Discovering Models of Parallel Workflow Processes
from Incomplete Event Logs
Julijana Lekic
1
and Dragan Milicev
2
1
University of Pristina in Kosovska Mitrovica, Faculty of Technical Sciences, Kneza Milosa 7,
38220 Kosovska Mitrovica, Serbia
2
University of Belgrade, Faculty of Electrical Engineering, Bulevar kralja Aleksandra 73, 11120 Beograd, Serbia
Keywords: Process Mining, Process Model Discovery, Parallel Business Processes, Incomplete Event Log,
α-algorithm.
Abstract: α-algorithm is able to discover a large class of workflow (WF) nets based on the behavior recorded in event
logs, with the main limiting assumption that the event log is complete. Our research has been aimed at
finding ways of business process models discovering based on examples of traces, i.e., logs of workflow
actions that do not meet the requirement of completeness. In this aim, we have modified the existing and
introduced a new relation between activities recorded in the event log, which has led to a partial correction
of the process models discovering techniques, including the α-algorithm. We have also introduced the
notions of causally and weakly complete logs, from which our modified algorithm can produce the same
result as the original algorithm from complete logs. The effect of these modifications on the speed of the
process model discovering is mostly evident for business processes in which many activities can be
performed in parallel. Therefore, this paper presents preliminary results obtained from the investigation of
opportunities to discover models of parallel processes based on incomplete event logs.
1 INTRODUCTION
Business process models play an important role in
large enterprises. Process mining (PM) techniques
help them to identify models of their actual business
processes based on examples of behavior, i.e., on
logs of performed activities. In process mining, an
event log is used for the implementation of three
techniques: discovering, described by Aalst van der,
Weijters and Maruster (2002), Aalst van der,
Weijters and Maruster (2004), Aalst van der (2011:
125-139), conformance, described by Rozinat and
Aalst van der (2008), Aalst van der (2011: 191-213),
and enhancement of the model, described by Aalst
van der (2011: 281-305).
A technique of interest in our research is
forementioned model discovering. This technique is
used to construct models of business processes only
on the basis of examples of behavior exhibited by
the processes i.e., traces recorded in the event log.
One of the basic and best known algorithms for
process model discovering, based on records in a
log, is the α-algorithm introduced by Aalst van der,
Weijters and Maruster (2002: 16), Aalst van der,
Weijters and Maruster (2004: 1135), Aalst van der
(2011: 133). From records in the event log, the α-
algorithm automatically generates a process model
that belongs to a subclass of Petri nets, according to
Aalst van der and Stahl (2011) known as workflow
(WF) nets.
Besides a number of other limitations, one of the
basic limiting assumptions of this algorithm is that
the event log needs to be complete. In addition, the
assumption of event log completeness requires that
all activities that potentially directly follow each
other, directly follow each other in some trace in the
log as it is defined by Aalst van der, Weijters and
Maruster (2002: 9), Aalst van der, Weijters and
Maruster (2004: 1132). The assumption of the
completeness of the event log requires that in the
traces recorded in it, there are all direct dependency
relations that may exist between the activities of the
observed business process. The property of
completeness of the log often requires a large
number of traces in the log on which the
"representative" model for the behavior seen in the
log has to be constructed. Therefore, our challenge
was to find logs with potentially much lower number
of traces, which may not be complete, but are
477
Lekic J. and Milicev D..
Discovering Models of Parallel Workflow Processes from Incomplete Event Logs.
DOI: 10.5220/0005242704770482
In Proceedings of the 3rd International Conference on Model-Driven Engineering and Software Development (MODELSWARD-2015), pages 477-482
ISBN: 978-989-758-083-3
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

