prime }>\right)$ needs to be added to I₇ in order to finally obtain the mapping I₇ in LPN₇;

Step 6 Mapping O composition: O₇ = O₆;

Step 7 Marking composition: $M_7=M_5\cup M_6={\left[1,0,1,1,0,0\right]}^T$;

Construct the logical incidence matrix of LPN₇: $R_7=\left[\begin{array}{cccccc}0& 1& 0& 0& 0& 1\end{array}\right]$; Construct the logical function matrix of LPN₇: $W_7=[{(-1,-1)}^T,{(*,*)}^T,{(-1,0)}^T,{(-1,-1)}^T,{(0,-1)}^T,\left(*,*{)}^T\right]$.

$<t_1,{t_1}^{\prime }>$ is enabled at $M_7={\left[1,01,1,0,0\right]}^T$, and it can be fired. Let p₁∧p₁′∧p₂′ be fired under the marking $M_7={\left[1,01,1,0,0\right]}^T$, then a vector $X_7=\left[{(1,0)}^T\right]$ is obtained.

Finally, the reachable marking functions of the LPN₇ is as follows: ${M_7}^{\prime }=M+\left(R_7^T\oplus W_7^T\right)\otimes X_7={\left[0,1,0,0,0,1\right]}^T$, the final marking ${M_7}^{\prime }={\left[0,1,0,0,0,1\right]}^T$ is obtained. It means that p₂ and p₄′ each contain a token, whereas p₁, p₁′, p₂′, p₃′ do not.

4.2.3. Synchronous Intersection Operation of Traditional and Logical-Output Transitions

For $\exists t_i\in T_1$ and $\exists t_j\in T_{2,O}$, ${\alpha }_1\left(t_i\right)=a_i={\alpha }_2\left(t_j\right)=a_j$, $a_i\in A_1$, $a_j\in A_2$, at this time, t_i and t_j can perform synchronous intersection operation to make t_i and t_j merge together, it can be recorded as $<t_i,t_j>$, and replace t_i and t_j. A new logical Petri net $LP{N}_8=\left(P_8,T_8;F_8,I_8,O_8,M_8,{\alpha }_8\right)$ is obtained by PN₁ and LPN₂ after synchronous intersection operation. The acquisition of the new net consists of several aspects: the set of activities composition, the set of places composition, the set of transitions composition, the set of directed arcs composition, mapping I composition, the logical output functions composition, mapping O composition, marking composition, mapping $\alpha$ composition.

Some of the steps and approaches in the synchronous intersection operation of traditional transition and logical output transition are consistent with the synchronous intersection operation of traditional transitions. The same aspects will not be introduced here, only the differences will be introduced. The same aspects mainly include the set of activities composition, the set of places composition, the set of directed arcs composition, marking composition, and mapping composition. The different aspects are as follows:

(1)

The set of transitions composition

$T_{8,D}=\left(T_1-t_i\right)\cup \left(T_{2,D}\right)$, $T_{8,O}=\left(T_{2,O}-t_j\right)\cup <t_i,t_j>$. Since $<t_i,t_j>$ obtained after synchronous intersection operation of t_i and t_j can be regarded as a logical output transition and belongs to the set of logical output transitions, it is necessary to delete t_i from T₁. Then, preforming the union operation on T₁ and T_2,D, T_8,D can be obtained. Finally, deleting t_j from T_2,O, and then preforming the union operation on T_2,O and $<t_i,t_j>$, $T_{8,O}$ can be obtained;

T_8,I = T_2,I. Because this part is the synchronous intersection of a traditional transition and a logical output transition, it does not include the logical input transitions composition. Assign T_2,I to T_8,I in order to get the set of logical input transitions of LPN₈;

(2)

Mapping I composition

I₈ = I₂. Because this part is the synchronous intersection of a traditional transition and a logical output transition, it does not include mapping I composition. Assign I₂ to I₈ in order to get the mapping I of LPN₈;

(3)

The logical output functions composition

$f_O\left(<t_i,t_j>\right)=f_O\left(t_i\right)\wedge f_O\left(t_j\right)$. Firstly, obtain $f_O\left(t_i\right)={\wedge }_{p_i\in t_i•.} p_i$ for t_i; then, intersect $f_O\left(t_i\right)$ with $f_O\left(t_j\right)$ of t_j to obtain $f_O\left(<t_i,t_j>\right)$ of $<t_i,t_j>$;

(4)

Mapping O composition

$O_8=O_2-O_2\left(t_j\right)$. Since PN₁ is a traditional Petri net, it does not involve mapping O. After deleting mapping $O_2\left(t_j\right)=f_O\left(t_j\right)$, O₂ is assigned to O₈ in order to obtain the mapping O₈ of LPN₈:

$O_8=O_8\cup \left(O_8\left(<t_i,t_j>\right)=f_O\left(<t_i,t_j>\right)\right)$. Since $<t_i,t_j>$ can be regarded as a logical output transition, the mapping of $<t_i,t_j>$ to the logical output function $f_O\left(<t_i,t_j>\right)$ needs to be added to O₈, which eventually obtains the mapping O₈ in LPN₈.

