Database Reference
In-Depth Information
D
→ ¬
d
|
d
.
For decidability reasons, roles in functional restrictions of the form £1
S
and their negation ³ 2
S
are restricted to be simple abstract roles.
A fuzzy TBox is a finite set of fuzzy concept axioms. Fuzzy concept axiom of the form
A
º are
called
fuzzy concept definitions
, fuzzy concept axiom of the form
A C
are called
fuzzy concept spe-
cializations
, and fuzzy concept axiom of the form
C
D
are called
general concept inclusion
(GCIs)
axioms.
A fuzzy ABox consists of fuzzy assertions of the form
C o
¢
( ) DD (
fuzzy concept assertions
),
R o o
n
( ,
n
)
(
fuzzy data type role assertions
), or
o
≠
′
(
inequality as-
(
fuzzy abstract role assertions
),
T o v
( ,
n
sertions
), where
o
,
′
∈
I
,
v
{ , >, , <}. We use
to denote ³ or >, and
to denote £ or <. We call ABox assertions defined by
positive assertions
,
while those defined by
negative assertions
. Note that, we consider only positive fuzzy role assertions,
since negative role assertions would imply the existence of role negation, which would lead to undecid-
ability (Mailis, Stoilos, & Stamou, 2007). An
f
-SHIF (
Î
I
, DD stands for any type of inequality, i.e., DD∈ ≥ ≤
c
D
knowledge base is a triple 〈
T R A
, ,
〉
with a TBox, a RBox and an ABox.
For a fuzzy concept
D
, we denote by
sub D
(
) the set that contains
D
and it is closed under sub-
concepts of
D
, and define
sub
(
as the set of all the sub-concepts of the concepts occurring in .
We abuse the notion
sub
(
D
to denote the set of all the data type predicates occurring in a knowledge
base.
The semantics of
f
-SHIF (
.
D .
Here D
is a non-empty set of objects, called the domain of interpretation, disjoint from D
D
, and
.
is
an interpretation function that coincides with
.
D
on every data value and fuzzy data type predicate, and
maps (
i
) different individual names into different elements in D
, (
ii
) a concept name
A
into a member-
ship function
A
: ∆
→ [0,1], (
iii
) an abstract role name
R
into a membership function
R
:∆ ∆
D
are provided by a fuzzy interpretation which is a pair
= (
,
)
× →
[0,1], (
iv
) a data type role
T
into a membership function
T
:∆ ∆
D
× →
[0,1] . The semantics of
f
-
SHIF (
D
concepts and roles are depicted as follows.
T
( ) =
o
^
( ) = 0
o
(
¬
C
) ( ) = 1
o
−
C o
( )
(
C D o
) ( )
=
min
{
C o D o
( ),
( )}
(
C D o
) ( )
=
max
{
C o D o
( ),
( )}
′
′
(
∀
R C
.
) ( )
o
=
inf
max
{
(1
−
R o o C o
( ,
),
(
))}}
′
∈
o
∆
′
′
(
∃
R C
.
) ( )
o
=
sup
min
{
(
R o o C o
( ,
),
(
))}
o
′∈
∆
(
∀
T d
. ) ( )
o
=
inf
max
{
(1
−
T o v d
( ,
),
( ))}}
v
∆
D
v
∈
