ρ-Algorithm: A SICN-oriented Process Mining Framework

This paper devises an algorithmic process mining framework characterized by the mathematical process model of structured information control nets (SICN) and the concept of mass-driven ρ-function as a decision-making criterion of structural process patterns. In order to prove the functional correctness of the proposed algorithmic framework, this paper also implements all the related algorithms as a process mining system and carries out an operational experiment on a typical synthetic dataset of process enactment event logs prepared and released in the 4TU Centre for Research Data. The core outcome of the algorithmic framework development is the ρ-Algorithm that ought to be a novel approach not only for mining all the primitive process patterns, such as linear (sequential), disjunctive (selective-OR), conjunctive (parallel-AND), and repetitive (iterative-LOOP) process patterns, with perfectly keeping the structural properties of matched pairing and proper nesting, but also for reasonably discovering structured (even unstructured) information control nets from such IEEE XES-formatted datasets of process enactment event logs. The discovery functionality of the ρ-Algorithm is made up of three stepwise algorithms: STEP-1, STEP-2 and STEP-3 algorithms, and these algorithms are formally described as an algorithmic framework supported by the conceptual process mining archiecture with a series of theoretical concepts with the temporal workcase model and the temporal loop-case model. Finally, we validate the functional correctness as well as the discovery perfectness of the proposed algorithmic framework named as ρ-Algorithm by deploying the implemented ρ-Algorithm on a synthetic, non-noise and IEEE XES-formatted dataset of process enactment event logs recorded from the 10,000 work-cases with 113 activities of the Petrinet-oriented process model named as the Large Bank Transaction Process Model.


I. INTRODUCTION
A workflow process (or business process, from now on, which is named as process) management system (WPMS or BPMS) is defined as a system that partially or fully automates the definition, creation, execution, and management of work procedures through the use of software that is able to interpret the procedure definition, interact with businesstask participants, and invoke their uses of IT tools and applications. Steps of a work procedure are called activities, and jobs or transactions that flow through the system are called work-cases or process instances. One of the recent research issues in the process management literature is the process intelligence coping with the process mining [1] [2] [3] [4] and its related process-aware knowledge [5] [6] [7] [8] [9] [10] discovery and rediscovery. The motivation of the paper is to develop an algorithmic process rediscovery framework (which is called a SICN-oriented process mining framework in this paper), in particular, that becomes an effective catalytic means for successfully accomplishing not only the organizational change management works but also the process automation works in all the phases of the process life-cycle, continuously. At this moment, we have to emphasize that the rediscovered process model ought not to be equivalent to its original process model in terms of its process aspect itself as well as almost all the other aspects like process resource allocation aspect, performer behavioral aspect and process relevant data aspect. Figure 1 is to illustrate a situational view of a certain process-aware enterprise (or organization) to define the conceptual difference between process-aware knowledge discovery and rediscovery. As you see, we assume that the enterprise has been supported from a variety of information systems that manage all the valuable assets of data, information and knowledge available and produced in the business and organizational environments. One of the recent trends in the traditional business and managerial information systems ought to be the adoption of process management systems and the deployment of the process automation methodologies as a means of monitoring and controlling business-activities and managements. We call those enterprises and organizations fortified with process automation and management methodologies, tools and systems as the process-aware enterprises and organizations, in general. The core competitiveness of the process-aware enterprises and organizations ought to be on the process mining functionality that is able to continuously manage as well as repeatedly evolve all the processes deployed in the corresponding enterprises and organizations. The process mining functionality is mainly composed of the process discovery [7] [11] [12] functionality and the process rediscovery [13] [1] [14] [5] functionality. The former is to discover business-activity processes from the event logs of the executions of the traditional information systems, while the later is to rediscover the enacted business-activity processes from the event logs stored whenever the corresponding business-activity processes are enacted and executed by their process management system. Especially, the rediscovered processes can be re-engineered and re-designed according to their performances measured and mined from the event logs by the process knowledge mining tools [15] and the process intelligence solutions as well. Note that we won't, from now on, differentiate the terminology of discovery from the the rediscovery in this paper.
Particularly, the paper proposes an algorithmic process mining framework for discovering structured process patterns from process enactment event logs and validates the proposed algorithmic framework by implementing all the related stepwise algorithms. Also, it carries practically out an experimental analysis by deploying the implemented framework onto the dataset provided by the BPI challenges of 4TU.Centre for Research Data [16] and formatted in the IEEE XES standardized format [17]. The following are the characteristics of the algorithmic process mining framework proposed in the paper. First, the algorithmic framework is able not only to discover the process patterns [18] but also to discover the enactment occurrences [19] [20] of the process patterns from a dataset of the IEEE XES-formatted enactment event logs of a corresponding process model. Second, the algorithmic framework is theoretically supported by the information control nets modeling methodology [21] of process models. Third, the essential algorithm of the algorithmic framework named as ρ-Algorithm (rho-Algorithm) is able to discover a structured information control net model with the enactment occurrences of the activities associated with an underlying process model. Fourth, the ρ-Algorithm is firstly developed in the process management and mining literature as the process mining algorithm that discovers a structured process model theoretically supported by the structured information control net modeling methodology. Fifth, the ρ-Algorithm is able to discover all the process patterns such as linear (sequential), conjunctive (parallel-AND), disjunctive (exclusive-OR), and repetitive (iterative-LOOP) process patterns and discover the enactment occurrences and proportions of each branch of the process patterns.
As the validation study of the proposed algorithmic framework (ρ-Algorithm), the paper applies the implemented framework and system to a real dataset that contains nonnoise and synthetic event logs from the enactment history of 10,000 process instance event traces of the Large Bank Transaction Process Model [16]. Through the feasibility study, the paper verifies the functional correctness of the ρ-Algorithm in discovering all the types of process patterns from the real dataset formatted in the IEEE XES standardized specification [17].
In terms of organizing the paper with the sections, the next section summarizes the related research works and scope as the algorithmic challenges of the conventional process discovery algorithms done in the business and workflow process intelligence literature. In the consecutive section, we describe the theoretical and algorithmic details of devising the ρ-Algorithm with three stepwise algorithms under the name of the SICN-oriented process mining framework. Subsequently, the implementation and operational details of validating the core functions of the ρ-Algorithm are explicated with focusing on how it works for discovering all the structural process patterns and finally building up a structured information control net from an exemplary dataset as well as the synthetic and non-noise dataset with the 10,000 instances' enactment histories of the Large Bank Transaction Process Model, respectively. Finally, the last section summarizes our research outcomes and experimental validation results with concluding our future works and remarks.

II. RELATED WORKS AND MOTIVATION
The main research challenge of this paper is to devise an algorithmic process mining framework for discovering a model of structured information control nets from a process enactment event log dataset. Therefore, the literature surveys of the challenge are summarized in this section of related works. The most popular approaches of the theoretical modeling methodologies to formally define and graphically represent business process models are the Petrinet-oriented model [22] [23] [24] of process modeling methodology and the information control net model [25] [26] [27] [10] of process modeling methodology. Both of them are based upon the mathematical representation and the graphical representation at the same time. The Petrinet-oriented process model has a strong advantage in terms of the mathematical and analytical power, whereas the information control net process model • Choice: x#y iff not x>y and not y>x.
• Parallel: x y iff x>y and y>x • Causality: x→y iff x>y and not y>x has a much stronger merit in terms of the expressiveness of the process domain. So far, there have been several process mining algorithms in the literature. One of the typical process mining algorithms for discovering the Petrinet-oriented process models is the α(alpha)-Algorithm [13] [28], whereas the typical process mining algorithm for discovering the information control net process models is the σ (sigma)-Algorithm [1]. Note that the name of the process mining algorithm proposed in the paper is the ρ(rho)-Algorithm, and the naming reason of the ρ-Algorithm will be explained later. The crucial idea and characteristics of these algorithms and their comparisons with the ρ-Algorithm are arranged in this section.

