# Translating Workflow Nets to Process Trees: An Algorithmic Approach

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Basic Notation

#### 2.2. Workflow Nets

**Definition**

**1**(Labeled Workflow net (WF-net))

**.**

- 1.
- $\u2022{p}_{i}=\varnothing \wedge \nexists p\in P\backslash \{{p}_{i}\}\left(\right)open="("\; close=")">\u2022p=\varnothing $;${p}_{i}$is the unique source place.
- 2.
- ${p}_{o}\u2022=\varnothing \wedge \nexists p\in P\backslash \{{p}_{o}\}\left(\right)open="("\; close=")">p\u2022=\varnothing $;${p}_{o}$is the unique sink place.
- 3.
- Each element$x\in P\cup T$is on a path from${p}_{i}$to${p}_{o}$.

**Definition**

**2**(Soundness)

**.**

- 1.
- $(W,[{p}_{i}])$is safe,i.e., $\forall M\in \mathcal{R}(W,[{p}_{i}])\left(\right)open="("\; close=")">\forall p\in P\left(\right)open="("\; close=")">M(p)\le 1$,
- 2.
- $[{p}_{o}]$can always be reached, i.e.,$\forall M\in \mathcal{R}(W,[{p}_{i}])\left(\right)open="("\; close=")">(W,M)\u21dd(W,[{p}_{o}])$.
- 3.
- Each$t\in T$is enabled, at some point, i.e.,$\forall t\in T\left(\right)open="("\; close=")">\exists M\in \mathcal{R}(W,[{p}_{i}])\left(\right)open="("\; close=")">M[t\rangle $.

#### 2.3. Process Trees

**Definition**

**3**(Process Tree)

**.**

- 1.
- $x\in \Sigma \cup \{\tau \}$; an (non-observable) activity,
- 2.
- $\oplus ({Q}_{1},\dots ,{Q}_{n})$, for$\oplus \in \u2a01$, $n\ge 1$, where${Q}_{1},\dots ,{Q}_{n}$are process trees;

**Definition**

**4**(Process Tree Language)

**.**

**Definition**

**5**(Process Tree Transformation Function)

**.**

## 3. Translating Workflow Nets to Process Trees

#### 3.1. Overview

#### 3.2. PTree-Nets and Their Unfolding

**Definition**

**6**(Process Tree-labeled Petri-net (PTree-net))

**.**

**Definition**

**7**(PTree-net Unfolding)

**.**

- 1.
- ${P}^{\prime}=P\cup {\displaystyle \bigcup _{t\in T}}{\widehat{P}}_{t}$,
- 2.
- ${T}^{\prime}={\displaystyle \bigcup _{t\in T}}{\widehat{T}}_{t}$,
- 3.
- ${F}^{\prime}={\displaystyle \bigcup _{t\in T}}{\widehat{F}}_{t}\cup {\displaystyle \bigcup _{t\in T}}\{(p,t)\mid p\in \u2022t\wedge t\in {p}_{{i}_{t}}\u2022\}\cup {\displaystyle \bigcup _{t\in T}}\{(t,p)\mid p\in t\u2022\wedge t\in \u2022{p}_{{o}_{t}}\}$,
- 4.
- $\ell ={\displaystyle \bigcup _{t\in T}}{\widehat{\ell}}_{t}$. (Since functions are binary Cartesian products, we write set operations here).

#### 3.3. Pattern Reduction

**Definition**

**8**(Feasible Pattern)

**.**

**Definition**

**9**(Pattern Reduction)

**.**

#### 3.3.1. Sequential Pattern

**Proposition**

**1**(→-Pattern)

**.**

- 1.
- $\forall 1\le i<n\left(\right)open="("\; close=")">|{t}_{i}\u2022|\ge 1\wedge {t}_{i}\u2022=\u2022{t}_{i+1}$, transition${t}_{i}$enables${t}_{i+1}$; and
- 2.
- $\forall 1\le i<n\left(\right)open="("\; close=")">\forall p\in {t}_{i}\u2022\left(\right)open="("\; close=")">\u2022p=\{{t}_{i}\}\wedge p\u2022=\{{t}_{i+1}\}$, enabling is unique,

