The trace consistency measurement between WFD-net systems

Introduction How to measure the trace consistency between process models in workflow net systems have become a crucial part of process mapping, process integration, and difference testing. Workflow net with data (WFD-net) is an effective language to describe a complete model of a workflow net system. Checking their trace consistency is a significant challenge in service-based software engineering. However, most of the existing research on trace consistency measurement has not taken the data into consideration. Therefore, they cannot accurately calculate the trace consistency degree of two WFD-net systems, especially when data operations and guard functions are complex. Methods We explore all traces in two WFD-net systems based on guard analysis and data dependencies. Results We point out that the trace consistency degree is uncertain when the values of guard functions are considered. Discussion We conduct experiments to evaluate our method and show its effectiveness.


. Introduction
Nowadays, business process models with data are very important in the development of business process systems (Song et al., 2016). They are very effective in reflecting the relations of activities. Here, we mainly focus on studying workflow net systems, which can be used to automate, optimize, and manage these business process models. They have become a major way to construct business processes in software systems (IEEE, 1990;Song et al., 2018). Software designers and business analysts may simulate workflows in the same granularity standards but with different extra activities (i.e., activities without correspondences). Meanwhile, service-based workflow net systems become more and more extensive, and the traces (i.e., firing sequences of transitions) of a system may be affected by data. Therefore, we should provide an effective way to measure the trace consistency of these workflow net systems.
Many models can be taken to describe workflow net systems, i.e., UML activity diagram (Fahland, 2008), Business Process Modeling Notation (BPMN) (van der Aalst et al., 2010;Zhu et al., 2014;Assy et al., 2015), Business Process Execution Language (BPEL) (Weidlich et al., 2014), Event-driven Process Chain (EPC) (Dijkman et al., 2011;Li et al., 2014), and Workflow net with data (WFD-net) (Trcka et al., 2009;Sidorova et al., 2011). Unfortunately, the first four modeling languages have not taken the delete operations into consideration. Compared with these modeling methods, WFD-net is a workflow net with data operations and guard functions. Therefore, we use WFD-net as an adequate formalism to model . /fcomp. . workflow net systems (Trcka et al., 2009;Sidorova et al., 2011) to analyze the trace consistency in this paper. At present, many methods are used to measure the trace consistency degree, e.g., DTC (Degree of Trace Consistency) (Weidlich et al., 2011) and CDAT (Consistency Degree of Aligned Traces) . Based on the sets of projected firing sequences, the first method can be used to obtain the trace consistency degree of two models without data. Based on the direct dependence relations and independence relations on transitions, the second method is used to calculate the trace consistency degree of two traces. Moreover, these methods only considered transitions with correspondence relations, while those without correspondence relations are ignored. However, they may affect the traces of transitions with correspondences. Therefore, these methods are ambiguous and cannot reflect the real traces in two models, especially when there are guard functions and data dependencies. They cannot accurately assess the trace consistency degree of systems with data information. Furthermore, calculating the trace consistency degree would require a lot of work, e.g., if there are k concurrently-executed transitions in a process, then there are k! different traces at most. Therefore, calculating the trace consistency degree would require a lot of work.
To solve these problems, a new behavioral profile relation is defined based on weak behavioral relations, guard functions, control and data dependency. After then, the formalization of correspondence relation and projected firing sequence are provided. Finally, we develop a new method to calculate trace consistency degree.
The contributions of this paper are shown in the following: 1) We reveal that the trace consistency degree of two WFDnet systems is non-uniqueness when their guard functions are uncertain since these guard functions may affect the real traces. 2) We point out that transitions without correspondence relations but with data dependencies should not be ignored as they may affect the traces of the systems. 3) We minimize the number of traces in WFD-net systems based on guard functions and data dependencies.
The rest of this paper is arranged as follows. Section 2 introduces some basic concepts used in this paper and provides a motivating example. Section 3 defines control dependency, data dependency and behavioral profile relations, order relation, firing sequence, trace, correspondence relation, projected firing sequence and trace consistency measurement method. Section 4 presents experiments to show that guard functions and data dependencies can affect the traces in WFD-net systems. Section 5 reviews the related work about the existing consistency measurement methods. Section 6 concludes this paper and presents future work.

