The Equivalency between Logic Petri Workflow Nets and Workflow Nets

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.


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][7][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][12][13][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][18][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

Logic Petri Workflow Nets
This section introduces some basic definitions about PNs, LPNs, and WF-nets.
Definition 3 (see [8]). LPN = ( , ; , , , ) is a logic Petri net where (1) is a finite set of places; (2) = ∪ ∪ is a finite set of transitions, ∪ ̸ = ⌀, ∩ = ⌀, for all ∈ ∪ :  (4) is a mapping from a logic input transition to a logic input expression; that is (2) (5) is a mapping from a logic output transition to a logic input expression; that is 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; ( ) = ( ); for all ∈ • and ∈ should satisfy ( ) = • • ; and for all ∈ • and ∉ , ( ) = ( ).
LPNs are the abstract and extension of IPNs and highlevel 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 ( ) = For a logic input transition , the standard form of ( ) = This definition puts forward the standard form of logic expression. and are called the standard minterms.
The Scientific World Journal From Definition 4, standard forms of ( 1 ) 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

Transforming an LPWN into an Equivalent WF-Net
This section puts forward isomorphism and equivalent definitions to prove the equivalence between LPWNs and WFnets. Based on Definitions 6 and 7, a theorem is given.

Theorem 8. For any LPWN, there exists an equivalent WFnet.
Proof. Consider the following.
For all ∈ , there are three conditions to transform a transition of Σ 1 into one or more corresponding transitions Σ 2 .
Step 1.2. For ∈ , let • = { 1 , 2 , . . . , }; ( ) = 1 ∨ 2 ∨ ⋅ ⋅ ⋅ ∨ ; is restricted by the standard logic input expression ( ). There are standard minterms of ( ), and each minterm corresponds to a transition of Σ 2 . That is, the logic input transition in Σ 1 can be represented equivalently by a set including traditional transitions in Σ 2 . The set is constructed in detail as follows.
Step 1.3. For ∈ , let • = { 1 , 2 , . . . , }; ( ) = 1 ∨ 2 ∨ ⋅ ⋅ ⋅ ∨ ; is restricted by the standard logic output expression ( ). There are standard minterms of ( ), and each minterm corresponds to a transition of Σ 2 . That is, the logic output transition in Σ 1 can be represented equivalently by a set including traditional transitions in Σ 2 . The set is constructed in detail as follows.
Step 2. Proof that the constructing WF-net Σ 2 is equal to Σ 1 . Based on Step 1, the place set and the initial marking 0 in Σ 1 are the same as those in Σ 2 ; that is, = , 0 = , but the transition set and the flow set are not; that is, ̸ = , ̸ = , and | | ≤ | |, | | ≤ | |, where | | denotes the size of set . Firing a transition of Σ 1 corresponds to firing a transition of Σ 2 ; that is, if a transition is enabled in Σ 1 , then there must be an enabled transition in Σ 2 and it is unique. Since Σ 1 and Σ 2 have the same initial marking, the equivalence between Σ 1 and Σ 2 is proved on the basis of the reachable marking graph.
In Σ 1 , for all 1 , 2 ∈ (Σ 1 ), ∈ ; if 1 [ > 2 , then there is a mapping function : The Scientific World Journal if ∈ , then ∃ ∈ : . is an identity mapping and satisfies injective and surjection requirements at ∈ ( 0 ). That is, Σ 1 and Σ 2 have the same behavior characteristics. Moreover, the structure of Σ 2 is unique since its standard form is only one. So is a bijective function, and RG(Σ 1 ) and RG(Σ 2 ) are isomorphic. Based on Definition 7, Σ 1 and Σ 2 are equivalent.
Based on Theorem 8 and the construction of Σ 2 , 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.

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 1 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.
In Figure 2, the logic input transition send to express can be transformed into traditional transitions shown in Figure 3 as follows.
In Figure 2, 1 , 2 , and 3 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.