An example of a traditional Petri net is shown in Figure 3a, denoted by PN₉. An example of a logical Petri net is shown in Figure 3b, denoted by LPN₁₀. $P{N}_9=\left(P_9,T_9;F_9,M_9,{\alpha }_9\right)$ is based on A₉; $LP{N}_{10}=\left(P_{10},T_{10};F_{10},I_{10},O_{10},M_{10},{\alpha }_{10}\right)$ is based on A₁₀, and $A_9\cap A_{10}\ne \phi$. For PN₉, $T_9=\left\{t_1\right\}$, $P_9=\left\{p_1,p_2\right\}$, and $M_9={\left[1,0\right]}^T$. For LPN₁₀, $T_{10,D}=\phi$, $T_{10,I}=\phi$, $T_{10,O}=\left\{{t_1}^{\prime }\right\}$, $P_{10}=\left\{{p_1}^{\prime },{p_2}^{\prime },{p_3}^{\prime },{p_4}^{\prime }\right\}$, $I_{10}=\phi$, $O_{10}\left({t_1}^{\prime }\right)=f_O\left({t_1}^{\prime }\right)=\left({p_2}^{\prime }\otimes {p_3}^{\prime }\right)\wedge {p_4}^{\prime }$, and $M_{10}={\left[1,0,0,0\right]}^T$.

Figure 3. Synchronous intersection of a traditional transition and a logical output transition. (a) The traditional Petri net model PN₉. (b) The logical Petri net model LPN₁₀. (c) The logical Petri net model LPN₁₁.

(1)

Reachable marking functions about PN₉

The reachable marking functions steps and results are the same as the reachable marking functions of PN₅ in Figure 2a, and the final marking ${M_9}^{\prime }={\left[0,1\right]}^T$ is obtained. It means that p₂ contains a token, whereas p₁ does not.

(2)

Reachable marking functions about LPN₁₀

Construct the logical incidence matrix of LPN₁₀: $R_{10}=\left[\begin{array}{cccc}-1& 0& 0& 0\end{array}\right]$; construct the logical function matrix of LPN₁₀: $W_{10}=[{(*,*)}^T,{(1,0)}^T,{(0,1)}^T,\left(1,1{)}^T\right]$. t₁′ is enabled at $M_{10}={\left[1,0,0,0\right]}^T$, and it can be fired. Let p₂′∧p₄′ be fired under the marking $M_{10}={\left[1,0,0,0\right]}^T$, then a vector $X_{10}=\left[{(1,0)}^T\right]$ is obtained.

Finally, the reachable marking functions of the LPN₁₀ is as follows: ${M_{10}}^{\prime }=M_{10}+\left(R_{10}^T\oplus W_{10}^T\right)\otimes X_{10}={\left[0,1,0,1\right]}^T$, and the final marking $M_{10}\prime ={\left[0,1,0,1\right]}^T$ is obtained. It means that both p₂′ and p₄′ contain a token, whereas p₁′ and p₃′ do not.

(3)

Synchronous intersection operation of PN₉ and LPN₁₀

By the synchronous intersection operation of PN₉ and LPN₁₀, LPN₁₁ can be obtained. The specific structure of LPN₁₁ is given in Figure 3c. The steps are as follows:

Step 1 The set of places composition:

$P_{11}=P_9\cup P_{10}=\left\{p_1,p_2,{p_1}^{\prime },{p_2}^{\prime },{p_3}^{\prime },{p_4}^{\prime }\right\}$;

Step 2 The set of transitions composition:

$T_{11,D}=\left(T_9-t_1\right)\cup \left(T_{10,D}\right)=\phi$, $T_{11,\mathrm{I}}=T_{10,I}=\phi$, $T_{11,O}=\left(T_{10,O}-{t_1}^{\prime }\right)\cup <t_1,{t_1}^{\prime }>$;

Step 3 The set of directed arcs composition:

$F_{11}=F_9-(t_1,p_{j_{p_j\in t_1•}})-(p_{i_{p_i\in •t_1}},t_1)$, $F_{11}=F_{11}\cup (F_{10}-({t_1}^{\prime },p_{j_{p_j\in {t_1}^{\prime }•}})-(p_{i_{p_i\in •{t_1}^{\prime }}},{t_1}^{\prime }))$, $F_{11}=F_{11}\cup (<t_1,{t_1}^{\prime }>,p_{j_{p_j\in t_1•}})\cup (p_{i_{p_i\in •t_1}},<t_1,{t_1}^{\prime }>)$, $F_{11}=F_{11}\cup (<t_1,{t_1}^{\prime }>,p_{j_{p_j\in {t_1}^{\prime }•}})\cup (p_{i_{p_i\in •{t_1}^{\prime }}},<t_1,{t_1}^{\prime }>)$;

Step 4 The mapping I composition: I₁₁ = I₁₀;

Step 5 The logical output functions composition:

Obtain $f_O\left(t_1\right)={\wedge }_{p_i\in t_1•}{p}_i=p_2$ for t₁, then intersect $f_O\left(t_1\right)$ with $f_O\left({t_1}^{\prime }\right)$ of t₁′ to obtain the logical output function $f_O\left(<t_1,t^{\prime }>\right)=f_O\left(t_1\right)\wedge f_O\left({t_1}^{\prime }\right)=p_2\wedge \left({p_2}^{\prime }\otimes {p_3}^{\prime }\right)\wedge {p_4}^{\prime }$ of $<t_1,{t_1}^{\prime }>$;

Step 6 Mapping O composition:

$O_{11}=O_{10}-O_{10}\left({t_1}^{\prime }\right)$. Since PN₉ is a traditional Petri net, it does not involve mapping O. After deleting mapping $O_{10}\left({t_1}^{\prime }\right)=f_O\left({t_1}^{\prime }\right)$, O₁₀ is assigned to O₁₁ in order to get the mapping O₁₁ of LPN₁₁;

$O_{11}=O_{11}\cup \left(O_{11}\left(<t_1,{t_1}^{\prime }>\right)=f_O\left(<t_1,{t_1}^{\prime }>\right)\right)$. Since $<t_1,{t_1}^{\prime }>$ can be regarded as a logical output transition, the mapping from $<t_1,{t_1}^{\prime }>$ to $f_O\left(<t_i,t_j>\right)$ needs to be added to O₁₁, which eventually obtains O₁₁ in LPN₁₁;

Step 7 The marking composition:

$M_{11}=M_9\cup M_{10}={\left[1,0,1,0,0,0\right]}^T$;

Construct the logical incidence matrix of LPN₁₁: $R_{11}=\left[-1,0,-1,0,0,0\right]$; construct the logical function matrix of LPN₁₁: $W_{11}=[{(\ast ,\ast )}^T,{(1,1)}^T,{(\ast ,\ast )}^T,{(1,0)}^T,{(0,1)}^T,\left(1,1{)}^T\right]$;

$<t_1,{t_1}^{\prime }>$ is enabled at $M_{11}={\left[1,01,0,0,0\right]}^T$, and it can be fired. Let p₂∧p₂′∧p₄′ be fired under the marking $M_{11}={\left[1,01,0,0,0\right]}^T$, then a vector $X_{11}=\left[{(1,0)}^T\right]$ is obtained.

Finally, the reachable marking functions of LPN₁₁ is as follows: ${M_{11}}^{\prime }=M_{11}+\left(R_{11}^T\oplus W_{11}^T\right)\otimes X_{11}={\left[0,1,0,1,0,1\right]}^T$, and the final marking ${M_{11}}^{\prime }={\left[0,1,0,1,0,1\right]}^T$ is obtained. It means that p₂, p₂′,and p₄′ each contain a token, whereas p₁, p₁′,and p₃′ do not.

5. Simulation Experiments

In Section 4, three cases of synchronous intersection operation of traditional and logical Petri nets are introduced and proved by examples. This chapter begins to take the emergency plan of a Provincial Earthquake Administration as an example to verify and simulate the synchronous intersection operation.

According to the information of the emergency plan, the subordinate units of the Provincial Earthquake Administration include the Bureau Office, the Emergency Rescue Department, the Earthquake Monitoring and Forecasting Center, the Earthquake Damage Defense Department, the Earthquake Disaster Defense Center, the Science and Technology Department (STD), the institutional service center, the development and finance department, and the Earth Observation Institute (EOI) affiliated with the National Earthquake Administration (NEA). These departments have corresponding work arrangements and a complete set of emergency response processes after an earthquake. Once the earthquake occurs, the above-mentioned departments will quickly start emergency response according to the established plan and complete a series of rescue work in the shortest time in order to shorten rescue time, improve rescue efficiency and reduce loss of life and property.

Since modeling experiments and simulations for the complete emergency plan and all departments in this part of the work would lead to a huge workload and a long article, this paper selects one of them, the EOI and the STD, as an example for modeling verification. In this way, it not only reduces the workload but also verifies the feasibility of synchronous intersection operations.

In this section, a traditional Petri net and a logical Petri net model are modeled for the EOI and the STD, respectively. Then, according to Section 4.2, synchronous intersection operation between traditional and logical Petri net model is conducted, and finally a complete, cross-departmental logical Petri net model is obtained. It also includes a series of emergency work completed by the EOI and the STD after the earthquake.

5.1. Emergency Planning Restatement

Through careful analysis of the emergency plan, this paper extracts tasks from the EOI and the STD, mainly including the following tasks and processes:

The EOI: after the earthquake, the EOI starts the emergency response and then mainly performs two tasks: (1) sending a team to the site to set up a mobile monitoring center station in the earthquake area, obtaining the mobile monitoring center station data in the earthquake area and reporting it to the NEA, and then obtaining the approval of the NEA; (2) starting the emergency response for strong earthquakes in key areas in collaboration with the STD, organizing relevant experts from the institute to work with the staff of the STD and combine with the approval of the NEA to come up with emergency recommendations for strong earthquakes in key areas, and reporting the recommendations to the NEA.