A. THE α-ALGORITHM FOR DISCOVERING THE PETRINET-ORIENTED PROCESSES
Through publishing the α-Algorithm, W.M.P. van der Aalst, et al. [13] firstly addressed the process mining problem. This problem was formulated as follows: Find a mining algorithm able to discover a large class of sound Petrinetoriented process model on the basis of complete process event logs. The α-Algorithm is able to discover a large and relevant class of structured processes. Through examples they showed that the algorithm provides interesting analysis results for processes and tackled the problem of short loops with focusing on hidden tasks, duplicate tasks, and advanced routing constructs. The discovery problem is not a goal by itself. They also assumed that the overall goal is to be able to analyze any process event logs without any knowledge of the underlying process and in the presence of noise. In VOLUME 4, 2016 L: case 1 : task A case 2 : task A case 3 : task A case 3 : task B case 1 : task B case 1 : task C case 2 : task C case 4 : task A case 2 : task B case 2 : task D case 5 : task E case 4 : task C case 1 : task D case 3 : task C case 3 : task D case 4 : task B case 5 : task F case 4 : task D a common sense in the literature, their works were accepted as a stepping stone for good and robust process mining techniques. Figure 2 shows the three types of Petrinet-oriented process patterns that can be discovered from the process event logs by the α-Algorithm, and the log-based ordering relations, such as causality, parallel and choice, that are used for deciding the type of process patterns by the α-Algorithm, respectively. Based upon these ordering relation operators, they applied the α-Algorithm to an exemplary dataset of the process enactment event logs as depicted in Figure 2, and proved that the α-Algorithm is reasonable and applicable in the real process intelligence arena.

B. THE σ -ALGORITHM FOR DISCOVERING THE INFORMATION CONTROL NET PROCESSES
Through publishing the σ (sigma)-Algorithm, K. Kim and C. A. Ellis [1] proposed a mining algorithm for discovering process models from the underlying process enactment event logs. Differently from the α-Algorithm dealing with the Petrinet-oriented process modeling methodology as introduced in the previous section, the σ -Algorithms theoretical basis is the structured information control net process modeling methodology. They understood process management systems as process-aware information systems that help to execute, monitor and manage work process flow and execution. These systems, as they are executing, keep a record of who does what and when (e.g. log of events). Also they defined the basic concept of the process mining as those activities of using computer software to examine these records and deriving various structural data results. They thought that the process mining activity needs to encompass behavioral (process/ control-flow), social, informational (data-flow), and organizational perspectives. Especially, they were insisted that process management systems are people systems that must be designed, deployed, and understood within their social and organizational contexts. The process mining activity ought to be planned and fulfilled under the preconditions of the people systems, too. They especially focused on the behavioral perspective of a structured process model that preserves the proper nesting and the matched pair properties and proposed a process mining algorithm that is able to discover a structured information control net process model, which is named as σ -Algorithm. The main reason of naming the σ -Algorithm is because it is incrementally amalgamating a series of temporal work-cases (process instance event traces) according to three types of basic merging principles. Figure  3 shows how the σ -algorithm works with temporal workcases by illustrating an example of the information control net process discovery using the σ -Algorithm.

C. THE MOTIVATION OF THE ρ-ALGORIT HM
The essential motivation of the paper is about the proof of concept and feasibility with the SICN-oriented process mining framework that is able to discover a process model of structured information control net from such a dataset of process enactment event logs formatted in IEEE XES standardized specifications. As stated in the previous subsections, for the sake of eventually supporting the model-log comparison, which is the process fidelity issue [13] [23] [29] [30] [31], the literature has produced so far the two concepts  and theories; one is the Petrinet-oriented process modeling methodology and the other is the information control net process modeling methodology. Also, the literature has published the α-Algorithm [2] [31] and the σ -Algorithm [1] for discovering Petrinet-oriented process models and information control net process models, respectively. However, all of these discovery approaches have the limitations that they are able to deal with only sequential, parallel and selective process patterns out of all the types of the process patterns such as sequential, parallel, selective and repetitive process patterns. In other words, both of the discovery algorithms have a limitation in dealing with discovering the repetitive-LOOP process patterns; the α-Algorithm, in principle, is implicitly dealing with the repetitive-LOOP process pattern as the selective process pattern. In recent, some [31] [32] [26] [33] [34] of the research groups tried to not only extend the α-Algorithm and the σ -Algorithm so as to explicitly deal with the repetitive-LOOP process patterns, but also expand the process mining usages to the predictive process managing and monitoring area.

III. DEVISING THE ρ-ALGORIT HM
In this section, a SICN-oriented process mining framework is proposed and formalized with a series of formal concepts and models. Especially the proposed algorithmic framework is characterized by the concept of mass-driven ρ-function as a decision-making criterion of process patterns. In other words, the proposed algorithmic framework is able to eventually discover 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 mass-driven ρ-function with their enactment event log occurrences. In this section we formally describes a series of conceptual components of the conceptual process mining architecture at first. Next, we also devise an algorithmic process mining framework with a series of algorithms for transforming and discovering all the process patterns and their enactment event log occurrences. Especially, the core component of the devised algorithmic framework is the ρ-Algorithm 1 , which is firstly introduced and so named by this paper, and the ρoperator of which gives the masses (occurrences) of all the activities in the event log dataset of a corresponding process model.

