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In-Depth Information
+
R s t
( , ),
if
Trans R
(
),
E
I
•
for each 〈
s t
,
〉 ∈ ×
S S
R
,
( , ) =
s t
where
T
I
(
R s t S
( , ),
( , )
s t
), otherwise.
T
max
E
*
,
S
R S R
≠
+
( , ) is the sup-min transitive closure of
R s t
R s t
( , ) =
(
〈
s t R
,
〉
,
)
, and
R s t
( , ) .
a
I
D
∆ ,
T
•
for each 〈
s t
,
〉 ∈ ×
S
T
( , ) =
s t
E
( , ) .
s t
c
D
KB
K
T R A
Lemma 4.
Let
T
be the fuzzy tableau of an
f
-SHIF (
= 〈
, , , then the canonical
〉
model
T
of
T
is a model of .
Proof.
Property 12 in Definition 10 ensures that
T
is a model of . For a detailed proof, see
Proposition 3 in (Stoilos et al., 2006). Property 1-11 and 13-17 in Definition 10 ensures that
T
is a
model of and . For a detailed proof, see Lemma 5.2 and 6.5 in (Stoilos et al., 2007).
Example 6.
By unraveling in Figure 1, we obtain a model I
F
which has as domain the infinite set
of paths from
o
to each
o
i
. Note that a path actually comprises a sequence of pairs of nodes, in order
to witness the loops introduced by the blocked variables. When a node is not blocked, like
o
1
, the pair
o o
1
2
tree-blocks
T
o
4
2
, each time a path reaches
o
6
, which is a leaf node
/ is added to the path. Since
T
o
1
of a blocked tree, we add
o o
3
/ to the path and 'loop' back to the successors of
o
3
. This set of paths
constitute the domain D
I
F
. For each concept name
A
, we have
A
I
F
(
p
) ³ , if 〈 ≥ 〉
n
A
, , occurs in
n
i
I
F
(
the label of the last node in
p
i
. For each role
R
,
R
p p
,
) ³ if the last node in
p
j
is an
R
n
n
i
j
³,
-successor of
p
i
. If role
R Trans
Î
, the extension of
R
is expanded according to the sup-min transitive
I
F
(
I
F
(
semantics. Therefore,
C
p
i
)
0.³ for
i
³0, and
R
p p
i
,
)
0.³ for 0
£
i
<
j
.
j
From a complete and clash-free completion forest , we can obtain a fuzzy tableau
T
, through
which a canonical model I
F
is constructed.
Lemma 5.
Let
F
Î
ccf
k
(
)
, then I K
F
, where
k
³1.
K
Proof.
It follows from Lemma 5.9 and 6.10 in (Stoilos et al., 2007) and Proposition 5 in (Stoilos et
al., 2006) that the induced tableau in Definition 11 satisfies Property 1-15 in Definition 10. By Lemma
4, the canonical model I
F
constructed from
T
is a model of .
Now we illustrate how to construct a mapping of
q
to from a mapping of
q
to I
F
.
Definition 13.
(Mapping graph) Let
F
with
k
³0
, and a fuzzy query
q
, such that
I
F
q
.
Î
ccf
k
(
)
K
p is a mapping in Definition 2, then the mapping graph
G
π
=
〈
V E
,
〉 is defined as:
I
F
D
V G
(
) = { ( )
π
x
∈
∆
∪
∆
|
x
∈
Terms q
( )}
,
π
I
F
I
F
E G
(
) = { ( ),
〈
π
x
π
(
y
)
〉 ∈
∆
×
∆
|
R x y
( ,
)
≥ ∈
n
q
}
∪
π