The STD: After the earthquake, the STD initiates emergency response for strong earthquakes in key areas in collaboration with the EOI, organizes experts in reservoirs, key buildings, strong vibration and other industries to conduct research, invites gravity experts to participate in research when necessary, and finally proposes emergency response recommendations for strong earthquakes in key areas in collaboration with the EOI and implements all emergency recommendations.

5.2. EOI Model and Its Reachable Markings

Based on the EOI emergency response workflow in Section 5.1, a set of activities can be extracted. A₁₂ = {Activate emergency, Dispatch on-site team, Set up stations, Obtain station data, Report to the NEA, Get approval, Activate strong earthquake response, Organize Institute experts, Make a suggestion, Presented to the NEA}; a traditional Petri net model $P{N}_{12}=\left(P_{12},T_{12};F_{12},M_{12},{\alpha }_{12}\right)$ about the EOI can be developed based on this and in conjunction with the emergency response workflow of the EOI. For $P{N}_{12}=\left(P_{12},T_{12};F_{12},M_{12},{\alpha }_{12}\right)$, $P_{12}=\left\{p_1,p_2,p_3,p_4,p_5,p_6,p_7,p_8,p_9,p_{10},p_{11},p_{12}\right\}$, $T_{12}=\left\{t_1,t_2,t_3,t_4,t_5,t_6,t_7,t_8,t_9,t_{10}\right\}$, and $M_{12}={\left[1,0,0,0,0,0,0,0,0,0,0,0,0\right]}^T$, Figure 4 shows the specific structure of the PN₁₂ model.

Figure 4. The traditional Petri net model PN₁₂ structure of the EOI.

The emergency response workflow for the EOI can be calculated for its marking. Firstly, construct the logical incidence matrix of PN₁₂:

$$R_{12}=\left[\begin{array}{cccccccccccc}-1& 1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 0& 0& -1& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 1\end{array}\right]$$

Since PN₁₂ is a traditional Petri net, it is not necessary to construct the logical function matrix.

Combining the emergency response workflow of the EOI in Section 5.1 and Figure 4, it can be seen that t₁ is enabled at $M_{12}$, and the new marking $M_{12}^{\left(1\right)}={\left[0,1,0,0,0,0,0,1,0,0,0,0\right]}^T$ is obtained after t₁ is fired. At this time, t₂ and t₇ are enabled, then t₂ and t₇ are fired to obtain the new marking $M_{12}^{\left(2\right)}={\left[0,0,1,0,0,0,0,0,1,0,0,0\right]}^T$. Next, t₃, t₄, t₅, t₆, and t₈ are fired in turn to obtain the marking $M_{12}^{\left(3\right)}={\left[0,0,0,0,0,0,1,0,0,1,0,0\right]}^T$.

Then, t₉ is enabled at $M_{12}^{\left(3\right)}$, and new marking $M_{12}^{\left(4\right)}={\left[0,0,0,0,0,0,0,0,0,0,1,0\right]}^T$ is obtained after t₉ is fired. t₁₀ is enabled at $M_{12}^{\left(4\right)}$, and the final marking $M_{12}^{\left(5\right)}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ is obtained after t₁₀ is fired. At this point, the relevant emergency response workflow of the EOI has been completed. Finally, p₁₂ contains a token, whereas p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈, p₉, p₁₀, and p₁₁ do not.

Accordingly, $X_{12}={\left[1,1,1,1,1,1,1,1,1,1\right]}^T$ can be constructed. Finally, the reachable marking function of PN₁₂ is as follows: ${M_{12}}^{\prime }=M_{12}+R_{12}^T\otimes X_{12}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$; by marking the calculation result, ${M_{12}}^{\prime }={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ can be judged to be consistent with the final marking $M_{12}^{\left(5\right)}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$.

According to the calculation results, it can be judged that ${M_{12}}^{\prime }={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ is consistent with the final marking $M_{12}^{\left(5\right)}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$.

5.3. STD Model and Its Reachable Markings

Based on the STD emergency response workflow in Section 5.1, a set of activities can be extracted. A₁₃ = {Initiate emergency, Activate strong earthquake response, Organize reservoir experts, Organize construction experts, Organize vibration experts, Organize gravity experts, Make a suggestion, Implementing recommendations}, a logical Petri net model $LP{N}_{13}=\left(P_{13},T_{13};F_{13},I_{13},O_{13},M_{13},{\alpha }_{13}\right)$ about the STD can be developed based on this and in conjunction with the emergency response workflow of the STD. For $LP{N}_{13}=\left(P_{13},T_{13};F_{13},I_{13},O_{13},M_{13},{\alpha }_{13}\right)$, $P_{13}=\left\{{p_1}^{\prime },{p_2}^{\prime },{p_3}^{\prime },{p_4}^{\prime },{p_5}^{\prime },{p_6}^{\prime },{p_7}^{\prime },{p_8}^{\prime },{p_9}^{\prime },{p_{10}}^{\prime },{p_{11}}^{\prime },{p_{12}}^{\prime }\right\}$, $T_{13}=T_{13,I}\cup T_{13,\mathrm{O}}\cup T_{13,\mathrm{D}}$, $T_{13,\mathrm{I}}=\left\{{t_7}^{\prime }\right\}$, $T_{13,\mathrm{O}}=\left\{{t_2}^{\prime }\right\}$, $T_{13,\mathrm{D}}=\left\{{t_1}^{\prime },{t_3}^{\prime },{t_4}^{\prime },{t_5}^{\prime },{t_6}^{\prime },{t_8}^{\prime }\right\}$, $O_{13}\left({t_2}^{\prime }\right)=f_O\left({t_2}^{\prime }\right)={p_3}^{\prime }\wedge {p_5}^{\prime }\wedge {p_7}^{\prime }\vee {p_9}^{\prime }$, $I_{13}\left({t_7}^{\prime }\right)=f_I\left({t_7}^{\prime }\right)={p_4}^{\prime }\wedge {p_6}^{\prime }\wedge {p_8}^{\prime }\vee {p_{10}}^{\prime }$, and $M_{13}={\left[1,0,0,0,0,0,0,0,0,0,0,0\right]}^T$. Figure 5 shows the specific structure of the PN₁₃ model.

