Abstract
Logic Petri nets (LPNs) can describe and analyze batch processing functions and passing value indeterminacy in cooperative systems. Logic Petri workflow nets (LPWNs) are proposed based on LPNs in this paper. Process mining is regarded as an important bridge between modeling and analysis of data mining and business process. Workflow nets (WF-nets) are the extension to Petri nets (PNs), and have successfully been used to process mining. Some shortcomings cannot be avoided in process mining, such as duplicate tasks, invisible tasks, and the noise of logs. The online shop in electronic commerce in this paper is modeled to prove the equivalence between LPWNs and WF-nets, and advantages of LPWNs are presented.
1. Introduction
Petri nets (PNs) [1] are a process modeling technique applied to the simulation and analysis of distributed systems, and PNs are also an effective description and analysis tool for many fields. With the continuous development of PN theory and the increasing popularity of its application, some of their extensions have been defined, such as colored [2], time [3], fuzzy [4], and stochastic PNs [5]. Logic Petri nets [6–8] are the abstract and extension of high-level PNs and have been applied efficiently to the modeling and analysis of Web services, cooperative systems, and electronic commerce. Transitions restricted by logic expressions are called logic transitions. The inputs and outputs can be described by logic transitions in LPNs. Based on LPNs, the definition of LPWNs is proposed in this paper. An LPWN is logic Petri net with a dedicated source place where the process starts and a dedicated sink place where the process ends. Moreover, all nodes are on at least a path from source to sink.
Larger online shops produce a great quantity of transaction records every day. How to find valuable information in these records is a meaningful task. These records are called event logs in process mining, which are the starting point of process mining [9, 10]. When modeling business processes in terms of Petri nets, a subclass of Petri nets known as Workflow nets is considered [11–14]. WF-nets are also a natural representation for process mining. Process mining [15, 16] is a young cross field and crosses the computational intelligence and data mining field to the modeling process and analysis area. Process mining is regarded as an important bridge between modeling and analysis of data mining and business process [17–19]. LPWNs and WF-nets are evolutions of PNs. The LPWN will be introduced into process mining in our later work, so the equivalency between LPWNs and WF-nets is firstly proved by an online shop model in this paper. Compared with WF-nets, LPWNs can well describe and analyze batch processing functions and passing value indeterminacy in cooperative systems and effectively alleviate the state space explosion problem to an extent.
The rest of this paper is organized as follows. Section 2 reviews definitions of PNs, WF-nets, and LPNs, and the standard forms of logic expressions and LPWNs are put forward. A simple LPWN model is given to explain how the LPWN works. In order to prove the equivalence between LPWNs and WF-nets, isomorphism and equivalent definitions are proposed in Section 3. Theorem 8 has been proved on the basis of isomorphism and equivalent definitions, and the constructing algorithm of an equivalent WF-net from an LPWN is presented. In Section 4, Theorem 8 and the algorithm are illustrated by an online shop model. Concluding remarks are made in Section 4.
2. Logic Petri Workflow Nets
This section introduces some basic definitions about PNs, LPNs, and WF-nets.
Definition 1 (see [8]). is a marked , where(1) is a net;(2) is a marking function, where is the initial marking and ;(3)transition firing rules are as follows:(a) is enabled at if for all , represented by ;(b)if is enabled, it can fire, and a new marking is generated from , represented by , where
Definition 2 (see [8]). Let be a Petri net and a fresh identifier not in . The PN is a workflow net (WF-net) if and only if (a) contains an input place (also called source place) such that ;(b) contains an output place (also called sink place) such that ;(c) is strongly connected.
There is a directed path between any pair of nodes in PN.
Definition 3 (see [8]). is a logic Petri net where(1) is a finite set of places;(2) is a finite set of transitions, , , for all , where(a) denotes a set of traditional transitions;(b) denotes a set of logic input transitions, where, for all , the input places of are restricted by a logic input expression ;(c) denotes a set of logic output transitions, where, for all , the output places of are restricted by a logic output expression ;(3) is a finite set of directed arcs;(4) is a mapping from a logic input transition to a logic input expression; that is (5) is a mapping from a logic output transition to a logic input expression; that is (6) is a marking function, where, for all , is the number of tokens in ;(7)Transition firing rules are as follows:(a)for all , the firing rules of are the same as in PNs;(b)for all , is enabled only if ; make , , where, for all and , ; for all and , ; for all , ; and, for all , ;(c)for all , is enabled only if for all . , where for all ; for all ; for all and should satisfy ; and for all and , .
