Shared P-Type Logic Petri Net Composition and Property Analysis: A Vector Computational Method

Petri net (PN) is an effective modeling and analysis tool for discrete event systems. By attaching a first-order logic predicate logic formula to a transition in a PN, a high-level Petri net named Logic Petri Net (LPN) is obtained. LPN has been proved to have equivalent modeling capability with inhibition Petri nets but keeps simpler net structures than the latter. It has the advantage of modeling the cooperative systems with the function to process batch and indeterminate resources. This paper proposes a vector computational method for LPN compositional analysis. It studies the composition of LPNs with the shared P-type subnets. Each logical expression is transformed into a unique disjunctive normal one and then into a unique set of vectors. A vector computational method is proposed such that the properties of the composition of the shared P-type subnets such as liveness, boundedness, and reversibility are verified. An E-commerce system with customers, merchants, and a third-party is constructed to illustrate the method. This paper can improve the state of the art in the theory of LPNs.


I. INTRODUCTION
Petri net (PN) proposed by C. A. Petri [1] is a modeling tool to describe discrete event systems and concurrent and parallel systems intuitively and effectively. In the past halfcentury, PN and its extensions such as colored PN (CPN) [2], [3], Timed PN (TPN) [4], and stochastic PN (SPN) [5] have been proposed and used for modeling, simulation, and analysis in a great number of application areas such as computer operating systems [6], flexible manufacturing [7], communications, business processes [8], and transportation systems [9]- [11]. Because of their well mathematical foundation and graphical expression, PN and its extensions are used to describe, analyze, and make sure the systems The associate editor coordinating the review of this manuscript and approving it for publication was Lu Liu . have desirable properties such as liveness, boundedness, and reversibility before the installment of the systems. High-level PNs such as CPN, TPN, and SPN can model complex systems with more concise net structures than the classical PNs. However, they have equivalent modeling capability with classical PNs. Thus, in order to strengthen the capability of the model, augmented PNs such as inhibition Petri nets (IPNs) [12] are proposed.
A cooperative system usually needs a sub-system to interact by exchanging messages with other sub-systems. During the intersection processes, a sub-system may process data from several other sub-systems simultaneously. Thus, it requires that the system is with a function of processing batch resources. Meanwhile, when a sub-system waits for feedback from the other sub-systems, the latter may not give a response. Another condition is that a system does not respond in a certain time interval which means a delay happens. Thus, there always exist indeterminate conditions. An example of the aforementioned situations is a basic E-commerce system that realizes the trade between a merchant and several customers as shown in Fig. 1. From the figure, we can see that the merchant system needs to deal with several customers simultaneously and handles a batch of resources from the customers in a time interval. In the trading process, indeterminate resource intersection from both the merchant and customers always exists. For example, customers can order products of different quantity of products that may be ordered by different customers, shortage of products could be faced by the merchant, or a customer may cancel an order during the whole trading progress event if they have received their products. Thus, we should design some special models to describe and analyze the batch and indeterminate processing of data, which is significant from theoretical and practical aspects.
