## 3 Preliminaries  
_Problem Definition_  
In this section, we introduce some basic concepts used in this paper. The full notations used in this paper is available in Table 1.  
**Definition 1** (Event Log, Trace, Activity, Event). : _An event log \(L\) is a collection of traces. A trace \(t\in L\) is a sequence of events. \(A\) is the set of activities, and an event \(e\) is an execution record of an activity \(a\in A\). \(\#_{n}(e)\) is a function that obtains attribute values recorded for an event \(e\). For example, \(\#_{activity}(e)\) obtains the activity name for an event, and \(\#_{attribute}(e)\) obtains the attribute value for an event._  
For example, let \(E=\{a,b,c,d\}\) be a set of activities. \(t=<e_{1},e_{2}>\) is a trace, where \(\#_{activity}(e_{1})=a\) and \(\#_{activity}(e_{2})=b\). \(L=\{<e_{1},e_{2}>,<e_{3},e_{4}>\}\) is an event log, where each \(e_{n}\) represents a unique execution record of a specific activity. For the sake of understanding, \(\#_{activity}(e_{n})\) will be used for each event for the rest of the paper. **Definition 2** (Ordering Relation). : _Let \(L\) be an event log and \(t\in T\) be a trace. For \(\forall a,b\in A\), the ordering relations between \(a\) and \(b\) are defined as follows:_  
* _Directly-follows relation:_ \(a>_{W}b\) _holds if there is a trace_ \(t=<e_{1},e_{2},e_{3},e_{4},...,e_{n}>\) _and_ \(i\in\{1,2,3,...,n-1\}\) _such that_ \(t\in L\) _and_ \(\#_{activity}(e_{i})=a\) _and_ \(\#_{activity}(e_{i+1})=b\)_._
* _Indirectly-follows relation:_ \(a>>_{W}b\) _holds if there is a trace_ \(t=<e_{1},e_{2},e_{3},e_{4},...,e_{n}>\) _and_ \(i<j\) _and_ \(i,j\in\{1,2,3,...,n-2\}\) _such that_ \(t\in L\) _and_ \(\#_{activity}(e_{i})=a\) _and_ \(\#_{activity}(e_{j})=b\)Because we are only interested in indirectly-follows relations with strong connections, we define the following measurement based on [6] to measure how reliable an indirectly-follows relation is. **Definition 3** (Long Distance Measure ). : \[a\Rightarrow_{W}b=\left(\frac{2(|a>>_{W}b|)}{|a|+|b|+1}\right)-\left(\frac{2 Abs(|a |-|b|)}{|a|+|b|+1}\right)\] (1)  
If the first part of the equation is close to 1, then activity \(a\) is always followed by activity \(b\). The second part of the equation measures the frequency distributions for activities \(a\) and \(b\). A value close to 0 indicates that the frequency of activities \(a\) and \(b\) is about equal [6]. Therefore, a value of the long-distance measure close to 1 means that the activities \(a\) and \(b\) have a strong indirectly-follows relation. Based on this, we define a strong indirectly-follows relation between two activities. **Definition 4** (Strong Indirectly-Follows Relation). : \(a>>_{W}b\) _holds between two activities \(a,b\in A\) if \(a\Rightarrow_{W}b\) is larger than a given threshold \(p\). In this paper, \(p\) is set to 0.9, as recommended in [6]._  
For example, \(L=\{<a,b,c,d>,<b,c,d>^{5}\}\) is an event log. The directly-follows relations are \(a>_{W}b\), \(b>_{W}c\), and \(c>_{W}d\). The indirectly-follows relations are \(a>>_{W}c\), \(a>>_{W}d\), and \(b>>_{W}d\). The long-distance measures for these three indirectly-follows relations are 0, 0, and 0.92. Thus, the strong indirectly-follows relation is \(b>>_{W}d\). **Definition 5** (Directly-Follows Graph). : _A directly-follows graph is defined as \(G=(A,K)\) where \(A\) is a finite set of activities in the event log (same as Definition 1), and \(K\subseteq A\times A\) is a set of directed arcs, which represent directly-follows relations (i.e., \(a>_{W}b\) exists if \((a,p)\in K\)). An example is shown in Figure 3._  
**Definition 6** (Pre-Sets and Post-Sets). : _Let \(G=(A,K)\) be a directly-follows graph. For \(a\in A\), we have_  
\[a\bullet=\{b|b\in A\land(a,b)\in K\}\]  
_and_  
\[\bullet a=\{b|b\in A\land(b,a)\in K\},\]  
_where \(a\bullet\) and \(\bullet a\) are called a post-set and a pre-set of \(a\), respectively.