sufficiently valid so that, using the appropriate
algorithm based on the evidences recorded in such
logs, a "representative" model can be obtained.
To achieve this, we have partially modified the
technique of process model discovering, and also the
α-algorithm itself by introducing the relation of
indirection as another basic relation between the
activities recorded in the event log. Our aim was to
determine how the existence of the relation of
indirection affects detection of concurrency
relationships, and how it affects the efficiency of
discovering the business process model. The
preliminary results have shown that, by a
generalization of the concurrency relation,
considering the relations of indirection between the
activities recorded in the event log, the concurrency
relation, and therefore business process models, can
be obtained from smaller, i.e., incomplete event
logs, which enables faster detection of business
processes models. This shows particularly good
results in processes with a lot of activities that can
be performed in a mutually independent order, i.e.,
concurrently. In order to illustrate the effects of
using the proposed modified technique of
discovering process models and make it more
noticeable, the results of applying the modified PM
technique for discovering a parallel processes model
will be shown on sample processes.
The paper is organized as follows. The next
section describes: the proposed modification of the
PM technique for business processes model
discovering, the concept of a parallel business
process and the proposed partial modification of the
α-algorithm (α
||
-algorithm). Section 3 presents the
application of the proposed modified PM techniques
to discovering parallel business process models
based on the so called causally complete and weakly
complete logs. Section 4 brings conclusions and
guidelines for a future work.
2 MODIFIED TECHNIQUE FOR
DISCOVERING PROCESS
MODELS
The subject of our research, presented in this paper,
is the problem of completeness of the event log, as it
was presented by Aalst van der (2011: 147). In the
basic α-algorithm, it is assumed that an event log L
is complete in terms of the relation >
L
, which was
basic assumption established by Aalst van der,
Weijters and Maruster (2002: 9), Aalst van der,
Weijters and Maruster (2004: 1132). This
assumption about the completeness of the log L
requires that if a process model allows an activity b
to be executed immediately after an activity a, then
the log must contains at least one trace where b is
really executed immediately after a, i.e. a >
L
b must
hold. In this case, b is said to be directly (or
immediately) following a in L, as it was defined by
Aalst van der, Weijters and Maruster (2002: 8),
Aalst van der (2011: 130).
Considering the fact that the existence of the
relation of direct following in the event log affects
the definition of the relation of parallelism (a ||
L
b iff
a >
L
b and b >
L
a), according to the definition
introduced by Aalst van der, Weijters and Maruster
(2002: 8), Aalst van der, Weijters and Maruster
(2004: 1132), Aalst van der (2011: 130), our aim
was to examine how the existence of the relation of
indirect following between activity affects the
inference of the relation of parallelism, and how it
reflects on the efficiency of discovering process
models. Due to this reason, we have introduced a
new basic relation a >>
L
b, which indicates that b
indirectly follows a. The log-based relations that are
used to indicate the relevant patterns in the log in
our modified technique of detecting process models
are defined by Definition 1 below.
According to Aalst van der (2011: 129) the event
log L is observed as a multiset of traces over a set of
actions A, i.e., L B(A*). Each trace corresponds to
a single executed process case (or scenario). The
elements of the set A are the activities that
correspond to transitions in the resulting Petri net,
and are denoted by lowercase letters (i.e., a, b, c, ...
 A), while the sets of activities are denoted by
uppercase letters (i.e., A, B, C, ...  A ).
Definition 1. (Log-based ordering relations, in
the modified PM technique of discovering process
models). Let L be an event log over A, i.e., L  (A*).
Let a, b ∈ A be two activities. Then, by definition:
 a >
L
b if and only if there is a trace σ = t
1
, t
2
, t
3
,
… , t
n
 and i  {1, …, n -1} such that σ  L and
t
i
= a and t
i+1
= b
 a >>
L
b if and only if there is a trace σ = t
1
, t
2
, t
3
,
… , t
n
 and there are i, j  {1, …, n} such that i +
2  j, where σ  L and t
i
= a, t
j
= b, and it is not
that a >
L
b
 a 
L
b if and only if a >
L
b, and it is not b >
L
a,
and it is not b >>
L
a
 a 
L
b if and only if a >>
L
b, and it is not b >
L
a,
and it is not b >>
L
a
 a #
L
b if and only if it is not a >
L
b, and it is not b
>
L
a, and it is not a >>
L
b, and it is not b >>
L
a
MODELSWARD2015-3rdInternationalConferenceonModel-DrivenEngineeringandSoftwareDevelopment
478