. Preliminaries and motivation
In this section, we first introduce some concepts about WF-net, weak order relation, weak behavioral profile, and WFD-net. Some symbols are demonstrated in Table 1. The related details can be found in Li and Liu (2009), Weidlich and van der Werf (2012), Liu et al. (2017), Liu et al. (2018), Xiang et al. (2021), and Zhao et al. (2022a). Furthermore, a motivating example is given.

. . Preliminaries
A Petri net N = (P, T, F) as a tool is often used to simulate and analyze the trace (i.e., firing sequences of transitions) of a workflow system (Liu et al., 2019;Zhang et al., 2022), where P is a finite non-empty set of places, T is a finite non-empty set of transitions, P ∩ T = ∅, and F ⊆ (P × T) ∪ (T × P) is a set of arcs. Given a node x ∈ P ∪ T, the preset of x is · x = y y ∈ P ∪ T ∩ y, x ∈ F , and the post-set of x is x · = {y|(y ∈ P ∪ T) ∩ (x, y) ∈ F}. If x is the initial place, it must satisfy · x = ∅; If x is the final place, it must satisfy x · = ∅. Given a set of nodes X ⊆ P ∪ T and x ∈ X, the preset of X is denoted by · X, and the post-set of X is denoted by Figure 1A, if X = {t 5 , t 6 }, we have · X = · t 5 ∪ · t 6 = {p 4 , p 5 }, and X · = t · 5 ∪ t · 6 = {p 5 , p 7 }. If we use k to denote the preset number, then we have (·) 1 t 5 = · t 5 = {p 4 }, and k = 1 (resp. (·) 3 t 6 = · ( · ( · t 6 )) = {p 4 }, and k = 3). For the post-set number of X, it can be defined analogously.
The following parts present the workflow net (WF-net), as a special Petri net satisfying the following conditions (Liu et al., 2013). Definition 1 [WF-net (Liu, 2014(Liu, , 2022]. Let N = (P, T, F) be a workflow net (WF-net), if the following conditions hold: (1) There exists only one place p i ∈ P as begin and only one place p o ∈ P as end in N; and (2) The short-circuit net A WF-net system (N, M 0 ) is a WF-net extended with an initial marking M 0 . The dynamic behaviors of workflow systems are decided by distributing different markings. Figures 1A, B depict two WF-net systems if the guard functions and data operations are ignored.
For the example of Figure 1A, we have t 1 ≻ c t 2 and t 2 ⊁ c t 1 . The set of all weak firing subsequences is denoted by T * , and Based on the above definitions, weak behavioral profile between two transitions is shown in the following. Definition 3 [Weak behavioral profile, bp (Weidlich et al., 2011)]. Given a WF-net system (N, M 0 ) = (P, T, F, M 0 ), a pair of transitions t x , t y ∈ T is in one of the following relations, where k ∈ {1, 3, · · · , n} , n = 2m − 1, and m∈ N : It contains the following two kinds: The above relations comprise the weak behavioral profile of (N, M 0 ), denoted as bp = bp 1 , bp 2 , bp 3 , where bp 1 =→ c , bp 2 = + c , and bp 3 = c . As for the inverse order relation of bp 1 [i.e., t x ⊁ c t y ∧ t y ≻ c t x ], we denote it as → (−1) c or bp −1 . If t x → c t y , then we have t y → (−1) c t x or (t y , t x ) = bp −1 . The weak behavioral profile relation is also called weak behavioral relation in this paper. In Figure 1A, we have t 1 → c t 2 , t 3 + c t 4 , and t 3 c t 5 . They are denoted as (t 1 , t 2 ) = bp 1 , (t 3 , t 4 ) = bp 2 , and (t 3 , t 5 ) = bp 3 , respectively.
A workflow net with data (WFD-net) is a WF-net with every transition t ∈ T labeled with four elements at most: (1) Some data items need to be read before firing t; (2) Some data items would be written after firing t; (3) Some data items may be deleted after firing t; (4) Some data items may be the guard functions of t (optional).
The update of data items can be seen as the combination of read/write operations on transitions. Furthermore, writing a data item on a transition for the first time can be known as creating the data item. We also assume that there are no read operations or guard functions on the initial transition (Trcka et al., 2009;Sidorova et al., 2011). Definition 4 [WFD-net (Trcka et al., 2009;Sidorova et al., 2011)]. A WFD-net ND = (P, T, F, D, Guard, R, W, De, Gd) is a workflow net (WF-net) extended with data, it consists of the following: (1) (P, T, F) is a WF-net; (2) D is a different set of data items; (3) Guard is a set of guards over D; Frontiers in Computer Science frontiersin.org . /fcomp. .
(4) R : T → 2 D is the read operation set; (5) W : T → 2 D is the write operation set; (6) De : T → 2 D is the delete operation set; and (7) Gd : T → Guard is a function that assign guards to transitions.
A WFD-net system is a WFD-net with an initial marking M 0 , it can be denoted as (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd). In the following, all workflow systems are in the form of WFD-net systems. Figures 1A, B depict two WFD-net systems.