Figure 5. The logical Petri net model LPN₁₃ structure of the STD.

The emergency response workflow for the STD can be calculated for Its marking. Firstly, construct the logical incidence matrix of LPN₁₃:

$$R_{13}=\left[\begin{array}{cccccccccccc}-1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -1& 1& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& -1& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& -1& 1& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& -1& 1& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& -1& 1\end{array}\right]$$

Construct the logical function matrix of LPN₁₃:

$$W_{13}={\left[\begin{array}{cccccccc}0& (*,*)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (*,*)& 0\\ 0& (1,1)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (-1,-1)& 0\\ 0& (1,1)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (-1,-1)& 0\\ 0& (1,1)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (-1,-1)& 0\\ 0& (1,0)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (-1,0)& 0\\ 0& (*,*)& 0& 0& 0& 0& (*,*)& 0\\ 0& (*,*)& 0& 0& 0& 0& (*,*)& 0\end{array}\right]}^T$$

Combining the emergency response workflow of the STD in Section 5.1 and Figure 5, it can be seen that t₁′ is enabled at $M_{13}$, and the new marking $M_{13}^{\left(1\right)}={\left[0,1,0,0,0,0,0,0,0,0,0,0\right]}^T$ is obtained after t₁′ is fired. At this time, t₂′ is enabled. Let p₃′∧p₅′∧p₇′∧¬p₉′ be fired under the marking $M_{13}^{\left(1\right)}$, and t₂′ are fired to get the new marking $M_{13}^{\left(2\right)}={\left[0,0,1,0,1,0,1,0,0,0,0,0\right]}^T$. Next, t₁′, t₃′, t₄′, and t₅′ are fired in turn to get the marking $M_{13}^{\left(3\right)}={\left[0,0,0,1,0,1,0,1,0,0,0,0\right]}^T$. The logical input function p₄′∧p₆′∧p₈′∧¬p₁₀′ for t₇′ is fired under $M_{13}^{\left(3\right)}$, whereas t₇′ is enabled. Then, the new marking $M_{13}^{\left(4\right)}={\left[0,0,0,0,0,0,0,0,0,0,1,0\right]}^T$ is obtained after t₇′ is fired. At this time, t₈′ is enabled at $M_{13}^{\left(4\right)}$, and the final marking $M_{13}^{\left(5\right)}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ is obtained after t₈′ is fired. At this point, the relevant emergency response workflow of the STD has been completed. Finally, p₁₂′ contains a token, whereas p₁′, p₂′, p₃′, p₄′, p₅′, p₆′, p₇′,p₈′, p₉′, p₁₀′, and p₁₁′ do not.

Accordingly, a vector $X_{13}={\left[1,{(0,1)}^T,1,1,1,0,{(0,1)}^T,1\right]}^T$ can be constructed. Finally, the reachable marking functions of LPN₁₃ is as follows: ${M_{13}}^{\prime }=M_{13}+\left(R_{13}^T\oplus W_{13}^T\right)\otimes X_{13}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$.

According to the calculation results, it can be judged that ${M_{13}}^{\prime }={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ is consistent with the final marking $M_{13}^{\left(5\right)}={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$.

5.4. Synchronous Intersection of PN₁₂ and LPN₁₃

For PN₁₂ and LPN₁₃, $\exists t_7,t_9\in P{N}_{12}$, $\exists {t_2}^{\prime },{t_7}^{\prime }\in LP{N}_{13}$, ${\alpha }_{12}\left(t_7\right)={\alpha }_{13}\left({t_2}^{\prime }\right)=‘\mathrm{Activate} \mathrm{strong} \mathrm{earthquake} \mathrm{response}’$ and ${\alpha }_{12}\left(t_9\right)={\alpha }_{13}\left({t_7}^{\prime }\right)=‘\mathrm{Make} \mathrm{a} \mathrm{suggestion}’$. At this time, $<t_7,{t_2}^{\prime }>$ and $<t_9,{t_7}^{\prime }>$ can be obtained by performing synchronous intersection operations of t₇ and t₂′, t₉ and t₇′. Since t₇ and t₉ belong to PN₁₂, and t₂′ and t₇′ belong to LPN₁₃, the synchronous intersection operations of t₇ and t₂′ and the synchronous intersection operations of t₉ and t₇′ can be performed in parallel. A new logical Petri net LPN₁₄ is obtained by synchronous intersection operations of PN₁₂ and LPN₁₃, and $LP{N}_{14}=\left(P_{14},T_{14};F_{14},I_{14},O_{14},M_{14},{\alpha }_{14}\right)$.