A. THE CONCEPTUAL PROCESS MINING ARCHITECTURE
In this section, we explicate the conceptual approach with formal notations for devising the ρ-Algorithm proposed in the paper, which is to formally define a conceptual process mining architecture to discover a structured information control net process model from its enactment event histories logged by executing the corresponding process model built from any combinations of all the four types of primitive 1 In the ρ-Algorithm, the symbol and name of rho (ρ) comes from the A programming language (APL) firstly released in 1960s. The function rho, coded like ρX in APL, implies that it gives the number of elements in X, from which the concept of mass comes. The central idea of the discovery algorithm of the framework is exactly same to the implication of the APL function, rho (ρ).   Figure 4 illustrates a series of formal components of the conceptual process mining architecture, which is revised from the original architecture [1] to emphasize the proposed algorithm's conceptual requirement with discovering all the possible combinations of the four types of primitive process patterns [35] [18], such as linear, disjunctive (exclusive-OR), conjunctive (parallel-AND) and repetitive (iterative-LOOP) process patterns, with their graphical representations given in Figure 5. The concrete algorithm is theoretically supported by a series of functional routines that are formalized by a series of formal concepts and their related algorithms, such as process enactment event log datasets, process warehouses, temporal work-cases, temporal loop-cases, temporal workcase model, temporal loop-case model and a process pattern rediscovery algorithm named as the ρ-Algorithm. Especially, the functional core of the ρ-Algorithm is to receive a bunch of temporal work-cases as inputs and to discover a structured information control net as an output. At the moment of composing a structured information control net as the output, the ρ-Algorithm has to precisely situate three types of special nodes, such as selective-XOR, parallel-AND and iterative-LOOP nodes, between those activity nodes having multiple incoming or outgoing transition relationships, as shown in Figure 5. Besides, these three types of special nodes are called gateway activity nodes; each type of the special nodes has to match a pair of OPEN and CLOSE gateway activities; simultaneously, the rule of properly nested formation must be kept so that multiple sets of the OPEN and CLOSE gateway activities are theoretically satisfied with the principles of safeness and soundness. Eventually, the discovered process model is graphically represented by the control flow aspect of the information control net that is built from a starting node, a terminating node, the enacted activities with their nodes, and the edges combining all these nodes with a set of sequential node, a set of pairs of OPEN and CLOSE parallel gateway nodes, a set of pairs of OPEN and CLOSE selective gateway nodes, and a set of pairs of OPEN and CLOSE iterative gateway nodes. Likewise, it is necessary to be graphically visualized as a graphical process model with keeping the structural properties of matched-pairing and proper-nesting according to the following graphical composition rules: The conceptual architecture is concretized by a series of formal concepts and their algorithms, such as process enactment event log datasets, process warehouses, temporal work-cases, temporal loop-cases, temporal work-case model, temporal loop-case model and a process pattern discovery algorithm named as the ρ-Algorithm. Especially, the figure highlights the conceptual role of the ρ-Algorithm that is the core component of the conceptual architecture. This section formally defines these formal components.

1) Process Enactment Event Logs
On the conceptual approach of Figure 4, the starting point is the process enactment event logs that are organized into a cubic structure of the process enactment event logs of each process package, the accumulated process instance enactment event traces of each process model and the process work-item enactment events of each process instance. In other words, the event logs imply the enactment histories of all the process models (or packages) managed by an underlying process management system. According as an instance of a process model executes, its temporal execution sequence is produced and logged into the backup databases or files; this temporal execution sequence is called a process enactment event trace. The process enactment event trace is made up of a temporal sequence of the activity event logs associated with a specific process instance. According as a process instance is executed, the logging and auditing component of the process enactment engine records its work-item (activity instance) execution events on a log repository, and those logged events are arranged in a form of temporal sequence of events. This execution sequence of a process instance is forming a process instance event trace. Here, we start to formally describe the conceptual architecture from defining a formal structure of the process enactment event logs with its three dimensional cube as defined in the following consecutive definitions from Definition 1 through Definition 4. Definition 1: Process Work-Item Enactment Event. Let pe = (α, pc, w f , wc, ac, p * , t, s) be a process work-item enactment event stored as logs, where − α is a work-item (activity instance) identifier, − pc is a package identifier, − w f is a process identifier, − wc is a process instance (work-case) identifier, − ac is an activity identifier, − p 2 is a participant (or performer) identifier, − t is a timestamp, and − s is a work-item's current state, which is one of the states such as ready, assigned, reserved, running, completed, and cancelled. where PET (c) = (pe 1 , . . . , which formally represents a temporally ordered work-activity event sequence of a specific work-case, which is built through preprocessing the process enactment event logs by considering the TIMESTAMP and the STATE attributes. .., c i m }, be a process enactment event log with a set of completed process instances (m is the number of the process instances) that have been instantiated from a process model, Definition 4: Process Package Cubic Warehouse. Let PW be a process package cubic warehouse consisting of a set of process enactment event logs, PL(I 1 ), . . . , PL(I n ), where PL(I i ) = ∀PET (c i ∈ I i ), and n is the number of process models in a process package managed in a system.
Basically, we build a series of the formal models of a process package cubic warehouse, PET , PL and PW , so far. Importantly, we remind that a process instance enactment event trace is composed of a temporally ordered group of the work-item enactment events having the same process instance identifier. Note also that the meaningful temporal order in managing process instances (work-cases) ought to be based upon one of the following instantaneous points of time, each of which is named as a timestamp-origin. Accordingly, we necessarily build several different types of process package cubic warehouses based upon the type of timestamp-origins as follows: • The Scheduled Point of Time: the event's timestamp is taken at when the state of a work-item is changed from READY 3 to ASSIGNED 4 : a work-activity event log with scheduledTimestamp, we t.s ⇒ (t = we.t ∧ s = we.s ∧ s = "assigned") • The Assessed Point of Time: the event's timestamp is taken at when the state of a work-item is changed from ASSIGNED to RESERVED 5 : a work-activity event log with assessedTimestamp, we t.e ⇒ (t = we.t ∧ e = we.s ∧ e = "reserved") • The Started Point of Time: the event's timestamp is taken at when the state of a work-item is changed from RESERVED to RUNNING 6 : a work-activity event log with runningTimestamp, we t.u ⇒ (t = we.t ∧ u = we.s ∧ u = "running") • The Completed Point of Time: the event's timestamp is taken at when the state of a work-item is changed from RUNNING to COMPLETED 7 : a work-activity event log with completedTimestamp, 2) Temporal Work-cases and Models In order to develop a formal algorithm for discovering a structured information control net model from the process enactment event logs, we need to build a formal concept and model of temporally ordered activity event sequences of process instances, which can be extracted from the formal model of process instance enactment event traces, PET (c). The formal model of temporally ordered activity event sequences is named as temporal work-cases. At this moment, we have to remind that the meaningful temporal order in managing process instances ought to be based upon one of the timestamp-origins like scheduled, assessed, started and completed timestamps. Accordingly, we necessarily build four species of temporal work-cases with holding one of the timestamp-origins. Figure 6 illustrates the formal concepts and models of the temporal work-cases including the temporal loop-cases associated with the repetitive-LOOP process patterns. The formal definition of the temporal work-cases is as follows: Definition 5: Temporal Work-case. Let TWC(c) be a temporal work-case of the activity event sequence of a specific process instance, c: which is a temporally ordered work-activity event sequence along with one of the timestamp-origins. Especially, each temporal work-case is formally defined as a temporal work-case model, and it is assumed that all the workitems (work-activity instances) in the corresponding temporal work-case are successfully completed, and their executions are running without being suspended, as well.
Based on Definition 5, we can interpret the formal definition as the conceptual implication that all of the workactivity events holding an identical INSTANCE ID are lined up in a temporal work-case along with its timestamp-origin. Consequently, from a process instance event trace, we produce a corresponding temporal work-case by extracting the activity identifiers and their timestamps, and we name it as a temporal work-case along with one of the timestamp-origins like the scheduled time, assessed time, started time, and completed time. For the sake of the formal representation, a temporal work-case is defined as a temporal work-case 7 The COMPLETED state of a work-item implies that the work-item has been fully executed and completed with either success or failure. 8  , on a temporal work-case, TWC(c), of a process instance, c, and a species K (= {s, e, u, o}) of the timestamp-origins, where • F c r is an activity or an activity-group linked from an external temporal work-case model; • T c o is an activity or an activity-group linked to an external temporal work-case model; • The species of temporal work-case models: The formal concept and the theoretical model of a temporal loop-case are given in Definition 7 and Definition 8, respectively. As shown in Figure 6, those temporal loopcases are extracted from their corresponding process enactment events, too. For instance, {α 11 , α 12 , . . . , α 1s(1) } is a temporal loop-case having an identical work-case identifier with others of the bag and being temporally ordered activity events with their timestamp-origin properties, because each temporal loop-case represents each of the iterative workactivities involved in a loop transition construct. Therefore, two or more identical temporal loop-cases can be iterated in a temporal work-case according to the number of iterations of a corresponding loop transition. In addition, we have to consider a work-case of nested loop transitions, and then its temporal loop-cases have to be unfolded with multiple temporal loop-cases from its inner loop transitions, recursively.    (1), . . . , s(k)]}, which is a bag of temporally ordered sub-sequences of workactivity events according to the timestamp-origin property. A work-activity iteration event trace is called a temporal loopcase.
Definition 8: Temporal Loop-case Model (TLCM). A temporal loop-case model is formally defined through 3-tuple L = (ω, P, S) over a bag of iterative activities' (∀α ∈ A) event logs on a process trace, where • P is a predecessor activity of the first activity of the loopcase model folded out from the first iteration; • S is a successor activity of the last activity of the loopcase model folded out from the last iteration; is a single-valued mapping function of a work-activity event log to its (immediate) successor in a work-activity iteration trace with one of the timestamp-origin types, and θ i : is a single-valued mapping function of a work-activity event log to its (immediate) predecessors in a work-activity iteration trace with one of the timestamp-origin types.
In the formal approach of Figure 4, there are four different process patterns expected to be discovered from a bunch of temporal work-cases associated with a specific process model through the formal discovery algorithm of the ρ-Algorithm. Also, there are four different temporal work-case types forming the temporal work-cases that are characterized by the types of timestamp-origins such as scheduled, assessed, started, and completed timestamps held by their work-activity event logs. The types of temporal work-cases are differentiated from each other according to such temporal information, i.e. the activity execution (work-item) events' timestamps, logged at the time when the corresponding activities work-items were performed. The enactment engine of a process management system maintains the four types of timestamp-origins when it stores the enactment event timestamps of activity execution work-items. Accordingly there possibly exist four types of temporal work-cases in the process enactment event log traces at the same time, and from which there also possibly exist four types of process warehouses, too. The concepts and their related algorithms in this paper do not differentiate from these specific types of the temporal work-case models in detail, and assume that the type of the completed timestamp-origin is the most appropriate timestamp type in terms of forming a dataset of process enactment event log traces. VOLUME 4, 2016 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.  The algorithmic process mining framework is based mainly upon the detailed implementation of the ρ-Algorithm simply stated in the conceptual architecture in the previous section. As shown in Figure 7, the algorithmic framework is composed of seven procedural concepts and six transformation algorithms, each of which conducts a transformation from one concept to another concept, one after another, to support the procedural process discovery and processaware knowledge mining experiments. These concepts are procedurally started from process enactment event log histories formatted in the IEEE XES standard: process instance event traces, temporal work-case models, pairs of temporally ordered adjacent-activity groups, weighted adjacent-activity set, weighted process pattern graph, and structured information control net process model, whereas the transformation algorithms are listed as a series of the functional analytics algorithms: the event trace mining algorithm, temporal workcase composing algorithm, adjacent-activity fragmentizing algorithm, adjacent-activity quantifying algorithm, process pattern discovering algorithm, and the structured information control net discovering algorithm. In this section, we axiomatically formalize the algorithmic framework with its core concepts and algorithms related with the ρ-Algorithm in particular.