. . Motivating example
This subsection takes two shipper process models as a motivating example to illustrate our work, as shown in Figures 1A, B. There are 5 same data items in these figures (i.e., cus, gds, des, price, and bon, where cus denotes the data of customers, gds stands for the data of goods, des represents the destination of goods, price means the price of goods, and bon denotes the bonus), and the set of guards is Guard = {isHigh price , ¬isHigh price }. For the transition t 5 , its data operations and guard function are Figure 1A describes a shipper process model in the shape of a WFD-net system. Firstly, shippers receive goods from customers and write 3 data items, i.e., the data of customers and goods, and the destination of goods. Then, shippers should calculate the price by reading the data of goods and generating the price. Afterwards, customer advisers and shipment departments execute their tasks concurrently. If the goods' price is high, express shipment is used by reading the data of goods and the destination of the goods. Under the same condition, customer advisors calculate the bonus for customers by reading the price and producing the bonus. If the price of goods is not high, normal shipment is taken by reading the data of goods and the destination of goods, and there is no bonus to calculate. If the customer advisors calculate the bonus for customers, then they should register the bonus by reading the data of customers and the bonus as well as generating new data of customers. At last, customers who send goods are informed by call. Figure 1B shows a similar model to Figure 1A. However, there are some extra transitions (i.e., t 9 and t 10 ) and data items (i.e., d q ) without any correspondences in Figure 1B. The transition t 9 represents Review Goods, the transition t 10 denotes Confirm Goods, and the data item d q stands for the quality of the goods.
The method of DTC ignores those transitions without correspondence, e.g., t 9 and t 10 in Figure 1B. Therefore, the method of DTC is insensitive to these extended elements, and the trace consistency degree of Figures 1A, B computed by this method is 1. This result means that the traces in the above figures are the same. However, these extensions may produce data dependencies (cf. Def. 6) and affect the real traces. Therefore, the DTC method may be inappropriate for analyzing the real trace consistency degrees of WFDnet systems.