Step 1 The set of places composition: P₁₄ = P₁₂∪P₁₃ = {p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈, p₉, p₁₀, p₁₁, p₁₂, p₁′, p₂′, p₃′, p₄′, p₅′, p₆′, p₇′, p₈′, p₉′, p₁₀′, p₁₁′, p₁₂′};

Step 2 The set of transitions composition:

$T_{14,D}=\left(T_{12}-t_7-t_9\right)\cup \left(T_{13,D}\right)=\left\{t_1,t_2,t_3,t_4,t_5,t_6,t_8,t_{10},{t_1}^{\prime },{t_3}^{\prime },{t_4}^{\prime },{t_5}^{\prime },{t_6}^{\prime },{t_8}^{\prime }\right\}$, $T_{14,I}=\left(T_{13,I}-{t_7}^{\prime }\right)\cup <t_9,{t_7}^{\prime }>$, $T_{14,O}=\left(T_{13,O}-{t_2}^{\prime }\right)\cup <t_7,{t_2}^{\prime }>$, $T_{14}=T_{14,D}\cup T_{14,I}\cup T_{14,O}$;

Since t₇ is a traditional transition and t₂′ is a logical output transition, $<t_7,{t_2}^{\prime }>$ can be regarded as a logical output transition. Since t₉ is a traditional transition and t₇′ is a logical input transition, $<t_9,{t_7}^{\prime }>$ can be regarded as a logical input transition.

Step 3 The set of directed arcs composition:

$F_{14}=F_{12}-(t_7,p_{i_{p_i\in {t_7}^{\prime }•}})-(p_{j_{p_j\in •{t_7}^{\prime }}},t_7)$, $F_{14}=F_{14}\cup (<t_7,{t_2}^{\prime }>,p_{i_{p_i\in {t_7}^{\prime }•}})\cup (p_{j_{p_j\in •{t_7}^{\prime }}},<t_7,{t_2}^{\prime }>)$,

$F_{14}=F_{14}\cup (F_{13}-({t_2}^{\prime },p_{i_{p_i\in {t_2}^{\prime }•}})-(p_{j_{p_j\in •{t_2}^{\prime }}},{t_2}^{\prime }))$, $F_{14}=F_{14}\cup (<t_7,{t_2}^{\prime }>,p_{i_{p_i\in {t_2}^{\prime }•}})\cup (p_{j_{p_j\in •{t_2}^{\prime }}},<t_7,{t_2}^{\prime }>)$,

$F_{14}=F_{14}\cup (F_{12}-(t_9,p_{i_{p_i\in t_9•}})-(p_{j_{p_j\in •t_9}},t_9))$, $F_{14}=F_{14}\cup (<t_9,{t_7}^{\prime }>,p_{i_{p_i\in {t_9}^{•}}})\cup (p_{j_{p_j\in •t_9}},<t_9,{t_7}^{\prime }>)$,

$F_{14}=F_{14}\cup (F_{13}-({t_7}^{\prime },p_{i_{p_i\in {t_7}^{\prime }•}})-(p_{j_{p_j\in •{t_7}^{\prime }}},{t_7}^{\prime }))$, $F_{14}=F_{14}\cup (<t_9,{t_7}^{\prime }>,p_{i_{p_i\in {t_7}^{\prime }•}})\cup (p_{j_{p_j\in •{t_7}^{\prime }}},<t_9,{t_7}^{\prime }>)$;

After the above operation the final set of directed arcs of the logical Petri net LPN₁₄ is obtained. The specific structure of LPN₁₄ is given in Figure 6.

Figure 6. The logical Petri net model LPN₁₄.

Step 4 The logical output functions composition:

$<t_7,{t_2}^{\prime }>$ can be regarded as a logical output transition. Therefore, it is necessary to obtain the logical output function $f_O\left(t_7\right)={\wedge }_{p_i\in t_7•}{p}_i=p_9$ for t₇, then intersect $f_O\left(t_7\right)$ with $f_O\left({t_2}^{\prime }\right)={p_3}^{\prime }\wedge {p_5}^{\prime }\wedge {p_7}^{\prime }\vee {p_9}^{\prime }$ of t₂′ to obtain the logical output function $f_O\left(<t_7,{t_2}^{\prime }>\right)=f_O\left(t_7\right)\wedge f_O\left({t_2}^{\prime }\right)=p_9\wedge {p_3}^{\prime }\wedge {p_5}^{\prime }\wedge {p_7}^{\prime }\vee {p_9}^{\prime }$ of $<t_7,{t_2}^{\prime }>$;

Step 5 Mapping O composition:

$O_{14}=O_{13}-O_{13}\left({t_2}^{\prime }\right)$; since PN₁₂ is a traditional Petri net, it does not involve mapping O. After deleting the mapping $O_{13}\left({t_2}^{\prime }\right)=f_O\left({t_2}^{\prime }\right)$, O₁₃ is assigned to O₁₄ in order to get the mapping O₁₄ of LPN₁₄;

$O_{14}=O_{14}\cup \left(O_{14}\left(<t_7,{t_2}^{\prime }>\right)=f_O\left(<t_7,{t_2}^{\prime }>\right)\right)$; since $<t_7,{t_2}^{\prime }>$ can be regarded as a logical output transition, the mapping from $<t_7,{t_2}^{\prime }>$ to $f_O\left(<t_7,{t_2}^{\prime }>\right)$ needs to be added to O₁₄, which eventually obtains O₁₄ in LPN₁₄;

Step 6 The logical input functions composition:

$<t_9,{t_7}^{\prime }>$ can be regarded as a logical input transition. Therefore, it is necessary to obtain the logical input function $f_I\left(t_9\right)={\wedge }_{p_i\in •t_9}{p}_i=p_7\wedge p_{10}$ for t₉, then intersect $f_I\left(t_9\right)$ with $f_I\left({t_7}^{\prime }\right)={p_4}^{\prime }\wedge {p_6}^{\prime }\wedge {p_8}^{\prime }\vee {p_{10}}^{\prime }$ of t₇′ to obtain the logical input function $f_I\left(<t_9,{t_7}^{\prime }>\right)=f_I\left(t_9\right)\wedge f_I\left({t_7}^{\prime }\right)=p_7\wedge p_{10}\wedge {p_4}^{\prime }\wedge {p_6}^{\prime }\wedge {p_8}^{\prime }\vee {p_{10}}^{\prime }$ of $<t_9,{t_7}^{\prime }>$;

Step 7 Mapping I composition:

$I_{14}=I_{13}-I_{13}\left({t_7}^{\prime }\right)$; since PN₁₂ is a traditional Petri net, it does not involve mapping I. After deleting the mapping $I_{13}\left({t_7}^{\prime }\right)=f_I\left({t_7}^{\prime }\right)$, I₁₃ is assigned to I₁₄ in order to get the mapping I₁₄ of LPN₁₄;

$I_{14}=I_{14}\cup \left(I_{14}\left(<t_9,{t_7}^{\prime }>\right)=f_I\left(<t_9,{t_7}^{\prime }>\right)\right)$; since $<t_9,{t_7}^{\prime }>$ can be regarded as a logical input transition, the mapping from $<t_9,{t_7}^{\prime }>$ to $f_I\left(<t_9,{t_7}^{\prime }>\right)$ needs to be added to I₁₄, which eventually obtains I₁₄ in LPN₁₄;

Step 8 The marking composition:

$M_{14}=M_{12}\cup M_{13}={\left[1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0\right]}^T$;

Construct the logical incidence matrix of LPN₁₄: Since the logical incidence matrix $R_{14}$ of LPN₁₄ is large, it is split into multiple row vectors.

$r\left[t_1\right]=\left(-1,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_2\right]=\left(0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_3\right]=\left(0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_4\right]=\left(0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_5\right]=\left(0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_6\right]=\left(0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[<t_7,{t_2}^{\prime }>\right]=\left(0,0,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[t_8\right]=\left(0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[<t_9,{t_7}^{\prime }>\right]=\left(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0\right)$,

$r\left[t_{10}\right]=\left(0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[{t_1}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0,0,0\right)$,

$r\left[{t_3}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,0\right)$,

$r\left[t_4\prime \right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0\right)$,

$r\left[{t_5}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0\right)$,

$r\left[{t_6}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0\right)$,

$r\left[{t_8}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1\right)$;

The final logical incidence matrix of LPN₁₄ can be obtained: R₁₄ = (r[t₁]^T,r[t₂]^T,r[t₃]^T, r[t₄]^T,r[t₅]^T,r[t₆]^T,r[<t₇,t₂′>]^T,r[t₈]^T,r[<t₉,t₇′>]^T,r[t₁₀]^T,r[t₁′]^T,r[t₃′]^T,r[t₄′]^T,r[t₅′]^T,r[t₆′]^T,r[t₈′]^T);

Construct the logical function matrix of LPN₁₄: Since the logical function matrix $W_{14}$ of LPN₁₄ is large, it is split into multiple row vectors.

