## 4 Process Mining Functionality
### The Process Mining Approach: \(\rho\)-Algorithm  
This subsection simply describes a process mining framework that is proposed and implemented by the authors' research group. As shown in Figure 4, the framework is able to rediscover all the four types of primitive process patterns in a structured information control net process model from a process enactment event log dataset by applying the concept of mass with their enactment event log occurrences. The conceptual initiation of the process mining framework is introduced in this subsection. Especially, the core component of the framework is the \(\rho\)-Algorithm [26; 29]. In the \(\rho\)-Algorithm, the symbol and name of _rho_ (\(\rho\)) comes from the A programming language (APL) firstly released in 1960s. The function \(\rho\), coded like \(\rho\)X in APL, implies that it gives the number of elements in X, from which the concept of _mass_ comes. The central idea of the rediscovery algorithm of the framework is exactly the same for the implication of the APL function, rho (\(\rho\)). The \(\rho\)-operator gives the masses (occurrences) of all the activities in the event log dataset of a corresponding process  
Figure 3: The structural control-flow primitives in structured information control net process models. model. In the paper, the emphasis is placed on rediscovering very large scale and massively parallel process models that are usually built by a combination of the four types of primitive process patterns, linear, disjunctive (exclusive-OR), conjunctive (parallel-AND), and repetitive (iterative-LOOP) process patterns, which were formally defined in the previous section. The overall algorithmic approach is a stepwise mining procedure with a series of graphical transformations to be used for rediscovering all the types of the primitive process patterns that constitute a structured information control net process model. That is, as shown in Figure 5, the \(\rho\)-Algorithm consists of three major transformation steps, STEP-1, STEP-2, and STEP-3, from forming temporal workcases out of the process enactment event logs to discovering a structured information control net process model as a result. The first transformation of the algorithm is to discover the enacted workcases from the event logs, each of which can be modeled into a temporal workcase model. At the same time, it is necessary to count the occurrence of each temporal workcase with its activities. The second transformation of the algorithm is to discover an activity-driven pattern graph by integrating all the members of the adjacent-activity set and calculating the occurrences of the temporal workcases. In terms of rediscovering the structured information control net process model from the corresponding activity-driven pattern graph, we develop the third transformation of the algorithm that is able to transform to any combinational number of AND/OR/LOOP primitive process patterns. * _STEP-1: Groups of Temporally Ordered Adjacent-Activity Pairs_: The first step of the \(\rho\)-Algorithm is to mine a group of temporally ordered adjacent-activities pairs from temporal workcases of the process instance event logs. Additionally, each of the temporal workcases is formally represented by one of the workcase model types introduced in the conceptual framework. That is, a temporal workcase represents an ordered enactment sequence of activity event logs, each of which is formed with its activity identifier and its time-stamp extracted from its corresponding process enactment event log. * _STEP-2: Quantitative Adjacent-Activity Set and Process Pattern Graph_: The STEP-2 of the \(\rho\)-Algorithm is to build all the groups of temporally ordered adjacent-activity pairs, each of which corresponds to a process instance event trace. The eventual output of this algorithm is a quantitative adjacent-activity set named adjacencyList \(\beta\). This set is built from all the groups of temporally ordered adjacent-activity pairs through an internal transformation procedure. Figure 4: The process mining framework.