. Trace consistency measurement based on data and behavioral profiles
Before introducing the trace consistency degree of two WFDnet systems, we present control dependency and data dependency of transition pairs in this section. Then, we illustrate trace and trace consistency degree. Definition 5 (Control dependency). Given two transitions t x and t y in When the specific read/delete operation of a transition is derived from the write operation of another transition, we say there exists a data dependency. The data dependency may affect the traces in a model. The specific details are shown as follows. Definition 6 (Data dependency). Given two transitions t x and t y in Let t x and t y be a pair of transitions derived from (ND, M 0 ). The control dependency indicates there is a weak strict order relation such that t x → c t y . In Figures 1A A transition t y is data-dependent on another transition t x implies that t y reads/deletes a data item written by t x . If there is no data dependency between t x and t y , then we have t x d t y . In Figure 1B, t 10 is data-dependent on t 9 as d q ∈ W(t 9 ) ∩ R(t 10 ) and there is no transition t w enabled between t 9 and t 10 , such that d q ∈ W(t w ) ∩ De(t w ). Definition 7 [Distance between transitions in weak circulation relation (Zhao et al., 2022a)]. Let (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd) be a WFD-net system, ∃t x , t y ∈ T : t x ↔ c t y , t x ∈ t (·) k1 , and t y ∈ t (·) k2 , where t is a transition outside the weak circulation relation with t x and t y , k 1 and k 2 are the minimum number of post-sets, such that k 1 , k 2 ∈ {2, 4, · · · , 2m}, and m∈ N. The distance between t x and t y is defined as δ t y , t x = k 2 − k 1 .
For the example of Figure 2A, we have t x ↔ c t y , and δ t y , t x = k 2 − k 1 = 2.

. . Transition pairs behavioral relations in a WFD-net system
Before formalizing the relation of transitions t x and t y in a WFD-net system, we first explain some relevant symbols as follows: (1) Rel(Gd (t x )) (resp. Rel(Gd t y )) is the set of data items in guards related to t x (resp. t y ), and Rel(Gd t x (d) ) (resp. Rel(Gd t y (d) )) is the value of d in guards related to t x (resp. t y ); and (2) Ŵ(Rel Gd (t x ) ) (resp. Ŵ(Rel Gd t y )) is the set of final values of data items in Rel Gd (t x ) (resp. Rel(Gd t y )), and Ŵ(Rel Gd t x (d) )(resp. Ŵ(Rel Gd t y (d) ) is the final value of d in Rel Gd (t x ) (resp. Rel(Gd t y )).
The transition pairs' behavioral relations in a WFD-net system depend on their weak behavioral relations, data dependencies, and guard functions. The following behavioral relations compose the behavioral profile of a WFD-net system. Definition 8 (Behavioral profile, BP). Given a WFD-net system (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd) and a pair of transitions t x ,t y ∈ T, we have (2) Exclusiveness relation (t x + cd t y ), if

and
Rel Gd t x d ∩ Rel Gd t y d = ∅. or (3) Interleaving order relation (t x cd t y ). It can be divided into the following two kinds: These relations constitute the behavioral profile of (ND, M 0 ), denoted as BP = {BP 1 , BP 2 , BP 3 }, where BP 1 =→ cd , BP 2 = + cd , and BP 3 = cd . We use BP −1 or → −1 cd to show the inverse order relation of BP 1 . It's worth noting that each transition pair can be in a single behavioral relation.
As we know, Rel Gd t 3 price = Rel Gd t 5 price = Rel Gd t 6 price = isHigh price and Rel Gd t 4 price = Rel Gd t 7 price = {[¬isHigh(price)]}. Therefore, we have Rel Gd t 3 price ∩Rel Gd t 7 price = ∅, Rel Gd t 4 price ∩ Rel Gd t 5 price = ∅, and Rel Gd t 4 price ∩ Rel(Gd t 6 price ) = ∅. According to Def. 8, we have (t 3 , t 7 ) = (t 4 , t 5 ) = (t 4 , t 6 ) = BP 2 , and (t 3 , t 5 ) = (t 4 , t 7 ) = BP 3 in Figures 1A, B after taking these guard functions into consideration. Without considering data dependencies of transitions without correspondences, we have (t 3 , t 6 ) = BP 3 in Figures 1A, B. After considering the data item d q , we know d q ∈ W(t 9 ) ∩ R(t 10 ), and t 9 → cd t 10 as t 10 is datadependent on t 9 in Figure 1B. Therefore, we have (t 3 , t 6 ) = BP 3 in Figure 1A and (t 3 , t 6 ) = BP 1 in Figure 1B. In Figure 2B, if there exists no data item d satisfying d ∈ Rel(Gd (t x )) ∩ Rel(Gd t y ), then we have t x ⇄ cd t y . Definition 9 (Order relation). Let (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd) be a WFD-net system, the order relation t x ≻ t y includes transitions t x and t y satisfying one of the following conditions: (1) t x → c t y ; (2) t x ⇄ c t y ; ( If some transitions lack data dependencies, there is a missing data. Based on the above relation, we further formalize missing data. Definition 10 (Missing data, MD). Given a WFD-net system (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd), T * is the set of all weak firing subsequences, ∃σ ∈ T * , and ∃t m ∈ σ : p i ∈t m . A data item d ∈ D is missing if one of the following conditions is satisfied: A missing data appears when a data item is read or deleted but not written before (Meda et al., 2010), as shown in Figures 3A-D. This error may affect the trace (Def. 12) in a WFD-net system. If the missing data is not an ending error, we can analyze the trace consistency of two WFD-net systems. As shown in Figure 3B, we have t 3 → cd t 2 as t 2 is data-dependent on t 3 .

