Database Reference
In-Depth Information
Fuzzy Query Language
We now provide the formal definition of the syntax and semantics of the fuzzy querying language used
in this paper, extending Mallis's work (Mailis et al., 2007) to allow for querying concrete domains.
Let
V
be a countable infinite set of variables and is disjoint from
A
,
R
,
R
c
,
I
, and
I
c
. A
termt
is either an individual name from
I
or
I
c
, or a variable name from
V
. A
fuzzy query atom
is an expres-
sion of the form 〈
′
≥ 〉
′
≥ 〉
( , ) with
C
a concept,
R
a simple abstract
role,
T
a data type role, and
t
,
t
terms. As with fuzzy assertions, we refer to these three different types
of atoms as
fuzzy concept atoms
,
fuzzy abstract role atoms
, and
fuzzy data type role atoms
, respectively.
The fuzzy abstract role atoms and the fuzzy data type role atoms are collectively referred to as
fuzzy role
atoms
.
C t
( )
≥ 〉
n
, 〈
R t t
( ,
)
n
, or 〈
T t t
n
Definition 1.
(Fuzzy Boolean Conjunctive Queries) A fuzzy boolean conjunctive query
q
is a non-
empty set of fuzzy query atoms of the form
q
= 〈 ≥ 〉
{
at
n
1
,
,
〈 ≥ 〉
at
n
}
. Then for every fuzzy
1
k
k
query atom, we can say 〈 ≥ 〉 ∈
at
n
q
.
i
i
We use
Vars
( ) to denote the set of variables occurring in
q
,
AInds
( ) and
CInds
( ) to denote
the sets of abstract and concrete individual names occurring in
q
,
Inds
( ) to denotes the union of
AInds
( ) and
CInds
( ) , and
Terms
( ) for the set of terms in
q
, i.e.
Terms q
( ) = ( ) ( È .
The semantics of a fuzzy query is given in the same way as for the related fuzzy DL by means of
fuzzy interpretation consisting of an interpretation domain and a fuzzy interpretation function.
Vars q
Inds q
Definition 2.
(Models of Fuzzy Queries) Let
D be a fuzzy interpretation of an
f
-SHIF (
D
KB,
q
a fuzzy boolean conjunctive query, and
t
,
t
terms in
q
. We say is a model of
q
, if there
exists a mapping π :
= (
,
.
)
D
such that p( )
a a
for each
a
( ( ))
Terms q
→ ∪
( )
∆
∆
Î ( ) ,
C
Ind q
π
t
≥
n
π
′
≥ (resp.
T
π
′
≥ ) for each
for each fuzzy concept atom
C t
( ) ≥ ∈ ,
R
n
q
( ( ),
π
t
( ))
t
n
( ( ),
π
t
( ))
t
n
′
≥ (resp.
T t t
′
≥ ) Î
q
.
fuzzy role atom
R t t
( ,
n
( ,
n
If I
p
at
for every atom
at
Î
q
, we write I
p
q
. If there is a p, such that I
p
q
, we say
satisfies
q
, written as I
q
. We call such a p a
match
of
q
in . If I
q
for each model of a
KB , then we say entails
q
, written as
K
q
. The
query entailment problem
is defined as follows:
given a knowledge base and a query
q
, decide whether
K
q
.
Example 2.
Considering the following fuzzy boolean CQ:
q
= { ( ,
R x y
)
³
0.6,
R y z
( ,
)
³
0.8,
T y y
( ,
)
³
1,
C y
(
)
³ .
0.6}
c
We observe that
K
q
. Given the GCI
C
$ . , we have that, for each model of ,
R C
holds. By the definition of fuzzy interpretation, there exists some
element
b
in D
, such that
R o b
∃
R C o
.
(
)
≥
C o
(
)
≥
0.8 > 0.6
(
,
)
³
0.8 > 0.6
and
C
( )
³
0.8 > 0.6
holds. Similarly, there is some
element
c
in D
, such that
R b c
( , )
( )
0.³ and
C c
0.³ holds. Since $
T
. , there is some