1) Data Formats for the ρ-Algorithm
In terms of the event format, we consider the work-item event being stored in a tag-based language (XML schema).
An XML-based process work-item event format, XWELL 9 [36], for the purpose of process mining had been studied and proposed by the authors' research group, and the WfMC ( Workflow Management Coalition) has released the standardized audit and log specification, BPAF 10 [37]. In recent, IEEE has released a standard tag-based language, XES 11 [17], which aims at providing the designers of information systems with a unified and extensible methodology for capturing systems' behaviors by means of event logs and event streams.
As the format of the process work-item event structure, we can use the "IEEE XES Schema" describing the structure of an XES event log/stream and the "XES extension" describing the structure of an extension of such a log/stream. Note that all the formal concepts and algorithms devised in the paper are assumed that the format of the process enactment event logs is the IEEE XES standardized format. We simply define the XML schema with the essential attributes as the process work-item event structure, as follows: • The EVENT attribute is used to specify an event identifier, which is assigned by the process enactment engine. • The WORK-ITEM attribute of a process activitylevel event represents a work-item identifier that is uniquely assigned by using those combined identifiers of PACKAGE ID, PROCESS ID, ACTIVITY ID, and INSTANCE ID. • The PARTICIPANT attribute is used to specify the performer who is in charge of enacting the work-item.
• The TIMESTAMP attribute specifies the time stamp of the event occurred. • Finally, the STATE attribute represents the work-item's runtime state maintained by the engine. Whenever the work-item' state is changed, it is logged with the event-code of WMChanged WorkitemState. It ought to be one of those states such as READY, ASSIGNED, RESERVED, RUNNING, COMPLETED, and CANCELLED.