Frontiers in Computer Science
frontiersin.org , there is a missing data. When the missing data is an ending error, there is no point in analyzing the traces in two WFD-net models.

. . Firing sequence and traces in a WFD-net System
Before defining the trace, we present the definition of firing sequence. Definition 11 (Firing sequence). Given a WFD-net system If t x → cd t y , then we denote it as t x t y . If t x cd t y , then we denote it as {(t x ) (t y )}. If t x cd t y , t x cd t r , and t y → cd t r , then we denote it as {(t x ) (t y t r )}. As shown in Figure 1A, t 1 t 2 {(t 4 ) (t 7 )}t 8 is a firing sequence. Definition 12 (Trace). Given a WFD-net system (ND, M 0 ) = (P, T, F, M 0 , D, Guard, R, W, De, Gd), a trace in (ND, M 0 ) is a firing sequence σ i = t 1 t 2 · · · t n (n∈ N). The set of all traces is defined as where k is the number of traces. The definition of a trace depends on the firing sequence, and there is no circulation relation by default. For the example of Figure 1A, the traces are t 1 t 2 {(t 3 ) (t 5 t 6 )}t 8 and t 1 t 2 {(t 4 ) (t 7 )}t 8 . After then, we provide a way to compute the trace consistency degree of two WFD-net systems.

. . Trace consistency degree of WFD-net systems
Notice that there are only 1:1 correspondence relation in this paper. The transitions with correspondence relations read, write and delete the same data items. Therefore, we use the same transitions to denote them (e.g., transitions t 1 ∼ t 8 in Figures 1A, B). A trace consistency guarantees that the essential with the sets of projected firing sequences ϕ ∼ 1 and ϕ ∼ 2 , and the sets of same projected firing sequences σ ∼ 1 and σ ∼ 2 . Output: Trace consistency degree of two WFD-net Algorithm . Trace consistency degree of two WFD-net systems.
rotes are reserved among transitions with correspondences in two WFD-net systems. According to our method, the inconsistencies manifest there exist traces without correspondence. According to Def. 15, we propose Algorithm 1 to calculate the trace consistency degree. This algorithm uses the sets of projected firing sequences ϕ ∼ 1 and ϕ ∼ 2 , and the sets of the same projected firing sequences σ ∼ 1 and σ ∼ 2 in two WFD-net systems as inputs, and the trace consistency degree TCDW ∼ as outputs.
Step 1 initializes TCDW ∼ , ϕ ∼ 1 , ϕ ∼ 2 , σ ∼ 1 , and σ ∼ 2 . ϕ ∼ 1 and ϕ ∼ 2 can be calculated by taking Steps 2-7. σ ∼ 1 and σ ∼ 2 are derived from Steps 8-21. Finally, the trace consistency degree TCDW ∼ is calculated according to ϕ ∼ 1 , ϕ ∼ 2 , σ ∼ 1 , and σ ∼ 2 , as shown in Step 22. The number of firing sequences in (ND 1 , M 01 ) is n 1 , and the number of firing sequences in (ND 2 , M 02 ) is n 2 . The time cost of finding projected firing sequences in (ND 1 , M 01 ) and (ND 2 , M 02 ) is n 1 + n 2 . Let n be the number of the max firing sequences in these two models [i.e., n = max(n 1 , n 2 )], and then their projected firing sequences can be found in O(n 2 ). The calculation of trace consistency degree is in linear time with the calculation of projected firing sequences. Therefore, the time complexity of this algorithm is O(n 2 ).
According to the DTC method, there are 10 traces in Figures 1A, B if data operations and guard functions are ignored. The traces in Figures 1A, B are t 1 t 2 {(t 3 )(t 5 t 6 )}t 8 , t 1 t 2 {(t 4 )(t 5 t 6 )}t 8 , t 1 t 2 {(t 3 )(t 7 )}t 8 , and t 1 t 2 {(t 4 )(t 7 )}t 8 , as shown in Figure 4A. Therefore, the trace consistency degree of these two models computed by the DTC method is 1.