Logic Petri net (LPN) is such a model that describes and analyzes the batch and indeterminate processing of data [12]. It is obtained by attaching a first-logic predicate logic formula to a transition in a classical PN in [13]. Du and Guo [12] verify that each LPN has an equivalent inhibition Petri net (IPN) model. In an LPN, transitions are divided into two groups, i.e., normal transitions and logic transitions. Normal transitions are the same as those in classical PNs, and a logic transition has a first-order predicate logic expression with connected input or output places as the predicate. In detail, logic transitions are further divided into logic input transitions and output transitions. The former has a logic expression of their input places named logic input expression while the latter has a logic expression of their output places named as a logic output transition. A logic input expression restricts the firing of the transition by checking if the tokens in the input places satisfy the expression. A logic output expression restricts the new state such that after the transition fires the token in its output places satisfy the logic transition. Fortunately, batch and indeterminate resource processing can be modeled by the logic transitions in LPNs. Thus, LPNs have enhanced capability for modeling more complex systems. Till now, many studies have used LPNs to model and analyze E-commerce systems [13] and business processes [14]. Sanders and Meyer propose stochastic activity networks that have predicates and functions on markings of places [15]. They can describe the processing of batch and indeterminate data. However, their analysis is very complicated because each transition could have two or more predicates or functions. In [16], we present a vector matching method to analyze the properties of LPNs such as reachability, liveness, and reversibility. In [17], we study interactive logic Petri nets and design some strategies to ensure good properties. One of the important properties is compatibility which concerns if the subsystems in the cooperative system have proper interactions with each other. It characterizes cooperative abilities in practice and analyzes the relationships of compatibility with liveness and boundedness. In [18], we present a systematic LPN synthesis approach for cooperative systems by gradually refining basic design modules. It designs some control strategies to guarantee that the synthesized LPNs have desirable properties. This work is of significance in the sense that it provides industrial engineers and academic researchers a methodology of applying LPNs to modeling and analysis of cooperative systems. Till now, there is no good method to verify the composition of LPN models. Thus, this paper proposes a vector computational method for analyzing the compositional properties of LPNs. It studies the composition of LPNs with the shared P-type subnets. Each logical expression is transformed into a unique disjunctive normal one and then into a unique set of vectors. A vector computational method is proposed such that the properties of the composition of LPN models with two types such as liveness, boundedness, and reversibility are verified.
The rest of the paper is organized as follows. Section II briefly reviews the basic concepts of LPNs. Section III shows a vector computational method where each logical expression is transformed into a unique disjunctive normal one and then turned into a unique set of vectors. Section IV describes the composition of shared P-type LPN models. In Section V, an E-commerce system composed of models of customers, merchants, and a third-party is constructed to illustrate the proposed approach. Section V concludes this paper and discusses future work.

II. PRELIMINARIES
In this section, some concepts of PNs [19]- [27] and LPNs [12] are briefly presented. Firstly, we give some notations that will be used in the paper. Let R 0 and N denote the real number set and the natural number set, respectively, and N + be the set of positive integer numbers. 3) I : P × T → N denotes a weight function on a directed arc from a place p ∈ P to a transition t ∈ T , where (a) I (p, t) = 0 if there is no arc from p to t; and (b) I (p, t) > 0 if there is an arc from p to t and its weight is I (p, t); 4) O: P × T → N denotes a weight function on a directed arc from a transition t ∈ T to a place p ∈ P, where We say that the new marking M is reachable from M . If there exists a transition sequence denoted by σ = t 1 . . . Before presenting the definition of LPN, we give the following notations by inducing first-order predicate logic to PNs. In a first-order predicate logic expression [28], [29], there are three kinds of connectives including a logic disjunction ''∨'', a conjunction ''∧'', and a negation ''¬''. • T • and We attach first-order predicate logic to PNs by treating each place as one-predicate logic as follows.
Definition 3: Given PN= (P, T , I , O, M , M 0 ), for ∀p ∈ P and ∀t ∈ T , p t represents a predicate logic of p on t, and a logic value of p t at M is denoted and defined as We can rephrase the transition firing rule that t is enabled at M denoted by M [t if for each p ∈ P: p t (M ) = • T • . Thus, for Now we attach a transition t ∈ T with a first-order predicate logic expression on its connected places. The expressions are divided into two categories, i.e., logic input expressions on the input places of t and logic output expressions on the output places of t, which are formally denoted by f ( • t) and g(t • ), respectively. f ( • t) has atomic predicates p t 11 , p t 12 , . . ., and p t 1k connected by connectives where • t = {p 11 , p 12 , . . . , p 1k }; g(t • ) has atomic predicates p t 21 , p t 22 , . . . , and p t 2l connected by connectives where t • = {p 21 , p 22 , . . . , p 2l }. Let L( • t) and L(t • ) denote all logic expressions on the input and output places of t and we have f , respectively, can be obtained given the logic values of all atomic predicates at M . Now we give the formal definition of logic Petri net (LPN) [12], [18].