2) Procedural Components of the ρ-Algorithm
The overall algorithmic framework, as illustrated in Figure 7, is a stepwise mining procedure with the functional components to be used for discovering all the types of the primitive process patterns and their enactment occurrences that eventually build a discovered process model. The algorithmic framework is supported by a series of stepwise transformation algorithms for discovering an enacted process model from a dataset of process enactment event logs. The first transformation algorithm is to discover the enacted work-cases from the dataset, each of which is modeled by a temporal work-case model. At the same time, it is necessary to count the number of occurrences of each temporal workcase in the dataset. The second transformation algorithm, which is just right the ρ-Algorithm, is to discover a process model, which is formally modeled by using the structured information control net methodology and the number of occurrences of the temporal work-cases. In terms of discovering the process model, the ρ-Algorithm is able to deal with any combinational number of AND/OR/LOOP process patterns in the structured information control net methodology. The figure shows an exemplary case of the discovered process model represented by a structured information control net consisting of two Exclusive-OR (disjunctive) process patterns.
(1) Groups of Temporally Ordered Adjacent-Activity Pairs From now on, the first step of the ρ-Algorithm in the algorithmic framework is to develop an algorithm that is able to mine a group of temporally ordered adjacent-activities pairs from a temporal work-case and its formal work-case model corresponding to each of the process instance event traces. Also, each of the temporal work-cases is formally represented by one of the work-case model types formalized in the previous section. Figure 8 depicts a conceptual procedure of the STEP-1 algorithm of the ρ-Algorithm, which is related with a single process instance event trace (PET 1 ). That is, a temporal work-case represents an ordered enactment sequence of work-activity event logs, each of which is formed with its work-activity identifier and its timestamp extracted from its corresponding process enactment event log. The extracted temporal work-case (TWC 1 ) is used for formally defining its work-case model (TWCM 1 ), from which the STEP-1 algorithm is able to mine a group (AAG 1 ) of temporally ordered adjacent-activity pairs belonging to a corresponding process instance event trace (PET 1 ). Each of the adjacent-activity pairs is formally represented according to the formal definition of Definition 9. Finally, Algorithm 1 in the next page algorithmically describes the pseudo-codes of the STEP-1 of the ρ-Algorithm.
Definition 9: Group of Adjacent-Activity Pairs. Let AAG T (c), s, b, o ∈ T, be a group of all the pairs of adjacent-activities in a work-case model of the process instance, c, where AAG T (c) = (aa T p 1 , . . . , aa T p m ). Each adjacent-activity pair, aa T p, is defined in to be (α a , α s ), ∀α ∈ A in PET, where α a and α s are adjacent, {α s ∈ ω T o (α a ) ∧ α a ∈ ω T i (α s )}, each other, in a work-case model. There are three types of adjacent-activity pair according to the time-stamp type: As you can read, Algorithm 1 needs a dataset of the process enactment event logs as inputs. Assume that the dataset is formatted in the IEEE XES (extensible event stream) format [17], the schema structure of which has a hierarchical inclusion relationship among Log, Trace and Event classes. Accordingly, the Log class is containing the Trace class of the process instance event traces in an XML tag form of (< trace > · · · < /trace >), and the Trace class contains the Event class of the process enactment event logs in an XML tag form of (< event > · · · < /event >). The STEP-1 algorithm is to conceptually extract a process instance event log (PET 1 ) from the XES-formatted dataset, and to conceptually fragmentize a temporal work-case model into a fragmentgroup of temporally ordered adjacent-activity pairs. At the same time, the implemented STEP-1 algorithm can extract practically an XES-formatted process instance event trace, and fragmentize the tagged trace into a fragment-group of event-level adjacent-activity pairs. As you can see in Figure  8, the conceptual procedure illustrates the detailed internal transformations of the Algorithm 1, which is a conceptual mining procedure eventually being connected to the STEP-2 algorithm.
(2) Weighted Adjacent-Activity Set and Weighted Process Pattern Graph The STEP-2 of the ρ-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 weighted adjacent-activity set named as ad jacencyList (β ). This set is built from all the groups of temporally ordered adjacent-activity pairs through an internal transformation procedure. Especially, the concept VOLUME 4, 2016  of weights on the edge and its activities of each pair implies the number of occurrences of the corresponding pair and its activities. Also, the set is used to produce a weighted process pattern graph with all the edges weights and their activities weights through an internal transformation procedure. The middle part of Figure 7 depicts the conceptual procedure of the STEP-2 algorithm. Finally, Algorithm 2 algorithmically describes the pseudo-codes of the STEP-2 of the ρ-Algorithm.
(3) Discovering Structured Information Control Net Model The final step (STEP-3) of the ρ-Algorithm is to build a structured information control net model from the weighted process pattern graph mined from all the groups of temporally ordered adjacent-activity pairs. The eventual goals of the ρ-Algorithm are to discover a structured information control net model as the process discovery aspect and to discover proportions of the process patterns as the process knowledge discovery aspect. Note that the structured information control net model must be satisfied with the proper nesting as well as the matched pairing properties in forming gateway activities in each process pattern. The right-most part of Figure 7 depicts the conceptual procedure of the STEP-3 algorithm that is able to completely transform from a weighted process pattern graph as the outcome of the STEP-2 algorithm to a structured information control net model on every branch of the associated gateway activities. Finally, Algorithm 3 algorithmically describes the pseudo-codes of the STEP-3 of the ρ-Algorithm.
Algorithm 3 performs two essential functions. One (STEP-3.1) is a graphical visualization function [38] [39] [28] to visualize the weighted process pattern graph in a form of graphical viewer using the Graph Stream Library. The other (STEP-3.2) is a visual transformation function to transform the weighted process pattern graph into a graphical viewer of the structured information control net model formed by a certain combination of the four types of primitive process patterns shown in Figure 5. As described in the previous subsection, the weighted process pattern graph is built by performing the internal transformations from ad jacencyList (β ). In terms of visualizing the weighted process patterns and its information control nets, the Graph Stream Library is used. The core part of the ρ-Algorithm is the STEP-3.2 function that decides open-gateways and close-gateways of each process patterns in the graph by using the concept of rho (ρ). That is, the rho (ρ) function gives the number of occurrences of its associated activity, and the principles of decision-makings are followings: The eventual goal of the algorithmic process mining framework proposed in this paper is to discover a structural information control net model being composed of a certain combination of all the process patterns from a dataset of process enactment event logs through deploying the ρ-Algorithm with a series of concepts and their related algorithms like STEP-1 (Algorithm 1), STEP-2 (Algorithm 2) and STEP-3 (Algorithm 3) algorithms. Until now, the paper has formally described those conceptual and procedural formal definitions, such as the group of temporally ordered adjacent-activity pairs, the weighted adjacent-activity set, the weighted process pattern graph and the structured information control net model. Therefore, it is necessary to summarily show that the major goals of the paper have been achieved by not only discovering a structural information control net model as one goal of the process mining aspect but also discovering its occurrences on each of the process patterns as the other goal of the process-aware knowledge discovery aspect. An operational example, the ρ-Algorithm is deployed on the same exemplary dataset in Figure 2 and Figure 3 that Line ← currentline; 5: if ( currentline == "< trace >" ) then Checking out the beginning trace-tag of a single process instance 6: startTrace ← true; 7: else 8: if ( currentline == "< event >" ) then Checking out the beginning event-tag of a single work-item 9: startEvent ← true; 10: endEvent ← false; 11: else 12: if ( currentline == "< /event >" ) then Checking out the ending event-tag of the work-item 13: startEvent ← false; 14: endEvent ← true; 15: else 16: if ( currentline == "< /trace >" ) then Checking out the ending trace-tag of the process instance 17: startTrace ← false; 18: ListXES.add( currentTrace ); Forming a single process instance event trace (PET) 19 if ( startTrace && startEvent ) then 25: store all contents of the event to currentTrace; Store all the information of activities, performers and timestamp inside each event to currentTrace 26: end if 27: seek to the next line; 28: end while 29: closefile a.xes; 30: return ListXES; Returning the formed list of all the process instance event traces 31: end procedure were used for proving the α-Algorithm and σ -Algorithm, respectively. Figure 9 illustrates all the intermediary results produced from the dataset (L) by applying all the subroutines of the ρ-Algorithm, like the STEP-1, STEP-2 and the STEP-3 algorithms. The Algorithm 1 extracts the 5 temporal workcases, each of which is corresponding to each of the process instances, from the process enactment event logs (L); the Algorithm 2 fragmentizes each temporal work-case into a group of temporally ordered adjacent-activity pairs and internally transforms all the groups into a weighted adjacentactivity set; the Algorithm 3 algorithm finally discovers a structured information control net model, ρ(L), based upon the principles of decision-making type of the process patterns and their open-gateways and close-gateways. Through this operational example on the exemplary dataset, we were able to manually verify that the ρ-Algorithm ought to be theoretically correct and functionally reasonable.
Finally, the time complexity of the Algorithm 1 and the Algorithm 2 in the ρ-Algorithm is O (N × M), respectively, where N is the number of temporal work-cases, which implies the number of process instance event traces in a dataset of process enactment event logs, and M is the number of activities associated with the corresponding process model. However, the time complexity of the Algorithm 3 in the ρ-Algorithm is O(M), where M is the number of activities associated with the weighted process pattern graph discovered from the underlying dataset. Consequently, in comparing with these conventional process mining algorithms, the functional merits and restrictions of the ρ-Algorithm proposed in the paper, which also imply the functional goals of the ρ-Algorithm, can be summarized as follows: • The ρ-Algorithm is able to explicitly deal with discovering not only the three types of the primitive process patterns supported by the conventional algorithms but also the most challenging type of the repetitive-LOOP process pattern. In principle, its discovered model is in a graphical form of structured information control VOLUME 4, 2016 Algorithm 2 STEP-2 of the ρ-Algorithm Require: A List of Tagged Traces of XES Event Logs, ListXES Ensure: A Weighted Adjacent-Activity Set, ad jacencyList (β ) Ensure: A Weighted Process Pattern Graph, G 1: procedure STEP-2 OF ρ -MAIN(ListXES) Fragmentizing a process enactment event trace into a set of temporally ordered adjacent-activity pairs and building a weighted adjacent-activity set as well as its process pattern graph 2: initalize β and G to Null-value; 3: openfile ListXES; 4: for For each process enactment event trace 5: for ( ∀ event j ∈ trace i ) do For each process enactment event in the trace 6: currentActivity = event j .GetActivityName( ); 7: if ( currentActivity / ∈ β ) then Checking out the existence of the current activity in the weighted adjacent-activity set 8: β .add( currentActivity ); Building the weighted adjacent-activity set 9: else 10: listOfVertex ← β .add( currentActivity ); 11: nextActivity ← event j+1 .GetActivityName( ); 12: end if 13: if ( nextActivity / ∈ listOfVertex ) then Checking out the existence of the next activity as a node in the weighted process pattern graph 14: listOfVertex.add( nextActivity ); Building the weighted process pattern graph 15 closefile ListXES; 22: return β and G; Returning the weighted adjacent-activity set and its process pattern graph 23: end procedure net that is satisfied with keeping the matched-pairing as well as the proper-nesting properties in forming all the primitive process patterns made up of the corresponding structured information control net process model. • The ρ-Algorithm is theoretically supported by the information control net modeling methodology; its final output model discovered is in a mathematical graph of the information control net process model, even though its input dataset is not originated from the information control net process model. • The ρ-Algorithm deals with discovering only the DO-WHILE type of the repetitive-LOOP process patterns, in principle. • The next node(s) selection rule of the iterative-LOOP CLOSE-gateway node in the discovered process model is the same as the inclusive-OR OPEN-gateway node's selection rule. • Finally, the discovered information control net process model by the ρ-Algorithm may be unstructured only if the original process model of the input dataset is unstructured, and it may also be unreasonably and improperly constructed only if the input dataset is not a non-noise dataset without any types of control-flow-related and timestamp-related noises.