. Experiments
To evaluate the trace consistency of eight pieces of pseudo-code (i.e., Tables 2, 3), we use eight WFD-net systems to model them, as shown in Figures 5, 8. Although the method of DTC (Weidlich et al., 2011) can be used to measure the trace consistency degree, we want to address the following three problems. RQ1: How do guard functions affect the traces in WFDnet systems? RQ2: How do data dependencies affect the traces in WFDnet systems? RQ3: Is our method more effective in measuring the trace consistency degree?
To answer RQ1-RQ3, we analyze the above-mentioned eight WFD-net systems. These systems include weak strict order relations, weak exclusiveness relations, and weak interleaving order relations. To compare our method with the DTC method, we analyze the trace consistency degree of these models with guard functions and data dependencies.

. . RQ : The e ect of guard functions on traces
As shown in Section 3, we know the guard functions may change the behavioral relations. Therefore, we discuss the traces in the general cases (e.g., Figure 5A) as follows: (1) If there exists no data item d satisfying d ∈ Rel(Gd(t x )) ∩ Rel(Gd(t y )), we have (t 2 cd t 4 )∧(t 2 cd t 5 )∧(t 3 cd t 4 )∧(t 3 cd t 5 ) in Figure 5A. The corresponding traces are illustrated in Figure 6A.
In our method, Figure 6H shows the traces of Figure 5B, Figure 6B shows the traces of Figure 5C, and Figure 6C shows the traces of Figure 5D. Therefore, the traces in a WFD-net system are more than one when their guard functions are uncertain, e.g., Figure 5A. Figure 7 shows the trace consistency degrees of models in Figures 5A-D.
. . RQ : The e ect of data dependencies on traces In this subsection, we discuss the traces that may be affected by data dependencies, as shown in Figure 8. Based on Def. 7, the data dependencies in Figures 8A, C, D cannot change the traces. Although we have d ⋖ MD in Figure 8B, the final place can be reached. The data dependencies in Figure 8B can change the traces if there is no locking mechanism. In a word, we have t 2 → cd t 3 in Figure 8B. Therefore, the corresponding traces of Figures 8A-D are shown in Figure 9. Figure 10 shows the trace consistency degrees of models in Figures 8A-D. . . RQ : Comparison between di erent trace consistency measurement methods In this subsection, we present a comparison between different methods for trace consistency measurements. The .

. Related work
Consistency checking has become a critical technology in checking how a process model is similar to another. It is significant in the matching process models . Existing studies on the measurements of consistency degree can Frontiers in Computer Science frontiersin.org . /fcomp. .