is a logic Petri net (LPN) where 1) P = P R ∪ P C is place set, P R and P C respectively represent a set of resource and control places, where P = ∅ and P R ∩ P C = ∅; 2) T is a transition set that can be divided into a set of traditional transitions, and logic transitions including a set of logic input transitions and a set of logic output transitions, denoted by T D , T I , and T O , respectively.
. . , p m }; and 8) τ : T → R + attaches a time delay to each transition where an enabled transition t ∈ T fires in time interval τ (t).
The transition firing rules is that for In the above definition, a transition in LPN is combined with a predicate logic. Specifically, if it is a logic input transition, it fires at a marking M only if its predicate logic of the transition's input places is true at M ; if it is a logic output transition, it fires at a marking M generating a new marking M and the predicate logic of the transition's output places is true at M after the transition fires. Thus, logic input and output transitions endorse the batch and indeterminate resource modeling in cooperative systems. Notice that Besides, time is always involved in real systems. Thus, we attach the time information to transitions. When a transition is enabled, the current state is kept for a certain time interval τ (t). After that, it fires. t is an immediate transition if τ (t) = 0, and otherwise, it is a deterministic transition. In LPNs, a token, resource place, control place, logic transition, ordinary transition, and directed arc are drawn in Fig. 2, respectively. In this paper, the LPNs are restricted with I : P × T → {0, 1} and O: P × T →{0,1}; and the capacity of a control place is 1, i.e., ∀M ∈ R(M 0 ) and ∀p ∈ P C : M (p) ≤1.
To ensure this, we give a capacity restriction to each place.
We give some examples of LPNs. In Fig. 3(a), the LPN has a logic input transition t with f We can see that though the markings of input places of t are different, after t fires the generated marking is the same, i.e., only p 4 has a token. In Fig. 3(b), the LPN has an output logic transition . Inhibition Petri nets (IPNs) [12] are an augmented PN by attaching a traditional PN with inhibitor arcs. An inhibitor arc is a non-directed arc. A transition can fire only if there is no token in the places connected with the inhibitor arcs. The equivalence between an LPNs and an IPN has been proved in [12]. Given an LPN, a corresponding equivalent IPN can be obtained. For example, LPNs in Fig. 3 (a) and (b) have their equivalent IPNs respectively in Fig. 3 (c) and (d). Thus, we can see that LPN and IPN are with equal modeling capability to model input and output indeterminacy. We prefer adopting LPNs to model such indeterminacy of the systems because of the following reasons: (1) LPN has compact net structure compared with its equivalent IPN, for example, it has 2 fewer transitions and 6 fewer arcs in the LPN model in Fig. 3 (a) than the one in Fig. 3 (c); and it has 2 fewer transitions and 4 fewer arcs in the LPN model in Fig. 3 (b) than the one in Fig. 3 (d); (2) There is no good property analysis tool or method for IPNs till now; and (3) We have designed some property analysis methods in our previous work [16]- [18].

III. LPN COMPOSITION A. VECTOR COMPUTATIONAL METHOD
A first-order logic expression can be transformed into an equivalent unique disjunctive normal logic expression. Each disjunctive item is a conjunctive expression of all atomic predicates. Each disjunctive item can be represented by a vector with the element equaling the logic value of atomic predicates. Then, a disjunctive normal logic expression can be transferred to a set of vectors. Thus, each transition is attached to a set of vectors. We now give the related definitions.