 a ||
L
b if and only if a >
L
b and b >
L
a, or a >
L
b
and b >>
L
a, or a >>
L
b and b >
L
a, or a >>
L
b
and b >>
L
a.
These relations can be represented with a matrix,
which is the footprint of the event log, as it will be
shown in the later examples.
2.1 a
||
-algoritam
The introduction of the relation of indirection, as an
additional basic relation between activities recorded
in the log, has led to a significantly faster
discovering of the concurrency relations between the
activities. The basic idea is the following: in order to
conclude that two activities a and b are mutually
independent, i.e., concurrent, it is enough to have a
trace in the event log in which a directly or
indirectly precedes b, and another trace where b
directly or indirectly precedes a. Discovering
concurrency relations from traces recorded in the
event log is one of the key elements to discovering
process models. Since concurrency is very important
in real business processes, in which many things
may occur simultaneously, our research has
primarily been focused on the effects of introducing
the relation of indirection to the speed of obtaining
business process models in which many activities
are performed in parallel. This is why we have
defined a particular subclass of business processes of
interest, parallel processes.
The first assumption is that, in parallel processes,
all the activities perform either sequentially or in
parallel, but not in alternative (optionally). The
second assumption is that during the execution of a
single instance of the process, each activity can be
performed at most once, which eliminates the
possibility of iterations in parallel process models.
For the presentation of the model, we will use a
variant of the classical Petri nets called sound WF-
nets, which belong to the set of WF-nets, denoted
with W, in accordance with Aalst van der, Weijters
and Maruster (2002: 7). Figure 1 shows our running
example of a parallel process.
The assumptions defined above led to a lack of
the relation #
L
defined by Aalst van der, Weijters
and Maruster (2002: 8), Aalst van der, Weijters and
Maruster (2004: 1132), Aalst van der (2011: 130), in
a part which referred to relations between different
activities of parallel processes. Since the α-algorithm
is based on the relations #
L
and 
L
(as it was cited
previously), the loss of the relation #
L
between
different activities will lead to changes in the α-
algorithm and its application to the discovery of a
parallel process model.
That modified α-algorithm will be denoted with
α
||
(L), where „
||
“ reflects the fact that the algorithm
targets parallel business processes, and is defined as
follows:
Definition 2. (α
||
-algorithm). Let L be an event
log over T  A. α
||
(L) is defined as follows:
(1) T
L
= {t  T | (σ  L) t  σ}
(2) T
I
= {t  T | (σ  L) t = first(σ)}
(3) T
O
= {t  T | (σ  L) t = last(σ)}
(4) X
L
= {(A, B) | A  T
L
 A ≠ Ø  B  T
L
 B ≠
Ø  (a  A) (b  B) (a 
L
b)}
(5) P
L
= {p
(A, B)
| (A, B)  X
L
}  {i
L
, o
L
}
(6) F
L
= {(a, p
(A, B)
) | (A, B)  X
L
 a  A} 
{(p
(A, B)
, b) | (A, B)  X
L
 b  B}  {(i
L
, t) | t T
I
}
 {(t, o
L
) | t  T
O
}
(7) α
||
(L) = (P
L
, T
L
, F
L
)
3 APPLYING THE MODIFIED
TECHNIQUE FOR
DISCOVERING PARALLEL
PROCESS MODELS FROM
INCOMPLETE EVENT LOGS
In PM techniques for model discovering (as it was
cited previously), a necessary condition for discove-
ring the original network by the α-algorithm is that
the log on which the algorithm is implemented needs
to be complete, where the condition of completeness
is based on the relation >
L
. The condition of
completeness in our modified PM technique for
model discovering is related to the causality relation