**Proof.**

**Lemma**

**1**(→-Pattern (Proposition 1) is Globally Language-Preserving)

**.**

**Proof.**

#### 3.3.2. Exclusive Choice Pattern

**Proposition**

**2**(×-Pattern)

**.**

- 1.
- $\u2022{t}_{1}=\u2022{t}_{2}=\cdots =\u2022{t}_{n}$, all pre-sets are shared among the members of the pattern;
- 2.
- ${t}_{1}\u2022={t}_{2}\u2022=\cdots ={t}_{n}\u2022$, all post-sets are shared among the members of the pattern; and
- 3.
- $\forall 1\le i\le n\left(\right)open="("\; close=")">\u2022{t}_{i}\ne {t}_{i}\u2022$, self-loops are not allowed,

**Proof.**

**Lemma**

**2**(×-Pattern (Proposition 2) is Globally Language-Preserving)

**.**

**Proof.**

#### 3.3.3. Concurrent Pattern

**Proposition**

**3**(∧-Pattern)

**.**

- 1.
- $\forall 1\le i<j\le n\left(\right)open="("\; close=")">\u2022{t}_{i}\cap \u2022{t}_{j}=\varnothing $, no interaction between the member’s pre-sets;
- 2.
- $\forall 1\le i<j\le n\left(\right)open="("\; close=")">{t}_{i}\u2022\cap {t}_{j}\u2022=\varnothing $, no interaction between the member’s post-sets;
- 3.
- $\forall 1\le i\le n\left(\right)open="("\; close=")">\forall p\in \u2022{t}_{i}\left(\right)open="("\; close=")">p\u2022=\left(\right)open="\{"\; close="\}">{t}_{i}$, pre-set places uniquely connect to a member;
- 4.
- $\forall 1\le i\le n\left(\right)open="("\; close=")">\forall p\in {t}_{i}\u2022\left(\right)open="("\; close=")">\u2022p=\left(\right)open="\{"\; close="\}">{t}_{i}$, post-set places uniquely connect to a member;
- 5.
- $\forall p\in \u2022T\left(\right)open="("\; close=")">\u2022p\cap \{{t}_{1},...,{t}_{n}\}=\varnothing $, members do not influence other members;
- 6.
- $\forall p,{p}^{\prime}\in \u2022T\left(\right)open="("\; close=")">\u2022p=\u2022{p}^{\prime}$, member’s pre-sets share their pre-set;
- 7.
- $\forall p\in T\u2022\left(\right)open="("\; close=")">p\u2022\cap \{{t}_{1},...,{t}_{n}\}=\varnothing $, member firing does not affect other members;
- 8.
- $\forall p,{p}^{\prime}\in T\u2022\left(\right)open="("\; close=")">p\u2022={p}^{\prime}\u2022$, member’s post-sets share their post-set;
- 9.
- $\forall t,{t}^{\prime}\in \u2022\left(\right)open="("\; close=")">\u2022T$, pre-sets of enablers are equal;
- 10.
- $\forall t,{t}^{\prime}\in \left(\right)open="("\; close=")">T\u2022$, post-sets of enablers are equal,

**Proof.**

**Lemma**

**3**(∧-Pattern (Proposition 3) is Globally Language-Preserving)

**.**

**Proof.**

#### 3.3.4. Loop Pattern

**Proposition**

**4**(⥀-Pattern)

**.**

- 1.
- $\u2022{t}_{1}={t}_{2}\u2022$, pre-set of${t}_{1}$is the post-set of${t}_{2}$;
- 2.
- ${t}_{1}\u2022=\u2022{t}_{2}$, pre-set of${t}_{2}$is the post-set of${t}_{1}$;
- 3.
- $\forall p\in \u2022{t}_{1}\left(\right)open="("\; close=")">p\u2022=\{{t}_{1}\}$, ${t}_{1}$is the only transition in the post-set of its pre-set;
- 4.
- $\forall p\in {t}_{1}\u2022\left(\right)open="("\; close=")">\u2022p=\{{t}_{1}\}$; ${t}_{1}$, is the only transition in the pre-set of its post-set,