Definition 5: Let f be a disjunctive normal logic expression of atomic predicates p 1 -p m and f 1 − f n be n disjunctive items, i.e., Given LPN = (P, T , I , O, M , f , g, τ ), if t ∈ T D ∪ T O , there is only one vector of f ( • t) with all elements being 1, and so is ). Notice that in the following content we do not consider a 0-vector which does not make sense in realsystems. We re-write transition firing rules of LPNs as that for ∀t ∈ In the following content, we only compute the vector set of the predicate logic expressions of the logic input and output transitions, denoted by V ( • t) and V (t • ), respectively.

B. LPN COMPOSITION WITH SHARED P-TYPE SUBNETS
This section will present the composition of LPNs where the components are P-type subnets. It realizes the composition of the subnets by common shared places which can be divided into resource places and control places. Given an LPN, some places are shared resource or control places while others are not. To distinguish them we call the control places which are not shared places as basic control places. In order to ensure the correctness of the composition of LPN subnets we firstly require the soundness of the subnets. We can get a subnet and an LPN on a set of places.
is remaining of f ( • t) by deleting p and its connected connectives, where p ∈ • t-P and P = P-P ; 7) g (t • -P ) is the remaining of g(t • ) by deleting p and its connected connectives, where p ∈ t • -P and P = P-P ; and 8) τ (t) = τ (t).
We then give the following definition of a P-type subnet. We now define the LPN composition with shared P-type subnets.
Definition 9: L 1 = (P 1 ∪ P I 1 ∪ P O1 , T 1 , I 1 , O 1 , M 1 , f 1 , g 1 , τ 1 ) and L 2 = (P 2 ∪ P I 2 ∪ P O2 , T 2 , I 2 , O 2 , M 2 , f 2 , g 2 , τ 2 ) are two P-type sound subnets, where P I 1 ∩P I 2 = P O1 ∩P O2 = ∅, (P O1 ∩ P I 2 ) ∪ (P O2 ∩ P I 1 ) = P S = ∅, and T 1 ∩ T 2 = ∅. Then, the composition of L 1 and L 2 is an LPN= (P ∪ P I ∪ P O , T , According to Definition 9, An LPN can be composed of n P-type sound subnets L 1 -L n in a pair-wise composition way through common places denoted by L = L 1 ⊕ L 2 ⊕. . . ⊕L m . Only shared input places can be combined with the output ones. When two P-type sound subnets are composed based on a set of shared places, these places cannot be used for other combinations in the future. Notice that a shared place belongs to at most two subsystems, so such shared places are usually regarded as resource places in cooperative systems. This kind of shared P-type LPN composition can describe the cooperative interaction such as data interchange and order between the sub-systems.

C. PROPERTY ANALYSIS
We now discuss property analysis including liveness, reachability, and reversibility of LPN composition with P-type subnets. It is realized by the proposed vector computational method in sub-section A. Firstly we give some related notations and definitions. An LPN can be treated as a directed graph including directed arc and places and transitions as nodes. Given an LPN, a node n 1 is directly reached from another node n 2 if there exists a directed arc (n 1 , n 2 ); and a node n 1 is in-directly reached from another node n k if there exist a sequence of nodes n 2 , . . . , n k−1 such that n j is directly reached from n j−1 , j = 2, . . . , k. Here, we denote C =< n 1 , n 2 , . . . , n k > as a path from node n 1 to n k . The alphabet of C is denoted by &(C) = {n 1 , n 2 , . . . , n k }. C is a basic path if n i , n j ∈ & (C), and i = j: n i = n j . n i > C n j denotes that n j is (directly or indirectly) reached from n i . A path C =< n 1 , . . . , n k > is a circle when < n 1 , . . . , n k−1 > is a basic path with n 1 = n k .
To do so, we should analyze if the connected transitions of the shared places are traditional transitions or logic transitions.   If they are the latter ones, we should consider their vector sets of predicate logic expressions. Based on these vector sets, we have the following conclusions.