$w\left[t_1\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_2\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_3\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_4\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_5\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_6\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

w[<t₇,t₂′>] = ((*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(1,1)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(1,1)^T,(*,*)^T,(1,1)^T,(*,*)^T,(1,1)^T,(*,*)^T,(1,0)^T,(*,*)^T,(*,*)^T,(*,*)^T),

$w\left[t_8\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

w[<t₉,t₇′>] = ((*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(*,*)^T,(−1,−1)^T,(*,*)^T,(*,*)^T, (*,*)^T,(*,*)^T,(−1,−1)^T,(*,*)^T,(−1,−1)^T,(*,*)^T,(−1,−1)^T,(*,*)^T,(−1,0)^T,(*,*)^T,(*,*)^T,(*,*)^T),

$w\left[t_{10}\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[{t_1}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[{t_3}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[t_4\prime \right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[{t_5}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[{t_6}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$,

$w\left[{t_8}^{\prime }\right]=\left(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0\right)$;

The final logical function matrix of LPN₁₄ can be obtained: W₁₄ = (w[t₁]^T,w[t₂]^T,w[t₃]^T, w[t₄]^T,w[t₅]^T,w[t₆]^T,w[<t₇,t₂′>]^T,w[t₈]^T,w[<t₉,t₇′>]^T,w[t₁₀]^T,w[t₁′]^T,w[t₃′]^T,w[t₄′]^T,w[t₅′]^T,w[t₆′]^T,w[t₈′]^T).

Based on the vector $X_{12}={\left[1,1,1,1,1,1,1,1,1,1\right]}^T$ constructed from the emergency response workflow of the EOI and the vector $X_{13}={\left[1,{(0,1)}^T,1,1,1,0,{(0,1)}^T,1\right]}^T$ constructed from the emergency response workflow of the STD, $X_{14}={\left[1,1,1,1,1,1,{(0,1)}^T,1,{(0,1)}^T,1,1,1,1,1,0,1\right]}^T$ constructed from the emergency response workflow of LPN₁₄ can be obtained.

The reachable marking functions of LPN₁₄ is as follows: ${M_{14}}^{\prime }=M_{14}+\left(R_{14}^T\oplus W_{14}^T\right)\otimes X_{14}$, and ${M_{14}}^{\prime }={\left[0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$, which means that both p₁₂ and p₁₂′ contain a token, whereas p₁, p₂, p₃, p₄, p₅, p₆, p₇, p₈, p₉, p₁₀, p₁₁, p₁′, p₂′, p₃′, p₄′, p₅′, p₆′, p₇′, p₈′, p₉′, p₁₀′, and p₁₁′ do not. The marking ${M_{14}}^{\prime }$ is consistent with the final marking ${M_{12}}^{\prime }={\left[0,0,0,0,0,0,0,0,0,0,0,1\right]}^T$ and ${M_{13}}^{\prime }={(0,0,0,0,0,0,0,0,0,0,0,1)}^T$ of PN₁₂ and LPN₁₃. Thus, LPN₁₄ is fully capable of representing all the reachable marking states of PN₁₂ and LPN₁₃, and synchronization of t₇, t₂′ and t₉, t₇′ is achieved.

In the synchronous intersection operation of PN₁₂ and LPN₁₃, two synchronous transition units $<t_7,{t_2}^{\prime }>$ and $<t_9,{t_7}^{\prime }>$ are obtained by synchronous intersection operations of the transitions t₇, t₂′ and t₉, t₇′. The synchronous intersection operation of PN₁₂ and LPN₁₃ achieves the collaboration between the EOI and the STD for the tasks of “Activate strong earthquake response” and “Make a suggestion”, which solves the problem of cross-departmental collaboration.

The above examples can illustrate that after the synchronous intersection of the original Petri nets and the original logical Petri nets, the new logical Petri nets can still represent the original Petri nets and the original logical Petri nets in each of the reachable marking states. It also realizes the cross-departmental collaboration, the synchronization of logical transition and traditional transition, and the intersection operation of related logical functions; meanwhile, it provides an approach for the collaboration between traditional and logical Petri nets for the relevant earthquake emergency departments.

6. Conclusions

This study proposes a synchronous intersection operation for traditional and logical Petri nets to solve the problem of the collaboration between different departments. It realizes the mergence and synchronization of transitions in the two kinds of Petri nets, and effectively solves the cooperation problem of earthquake emergency departments in practical applications.

Firstly, this paper introduces the related knowledge of traditional and logical Petri nets. Secondly, the definition of synchronous transition unit, logical functions of synchronous transition unit, and the approaches to extraction logical functions of traditional transition are proposed. Then, three cases of synchronous intersection operations in traditional and logical Petri nets are introduced and exemplified, respectively.

Finally, this paper applies the modeling approaches of traditional and logical Petri nets to the field of earthquake emergency response. Through the model extraction of the earthquake emergency plan and the verification of relevant examples, it can be concluded that the synchronous intersection operation of traditional and logical Petri nets can solve the synchronization problem with the two kinds of Petri nets in order to realize the collaboration between two departments, especially the earthquake emergency department.

At present, there are still a lot of research issues for the synchronous intersection operation of traditional and logical Petri nets. The future research mainly includes several aspects as follows: (1) it is meaningful to extend the synchronous intersection operation to synchronize multiple traditional and logical Petri nets; (2) the synchronous intersection operation among the logical Petri nets is further considered; (3) the approaches to compute the reachable markings lay the foundation for the development of synchronous intersection operations of the logical-input and logical-output transitions; (4) in order to verify the availability and soundness of the approach proposed in this paper, we will prove the reachability, boundedness, safety, liveness and fairness of the composition models in the future.