LPNs are the abstract and extension of IPNs and high-level PNs. In Definition 3, a logic input/output transition is restricted by the logic input/output expression in LPNs. All logic input/output transitions are called logic transitions. The logic expressions can describe the indeterminacy of values in input and output places. and represent input and output ways of logic transitions, respectively. They are not the disjunctive normal of .
Definition 4. Suppose that a logic input/output transition is restricted by , and the standard form is as follows.
For a logic input transition , the standard form of can be obtained by
For a logic input transition , the standard form of can be obtained by
This definition puts forward the standard form of logic expression. and are called the standard minterms.
Definition 5. Let be a logic Petri net, and the LPN is a logic Petri workflow net (LPWN) if and only if(a)LPN has , where are control/data place sets;(b)there is a source place such that ; there is a sink place such that ;(c)there is a directed path between the source place and sink place.From Definitions 3 and 5, the LPWNs and WF-nets have the same kind of starting place with a token and an ending place.
Figure 1 shows a simple LPWN model. , , and are three traditional transitions. restricted by is a logic output transition; , where , ; restricted by is a logic input transition; , where , .
From Definition 4, standard forms of and are and , respectively. Note that each place of a logic expression has a logic value at marking in an LPWN, and, by substituting the values of all places into the logic expression, the expression corresponds to a logic value.
In the LPWN model of Figure 1, , , and . For the source place having a token, the transition can fire. Suppose that fires, has a token, and . From Definition 3, , having , , , and . The logic expression is satisfied, is enabled, and . From condition (b) of Definition 3, .
3. Transforming an LPWN into an Equivalent WF-Net
This section puts forward isomorphism and equivalent definitions to prove the equivalence between LPWNs and WF-nets.
Definition 6. Let be an LPWN and a WF-net. is the reachable tree of , and is the node set of , . If there exists a bijective function , such that, for all , , , , . Then, and are isomorphic.
Definition 7. Let be an LPWN and a WF-net. and are equivalent if and only if RG() and RG() are isomorphic.
Based on Definitions 6 and 7, a theorem is given.
Theorem 8. For any LPWN, there exists an equivalent WF-net.
Proof.
Consider the following.
Step 1. Constructing an equivalent WF-net is as follows.
Let be an LPWN, and the deterministic WF-net being equivalent to should be constructed at the very start.
For all , there are three conditions to transform a transition of into one or more corresponding transitions .
Step 1.1. For , let ; for all , if , then ; and if , then .
Step 1.2. For , let ; ; is restricted by the standard logic input expression . There are standard minterms of , and each minterm corresponds to a transition of . That is, the logic input transition in can be represented equivalently by a set including traditional transitions in . The set is constructed in detail as follows.
For any standard minterm , where , assume that corresponds to the transition in ; that is, . Then, the arc set related to is defined. For all , where , if in is , we have ; for all , we have , where .
Step 1.3. For , let ; ; is restricted by the standard logic output expression . There are standard minterms of , and each minterm corresponds to a transition of . That is, the logic output transition in can be represented equivalently by a set including traditional transitions in . The set is constructed in detail as follows.
For any standard minterm , where , assume that corresponds to the transition in ; that is, . Then, the arc set related to is defined. For all , where , if in is , we have ; for all , we have , where .
Step 2. Proof that the constructing WF-net is equal to .
Based on Step 1, the place set and the initial marking in are the same as those in ; that is, , , but the transition set and the flow set are not; that is, , , and , , where denotes the size of set . Firing a transition of corresponds to firing a transition of ; that is, if a transition is enabled in , then there must be an enabled transition in and it is unique. Since and have the same initial marking, the equivalence between and is proved on the basis of the reachable marking graph.
