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AB
||
ĺ
+
ĺ
AB
C
B
A
A
B
A
B
+
A
B
+
ĺ
ĺ
||
||
C
C
C
+
+
C
||
||
(a)
(b)
Fig. 4.
Process models with complementary behavioural profiles
Complementation.
The complement operation is defined for a single
behavioural profile and returns a profile that specifies reverse relations for all
pairs of activities of the original behavioural profile.
Definition 7 (Complement).
Let
B
=
{
,
+
,
||}
be a behavioural profile over
a set of activities
A
.The
complement
B
=
{
C
,
+
C
,
||
C
}
of
B
is a behavioural
profile over
A
, such that for all pairs of activities
(
a
1
,a
2
)
∈
A
×
A
it holds:
◦
a
1
+
C
a
2
if and only if
a
1
||
a
2
,
◦
a
1
C
a
2
if and only if
a
1
−
1
a
2
,
◦
a
1
||
C
a
2
if and only if
a
1
+
a
2
.
The complement operation is illustrated in Fig. 4. Model 4(a) and model 4(b)
show complementary behavioural profiles (neglecting start and end nodes) - the
profile of the left model is the complement of the profile of the right model,
and vice versa. For instance, there is a strict order constraint between activities
A
and
B
in model 4(a), whereas both activities are in reverse strict order in
model 4(b). Further, activity
A
may occur multiple times in model 4(a), whereas
it is exclusive to itself in model 4(b).
Intersection.
Given two behavioural profiles, the intersection operation yields a
third behavioural profile that combines the strictest relations of the behavioural
profile for all shared pairs of activities. Therefore, the intersection represents the
behavioural constraints that are shared by both profiles.
Definition 8 (Intersection).
Let
B
1
=
{
1
,
+
1
,
||
1
}
and
B
2
=
{
2
,
+
2
,
||
2
}
be two behavioural profiles over a set of activities
A
.The
intersection
∩
of these
profiles is a behavioural profile
B
3
=
{
3
,
+
3
,
||
3
}
over
A
, denoted by
B
1
∩B
2
=
B
3
, such that for all pairs of activities
(
a
1
,a
2
)
∈
A
×
A
it holds:
a
1
−
1
◦
a
1
+
3
a
2
if and only if either
a
1
+
1
a
2
,
a
1
+
2
a
2
,
(
a
1
1
a
2
∧
a
2
)
,or
2
(
a
1
−
1
a
2
∧
a
1
2
a
2
)
,
1
◦
a
1
3
a
2
if and only if either
(
a
1
1
a
2
∧
(
a
1
2
a
2
∨
a
1
||
2
a
2
))
or
(
a
1
2
a
2
∧
(
a
1
1
a
2
∨
a
1
||
1
a
2
))
,
◦
a
1
||
3
a
2
if and only if
a
1
||
1
a
2
and
a
1
||
2
a
2
.
We illustrate the intersection of behavioural profiles with the models in Fig. 5. The
lower model (c) shows a behavioural profile that corresponds to the intersection
of the behavioural profiles of the upper two models (a) and (b). Consider, for in-
stance, activities
A
and
B
. While model (a) allows for interleaving order between
both activities, model (b) is more restrictive and enforces strict order. Hence, the
intersection also defines strict order for both activities. Due to the assumed be-
havioural abstraction, again, model (c) does not represent the intersection of the
