## 5 Process Mining for EIS  
Process mining insights support process discovery; operational processes monitor transform into automated actions execution, aim to discover, monitor, and improve business processes by extracting knowledge from event logs available in EIS. It also offers techniques for automatic discovery of process models from event logs, which enhance discovered processes with event data. Discovered process models become more available and understandable, which can be imported/exported from/to most modeling tools to simulate. Moreover, process mining techniques can be easily integrated into the existing suites, and process discovery models with different representations, such as Petri net, transition systems, and BPMN, allow for the combination of different perspectives varying from control flow to the perspective of resources. The model with process mining for EIS mainly includes three parts: the event logs, rule monitor (similar to Kanban information in the previous section) and service engine (as work nodes in the previous section), which are defined as follows:  
* Event logs: They are considered a starting point in the context of process mining. Each event may refer to different objects from different object classes. A conventional event log (trace) is a special case of this event data notion. Their formal definition is defined as follows: Let \(A\subseteq\mu_{A}\) be a set of activity labels. Where A trace \(\sigma\in A^{*}\) is a sequence of activity labels. \(L\in\beta(A^{*})\) is an event log, i.e., a multi-set of traces. Event projection can be expressed as \(\epsilon=(Trace,name,org,time,state)\in\mu_{event}\)(See Table 2). * Rule Monitor: Abstracted from context, formula, rule, and instance, it is able to analyze future events. Let \(R\subseteq\mu_{r}\) be a set of rules to be used for monitoring. \(rm_{R}\in\mu_{log}\rightarrow\mu_{RL}\) is the rule monitor, \(\forall L\in\mu_{log}\). * Service Engine: Based on actions (activity/operation), formula, and instance, it is able to assess future rule instances. Let \(A\subseteq\mu_{a}\) be a set of actions used by the service engine. \(se_{A}\in\mu_{log}\rightarrow\mu_{AL}\) is the service engine, \(\forall L\in\mu_{log}\). The architecture of process mining for EIS is shown in Figure 4:  
There are two major components of rule monitoring (with the model, knowledge, and rule) and service engine (with a series of actions) defined above. The sound structure of an SWF-net reflects its behavior and vice versa, which includes routing constructs (sequential, parallel, conditional, iterative) and workflow building blocks (AND-split, AND-join, OR-split, OR-join). The mining algorithm, Algorithm \(\alpha\), is able to discover a large class of sound WF-nets/SWF-nets on the basis of complete event logs, whose basic relations are expressed as follows:  
* Direct succession: \(x>y\) iff for case \(x\) is directly followed by \(y\);
* Causality: \(x\to y\) iff \(x>y\) and not \(y>x\);* Parallel: \(x||y\) iff \(x>y\) and \(y>x\);
* Choice: \(x\#y\) iff not \(x>y\) and not \(y>x\). Notation \(A,\sigma,L,e\), and \(R\) are the element symbols of process mining, which constitute the basis for describing the model. The service business processes with the basic relations \((>,\rightarrow,||,\#)\) can be mined by importing set \(L\) with \(\sigma\) and \(e\) into ProM software, combining it with EIS rule R and applying algorithms (such as \(\alpha\) Miner). The process mining approach starts with a subset of model formal semantics (based on token includes activities, start and end events, exclusive and parallel gateways), applying the \(\alpha\) mining algorithm on the original event log L (a concrete case for EIS given in Table 2). Therewith, a process model is discovered in the form of a well-defined control flow modeling formalism, such as Petri nets, transition system, and causal nets process. \[L=[trace1^{i},trace2^{j},trace3^{k}\ldots] \tag{1}\]  
An event log is represented as a multi-set of traces, in which one trace can appear multiple (n) times.