Theorem 1: Proof: Because L 1 is a P-type sound subnet, the subnet on P 1 \P S = {p s11 , p s21 } denoted by L 1 is live and safe. Suppose L 1 at M in1 = p in1 is not reversible. Because of the liveness of L 1 , there exists and because L 1 is not reversible, M 2 = p in1 . Suppose that M 2 (p) > 0, and according to the safeness of L 1 , M 2 (p) = 1. Similarly, we have M 2 [σ t * 1 M 3 , and M 3 (p) = 2, which contradicts the safeness of L 1 . Thus, Theorem 2: Given that M in = p in1 + p in2 , C =< t 11 , p s11 , t 21 , . . . , t 22 , p s21 , t 12 > is a basic path, t 11 = t 12 , and there is at most one logic transition in T S = {t 11 , t 21 , t 22 , t 12 }. L is live, safe, and reversible iff t ∈ T S , p ∈ P S , and ∀v ∈ V Proof: (Sufficiency) If t ∈ T S , p ∈ P S , and ∀v ∈ V ( • t) or V (t • ): v| p = 1, it means that the logic expression ensures that: 1) if t is a logic input transition, it requires the input shared place has a token to be enabled; 2) if it is a logic output transition, it requires when it fires, its output shared place contains a token. According to Definition 8 and Theorem 1, L i on P i -P S is live, safe, and reversible, where i = 1, 2. In order to verify that L is live, and safe, we need to consider the shared places and their connected transitions. We can easily verify that p s11 and p s21 are safe and t 11 , t 21 , t 22 , and t 12 are live. Similar to the proof of Theorem 1, we can prove that L is reversible.
(Necessity) L is live and safe and C is a basic path but not a circle. We can use a contradiction approach and suppose that t 11 is a logic output transition. Suppose that there exists v ∈ V ( • t) or V (t • ) such that v| p = 0. As there is at most one logic transition, p • ∈{t 21 , t 12 } is not live. Thus, v| p = 0 and the conclusion holds.
Theorem 3: Given that M in = p in1 + p in2 , C =< t 11 , p s11 , t 21 , . . . , t 22 , p s21 , t 12 > is a basic path, t 11 = t 12 , and there is more than one logic transition in T S = {t 11 , t 21 , t 22 , t 12 }. L is live, safe, and reversible iff when two logic transitions it means that the logic input and output expressions ensure that: 1) if v 1 | p = v 2 | p = 0, t 1 is a logic output transition, and it fires generating no tokens in p; and 2) t 2 is a logic input transition and requires the input shared place has a token to be enabled. According to Definition 8 and Theorem 1, L i on P i -P S is live, safe, and reversible, where i = 1, 2. In order to verify that L is live, and safe, we need to consider the shared places and their connected transitions. We can easily verify that p s11 and p s21 are safe and t 11 , t 21 , t 22 , and t 12 are live. Similar to the proof of Theorem 1, we can prove that L is reversible.
(Necessity) L is live and safe and C is a basic path but not a circle. We can use a contradiction approach and suppose that t 11 is a logic output transition. Suppose that when two logic transitions t 1 , t 2 ∈ T S : and v 2 | p = 1, t 2 becomes dead; and 1) if v 2 | p = 0 and v 1 | p = 1, after t 1 fires, a token is generated in p which will stay in the place and make t 1 become dead. Thus, there must exist v 2 ∈ V ( • t 2 ) and v 1 ∈ V (t • 1 ) such that v 1 | p = v 2 | p and the conclusion holds.
Notice that, there is no circles that contain a shared place as a node. The following analysis will consider the composition of two P-type sound subnets with circles. Let two P-type sound subnets L 1 and L 2 be as shown in Fig. 6(a) and (b), respectively, where P S = {p s11 , p s21 }, P O1 ∩ P I 2 = {p s11 }, and P O2 ∩ P I 1 = {p s21 }; | • p s11 | = |p • s11 | = 1; and L = L 1 ⊕ L 2 as shown in Fig. 7. Then we discuss the properties such as liveness, safeness, and reversibility of the composed LPN of L 1 and L 2 . Similarly, we analyze the properties by considering if the connected transitions of the shared places are traditional VOLUME 8, 2020 transitions or logic transitions. We then compute the vector sets of logic expressions. According to the obtained vector sets, we have the following conclusion.
