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we describe
f
Φ
and
f
Ω
by complex class expressions in DL. Additionally, we
introduce a class expression
f
Ψ
which will be used in the next step (Def. 4).
Depending on the constraint type (
Conjunct
,
Required
or
Exclude
)oftheel-
ement
E
, the expression
f
Ω
is built either as intersection, implication or exclusive
class expression. However, instead of the element
E
, we use the corresponding
mapping class
Map
E
(cf. Axiom 9) of the mapping model and the property
feature
that is a superproperty of
hasF eature
(cf. Axiom 10). This guarantees
the alignment of classes and properties between
Σ
Φ
and
Σ
Ω
.
For the feature model,
f
Φ
is the intersection of parents and cross-tree con-
straints of all mapped features (
)of
E
. The absence of optional child features
in the parent definition of the feature model (Axioms 1-3) directly meets the need
of the feature representation in
f
Φ
, since for an optional mapped feature, we can
not guarantee the appearance of the corresponding element in each business
process model. The set
F
F
of mapped features
F
is obtained from the mapping
knowledge base
Σ
M
feature.F
. Cross-tree constraints
are captured by the expression
cr
(
F
). To allow a subsumption checking of the
expressions in
cr
(
F
), we define the properties
includes
and
excludes
as sub-
properties of
feature
. The functions
elements
and
element
are abbreviations.
The element(s) is/are either the element that require another element or sibling
elements, they can be found by using the introduced properties
isRequired
and
sibling
.
Definition 4.
The final knowledge base
Σ
is constructed from the problem, so-
lution and mapping space knowledge bases, i.e.,
Σ
:=
Σ
Φ
∪
by axioms like
Map
E
∃
Σ
Ω
∪
Σ
M
. Moreover,
for each mapped and constrained element, an axiom
f
Ψ
≡¬
f
Φ
f
Ω
is added to
Σ
,where
f
Φ
and
f
Ω
are defined as follows:
-
for each element
E
Conjunct
and
S
:=
elements
(
sibling, E
)
:
f
Ω
≡
E
∈S
Map
E
and
f
Φ
≡
F∈F
cr
(
F
)
-
for each element
E Required
and
E
:=
element
(
isRequired, E
):
f
Ω
≡¬
Parent
(
F
)
Map
E
and
f
Φ
≡
F∈F
Map
E
Parent
(
F
)
cr
(
F
)
-
for each element
E
Exclude
and
S
:=
elements
(
sibling, E
)
:
f
Ω
≡
E
∈S
Map
E
¬
(
Map
E
Map
E
)
for (
E
,
E
∈S
)
and
f
Φ
≡
F∈F
Parent
(
F
)
cr
(
F
)
Implication Checking.
We reduce subsumption checking
f
Φ
f
Ω
toaclas-
sification problem by introducing
f
Ψ
. According to Def. 4, we add for each sub-
sumption checking problem the corresponding axiom
f
Ψ
≡¬
f
Φ
f
Ω
. Finally,
we check for each class expression
f
Ψ
whether it is equivalent with the top class
(
f
Ψ
≡
). In this case, the solution space constraints (
f
Ω
) are satisfied, otherwise
there is a violation. Moreover,
f
Ω
is a subclass of the corresponding mapping
class
Map
E
. This directly indicates the element that violates the constraint.
The validation effort is determined by the number of mappings and the num-
ber of elements that are involved in one of the constraints (
Required
,
Conjunct
and
Exclusive
). There are two steps where reasoning is applied. (i) We classify
the knowledge base in order to find the constrained elements. To build the class