**Proof.**

**Lemma**

**4**(⥀-Pattern (Proposition 4) is Globally Language-Preserving)

**.**

**Proof.**

#### 3.4. Algorithm

Algorithm 1: WF-net reduction |

**Lemma**

**5**(Pattern Reduction is Soundness Preserving)

**.**

**Proof.**

**Lemma**

**6**(Pattern Reduction is Language Preserving in $\mathrm{\Lambda}$)

**.**

**Proof.**

**Theorem**

**1**(Algorithm 1 is able to find Language-Equal Process Trees)

**.**

**Proof.**

## 4. Evaluation

#### 4.1. Implementation

#### 4.2. Experimental Setup

#### 4.3. Results

## 5. Related Work

## 6. Discussion

#### 6.1. Extensibility

#### 6.2. Relation to Refined Process Structure Tree

#### 6.3. Reducibility of WF-Nets

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A simple example process tree [1], describing all basic control-flow constructs.

**Figure 2.**The same process model, obtained by applying the Inductive Miner [12] implementation of ProM [13] on a real event dataset [14], in different process modeling formalisms. Because of its hierarchical nature, the process tree formalism easily allows us to spot the main control-flow behavior. (

**a**) The process, represented as a WF-net; (

**b**) The process, represented as a process tree.

**Figure 5.**Instantiations of $\lambda $ (5). The $\lambda $-functions for operators are defined recursively, using the $\widehat{\lambda}$-values of their children, i.e., a place “entering”/“exiting” a $\widehat{\lambda}({Q}_{i})$ fragment, connects to ${p}_{i}\u2022$/$\u2022{p}_{o}$ (respectively) of $\lambda ({Q}_{i})$.

**Figure 6.**Application of the algorithm on the running example, i.e., ${W}_{1}$. The label of ${t}_{6}^{\prime}$, i.e., $\kappa ({t}_{6}^{\prime})$, depicted in Figure 6e, is the resulting process tree. The resulting process tree, i.e., $\to (a,\u2940(\to (\wedge (\times (b,c),d),e),f),\times (g,h))$, is equal to Figure 4. (

**a**) Result of the first two rounds of the algorithm. The first two patterns that can be found are choice constructs, between b and c and between g and h, respectively. (

**b**) Result of the third round of the algorithm on the running example. We find a concurrent construct between transition ${t}_{2}^{\prime}$ and ${t}_{4}$. (

**c**) Result of the fourth round of the algorithm. We find a sequential construct. (

**d**) Result of the fifth round of the algorithm. We find a loop construct. (

**e**) Result of the final round of the algorithm. We find a sequence construct.

**Figure 7.**Example WF-net (and a corresponding reduction) in which we are able to detect the feasible pattern $\to (a,c)$. The language of the original net (Figure 7a) is $\{\langle {t}_{1},{t}_{2},{t}_{3},{t}_{4},{t}_{5}\rangle ,\langle {t}_{1},{t}_{3},{t}_{2},{t}_{4},{t}_{5}\rangle ,\langle {t}_{1},{t}_{2},{t}_{4},{t}_{3},{t}_{5}\rangle \}$. The language of the reduced net (Figure 7b) is $\{\langle {t}_{1},{t}_{2}^{\prime},{t}_{3},{t}_{5}\rangle ,\langle {t}_{1},{t}_{3},{t}_{2}^{\prime},{t}_{5}\rangle \}$. Applying the label functions on the firing sequences yields different labeled languages. (

**a**) A WF-net describing concurrent behavior between a sequential construct between a and c and activity b. Observe that the fragment formed by ${p}_{1}$, ${p}_{3}$, ${p}_{5}$, ${t}_{2}$, and ${t}_{4}$ is a feasible sequence pattern. (

**b**) The (PTree)WF-net after reduction of the sequential pattern between ${t}_{2}$, and ${t}_{4}$.

**Figure 8.**Example WF-net (and a corresponding reduction) in which we are able to detect feasible patterns ($\u2940(a,b)$ and $\u2940(c,d)$) that are not globally language preserving. In the exemplary reduced net (Figure 8b), once we have executed ${t}_{2}^{\prime}$, we are only able to execute the loop construct between ${t}_{4}$, and ${t}_{5}$. (

