Database Reference
In-Depth Information
Lemma 1 8
.
∀
L
, ...,
L L
,
, ...,
L and G
,
=
merge L
(
, ...,
L
),
1
j
nj
1
k
nk
j
1
j
nj
f
f
fs
sameDB
G
=
merge L
(
, ...,
L G
),
G
=
merge L
(
L
,
...,
L
L
).
−
−
−
k
1
k
nk
j
α β
,
k
1
j
α β
,
1
k
nj
α β
,
nk
fs
sameDB
Proof:
Let G
=
(
G
G
)
−
j
α β
,
k
f
and Let L
L
g G g s DB for some i
=
(
L
).
−
i
j
α β
,
ik
∈ ∧
.
=
∈
{ , ..., }
1
n
⇔
i
fs
sameD
B
g G
∈
(
G
)
∧
g s DB for some i
.
=
∈
{ , ..., }
1
n
⇔
−
j
α β
,
k
i
g A
g A
g A g A
(
µ
( .
g A g A
,
. )
α µ
∧
( . )
g A
=
max(
µ
( . ),
µ
(
. ))
β
)
∨
(
µ
( .
,
. )
<
EQ
Gj
G
G
Gj
Gj
EQ
Gk
α
∨
max(
µ
( . ),
g A
µ
(
g A
. ))
< ∧
β
)
g s DB for some i
.
=
∈
{ , ..., }
1
n
⇔
G
Gk
Gk
i
(
µ
( .
g A g A
ij
,
. )
α µ
∧
( . ) max(
g A
=
µ
( . ),
g A
µ
(
g A
. ))
β
)
∧
(
µ
( .
g A L A
,
. )
<
EQ
L
L
L
L
Lij
EQ
ik
ij
α
∨
max(
µ
( . ),
g A
µ
(
g A
. ))
<
β
)
for some i
∈
{ , ..., }
1
n
⇔
L
L
L
ik
ik
f
g A
.
∈
(
L
L
)
for some i
∈
{ , ..., }
1
n
⇔
−
ij
α β
,
ik
f
f
merge L
(
L
, ...,
L
L
nk
).
−
−
ij
α β
,
ik
nj
α β
,
Having proved the consistency of the above FTS relational operations, the following corollary be-
comes apparent.
fs
sameDB
fs
sameDB
fs
sameDB
fs
sameDB
fs
Corollary:
The set of FTS relational operations {
σ π
,
D
fs
sameDB
∪
}
,
,
∩
,
and
−
is consistent with respect to the export fuzzy relational operations.
Algebraic Properties of FTS Relational Model
Given a global multidatabase fuzzy query written in FTS-SQL, a distributed fuzzy query processor derives
its equivalent FTS relational expression and evaluates the FTS relational operations in the expression.
As part of the fuzzy query evaluation process, the fuzzy query processor may have to transform the
relational expression in order to obtain an algebraically equivalent relational expression that requires
the least evaluation cost (in terms of disk and communication overheads). Such transformation of FTS
queries can only be possible when the algebraic properties of FTS relational model are known. In this
section, we will present a few important algebraic properties related to the
Union
operation in the FTS
relational model.
fs
sameDB
α β
,
is commutative.
Proof:
The definition is independent of the ordering of operands, so it can be proved.
Theorem:
È
fs
sameDB
α β
fs
sameDB
fs
sameDB
fs
sameDB
fs
sameD
B
T
)
Proof:
Let S, R and T be FTS relations of same arity and domain of
i
th
attribute of S,R,T is the same.
In case of
sameDB
option, result must have non-`*' source attribute value. Let FTS relation
Theorem:
È
is associative. i.e.
Z
= ∪
(
R
S
)
∪
T
= ∪
R
(
S
∪
,
α β
,
α β
,
α β
,
α β
,
fs
sameDB
= ∪
fs
sameDB
= ∪
α β
X
(
R
S
)
and FTS relation
Y
S
T
)
. Hence,
α β
,
( .
t A
,
µ
µ
( . ), . )
t A t s
∈ ⇔
∈
Z
Z
fs
sameDB
( .
t A
,
( . ), . )
t A t s
(
X
T
)
⇔
∪
α β
Z
,
(
µ
Q
( .
t A t A
,
. )
α µ
∧
( . ) max(
t A
=
µ
( . ),
t A
µ
(
t A
. ))
β
∧
( .
t s
=
t s
.
≠ ∗
))
E
X
Z
Z
X X
X