L
, and accordingly new types of completeness
are defined: causal completeness and weak
completeness, which will be defined in this section.
3.1 Discovering Original Networks
from Causally Complete Logs
For a particular process model to be discovered,
there may be a large (in general, an unlimited)
number of different complete logs. However, all
these complete logs have the same footprint, i.e., the
same causality relation. We call this relation the
basic causality relation. On the other hand, there
may exist other logs for the same process model that
are not complete, but which have the same footprint,
i.e., the same causality relation obtained from those
logs. We are focused on investigating such logs,
which we refer to as causally complete logs (L
c
).
Obviously, the idea is to find causally complete logs
DiscoveringModelsofParallelWorkflowProcessesfromIncompleteEventLogs
479

o
L
p
({g},{h})
p
({f},{g})
p
({a},{f})
p
({c},{e})
p
({d},{h})
h
a
b
f
c
d
e
g
p
({b},{h})
p
({c},{d})
i
L
p
({a},{c})
p
({a},{b})
p
({e},{h})
Figure 1: Example of a parallel process model.
that may be, in general, much smaller than fully
complete logs (in the terminology of the original α-
algorithm).
Definition 3. (The basic causality relation) Let N
= (P, T, F) be a sound WF-net, i.e., N  W, and let L
be a complete workflow log of N. 
B
N
is the basic
causality relation of network N iff 
B
N
= 
L
.
Considering the so defined basic causality
relation, a causally complete log is defined as
follows.
Definition 4. (Causally complete log) Let N =
(P, T, F) be a sound WF-net, i.e., N  W, and let

B
N
be the basic causality relation of N. L
c
is a
causally complete workflow log of N iff:
1) 
Lc
= 
B
N
, and
2) for any t  T there is
  L
c
so that t   .
From Definition 2 it can be seen that the α
||
-
algorithm is based on the causality relation.
Preliminary results have shown that by the modified
PM technique and the α
||
-algorithm, the network can
be rediscovered from any incomplete log in which

L
= 
B
N
, i.e., from any causally complete log, as
the example that follows will show.
Let us observe a parallel process model shown in
Figure 1, and a log L
1
obtained after several
executions of the process.
L
1
= [< a, b, c, d, e, f, g, h >
4
, < a, f, g, b, c, e, d, h
>
2
, < a, c, d, e, f, g, b, h >, < a, b, f, g, c, d, e, h >]
The basic causality relation for this example is:

B
N
= {(a, b), (a, c), (a, f), (b, h), (c, d), (c, e),
(d, h), (e, h), (f, g), (g, h)}
The footprint of the event log L
1
is given in
Table 1.
From the footprint shown in Table 1, it can be
seen that the causality relation of L
1
is:

L1
= {(a, b), (a, c), (a, f), (b, h), (c, d), (c, e),
(d, h), (e, h), (f, g), (g, h)}
It can be observed that the causality relation of
L
1
is equal to the basic causality relation, i.e., 
L1
=

B
N
, which makes the log L
1
causally complete.
By applying the α
||
-algorithm to the given log L
1
,
we obtain the following:
(1) T
L1
= {a, b, c, d, e, f, g, h}
(2) i
L1
= a
(3) o
L1
= h
(4) X
L1
= {({a},{b}), ({a},{c}), ({a},{f}),
({b},{h}), ({c},{d}), ({c},{e}), ({d},{h}), ({e},{h}),
({f},{g}), ({g},{h})}
(5) P
L1
= {p
({a},{b})
, p
({a},{c})
, p
({a},{f})
, p
({b},{h})
,
p
({c},{d})
, p
({c},{e})
, p
({d},{h})
, p
({e},{h})
, p
({f},{g})
, p
({g},{h})
,
i
L1
, o
L1
}
(6) F
L1
= {(a, p
({a},{b})
), (p
({a},{b})
, b), (a, p
({a},{c})
),
(p
({a},{c})
, c), (a, p
({a},{f})
), (p
({a},{f})
, f), (b, p
({b},{h})
),
(p
({b},{h})
, h), (c, p
({c},{d})
), (p
({c},{d})
, d), (c, p
({c},{e})
),
(p
({c},{e})
, e), (d, p
({d},{h})
,), (p
({d},{h})
, h), (e, p
({e},{h})
),
(p
({e},{h})
, h), (f, p
({f},{g})
), (p
({f},{g})
, g), (g, p
({g},{h})
),
(p
({g},{h})
, h), (i
L1
, a), (h, o
L1
)}
(7) α
||
(L
1
) = (P
L1
, T
L1
, F
L1
)
Table 1: Footprint of the event log L
1
.
a b c d e f g h
a #