Theorem 4: Given that M in = p in 1 + p in 2, C =< t 12 , p s 11, t 21 , . . . , t 22 , p s 21, t 11 , . . . , t 12 > is a circle. L is live, safe, and reversible iff 1) there is a logic input transition t ∈ {t 11 , t 21 } and ∃v 1 ∈ V ( • t) and where t 1 , t 2 ∈ T S : p = t • 1 = • t 2 and p ∈ {p s 11, p s 21}\p. Proof: (Sufficiency) In order to verify that L is live, and safe, we need to consider the shared places and their connected transitions. Because there is a logic input transition t ∈ {t 11 , t 21 }, and ∃v 1 ∈ V ( • t), v 1 | p = 0, where p ∈ • t∩{p s 11, p s 21}, t is enabled without a token in p. Suppose that t = t 11 and p = p s 21. Notice that if t = t 21 and p = p s 11, we have the same proof process below. We can easily verify that t 11 is enabled even if p s 21 has no tokens. According to Definition 8 and Theorem 1, L 1 on P 1 -P S is live, safe, and reversible. Thus, t 12 can be enabled and fire. Because there where p = p s 11, so t 21 can be enabled and fire. Then, Thus, t 22 can be enabled and fire. After the firing of t 12 , t 21 , t 22 , and t 11 , successively, no tokens left in p s 21 and p s 21. Still, according to that L i on P i -P S is live, safe, and reversible, where i = 1, 2, we can prove that L is live, safe, and reversible.
(Necessity) L is live and safe and C is a basic path but not a circle. We can use a contradiction approach to prove both 1) and 2). Firstly, suppose that ∀t ∈ {t 11 , t 21 } and v 1 ∈ V ( • t) such that v 1 | p = 1. In this situation, t 11 and t 21 are dead because each transition's waiting for the token generated from another. Thus, there exists a transition that needs no token from the connected input shared place and 1) holds. Secondly, when two logic transitions t 1 , t 2 ∈ T S : . LPN composed of two shared P-type subnets.
and v 4 | p = 1, t 2 becomes dead; and 1) if v 4 | p = 0 and v 3 | p = 1, after t 1 fires, a token is generated in p which will stay in the place and make t 1 become dead. Thus, there must exist v 3 ∈ V ( • t 2 ) and v 4 ∈ V (t • 1 ) such that v 3 | p = v 4 | p , and condition 2) holds. Next, we consider an LPN that is composed of more than two P-type sound subnets. Let L = (P, T , I , O, M , f , g, τ ) be the composition of L i which is the i-th P-type sound subnet of the i-th sub-system, as shown in Fig. 7, the set of shared places between L 0 and L i be P Si = {p s 0i, p s 1i}, P I 0 ∩ P Oi = {p s 0i}, and P O 0 ∩ P Ii = {p s 1i}; | • p s 1i| = |p s 1i • | = 1; and L = ⊕ m i=0 L i . Then, we have the following conclusion. Corollary 1: Given that, M in = i j=0 p inj is live, safe, and reversible iff the composition of any two models of subsystems L 1 and L 2 is live, safe, and reversible.
The proof can be verified according to Theorems 2-4. Notice that in L, P S = ∅, which means that there is no additional shared place after the accomplishment of the composition. For example, L 01 = L 0 ⊕ L 1 is live, safe, and reversible according to Theorem 4, L 01 ⊕ L i is live, safe, and reversible according to Theorems 2 and 3, and similarly, L in Fig. 8 is live, safe, and reversible.
