## 4 Process Mining Functionality
### Structural and Behavioral Process Patterns  
The structured information control net (SICN) process model [6; 14] is a mathematical and graphical graph model to build the theoretical formalism of process-aware business activities. It is well known as the fittest methodology to describe and analyze control and information flows among the process-aware business activities by capturing temporal transitions and associations among the essential entities, such as business activities, roles, actors, applications, and repositories. Furthermore, it has been used within actual and hypothetical automated offices to yield a comprehensive description of activities, to test the underlying office description for certain flaws and inconsistencies, to quantify certain aspects of office information flows, and to suggest possible office restructuring permutations. In this subsection especially, we focus on the process-centered (control-flow) structural attributes in a process model of the structured information control nets.  
The structural and behavioral complexity of a SICN process model is determined by the control-flow attributes that are built through the combinational permutation of the four types of the structural transition patterns, such as sequential, disjunctive, conjunctive, and iterative transition patterns, as graphically shown in Figure 3 and formally defined in Definition 1. That is, the structure of a SICN process model is built by a set of business activities connected by temporal orderings (control flow) called activity transitions. The business activities can be temporally related to each other by combining sequential transition type, disjunctive transition type (after activity \(\alpha_{A}\), do activity \(\alpha_{B}\), or \(\alpha_{C}\) alternatively) with predicates attached, conjunctive transition type (after activity \(\alpha_{A}\), do activities \(\alpha_{B}\) and \(\alpha_{C}\) concurrently), and iterative transition type. Naturally, the composite transition types (except the sequential transition type) are represented by the so-called gateway activity types with split and  
Figure 2: The ultimate architecture of the functional integration of process mining and process analyzing.  
join properties. In the SICN process model, the gateway activity types must keep the properties of matched pairings and proper-nestings with splits and joins.  
**Definition 1**.: _Structural control-flow patterns in structured information control nets. The basic control-flow structure of a SICN process model is formally defined through 4-tuple \(\Gamma=(\delta,\kappa,\mathbf{I},\mathbf{O})\) over a set of \(\mathbf{A}\) activities (including a set of group activities), a set \(\mathbf{T}\) of transition conditions, where_  
* \(\mathbf{I}\) _is a finite set of initial input repositories, assumed to be loaded with information by some external process before execution of the model;_
* \(\mathbf{O}\) _is a finite set of final output repositories, which contains information used by some external process after execution of the model;_
* \(\delta=\delta_{i}\cup\delta_{\partial}\)_: control-flow structural attributes where,_ \(\delta_{\partial}:\mathbf{A}\longrightarrow\wp(\mathbf{A})\) _is a multivalued mapping function of an activity to its set of (immediate) successors, and_ \(\delta_{i}:\mathbf{A}\longrightarrow\wp(\mathbf{A})\) _is a multivalued mapping function of an activity to its set of (immediate) predecessors;_
* \(\kappa=\kappa_{i}\cup\kappa_{\partial}\)_: transition-condition associative attributes where_ \(\kappa_{i}:\mathbf{T}\longrightarrow\wp(\mathbf{A})\) _is a multivalued mapping function of an activity_ \((\alpha\in\mathbf{A})\) _to its incoming transition-conditions on each arc,_ \((\delta_{i}(\alpha),\alpha)\)_; and_ \(\kappa_{\partial}:\mathbf{T}\longrightarrow\wp(\mathbf{A})\)_: is a multivalued mapping function of an activity_ \((\alpha\in\mathbf{A})\) _to its outgoing transition-conditions on each arc,_ \((\alpha,\delta_{\partial}(\alpha))\)_._