Database Reference
In-Depth Information
Example 10
Let us show how we construct the set
S
and the compacting of tuples
given by Definition
11
above:
Let us consider an operad
q
i
∈
MakeOperads(
M
AB
)
, obtained from a nor-
.
=
malized implication
φ
Ai
(
x
)
⇒
r
B
(
t
)
in
M
AB
,
((y
f
1
(x,z))
∧
r
1
(x,y,z)
∧
(f
r
3
(y,z,w
,w)
.
r
2
(v,x,w)
∧
=
1
))
⇒
r
B
(x,z,w,f
2
(v,z))
, so that
q
i
is equal to
x,y,z,v,w,w
the expression
(e
(the
ordering of variables in the atoms (with database relational symbols) from left
to right),
t
⇒
(
_
)(
t
))
∈
O(r
1
,r
2
,r
3
,r
B
)
, where
x
=
.
=
=
x,z,w,f
2
(v,z)
, and the expression
e
equal to
(y
f
1
(x,z))
∧
(
_
)
1
(
t
1
)
∧
(
_
)
2
(
t
2
)
∧
(
_
)
3
(
t
3
)
, with
t
1
=
x,y,z
,
t
2
=
v,x,w
and
t
3
=
y,z,w
,w
.
Consequently, we obtain
=
(
1
,
1
),(
2
,
2
)
,
(
2
,
1
),(
1
,
3
)
,
(
3
,
1
),(
2
,
3
)
,
(
3
,
2
),(
4
,
3
)
S
that are the positions of duplicates (or joined variables) of
x,y,z
, and
w
, respec-
tively.
Thus, for given tuples
d
1
=
a
1
,a
2
,a
3
∈
α(r
1
)
,
d
2
=
b
1
,b
2
,b
3
∈
α(r
2
)
and
d
3
=
c
1
,c
2
,c
3
,c
4
∈
α(r
3
)
, the statement
π
j
h
(
d
j
)
|
(j
h
,j),(n
h
,n)
∈
S
=
π
n
h
(
d
n
)
is equal to
π
1
(
d
1
)
π
2
(
d
2
)
∧
π
2
(
d
1
)
π
1
(
d
3
)
∧
π
3
(
d
1
)
π
2
(
d
3
)
∧
π
3
(
d
2
)
π
4
(
d
3
)
,
=
=
=
=
which is true when
a
1
=
b
2
,
a
2
=
c
1
,
a
3
=
c
2
, and
b
3
=
c
4
.
The compacting of these tuples is equal to
Cmp
S,
d
1
,
d
2
,
d
3
=
d
=
a
1
,a
2
,a
3
,b
1
,b
3
,c
3
,
w
/c
3
]
with the assignment to variables
[
x/a
1
]
,
[
y/a
2
]
,
[
z/a
3
]
,
[
v,b
1
]
,
[
w/b
3
]
, and
[
.
x/a
1
,y/a
2
,z/a
3
,v/b
1
,w/b
3
,w
/c
3
]
is obtained by this assign-
ment
g
to the tuple of variables
x
, so that the sentence
e
That is,
d
=
x
[
[
(
_
)
n
/r
n
]
1
≤
n
≤
k
/g
is well
defined and equal to
a
2
=
I
T
(f
1
)(a
1
,a
3
)
∧
r
1
(a
1
,a
2
,a
3
)
∧
r
2
(b
1
,a
1
,b
3
)
∧
r
3
(a
2
,a
3
,c
3
,b
3
),
that is, to
a
2
=
I
T
(f
1
)(a
1
,a
3
)
∧
r
1
(
d
1
)
∧
r
2
(
d
2
)
∧
r
3
(
d
3
),
and if this formula is satisfied by such an assignment
g
, i.e.,
I
T
(e
[
(
_
)
n
/r
n
]
1
≤
n
≤
k
/g)
=
1, then
=
g(x),g(z),g(w),g
∗
f
2
(v,z)
=
a
1
,a
3
,b
3
,I
T
(f
2
)(b
1
,a
3
)
,
g
∗
(
t
)
f(
d
1
,
d
2
,
d
3
)
=