IV. VALIDATING THE ρ-ALGORIT HM
As the final stage of designing the algorithmic process mining framework, it is necessary to validate the functional correctness and feasibility of the ρ-Algorithm by implementing the ρ-Algorithm and deploying onto a process enactment event log dataset especially formatted in a form of the IEEE-XES 12 standardized log format [17]. The authors' group has successfully implemented a process mining system 13 theoretically supported by the ρ-Algorithm. By using the implemented system, we carry out an experimental analysis on the synthetic dataset of process enactment event logs, which was released to the public by the 4TU.Centre for Research Data [16] and named as the Large Bank Transaction Process Dataset. This validation with an experimental analysis aims at proving that the ρ-Algorithm is able to discover all the four types of primitive process patterns [20] such as linear, for ∀α ∈ G do α is a member node of the node-set in G.

A. IMPLEMENTATION OF THE ρ-ALGORIT HM
For the sake of confirming the feasibility and the applicability of the devised ρ-Algorithm, it ought to be implemented as a specific and tangible process mining system supporting the structured information control net modeling methodol- ogy. Consequently, we have successfully implemented the ρ-Algorithm, which is named as the SICN-oriented process mining system and extended from the process mining system [40] previously developed by the authors' research group, and the system's dashboard with a list of core-functional components are illustrated in Figure 10, on which the inline and dashboard prompts are displaying the experimental processing status like choosing the dataset file-name of the Large Bank Transaction process dataset, inputting the eventtrace identifiers, clicking the functional operations to get the eventual discovery result of the SICN-oriented Large Bank Transaction process model, and so on. In this subsection, we briefly introduce the functional descriptions of all the essential components of the implemented ρ-Algorithm, as followings: • Log analysis: This function is operated as the entrance gate of the system. The raw input dataset, formed in the different formats (e.g., XES, MXML, etc.), will be read using this function. Depending on the different input data types, the log analysis function analyzes their structures and entities. It then stores the analyzed information to the system's memory and represents in two different types of data formation: Table and Graph.
The key objective is to transform the raw event log data into the efficient managing form of Table with traces, events, and event's properties. In the Table structure, each row is assigned for an event's properties in each event-trace through the Trace ID and Event ID. Therefore, a complete event-trace holds a temporal sequence of associated events as each row of the Table through the same Trace ID. In the Graph structure, each vertex represents an event holding information with activity, performer, timestamp, and other additional information. An edge connecting two vertices represents the directed transition between the associated two events. The weights of vertex and edge represent the numbers of occurrences that the events and their transitions were operated, respectively. • Data table analysis: This function analyzes data about traces and events stored in the table after being imported from the raw data. Each row in the table contains information about event-operated time, information about the activity, performer, and other related information. Different rows are differentiated by Trace ID and Event ID, and hence we can perform different aspects of the data aggregation functions. For example, we can summarize how many times a specific activity occurred in a trace, and how many times it was operated by filtering the table with the activity among different Trace IDs. Similarly, we may produce some statistical information about a specific performer from the input dataset, like how many times the performer was participated in the instances' operations.
• Graph manipulating: This function is to manipulate the elements on a graph. That is, the graph manipulation operations are provided not only for adding, editing, and deleting vertices and edges, but also for updating the weights of vertices and edges on a graph. Additionally, this function also includes a series of operations like searching and traversing graphs with breadth-first search algorithm, spanning tree algorithm, binary search tree algorithms, and so on. Using the graph manipulating function, moreover, we are able to visualize whatever actually happened during the discovery algorithm's operations and whichever different control-flow paths identified from all the traces in the event log dataset. This function's main task is to support visualizing all the graph-formed outcomes, such as the adjacent-activity pair sets, the weighted process pattern graph, and the information control net graph, to be produced through the stepwise discovery operations being conducted with the implemented system. • Deliberate noises removing: In forming the weighted process pattern graph as one of the implemented system's outcomes, we realize that there exist some phantom edges among the activities involved in the conjunctive transitions, which is named as deliberate noises. This function is so for removing all the phantom edges existing among the activities involved in a single conjunctive (parallel-AND) transition group. The implemented system executes this function on every group of the conjunctive transitions and produces a weight process pattern graph satisfying the completeness and safeness requirements. • ρ-Algorithm: This function is to implement the ρ-Algorithm devised in the previous section and also extended from the proportional process mining algorithm proposed in [26]. The algorithm's idea is to discover all the disjunctive (exclusive-OR), conjunctive (parallel-AND), and repetitive (iterative-LOOP) process patterns from a dataset of process enactment event logs by considering the occurrences of activities and their controltransition relationships. This function is completed with the supports of the three steps of the algorithms (STEP-1, STEP-2, and STEP-3 Algorithms) and the graph manipulating and deliberate noises removing functions as described above. • Data filtering: In the historical event logs of the business process management system, the implemented ρ-Algorithm needs to analyze thousands of traces enacted, hundreds of thousands of events operated, and even a much larger number of the event properties' values. Therefore, the data filtering function ought to be a vital component in managing and manipulating the hugesized dataset in the most detailed and effective way. This function allows the users to filter the input dataset out onto tables or graphs according to the intentioned criteria made by using such events, activities, performers, and/or relevant information like timestamps.
• Graph visualizing: Graph visualization brings knowledge of what occurred inside the log in an intuitive manner. This function uses the Graphstream library [41] for visualizing and analyzing the graph. By using this library, not only we can generate, measure, visualize all the graph-formed outcomes of the implemented ρ-Algorithm, but also we can provide the users very useful and easy ways of interactive user-interfaces. • Data statistics: This function performs and calculates data statistics based on the information managed on the implemented ρ-Algorithm. By combining the Data table analysis and Data filtering functions, we can perform statistical measurements and generate reports related with traces, events, activities, and the others inside the historical event log dataset. • Export XPDL, GraphViz, GraphML: The SICN-oriented process mining system need to provide the exporting function that transforms the discovered processes and process models into any other different data formats to be used in different scenarios. Currently, the users can simply export the discovered SICN-oriented process model to any one of the XPDL, GraphViz, and GraphML formats.

