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3.1 Strictness of Behavioural Relations
The relations of the be-
havioural profile can be clas-
sified according to their
strictness
, as they allow differ-
ent levels of freedom for the
occurrences of activities in a
trace. Interleaving order can
be seen as the absence of any
restriction on the order of po-
tential occurrence - the activities are allowed to appear in an arbitrary order.
In contrast, (reverse) strict order defines a particular order of execution for two
activities, whereas exclusiveness completely prohibits the occurrence of two ac-
tivities together in one trace. Therefore, we consider interleaving order to be
the weakest relation, while exclusiveness is the strictest relation. For a dedicated
pair of activities (
x, y
), this strictness is also reflected in the containment hierar-
chy of traces that show either none, one, or both activities, illustrated in Fig. 2.
Here, all traces that conform to a specific behavioural relation are part of the
encircled set of traces. A process model that defines interleaving order between
two activities
x
and
y
allows for any trace, i.e., it may contain none, one, or both
activities in any order. A model that imposes exclusiveness for both activities
is most restrictive. It allows for traces that comprise none or only one of the
activities. This set is a proper subset of the traces induced by interleaving order.
This notion of strictness is the foundation for the definition of a set algebra for
behavioural profiles. Our operations and relations are
not
defined based on sets
of traces, but on the relations of the behavioural profile. Still, the strictness of
behavioural relations illustrated above with sets of traces is taken into account.
<…x...y...x…y…>
<…x...y...y…x…>
(x||y)
<…y…x…>
(x+y)
<…x…x…>
<…x…>
<…y…>
<…x…y…>
(x->
-1
y)
(x->y)
<…>
<…y…x...x…>
<…x…y...y…>
<…y...x…y…
<…x...y…x…x…y...y>
Fig. 2.
Sets of traces induced by the relations of the
behavioural profile for two activities
x
and
y
3.2 Set-Theoretic Relations
We start by introducing three set relations, i.e., equivalence, inclusion, and empti-
ness for behavioural profiles. Most use cases require the application of these con-
cepts for behavioural profiles of different models for which a separate relation
identifies corresponding pairs of activities. To keep the formalisation concise, we
abstract from such correspondences and assume corresponding activities to be
identical. As partially overlapping sets of activities do not impose serious chal-
lenges, we restrict the discussion to the behavioural aspects and assume identical
sets of activities once more than one behavioural profile is considered.
Equivalence.
Two behavioural profiles are equivalent, if they enforce equal be-
havioural constraints for the shared activities. Equivalence of behavioural profiles
does not imply equal trace semantics for the shared activities, cf., [16].
Definition 4 (Equivalence).
Let
B
1
=
{
1
,
+
1
,
||
1
}
and
B
2
=
{
2
,
+
2
,
||
2
}
be two behavioural profiles over a set of activities
A
.
B
1
equals
B
2
, denoted by
B
1
=
B
2
, if and only if their relations are equal for all activities, i.e.,
1
=
2
,
+
1
=+
2
,and
||
1
=
||
2
.
