A novel approach for process mining based on event types

Abstract

Despite the omnipresence of event logs in transactional information systems (cf. WFM, ERP, CRM, SCM, and B2B systems), historic information is rarely used to analyze the underlying processes. Process mining aims at improving this by providing techniques and tools for discovering process, control, data, organizational, and social structures from event logs, i.e., the basic idea of process mining is to diagnose business processes by mining event logs for knowledge. Given its potential and challenges it is no surprise that recently process mining has become a vivid research area. In this paper, a novel approach for process mining based on two event types, i.e., START and COMPLETE, is proposed. Information about the start and completion of tasks can be used to explicitly detect parallelism. The algorithm presented in this paper overcomes some of the limitations of existing algorithms such as the α-algorithm (e.g., short-loops) and therefore enhances the applicability of process mining.

Appendix

Theorem 1

Let N = (P,T,F) be a sound WF-net and let W be a complete event log of N . For any a,b ∈ T: a→ _W b implies a ∙ ∩ ∙ b ≠ ∅.

Proof

Assume a→ _W b and a ∙ ∩ ∙ b = ∅. We will show that this assumption leads to a contradiction and thus prove the theorem. From Definition 13, we know that a→ _W b implies a >  _W b and $\neg(a\times_W b)$. Since a >  _W b there exists at least one trace σ = e ₁ e ₂ e ₃ ⋯ e _n  ∈ W such that $\exists_{i,j} 2\leq i\leq n-2\wedge i<\!j\!<\!n$ such that e _i .type=COMPLETE, e _i .task=a, e _j .type=START, e _j .task=b and there is not any task occurrence between e _i and e _j . For ∀  _k i < k < j and e _k .type=COMPLETE, we know that e _k can occur before e _i in some traces. Similarly, for ∀  _m i < m < j and e _m .type=START, we know that e _m can wait until e _j occurs. Thus we can get a marking M of N, under which a can complete and after a completes, b can start immediately. Because a ∙ ∩ ∙ b = ∅, a does not produce tokens for any input place of b. So under the marking M, b can start before a completes. Therefore, we can find a× _W b from the log and a ∥  _W b holds. This result contradicts a→ _W b and we conclude that a→ _W b implies a ∙ ∩ ∙ b ≠ ∅. □

Theorem 2

Let N = (P,T,F) be a sound DWF-net and let W be a complete event log of N . For any a,b ∈ T: a ∙ ∩ ∙ b ≠ ∅ implies a→ _W b.

Proof

Because a ∙ ∩ ∙ b ≠ ∅, we assume a place p ∈ a ∙ ∩ ∙ b. According to the definition of DWF-net, b can start after a completes and a >  _W b holds in the log. Remains to prove $\neg(a\times_W b)$. Assume b can start before a completes and a× _W b holds. In this case, there should be one token in p for some marking M, under which a is fired and b is enabled. If a completes under M, a will produce one token for p and there would be two tokens in p. We get a contradiction, thus $\neg(a\times_W b)$ holds. Since a >  _W b and $\neg(a\times_W b)$, we conclude a→ _W b. □

Theorem 3

Let N = (P,T,F) be a sound WF-net and let W be a complete event log of N . For any a,b ∈ T:

• 1. If a ∙ ∩ b ∙ ≠ ∅, then $a\nparallel_W\!b$.

• 2. If ∙ a ∩ ∙ b ≠ ∅, then $a\nparallel_W\!b$.

Proof

Assume a ∥  _W b in both situations, we will show that this can lead to a contradiction respectively for the following two parts.

• 1. Assume a place p ∈ a ∙ ∩ b ∙. For a ∥  _W b, there should at least be one “overlapping sequence” in the log, i.e., a× _W b. Since the COMPLETE event of a task may occur at any time after the corresponding START event, there may be a snippet (a,1) (b,1) or (b,1) (a,1) in some trace. In this case, there will be a marking M that does not cover p, under which both a and b have started and can complete immediately. Thus p will contain at least two tokens after a and b complete and the net is not 1-safe. So we get a contradiction and $a\nparallel_W b$ holds.

• 2. Assume a place p ∈ ∙ a ∩ ∙ b. For a ∥  _W b, there should be at least a sequence a× _W b in the log. There will be a marking M of the net under which p is covered and a or b can start. But after a or b starts, the only token in p is consumed and the other transition can not start. The other one will wait until there is a token in p again. So a snippet of (a,0) (c,1) (b,0) or (b,0) (c,1) (a,0) will appear in some trace in the log. However, the COMPLETE event of c may start before the START event of a or b. Under the marking M, (c,1) may occur just before (a,0) or (b,0) and thus there will be two tokens in the place p, i.e., the net is not 1-safe. So we get a contradiction and $a\nparallel_W b$ holds. □

Theorem 4

Let N = (P,T,F) be a sound DWF-net and let W be a complete event log of N . For any a,b,c ∈ T:

• 1. If a→ _W c, b→ _W c and $a\nparallel_W\!b$ , then a ∙ ∩ b ∙ ∩ ∙ c ≠ ∅.

• 2. If c→ _W a, c→ _W b and $a\nparallel_W\!b$ , then c ∙ ∩ ∙ a ∩ ∙ b ≠ ∅.

Proof

Now we should prove the above two sub theorems respectively.

• 1. From Theorem 1 and a→ _W c, we deduce a ∙ ∩ ∙ c ≠ ∅. Similarly, $b\bullet\cap\bullet c\ne\emptyset$. Because $a\nparallel_W b$, there will be no chance that a and b occur concurrently. In other words, there will be no marking M of N, under which a has started and b is enabled. According to the requirement two of a DWF-net, a conclusion is drawn that $a\bullet\cap b\bullet\cap\bullet c\ne\emptyset$.

