# Alpha algorithm

The α-algorithm is an algorithm used in process mining, aimed at reconstructing causality from a set of sequences of events. It was first put forward by van der Aalst, Weijters and Măruşter. Several extensions or modifications of it have since been presented, which will be listed below.

It constructs P/T nets with special properties (workflow nets) from event logs (as might be collected by an ERP system). Each transition in the net corresponds to an observed task.

## Short description

The algorithm takes a workflow log $W\subseteq T^{*}$ 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.

### Definitions used

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

## Description

Declaratively, the algorithm can be presented as follows. Three sets of tasks are determined:

• $T_{W}$ is the set of all tasks which occur in at least one trace
• $T_{I}$ is the set of all tasks which occur trace-initially
• $T_{O}$ is the set of all tasks which occur trace-terminally

Basic ordering relations are determined ($\succ _{W}$ first, the latter three can be constructed therefrom)

• $a\succ _{W}b$ iff $a$ directly precedes $b$ in some trace
• $a\rightarrow _{W}b$ iff $a\succ _{W}b\wedge b\not \succ _{W}a$ • $a\#{}_{W}b$ iff $a\not \succ _{W}b\wedge b\not \succ _{W}a$ • $a\Vert _{W}b$ iff $a\succ _{W}b\wedge b\succ _{W}a$ 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 $A\times A$ and $B\times B$ contain any members of $\succ _{W}$ and
• $A\times B$ is a subset of $\rightarrow _{W}$ • $P_{W}$ 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:

• $\{(a,p_{(A,B)})|(A,B)\in Y_{W}\wedge a\in A\}$ • $\{(p_{(A,B)},b)|(A,B)\in Y_{W}\wedge b\in B\}$ • $\{(i_{W},t)|t\in T_{I}\}$ • $\{(t,i_{O})|t\in T_{O}\}$ The result is

• a petri net structure $\alpha (W)=(P_{W},T_{W},F_{W})$ • 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.

## Properties

It can be shown  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 $\succ _{W}$ relation is maximal. It is not required that all possible traces be present (which would be countably infinite for a net with a loop).

## Limitations

General workflow nets may contain several types of constructs  which the α-algorithm cannot rediscover.

Constructing $Y_{W}$ takes exponential time in the number of tasks, since $\succ _{W}$ is not constrained and arbitrary subsets of $T_{W}$ must be considered.

## Extensions

for example