In , for all , ; if , then there is a mapping function based on Step ; we have and . In , if , then : and ; if , then ; we have . is an identity mapping and satisfies injective and surjection requirements at . That is, and have the same behavior characteristics. Moreover, the structure of is unique since its standard form is only one. So is a bijective function, and RG() and RG() are isomorphic. Based on Definition 7, and are equivalent.
Based on Theorem 8 and the construction of , the construction algorithm of an equivalent WF-net from an LPWN can be obtained.
In Algorithm 1, the equivalent WF-net has the same place set and traditional transitions compared with its corresponding LPWN. Their differences are the logic transitions and flows. Next, an example is used to prove the correctness and appropriateness of Theorem 8 and Algorithm 1.
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4. A Case
In this section, the work processes of an online shop in electronic commerce shown in Figure 2 are modeled by the LPWN, and the validity and usefulness of the presented method are illustrated based on the analysis of the model. Functions of the online shop are modeled by transitions. For example, the transition receive_order represents that the shop owner will get an order from the client, and it is limited by the logic expression (receive_order). Based on Definition 4, all logic transitions and their standard items are shown in Table 1.
Next, the LPWN shown in Figure 2 will be transformed into its equivalent WF-net.
In Figure 2, the logic input transition receive_order can be transformed into three traditional transitions as follows.
The receive_order is a logic input transition restricted by , where , , and . The receive_order has three ways to transform tokens. For example, represents and order_1 loses a token and order_2 does not lose a token after the receive_order fires. From Algorithm 1, in the equivalent WF-net, the transition receive_order can be transformed into , , and , and they are three traditional transitions. Flows (, receive_order), (order_1, receive_order), and (order_2, receive_order) are transformed into seven flows (, ), (, ), (, ), (order_1, ), (order_1, ), (order_2, ), and (order_2, ). The flow (receive_order, ) is transformed into three flows , , and . The input transition receive_payment can also be transformed by this method.
In Figure 2, the logic input transition send_to_express can be transformed into traditional transitions shown in Figure 3 as follows.
The send_to_express is a logic output transition restricted by , where , , and . The send_to_express has three ways to transform tokens. For example, represents that gets a token and does not get a token after the send_to_express fires. From Algorithm 1, in the equivalent WF-net, the logic output transition send_to_express can be transformed into ste1, ste2, and ste3, and they are three traditional transitions. Flows (send_to_express, ), (send_to_express, ) are transformed into four flows (ste1, ), (ste3, ), (ste3, ), and (ste2, ). The flow (, send_to_express) is transformed into three flows (, ste1), (, ste2), and (, ste3). Other output transitions confirm_refuses, confirm_goods, and send_money can be transformed by this method.
In Figure 2, , , and are three traditional transitions, and places, transitions, and flows related to them do not change. Based on the above method, the equivalent WF-net can be obtained in Figure 3.
From Figures 2 and 3, the WF-net consists of 21 transitions and 58 flows while its equivalent LPWN model has 9 transitions and 30 flows, and the number of their places is the same. The rates of transitions and flows descending from its WF-net to its LPWN in the example are 42.86% and 51.72%, respectively. There is a conclusion that LPWNs and WF-nets are equal to modeling, and LPWNs compared with WF-nets can alleviate the state space explosion problem.
5. Conclusions
Based on the definition of LPNs, LPWNs are proposed in this paper. An LPWN is logic Petri net with a dedicated source place where the process starts and a dedicated sink place where the process ends. Moreover, all nodes are on a path from source to sink. Theorem 8 has been proved, and Algorithm 1 used to construct an equivalent WF-net from an LPWN is put forward. Effectiveness and practicality of the proposed algorithm have been exemplified by the online shop model.
In further work, the fundamental properties of LPWNs will be investigated according to the results proposed in this paper, such as state equivalency, liveness, and reachability. The LPWN will be applied efficiently to progress mining.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by the National Natural Science Foundation of China under Grants 61170078 and 61173042; the National Basic Research Program of China under Grant 2010CB328101; the Doctoral Program of Higher Education of the Specialized Research Fund of China under Grant 20113718110004; Basic Research Program of Qingdao City of China under Grant no. 13-1-4-116-jch; the SDUST Research Fund of China under Grant 2011KYTD102; and Graduate Innovation Foundation of Shandong University of Science and Technology under Grant YC140360.