• 2. From Theorem 1 and c→ _W a, we deduce c ∙ ∩ ∙ a ≠ ∅. Similarly, $c\bullet\cap\bullet b\ne\emptyset$. Because $a\nparallel_W b$, there will be no chance that a and b occur concurrently. In other words, there will be no marking M of N, under which a has started and b is enabled. According to the requirement two of a DWF-net, a conclusion is drawn that $c\bullet\cap\bullet a\cap\bullet b\ne\emptyset$. □

Theorem 5

Let N = (P,T,F) be a sound DWF-net and let W be a complete event log of N . For any two task sets PS and SS , such that $PS\subseteq T$ , $SS\subseteq T$ : ∀  _{a ∈ PS,b ∈ SS} a→ _W b, $\forall_{a1,a2\in PS}a1\nparallel_W a2$ and $\forall_{b1,b2\in SS}b1\nparallel_W b2$ if and only if $\exists_{p\in P} \forall_{a\in PS,b\in SS}p\in a\bullet\cap\bullet b$.

Proof

We should prove the theorem in both directions.

• 1. Assume $\exists_{p\in P} \forall_{a\in PS,b\in SS}p\in a\bullet\cap\bullet b$. Using Theorem 2, it is easy to see that ∀  _{a ∈ PS,b ∈ SS} a→ _W b. Using Theorem 3 it is also shown that the elements of PS and the elements in SS cannot be in parallel (i.e., ∀  _{a1,a2 ∈ PS} p ∈ a1 ∙ ∩ a2 ∙ and ∀  _{b1,b2 ∈ SS} p ∈ ∙ b1 ∩ ∙ b2). Thus the necessity of the theorem holds.

• 2. Assume ∀  _{a ∈ PS,b ∈ SS} a→ _W b, $\forall_{a1,a2\in PS}a1\nparallel_W a2$ and $\forall_{b1,b2\in SS}b1\nparallel_W b2$. According to the requirement two of a DWF-net, there exists a place p ∈ P such that $PS\subseteq\bullet p$ and $SS\subseteq p\bullet$. Therefore a conclusion can be drawn that $\exists_{p\in P}\forall_{a\in PS,b\in SS}p\in a\bullet\cap\bullet b$. Thus the sufficiency of the theorem holds. □

Theorem 6

Let N be a sound DWF-net and let W be a complete event log of N. β(W) = N modulo renaming of places, i.e., the discovered model matches the original model after renaming places.

Proof

Let N = (P,T,F) and β(W) = N _W  = (P _W ,T _W ,F _W ). Based on the completeness of W and mining step 1 of the β algorithm, we get T _W  = T. For the source and sink places (i.e., i and o) of N, there are the source and sink places i _W and o _W of N _W such that i _W  ∙ = i ∙ ∧ ∙ o _W  = ∙ o and vice versa. Remains to prove that the “internal places” of the two Petri nets N and N _W match.

• 1. First we prove that $\forall_{p\in\!P\backslash\{i,o\}} \exists_{p_W\!\in\!P_W\backslash\{i_W,o_W\}}\bullet p_W=\bullet p\wedge p_W\bullet=p\bullet$. According to the β-algorithm and Theorem 5, we know that < ∙ p,p ∙ > ∈ X _W and $\exists_{p_W\in P_W}\bullet p\subseteq\bullet p_W\wedge p\bullet\subseteq p_W\bullet$, i.e., < ∙ p _W ,p _W  ∙ > ∈ Y _W . Assume that $\exists_{t'\in T} t'\in\bullet p_W\wedge t'\notin\bullet p$. Using Theorem 2 and Theorem 3, we can show that $\forall_{t\in\bullet p}t'\nparallel_W t$ and ∀  _{t ∈ p ∙} t′→ _W t. According to the second requirement, there exists a place p′ such that ∙ p ⊂ ∙ p′ and $p\bullet\subseteq p'\bullet$. This violates the third requirement of a DWF-net and we get a contradiction. If we assume that $\exists_{t'\in T} \ t'\!\in\! p_W\bullet\wedge t'\!\notin\! p\bullet$, we can still get a similar contradiction. Therefore we prove the result in one direction.

• 2. Finally, we prove $\forall_{p_W\!\in\! P_W\backslash\{i_W,o_W\}} \exists_{p\in\! P\backslash\{i,o\}}\ \bullet p=\bullet p_W \wedge p\bullet=p_W\bullet$. According to the β-algorithm and p _W  ∈ P _W , we know that < ∙ p _W ,p _W  ∙ > ∈ Y _W . Theorem 5 can be used to show that $\exists_{p\in P}\forall_{a\in\bullet p_W,b\in p_W\bullet}a\bullet\cap\bullet b\!\!=\!\{p\}$. Therefore we deduce $\bullet p_W\!\subseteq\!\bullet p\wedge p_W\bullet\subseteq p\bullet$. Assume that $\exists_{t'\in T}t'\in\bullet p\wedge t'\!\notin\!\bullet p_W$. Using Theorem 2 and Theorem 3 we can show that $\forall_{t\in\bullet p_W} \ t'\!\nparallel_W\!t$ and $\forall_{t\!\in\! p_W\bullet} \ t'\!\rightarrow_W\!t$. Therefore we can prove that < ∙ p _W  ∪ {t′},p _W  ∙ > ∈ Y _W and thus obtain a contradiction. If we assume that $\exists_{t'\in T}t'\!\in\!p\bullet\wedge t'\!\notin\!p_W\bullet$, we can also get a contradiction, thus complete the proof. □