FIGURE
Trace consistency degree measurement results by taking DTC (Weidlich et al., ) and TCDW methods.  Table . be classified into two categories: with or without data. The following parts review some related work about the existing measuring methods.
. . Consistency degree of process models without data Consistency measurement is very important in process alignment and process integration. It can be taken to detect the deviations between the process models and corresponding event logs. Recently, many researchers have focused on measuring the consistency degrees of process models from the angle of control flow. Decker and Weske (2007) presented the standard for an optimum consistency relation regarding a given compatibility relation. They guaranteed that the implemented services' behaviors were compatible with others. Therefore, they checked the consistency degree of specifications and service implementations. After then, van der Aalst (2011) and van der Aalst et al. (2007) depicted the conformity relationship between event logs and process models. The consistency measurement techniques can be used to compare the modeled behavior in the initial process model with the observed behavior in event logs. The consistency guarantees there is no contradiction between the two models (Rull  (2012), and Weidlich et al. (2011) solved this problem. They defined behavioral profile relations that can capture the behavioral restrictions of a model. Moreover, they put forward the trace consistency as the number of the same traces in two models to the number of traces with transitions in correspondences. Additionally, based on correspondence relations, they defined the profile consistency degree. They also illustrated that consistency-checking technology states how the real behavior is consistent with the observed. After then, Nagel et al. (2013) described process model quality constraints that can be taken to obtain consistency among the goals and the process models. van der Aa et al. (2015) used natural language processing techniques to recognize the inconsistencies between textual descriptions and relevant process models. Awadid and Nurcan (2016) reviewed the existing consistency measurement methods. One aim of their research is to master the inter-model consistency issue. Besides, Wang et al. (2018) proposed relation profile methods to analyze the consistency degrees of different process models. Although these consistency measurement methods have been proposed, they mostly rely on detecting the consistency decrease caused by the difference in process models. They have not taken the data dependencies or guard functions into consideration.
. . Consistency degree of process models with data Generally speaking, the consistency degree of process models in workflow systems would be affected by the data. Song et al. (2013) and Song and Jacobsen (2015) proposed a way to calculate the consistency degrees between conceptual and executable process models. At the same time, they developed a tool named COCO to quantitatively calculate the consistency degree. After then, they measured the trace consistency degree between the two models on account of control dependence and data dependence and provided a new computing method to quantify the trace consistency (Song et al., 2021). Based on the program dependency graphs, Zhang et al. (2018) and Zhang et al. (2019) proposed a systematic method to check consistencies between different BPEL processes. They identified that the data dependencies can affect the consistency degrees of different models. They took advantage of activity constraint graphs to calculate the consistency degree of models with control flow and data-flow information. After taking their methods, we can distinguish different event constraints and check the consistency of conceptual and logical process models.
However, these studies are mainly based on analyzing the consistency of physical processes with read/write operations. They have not considered that the guard functions may affect the trace consistency degree. Usually, the transitions without correspondence relations but with data dependencies are ignored by them, too. However, all of them may affect the real trace consistency degree. By comparison, our measurement methods can solve these problems.

. Conclusions and future work
The trace consistency measurement is important to servicebased workflow net systems. This paper presents the behavioral profile relation in a WFD-net system, and provides a new way to calculate the trace consistency of two WFD-net systems. Compared with the DTC method, the advantages of our method are shown as FIGURE Trace consistency degree measurement results by taking DTC (Weidlich et al., ) and TCDW methods.
follows. Firstly, we illustrate that the guard functions may affect the traces in two WFD-net systems. Therefore, the trace consistency degree is non-uniqueness when the values of guard functions are uncertain. Secondly, we point out that the transitions with or without correspondence but with some data dependencies can also affect the traces and the real trace consistency degree. Some experiments on WFD-net systems above demonstrate that our method is more efficient in measuring trace consistency than the existing method. This paper shows some directions for future research: (1) Our methods need to be improved so as to measure the trace consistency between transitions in WFD-net systems extended with complex correspondence relations; (2) Some measures should be taken to repair control/data-flow errors as they may affect the actual traces; and (3) We need to develop a tool to help us calculate trace consistency automatically.

Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.