Database Reference
In-Depth Information
(
∃
T d
. ) ( )
o
=
sup
min
{
(
T o v d
( ,
),
( ))}
v
∆
D
v
∈
2
(
≥
2 ) ( )
S
o
=
{
{
R o o
( ,
)}}
sup
min
i
i
=1
o o
1
,
∈
∆
2
2
( 1 ) ( )
≤
S
o
=
{
{1
−
R o o
( ,
)}}
inf max
i
=1
i
o o
1
,
∈
∆
2
−
′
=
′
(
R
) ( ,
o o
)
R o o
(
,
)
A fuzzy interpretation satisfies a fuzzy concept specification
A C
, if
A o
( )
£
C o
( )
for any
o
∈ ∆
, written as I
A C
( ) for any
o
∈ ∆
, and
. Similarly, I
A
º if
A o
( ) =
C o
I
C
D
, if
C o
( )
£
D o
( )
for any
o
∈ ∆
. For ABox assertions, I
C o
( ) DD (resp.
n
¹
'
. If an
interpretation satisfies all the axioms and assertions in a KB , we call it a model of . A KB is
satisfiable
iff it has at least one model. A KB
entails
(logically implies) a fuzzy assertion j, iff all
the models of are also models of j, written as
K
j.
Given a KB , we can w.l.o.g assume that
¢
, iff
C o
(
¢
, and I
o
≈ / iff
o
I
R o o
( ,
)
n
)
) DD (resp.
R o o
n
(
,
)
n
)
o
1. all concepts are in their negative normal forms (NNFs), i.e. negation occurs only in front of concept
names. Through de Morgan law, the duality between existential restrictions ($
R
. ) and universal
restrictions ("
R
. ), and the duality between functional restrictions (£1
S
) and their negations (
³2
S
), each concept can be transformed into its equivalent NNF by pushing negation inwards.
2. all fuzzy concept assertions are in their positive inequality normal forms (PINFs). A negative
concept assertion can be transformed into its equivalent PINF by applying fuzzy complement
operation on it. For example,
C o
( ) > 1 .
3. all fuzzy assertions are in their normalized forms (NFs). By introducing a positive, infinite small
value e, a fuzzy assertion of the form
C o
( ) < is converted to ¬
n
C o
−
n
( ) ≥ + ε . The model
equivalence of a KB and its normalized form was shown to justify the assumption (Stoilos,
Straccia, Stamou, & Pan, 2006).
4. there are only fuzzy GCIs in the TBox. A fuzzy concept specification
A C
( ) > can be normalized to
C o
n
n
can be replaced by
≡
′
(Stoilos et al., 2007), where
A
is a new concept name,
which stands for the qualities that distinguish the elements of
A
from the other elements of
C
. A
fuzzy concept definition axiom
A
º can be eliminated by replacing every occurrence of
A
with
C
. The elimination is also known as
knowledge base expansion
. Note that the size of the
expansion can be exponential in the size of the TBox. But if we follow the principle of “Expansion
is done on demand” (Baader & Nutt, 2003), the expansion will have no impact on the algorithm
complexity of deciding fuzzy query entailment.
a fuzzy concept definition
A
A
C
Example 1.
As a running example, we use the
f
- SHIF (
D
KB
K
= 〈
T R A
, , with
〉
T
=
{
C
∃
R C
.
,
∃
T d
. }
, = Æ, and = { ( )
C o
³ .
0.8}
