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≥ , then for all
t
∈ ∆
D
,
c
D
( ) ³ ;
16.
if
( ,
s T d
∀
. )
n
(
〈
s t T
,
〉
,
)
≤ − or
d
1
n
t
n
≥ , then there exists
t
∈ ∆
D
, such that
c
D
( ) ³ .
17.
if
( ,
s T d
∃
. )
n
(
〈
s t T
,
〉
,
)
≥ and
d
n
t
n
The process of inducing a fuzzy tableau from a completion forest is as follows.
Each element in
S
corresponds to a path in . We can view a blocked node as a loop so as to define
infinite paths. To be more precise, a path
p
= [
v
/
v
¢
,
,
v
/
v
]
is a sequence of node pairs in .
0
0
n
n
'
We define
Tail p
(
) = , and
Tail p
v
n
'(
) =
v
n
¢
. We denote by [
p v
|
/
v
']
the path
n
+
1
n
+
1
and use
[
as the abbreviation of
[
v
/
v
,
,
v
/
v
,
v
/
v
']
p v
|
/
v
'
,
v
/
v
']
0
0
′
n
n
′
n
+
1
n
+
1
n
+
1
n
+
1
n
+
2
n
+
2
[
[
p v
|
/
v
'
],
v
/
v
']
. The set Paths (
of paths in is inductively defined as follows:
n
+
1
n
+
1
n
+
2
n
+
2
•
if
v
is a root node in , then [
v v
/ ÎPaths( );
']
•
if
p
ÎPaths( ), and
w
ÎNodes( ),
◦
if
w
is the
R
-successor of
Tail p
(
) and is not
k
-blocked, then [
p w
/ ÎPaths (
|
]
;
if there exists
′
w
Nodes (
◦
and is the
R
-successor of
Tail p
(
) , and is directly
k
-blocked
p w
/
′
∈Paths (
by
w
, then [
|
]
.
Definition 11.
(Induced fuzzy tableau) The fuzzy tableau
T
=
〈
S
,
H E E V
,
,
,
〉
induced by is as
F
a
c
follows.
•
S
=Paths (
,
•
H
(
p C
,
)
≥
sup
{
n C
|
〈 ≥ 〉 ∈
,
,
n
L
(
Tail p
(
)) },
i
i
′
〉
′
〉 ,
•
E
(
〈
p p
, [
| [
p w w
|
/
]] ,
R
)
≥
sup
{
n
|
〈 ≥ 〉 ∈ 〈
R
,
,
n
L
(
Tail p w
(
),
)}
a
i
i
′
′
〉 ,
E
( [
〈
p
| [
p w w
|
/
]],
p R
〉
,
)
≥
sup
{
n
|
〈
Inv R
(
),
≥ 〉 ∈ 〈
,
n
L
(
Tail p w
(
),
)}
a
i
i
*
E
( [
〈
v
/
v w w R
,
/
] ,
〉
)
≥
sup
{
n
|
〈 ≥ 〉 ∈ 〈
R
,
,
n
L
(
v w
,
〉
)}
, where
v
, are root nodes,
v
is the
a
i
i
R
*
-neighbour of
w
, and
R
*
denotes
R
or
Inv R
(
) ,
•
c
(
〈
p p v
, [
|
/
v
] ,
〉
T
)
≥
n
with 〈 ≥ 〉 ∈ 〈
T
,
,
n
(
Tail p v
c
(
),
)ñ , where
v
c
is a concrete node,
c
c
[
a
/
/
a
],
if
a
a
is a root node and
,
L
(
a
)
≠ ∅
,
i
i
i
i
•
V
(
a
)
=
[
a
a
],
if
is a root node and
,
L
(
a
)
=
∅
,
From the fuzzy tableau of a fuzzy KB , we can obtain the canonical model of .
i
j
j
i
i
with
a
≈
a
and
L
(
a
)
≠ ∅
.
i
j
j
Definition 12.
(Canonical model) Let
T
= 〈
S
, , , ,
H E E V
be a fuzzy tableau of , the canonical
〉
a
c
.
model of
T
,
= (
D
T
,
T
)
, is defined as follows.
T
• D
T
=
S
;
•
I
for each individual name
o
in
I
,
o
= ( ) ;
o
I
•
for each
s
Î
S
and each concept name
A
,
A
( ) =
s
H
( ,
s A
) ;
