alpha algorithm

The algorithm takes a workflow log $W\backslash subseteq\; T^\{*\}$ as input and results in a workflow net being constructed.

It does so by examining causal relationships observed between tasks. For example, one specific task might always precede another specific task in every execution trace, which would be useful information.

• A workflow trace or execution trace is a string over an alphabet $T$ of tasks.
• A workflow log is a set of workflow traces.

Event log is the primary requirement for applying any process discovery algorithm. An event log consists of a unique identifier for a case, activity name describing the action occurring in the process and timestamp. An event log can be represented as a multi-set of activities. For the sake of simplicity the following example would use alphabetic letter to represent an activity. Consider an example event log shown in the following figure:

An event log is a multi set of traces, and a trace is a sequence of activities. Thus, an event log such as above can be represented using the following notation:

Every event log can be boiled down into a multi-set of traces, and such traces can be further used to break down relationships between various activities in the process. According to the rules of alpha miner, activities belonging to various cases can have 4 types of relationships between them:

• Direct Succession: x > y if and only if some relation x is directly following by y. In our example, we can consider that A > B, A > E, A > C.
• Causality: x → y iff x > y and not y > x. In our example, we can consider that A → E.
• Parallel: x || y iff x > y and y > x. In our example, we have B || C.
• Choice: x # y iff not(x > y) and not(y > x). In our example, we have A # D.

File:Sequence_pattern_for_petri_nets.png Sequence Pattern: A → B

File:Xor-split_pattern_for_petri_nets.png XOR-split Pattern: A → B, A → C, and B # C

File:And-split_pattern_for_petri_nets.png AND-split Pattern: A → B, A → C, and B || C

The alpha miner starts with converting an event log into directly-follows, sequence, parallel, and choice relations, and using them to create a petri net describing the process model. Initially the algorithm constructs a footprint matrix. Using the footprint matrix and the above shown pattern, one can construct a process model.

Based on the four relations described earlier a footprint based matrix is first discovered. Using the footprint based matrix places are discovered. Each place is identified with a pair of sets of tasks, in order to keep the number of places low.

• $Y\_W$ is the set of all pairs $(A,B)$ of maximal sets of tasks such that
• Neither $A\; \backslash times\; A$ and $B\; \backslash times\; B$ contain any members of > and
• $A\; \backslash times\; B$ is a subset of →
• $P\_W$ contains one place $p\_\{(A,B)\}$ for every member of $Y\_W$, plus the input place $i\_W$ and the output place $o\_W$

The flow relation $F\_W$ is the union of the following:

• $\backslash \{(a,p\_\{(A,B)\})\; |\; (A,B)\; \backslash in\; Y\_W\; \backslash wedge\; a\; \backslash in\; A\backslash \}$
• $\backslash \{(p\_\{(A,B)\},b)\; |\; (A,B)\; \backslash in\; Y\_W\; \backslash wedge\; b\; \backslash in\; B\backslash \}$
• $\backslash \{(i\_W,t)\; |\; t\backslash in\; T\_I\backslash \}$
• $\backslash \{(t,i\_O)\; |\; t\backslash in\; T\_O\backslash \}$

The result is

• a Petri net structure $\backslash alpha(W)\; =\; (P\_W,T\_W,F\_W)$
• with one input place $i\_W$ and one output place $o\_W$
• because every transition of $T\_W$ is on a $F\_W$-path from $i\_W$ to $o\_W$, it is indeed a workflow net.

For the example given above, the following petri net would be resultant of the application of alpha miner.

File:Petri_net_for_alpha_miner_example.png

It can be shown van der Aalst et al. 2003 that in the case of a complete workflow log generated by a sound SWF net, the net generating it can be reconstructed. Complete means that its $\backslash succ\_W$ relation is maximal. It is not required that all possible traces be present (which would be countably infinite for a net with a loop).

• Implicit places: Alpha miner cannot distinguish between implicit and required places and thus might result in additional non required places in the discovered petri net.{{Cite web|title=Discovering Petri Nets from Event Logs|url=https://www.researchgate.net/publication/265764352|access-date=2021-08-31|website=ResearchGate|language=en}}
• Loops: Alpha miner cannot discover loops of the length 1 and 2 in the process model.{{Cite web|title=Limitations of Alpha miner|url=https://courses.edsa-project.eu/pluginfile.php/281/mod_resource/content/0/17%20Alpha%20Algorithm%20-%20Limitations.pdf|url-status=live|archive-url=https://web.archive.org/web/20210831175934/https://courses.edsa-project.eu/pluginfile.php/281/mod_resource/content/0/17%20Alpha%20Algorithm%20-%20Limitations.pdf |archive-date=2021-08-31 }}
• Local dependencies are often missed in alpha miner.
• Representational bias: Alpha miner can only discover petri net thus adding representational bias such as requirement on unique visible labels for every transition.

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Category:Process mining

Category:Data mining algorithms