      
b


# || || || || ||

c


|| #

 
|| ||

d


||


# || || ||

e


||


|| # || ||

f


|| || || || #
 
g


|| || || ||

#

h

      
#
The network N
||
= α
||
(L
1
) obtained by applying
the α
||
-algorithm to the log L
1
is equal to what is
shown in Figure 1. The log L
1
is not complete,
because there are missing elements in the relation
>
L
: b > d, b > e, b > g, c > b, c > f, c > g, d > b, d > f,
d > g, e > b, e > g, f > b, f > c, f > d, f > e, g > d and
g > e, which could be potentially performed on the
MODELSWARD2015-3rdInternationalConferenceonModel-DrivenEngineeringandSoftwareDevelopment
480

basis of the process model given in Figure 1, and the
resulting WF-net N
||
.
3.2 Discovering Original Networks
from Weakly Complete Logs
From the said above, it can be concluded that the
main task is to find a log with the causality relation
that is equal to the basic causality relation, and then
apply the α
||
-algorithm, which then leads to the
original network of a parallel processes. During our
research, we have found that there are logs from
which one can discover a causality relation equal to

B
N
, but one cannot come to it only from the
evidence recorded in the log, but the individual
elements of the causality relation can be inferred in
the process of applying the α
||
-algorithm of them. We
call a log with such properties a weakly complete log
(L
w
). Weakly complete logs can be significantly
smaller than complete logs as well as than causally
complete logs. Examples that we have analyzed
show that the original network can be discovered
from an incomplete log L for which the following
holds: 
B
N
 (
L
 
L
), and 
L
 
B
N
; such a
log we call a weakly complete log.
Definition 5. (Weakly complete log) Let N = (P,
T, F) be a sound WF-net, i.e., N  W, and let 
B
N
be the basic causality relation of N. L
w
is a weakly
complete workflow log of N iff:
1) 
B
N
 (
Lw
 
Lw
), and 
Lw
 
B
N
, and
2) for any t  T there is
  L
w
so that t   .
A network obtained from a weakly complete log
often contains dangling nodes, i.e., activities without
predecessors and/or successors. The definition of a
WF-net established by Aalst van der, Weijters, and
Maruster (2002: 7), includes an assumption of
network connectivity, which means connectivity of
all nodes in the network, and which prohibits the
existence of dangling nodes. In attempt to overcome
the problem of dangling nodes, we observed the
relations in footprints, and based on these we have
defined the rules of inference of direct from indirect
successors and predecessors. Thus for each activity
that is a dangling node, a successor and/or
predecessor can be found.
Direct successors obtained this way constitute
the elements of the causality relation that we call
inferred causality relations, 
I
Lw
. The causality
relation to be inserted into the final footprints
(denoted with 
Lf
), and over which the α
||
-
algorithm is applied becomes: 
Lf
= 
Lw
 
I
Lw
,
which gives 
Lf
= 
B
N
. The use of these rules in
order to find a causality relation equal to the basic
causality relation, and thus to discover the original
network, is shown in the following example.
Let us consider a parallel process model shown
in Figure 1, and a log L
2
with records obtained after
several executions of the process.
L
2
= [< a, b, c, d, e, f, g, h >
3
, < a, f, g, c, e, d, b,
h >
2
]
The basic causality relation for this example is:

B
N
= {(a, b), (a, c), (a, f), (b, h), (c, d), (c, e),
(d, h), (e, h), (f, g), (g, h)}
The footprint of the event log L
2
is given in
Table 2.
From the footprints shown in Table 2, it can be
seen that the causality relation of L
2
is:

L2
= {(a, b), (a, f), (b, h), (c, d), (c, e), (f, g),
(g, h)}
It can be noted that the causality relation of log
L
2
is not equal to the basic causality relation, i.e.,

