Databases Reference
In-Depth Information
and
map
p
, on the other hand, model the sought-after concept and property
correspondences, respectively. Given the state of the observable predicates, we
are interested in determining the state of the hidden predicates that maximize
the a-posteriori probability of the corresponding possible world. The ground
atoms of these hidden predicates are assigned the weights specified by the a-
priori similarity
σ
. The higher this value for a correspondence the more likely
the correspondence is correct
a-priori
. Hence, the following ground formulas are
added to the set of formulas:
(
map
c
(
c, d
)
,σ
(
C,D
))
if C and D are concepts
(
map
p
(
p,r
)
,σ
(
P,R
))
if P and R are properties
Notice that the distinction between
m
c
and
m
p
is required since we use typed
predicates and distinguish between the
concept
and
property
type.
Cardinality Constraints.
A method often applied in real-world scenarios is
the selection of a functional one-to-one alignment [17]. Within the ML frame-
work, we can include a set of hard cardinality constraints, restricting the align-
ment to be functional and one-to-one. In the following we write
x, y, z
to refer to
variables ranging over the appropriately typed constants and omit the universal
quantifiers.
map
c
(
x, y
)
∧
map
c
(
x, z
)
⇒
y
=
z
map
c
(
x, y
)
∧
map
c
(
z,y
)
⇒
x
=
z
Analogously, the same formulas can be included with hidden predicates
map
p
,
restricting the property alignment to be one-to-one and functional.
Coherence Constraints.
Incoherence occurs when axioms in ontologies lead
to logical contradictions. Clearly, it is desirable to avoid incoherence during the
alignment process. Some methods of incoherence removal for ontology align-
ments were introduced in [57]. All existing approaches, however, remove corre-
spondences
after
the computation of the alignment. Within the ML framework
we can incorporate incoherence reducing constraints
during
the alignment pro-
cess for the first time. This is accomplished by adding formulas of the following
type to set of hard formulas.
dis
1
(
x, x
)
sub
2
(
x, x
)
map
c
(
x
,y
))
∧
⇒¬
(
map
c
(
x, y
)
∧
dis
1
(
x, x
)
sub
2
(
y,y
)
map
c
(
x
,y
))
∧
⇒¬
(
map
p
(
x, y
)
∧
The second formula, for example, has the following purpose. Given properties
X,Y
and concepts
X
,Y
. Suppose that
X
and
O
1
|
=
∃
X.
¬
O
2
|
=
∃
Y.
Y
.Now,if
X
,Y
,
X,Y,
≡
and
≡
were both part of an alignment the merged
X
and
X
and, therefore,
ontology would entail both
∃
X.
∃
X.
¬
∃
X.
⊥
. The specified formula prevents this type of incoherence. It is known that such
constraints, if carefully chosen, can avoid a majority of possible incoherences [56].