From the above theorems and the corollary, based on the vector computational method, we can verify the properties including the liveness, safeness, and reversibility of the composition of two P-type sound subnets or the composition of more than two P-type sound subnets. We only consider the places shared by only two sub-systems, which can well describe many cooperative systems such as the E-commerce systems.

IV. AN E-COMMERCE SYSTEM MODEL A. SYSTEM DESCRIPTION
We now use case studies to show the effectiveness of the modeling method proposed in the work. By using ecommerce systems, trades happen between merchants and buyers. As described in Fig. 1, the merchant sub-system is dealing with several buys simultaneously and should have the function of batch handling requests from buyers during a certain time interval. Simultaneously, indeterminate intersections between the two sub-systems always occur. We now design the LPN models for such cooperative systems. In detail, the models of three main sub-systems, i.e., customers, merchants, and a third-party, are designed. In order to finish trading, each system should cooperate with others correctly. Customers mainly have the main operations including order requests, payment, and getting the ordered products, sequentially; Merchants should do order processes, payment checking, mailing products, and getting paid. During the trading process, the benefits and profits of both merchants and customers are protected by a third-party, which acts as a very important organization in current E-commerce systems. Usually, the third-party will keep the payment of the customer for the merchant; after the customers ensure the correct receiving of their ordered product, payments are transferred from the third-party to the corresponding merchant. Notice that, the activities can be refined by some more-detailed processes such as checking the detailed order information by the merchant, ensuring the detailed address by the customers, etc. During any process, time acts in a very important role. For example, if customers do not trigger the finish of the payment transfer from the third-party to the merchant by ensuring the checking of the received products after getting the products in a certain time interval, the payment will automatically be transferred successfully. In the following sub-section, we will model the aforementioned cooperative system and analyze the properties of the model based on the proposed vector computational method.

B. MODELING PROCESS
Three sub-systems in the e-commerce system are modeled by LPNs as shown in Fig. 9: (a) the i-th customer sub-system   Figure 9.   Tables 1 and 2, respectively.
In Fig. 9 (a), initially, p in_i contains a token meaning that customer i is ready for purchasing a product. Then it delivers order and waits for the merchant to ensure it, which is modeled by the firing of t order_i . The merchant checks the order and chooses to accept or reject it. When the order is VOLUME 8, 2020 accepted, the customer continues to do the payment modeled by a transition t payment_i ; or it may be rejected by the merchant for some reasons such as shortage of products and t refused_i fires. If the order is accepted, customers sequentially make the payment and then can receive the ordered product, which are modeled by t payment_i and t product_i , respectively. The models of the merchant and the third-party are shown in Fig. 9 (b) and (c), respectively. The merchant processes the order processing, payment checking, and product delivery, modeled by transitions t order , t payment , and t product , respectively. The third-party mainly has four operations including receiving the payment of customers, and ensuring and keeping the payment, ensuring that the customer has received the products, and transferring the payment to the merchant, which are respectively denoted by t step10 , t step11 , t step20 , and t step21 . Predicate logic expressions to logic transitions in the models in Fig. 9 are designed as shown in Table 2. According to Theorems 5, we can conclude that the composition of LPN models in Fig. 9 is live, safe, and reversible.

V. CONCLUSION
This paper proposes a vector computational method for LPN composition and property analysis. It studies the composition of LPNs with the shared P-type subnets. Each logical expression can be transformed into a unique disjunctive normal one and then to a unique set of vectors. A vector computational method is proposed such that the properties of the composition of the shared P-type subnets such as liveness, boundedness, and reversibility are verified. An E-commerce system is modeled to show the proposed method. This paper can improve the state of the art in the theory of LPNs. Future work will study of composition of LPNs for other systems such as manufacturing systems [30] [31], knowledge-based systems [32], and transportation systems [33]. His current research interests include Petri nets, remanufacturing, recycling and reuse of automotive, and intelligent optimization algorithm. VOLUME 8, 2020