B. DATASET OF THE PROCESS ENACTMENT EVENT LOGS
For carrying out the validation analysis of the ρ-Algorithm, we chose a synthetic dataset from the 4TU Centre's archive, the name of which is 10000 − all − nonoise − 150MB.xes This dataset ought to be a typical dataset to usefully validate the functional correctness of such a process mining algorithm, because it is a synthetic as well as non-noise dataset, and it is also built from enacting a large-enough number of activities with 113 different activities, and their enactment VOLUME 4, 2016 event histories embedding all the possible process enactment sequences. Finally, the ultimate advantage of the dataset is based upon the various and different control-flow patterns with combining all the four types of primitive process patterns. Therefore, by using this dataset we can effectively and efficiently verify and validate the functional correctness and the operational performance of the ρ-Algorithm. Fortunately for the experimental validation, the original process model of the dataset is released in a graphical form of the Petrinetoriented process model with those names of the involved activities. Figure 11 is illustrated with a Petrinet-oriented graphical form of a process building block and a table form listing the abbreviated and full names of those 16 activities involved. Additionally, the formal ID of each activity is listed in the table and these are arranged for depicting the formal graph models to be produced by the ρ-Algorithm. As you see, the process building block, which is chosen for validating the functional correctness of the disjunctive and conjunctive process pattern discovery, is made up of two pairs of Exclusive-OR gateway nodes and two pairs of Parallel-AND gateway nodes and these gateway nodes are properly nested. And moreover, Figure 13 shows a Petrinet-oriented graphical form of another process building block that is projected from the Large Bank Transaction process model and chosen for validating the repetitive process pattern discovery functionality of the ρ-Algorithm. Note that a Petrinet graph model is built from a group of rectangular transitions and a group of circled places and the directed edges connecting transitions and places; in a Petrinet-oriented process model, transitions and places imply activities and pre-/post-conditions (before/after enacting the activities), respectively.

C. DISCOVERING THE DISJUNCTIVE (EXCLUSIVE-OR), CONJUNCTIVE (PARALLEL-AND) AND REPETITIVE (ITERATIVE-LOOP) PROCESS PATTERNS
This section deploys the implemented process mining system, which is based upon all the algorithmic functions (STEP-1, STEP-2 and STEP-3) of the ρ-Algorithm, onto the typical dataset that is prepared in the previous section, and describes its outcomes and artifacts that are produced from this validation and feasibility analysis. Fortunately, the provider of the dataset released a supplementary file containing the graphical representation of the Petrinet-oriented process model as the original process model of the Large Bank Transaction process dataset, which ought to be very useful to check up on verifying and validating the functional correctness of the process mining algorithm and system proposed in this paper. Accordingly, in this section we carry out two cases of experimental analyses applying each step of the algorithmic process mining framework and validating their experimental outcomes by comparing this supplementary model with the discovered SICN-oriented process model by the implemented ρ-Algorithm. In other words, the first experimental analysis is focusing on the discovery experiment of disjunctive and conjunctive process patterns with a building block of the Sender Authentification Subprocess Model introduced in Figure 11, and the second experimental analysis is concentrating on the discovery experiment of repetitive process pattern with another building block of the Account Payment Processing Subprocess Model shown in Figure 13.

1) Mining Groups of Adjacent-Activity Pairs by the STEP-1 Algorithm
From now on, the STEP-1 algorithm launches to mine all the groups of temporally ordered adjacent-activity pairs from all the event traces on the Large Bank Transaction process dataset. We probably suppose that the dataset is recorded by enacting the 10,000 process instances of the Large Bank Transaction process model. In the previous section, we have already defined a formal concept of the temporal workcase model that is the formal representation of an event trace of a process instance. From all the temporal workcase models, the algorithm mined a large number of groups (10,000 AAGs) of temporally ordered adjacent-activity pairs through a series of the internal transformations of the STEP-1 algorithm.
2) Mining a Weighted Adjacent-Activity Set and its Weighted Process Pattern Graph by the STEP-2 Algorithm The first internal transformation of the STEP-2 algorithm is to build a weighted adjacent-activity set from all the groups (10,000 AAGs) of temporally ordered adjacent-activity pairs. This internal transformation algorithm fulfills integrating all of the groups and regrouping same adjacent-activity pairs with calculating their occurrences. To clearly validate this algorithm, we also scrutinize every single-case in the output and confirmed that the implemented ρ-Algorithm works correctly. The second internal transformation of the STEP-2 algorithm is to build a weighted process pattern graph by combining all the elements in the weighted adjacentactivity set. The devised and implemented functions of this combining step are properly connecting the source-nodes and the destination-nodes in the weighted adjacent-activity set along with the matched activity ID. Right after building the weighted process pattern graph, it is necessary to eliminate all the deliberate noises in the graph, which are naturally created in forming the temporal work-cases and their models from all the event traces. Figure 12 and Figure 14 depict the situational experiments with the discovered outcomes of the STEP-2 and the STEP-3 algorithms in the ρ-Algorithm and the implemented ρ-Algorithm, as well. The captured screens in the lefthand side of Figure 12 and Figure 14 are the projected outcomes of the STEP-2 algorithm of the implemented system, which are the weighted process pattern graphs corresponding to the building block of Figure 11 and the building block of Figure 13, respectively. Moreover, the hand-made graph in the leftmost of Figure 12 shows the weighted process pattern graph and the occurrence numbers of activity nodes and edges after removing the deliberate noises, which is the eventual weighted process pattern graph generated by the STEP-2 algorithm, and the hand-made graph-fragments and activities' total occurrences in the leftmost of Figure 14 represent the weighted adjacent-activity set with occurrences generated by the STEP-2 algorithm and corresponding to the subprocess model of Figure 13.