**a**) The WF-net containing two local language equivalent feasible patterns. (

**b**) The (PTree)WF-net after reduction of the loop pattern between ${t}_{2}$, and ${t}_{3}$.

**Figure 9.**Schematic visualization of the →-pattern reduction (dashed arcs are allowed to be part of the pattern, solid arcs are required). The post-set of each transition ${t}_{i}$ acts as the pre-set of ${t}_{i+1}$ ($1\le i<n$). The transition ${t}^{\prime}$ replacing the identified pattern inherits $\u2022{t}_{1}$ and ${t}_{n}\u2022$ (these corresponding places are not explicitly visualized in this figure). The label of ${t}^{\prime}$ is formed by the sequence operator defined on top of the labels of ${t}_{1},\dots ,{t}_{n}$, respectively.

**Figure 10.**Visualization of the ×-pattern reduction (dashed arcs are allowed to be part of the pattern, while solid arcs are required). All transitions in the pattern share the same pre- and post-set. The replacing transition inherits the aforesaid pre- and post-set.

**Figure 11.**Visualization of the ∧-pattern reduction. Transitions ${t}_{1},\dots ,{t}_{n}$ have disjunct pre-sets, yet, their pre-sets have the exact same pre-sets. The same holds for the post-sets of transitions ${t}_{1},..,{t}_{n}$. The replacing transition inherits all pre- and post-sets of ${t}_{1},..,{t}_{n}$.

**Figure 12.**Visualization of the ⥀-pattern reduction. The pre-set of transition ${t}_{1}$ equals the post-set of ${t}_{2}$ and vice versa. The replacing transition inherits the pre- and post-set of transition ${t}_{1}$.

**Figure 14.**Average time performance of the implementation. A quadratic relation, in computation time measured in micro-seconds ($\mu $-seconds), with respect to the size of the WF-net, is observable.

**Figure 15.**Schematic visualization of the ${\u2940}_{s}$-pattern reduction. The pre- and post-set of transitions ${t}_{1},{t}_{2},\dots ,{t}_{n}$ are the same. In the reduction, the places are “split” into two groups, one copying all dashed incoming arcs, one copying all dashed outgoing arcs. The newly added transition ${t}^{\prime}$ is placed in between with label $\u2940(\tau ,\times (\kappa ({t}_{1}),...,\kappa ({t}_{n})))$.

**Figure 16.**Example application of the self-loop pattern reduction. (

**a**) Simple WF-net, having self loop transitions ${t}_{2}$, and ${t}_{3}$. (

**b**) Self-loop reduction where ${T}^{\prime}=\{{t}_{2}\}$. (

**c**) Self-loop reduction where ${T}^{\prime}=\{{t}_{2},{t}_{3}\}$.

**Figure 18.**Example of an RPST decomposition (Figure 18c) based on the workflow graph (Figure 18b) of a simple sound WF-net ${W}_{2}$ (Figure 18a). (

**a**) Simple unsound (not well-handled) WF-net ${W}_{3}$. (

**b**) The workflow graph of ${W}_{3}$, including its canonical fragments. (

**c**) The RPST of ${W}_{3}$.

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**MDPI and ACS Style**

van Zelst, S.J.; Leemans, S.J.J.
Translating Workflow Nets to Process Trees: An Algorithmic Approach. *Algorithms* **2020**, *13*, 279.
https://doi.org/10.3390/a13110279

**AMA Style**

van Zelst SJ, Leemans SJJ.
Translating Workflow Nets to Process Trees: An Algorithmic Approach. *Algorithms*. 2020; 13(11):279.
https://doi.org/10.3390/a13110279

**Chicago/Turabian Style**

van Zelst, Sebastiaan J., and Sander J. J. Leemans.
2020. "Translating Workflow Nets to Process Trees: An Algorithmic Approach" *Algorithms* 13, no. 11: 279.
https://doi.org/10.3390/a13110279