L2
≠ 
B
N
, but also it holds: 
B
N
 (
L2
 
L2
),

L2
 
B
N
, which makes L
2
a weakly complete log.
It can also be seen from the footprints that the
activity c does not have its direct predecessors (the
row which relates to c does not contain a ), and
the activities d and e do not have direct successors
(the rows d and e do not have ), which means that
there will be dangling nodes in the network.
According to the rules of inference of direct from
indirect successors and predecessors in networks
with dangling nodes, we get:
c 
L2
a, b 
L2
a, c ||
L2
b then c 
I
L2
a, or c 
L2
a, f 
L2
a, c ||
L2
f then c 
I
L2
a i.e., a 
I
Lw
c
Table 2: Footprint of the log L
2
.
a b c d e f g h
a #
      
b

# || || || || ||

c


|| #
 
|| ||

d


||

# || || ||

e


||

|| # || ||

f

|| || || || #
 
g


|| || || ||

#

h

      
#
d 
L2
h, b 
L2
h, d ||
L2
b then d 
I
L2
h, or d 
L2
h, g 
L2
h, d ||
L2
g then d 
I
L2
h
e 
L2
h, b 
L2
h, e ||
L2
b then e 
I
L2
h, or e 
L2
h, g 
L2
h, e ||
L2
g then e 
I
L2
h
Thus: 
I
L2
= {(a, c), (d, h), (e, h)}, i.e.:

Lf
= 
L2
 
I
L2
= {(a, b), (a, c), (a, f), (b, h),
(c, d), (c, e), (d, h), (e, h), (f, g), (g, h)}
DiscoveringModelsofParallelWorkflowProcessesfromIncompleteEventLogs
481

It can be noted that now it holds 
Lf
= 
B
N
.
As it was shown in the previous example, using
the α
||
-algorithm over the log whose causality
relation is equal to the basic causality relation (
Lf
= 
B
N
), the obtained network will be the same as the
original network.
From the preliminary results presented in this
paper, it can be seen that our assumption that the
model of a parallel process can be obtained from the
logs that do not meet the requirement of
completeness is valid, and we are working on its
formal proof or a counterexample (in the latter case,
we will work on identifying the conditions in which
the property still holds and on its experimental
evaluation).
4 CONCLUSIONS
In this paper we were faced with one of the biggest
challenges in the research process mining, which is
the problem of completeness of the logs in the
discovering process model based on the example of
the process of behavior recorded in the workflow
logs. Solving problems of completeness logs in
parallel processes, presented in this paper, has led to
change in the technique discovering the process
model as well as in the α-algorithm, (as it was cited
previously).
Although the α-algorithm is basically simple, it
has not been particularly practical because of many
problems that it cannot overcome, as it was
described by Aalst van der (2011: 129). Besides the
basic α-algorithm, the examples of variation α-
algorithm given by L. Wen et al (2007), heuristic
mining, given by A.J.M.M. Weijters and J.T.S.
Ribeiro (2010), a genetic process mining, given by
A.K.A de Medeiros (2006), fuzzy mining, given by
Guenther and Aalst van der (2007), process mining
from a basis of regions, given by M. Sole and J.
Carmona (2010), are known, also. Most of these
algorithms are emerged in order to overcome
problems encountered in using the basic α-
algorithm, but the problem of completeness of logs
has not been overcome, and it still remains a
challenge for future researchers.
As such, it became the subject of our
observations and modifications as a part of a broader
research on discovering business process models by
examples. Preliminary results presented in this paper
address concurrent processes without loops. The
examples show that with our modification of the
discovering technique, we are able to overcome the
problem of completeness of logs in parallel
processes that occurs in the basic α-algorithm, and to
improve the efficiency of obtaining the process
model. Our future work will be focused on finding
the theoretical or empirical confirmation of the
obtained results. Also, the obtained results
encourage us to continue to investigate the effects of
introduced modifications to other types of processes.
REFERENCES
Aalst van der, W.M.P., 2011. Process Mining: Discovery,
Conformance and Enhancement of Business
Processes. Springer-Verlag, Berlin.
Aalst van der, W.M.P., Stahl, C., 2011. Modeling
Business Processes: A Petri Net Oriented Approach.
MIT press, Cambridge, MA.
Aalst van der, W.M.P., Weijters, A.J.M.M., Maruster, L.,
2002. Workflow Mining: Which Processes can be
Rediscovered? BETA Working Paper Series, WP 74,
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MODELSWARD2015-3rdInternationalConferenceonModel-DrivenEngineeringandSoftwareDevelopment
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