3) Discovering a Structured Information Control Net by the STEP-3 Algorithm
The last step of the validation is to discover a structured information control net from the weighted process pattern graph with the STEP-3 algorithm. The captured screens in the right-hand side of Figure 12 and Figure 14 display the discovered structured information control nets generated by the STEP-3 algorithm of the implemented system, which are transformed from the weighted process pattern graphs corresponding to Figure 11 and Figure 13, respectively. Finally, both of the structured information control nets discovered by the implemented system are neatly visualized by the graph visualization software, GraphViz, as attached on the rightmost part of Figure 12 and Figure 14, respectively. The detailed decision-making procedure of the STEP-3 algorithm is applied to the weighted process pattern graph with activities' enactment occurrences to finally discover all the primitive process patterns and form a structured information control net, as follows: • Discovering the Disjunctive (Exclusive-OR) and Conjunctive (Parallel-AND) Process Patterns: First of all, on Figure 12we elucidate all the stepwise outcomes on discovering the disjunctive and the conjunctive process patterns in the Sender Authentification Subprocess Model that is graphically represented in a form of the Petrinet-oriented process model as shown in Figure 11. Additionally, Figure 12 illustrates the de-tailed outcomes of the decision-making principle for disjunctive and conjunctive process patterns and displays the final outcome of the SICN-oriented process model and its GraphViz visualization. As you can see, the implemented ρ-Algorithm operates perfectly the decision-making principles of the open-gateways and the close-gateways with the disjunctive and the conjunctive process patterns. In the figure, the final outcome of the SICN-oriented process model holds two disjunctive pairs of open-gateway nodes and close-gateway nodes and two conjunctive pairs of open-gateway nodes and close-gateway nodes, and all of these disjunctive and conjunctive pairs are in keeping the matched pairing and proper nesting properties. Consequently, we can clearly declare that the functional correctness of the ρ-Algorithm and its implemented system in discovering disjunctive and conjunctive process patterns is successfully verified as well as validated. The second experimental analysis is done for validating the functional correctness of the ρ-Algorithm and its implemented system in terms of the discovering perfectness of repetitive process patterns with keeping the matched pairing and proper nesting properties. In this second experimental analysis, we use a specific Petrinet-oriented process model of the Account Payment Processing Subprocess containing a single iterative-LOOP construct and two parallel-AND constructs, as shown in Figure 13. On Figure 14, we illustrate all the outcomes of the detailed decision-making procedure of the STEP-3 algorithm and its eventual formation of the discovered structured information control net containing a repetitive pair of open-LOOP gateway and close-loop gateway nodes and two conjunctive pairs of open-AND gateway and close-AND gateway nodes with keeping the matched pairing and proper nesting properties. Additionally, we attach the GraphViz visualization of the discovered one to the rightmost part of Figure 14. As you can see, the discovered SICNoriented process model looks exactly like the structural formation of the Petrinet-oriented process model of the Account Payment Processing Subprocess, and it is also satisfied with the structural requirement of the match pairing as well as proper nesting properties. Through this experimental analysis, we can conclude that he STEP-3 algorithm and its implemented system work perfectly in terms of discovering the repetitive (iterative-LOOP) process patterns with keeping the structural properties.

D. SUMMARY OF THE ρ-ALGORIT HM VALIDATION
These two cases of experimental analysis for validating the ρ-Algorithm and its implemented system have been successfully done as presented in the previous section. In the first experimental analysis, we have proved that the ρ-Algorithm is able to discover the disjunctive process pattern type and the conjunctive process pattern type; in the second experimental analysis, we have also confirmed that the ρ-Algorithm can discover the repetitive process pattern type, too; eventually, we have so corroborated that the ρ-Algorithm works the discovery perfectness on all the primitive process patterns with keeping the structural requirement of matched pairing and proper nesting properties. Actually, we carried out a single experiment on the synthetic dataset for validating the STEP-1, STEP-2 and STEP-3 algorithms of the ρ-Algorithm.
Through this experiment, we found out, as shown in Figure  15, that the dataset contains all the enactment event histories recorded from enacting 10,000 process instances of the Large Bank Transaction Process Model with the total number of activities involved in the enactments is 113 different activities, and that the largest number of activity enactment occurrences is 18,458 times. Note that the titles of the activities having the largest occurrences are RBPC (Receiver Bank Profiling Checking Activity), REPC (Receiver External Profiling Checking Activity), SRPP (Start Receiver Pre Profiling Activity), RIBPC (Receiver Inter-Bank Profiling Checking Activity) and FRPP (Finish Receiver Pre Profiling Activity), and all of which are associated with the Receiver Processing Subprocess Model of the Original Model of the Large Bank Transaction Process Model. Figure 16 displays two of the captured-screens from the experimental results of the implemented ρ-Algorithm: one is the weighted process pattern graph from the STEP-1 and STEP-2 algorithms and the other is the structured information control net from the STEP-3 algorithm. In addition, in order to emphasize the functional correctness and perfectness of the proposed algorithm, we display the enlarged building block with being indicated by three boxes, each of which is marked with the corresponding primitive process patterns. All of these graph models in the captured-screens are in a visual expression of the ρ-Algorithm and mined from all the 10,000 temporal work-cases (process instance event traces) in the enactment event log dataset of the Large Bank Transaction Process Model. Finally, Figure 17 is the GraphViz visualization form of the SICN-oriented process model discovered from the IEEE XES-formatted synthetic dataset, at last. Based upon these results of the experiment, the ρ-Algorithm of the conceptual and theoretical approach and its implemented process mining system proposed in the  as the STEP-1, STEP-2 and STEP-3 algorithms, under the name of ρ-Algorithm. To validate the proposed algorithmic framework, we successfully implemented all the stepwise algorithms and carried out an experiment based upon the IEEE XES-formatted synthetic dataset of process enactment event logs recorded from simulating and enacting the Large Bank Transaction Process Model. The theoretical background of the proposed algorithmic framework stems from the conceptual discovery approach dealing with the SICN-oriented process patterns, such as linear, disjunctive, conjunctive and repetitive process patterns, whereas the implementable background of the proposed algorithmic framework is supported by a series of the procedural mining functions being concretized as the Algorithm 1, Algorithm 2 and Algorithm 3, which are the stepwise algorithmic components of the ρ-Algorithm, for discovering the structured information control nets from an IEEE XES-formatted dataset of process enactment event logs. Based upon these theoretical and algorithmic approaches, the paper devised, implemented and developed these concepts, algorithms and systems, respectively. By using the implemented ρ-Algorithm, the paper carried out two cases of experimental analysis to help for validating the discovery perfectness of the ρ-Algorithm, effectively and efficiently, and both cases of which are related with a specific subprocess building block with the disjunctive and conjunctive process pattern types as well as another subprocess building block with the repetitive process pattern type chosen from the Large Bank Transaction Process Model, respectively.
Summarily, the characteristics of the SICN-oriented process mining framework proposed and implemented in the paper are as followings: First, the proposed algorithmic framework is able not only to discover the SICN-oriented process patterns but also to mine the enactment occurrences of the process patterns from an IEEE XES-formatted dataset of process enactment event logs. Second, the proposed algorithmic framework is theoretically supported by the information control nets modeling methodology of workflow and business process models. Third, the essential algorithm of the proposed algorithmic framework is named as ρ-Algorithm that is able to properly deal all the structural process patterns, such as linear (sequential), conjunctive (parallel-AND), disjunctive (exclusive-OR), and repetitive (iterative-LOOP) process patterns, with keeping the structural properties of matched-pairing and proper-nesting by using the concept of ρ-function returning the enactment occurrences of the associated activities. Fourth, the ρ-Algorithm is the first SICN-oriented process mining framework successfully developed and experimentally proved in the workflow and business process management and mining literature. Fifth,