Database Reference
In-Depth Information
Let us show this algorithm in the case of Example
16
in Sect.
2.5
:
Example 35
Consider the following three cases:
1. The mapping
M
AD
={
Φ
}:
A
→
D
in Example
5
, with
Φ
equal to SOtgd
∃
⇒
Enrolment
(f
1
(x
1
),x
2
)))
. With the decompo-
sition we obtained
Φ
E
equal to
∀
x
1
∀
f
1
(
x
2
(
Takes
(x
1
,x
2
)
∀
x
1
∀
x
2
(
Takes
(x
1
,x
2
))
⇒
r
q
1
(x
2
)
, and
Φ
M
equal to
f
Ta k e s
∀
x
2
r
q
1
(x
2
)
∧
f
Ta k e s
(x
1
,x
2
)
.
1
⇒
Enrolment
f
1
(x
1
),x
2
.
∃
f
1
∃
x
1
∀
=
By application of the algorithm
VarTransform
to the implications in
Φ,Φ
E
and
Φ
M
, we obtain the followin
g
respe
ct
ive implic
at
i
on
s:
1.1.
(
Takes
(x
1
,x
2
)
.
=
∧
(
1
1
)
∧
r
(
0
,
0
)
∧
(z
1
=
f
1
(x
1
)))
⇒
Enrolment
(z
1
,x
2
)
.
1.2.
(
Takes
(x
1
,x
2
)
(
1
.
∧
=
1
)
∧
r
(
0
,
0
)
∧
r
(
1
,
1
))
⇒
r
q
1
(x
2
)
.
.
=
1.3.
(r
q
1
(x
2
)
∧
Takes
(x
1
,y
1
)
∧
(
1
1
)
∧
(y
1
=
x
2
)
∧
(z
1
=
f
1
(x
1
)))
⇒
Enrolment
(z
1
,x
2
)
.
2. The mapping
M
AD
={
}:
A
→
D
Φ
in Example
5
, with
Φ
equal to SOtgd
f
Student
∀
y
3
Takes
(x
1
,x
2
)
∧
f
Student
(x
1
,y
3
)
.
1
⇒
Enrolment
(y
3
,x
2
)
.
∃
x
1
∀
x
2
∀
=
We obtained
Φ
E
equal to
f
Student
∀
y
3
Takes
(x
1
,x
2
)
∧
f
Student
(x
1
,y
3
)
.
1
⇒
r
q
2
(y
3
,x
2
)
∃
x
1
∀
x
2
∀
=
and
Φ
M
equal to
f
Student
∀
y
3
r
q
2
(y
3
,x
2
)
∧
f
Student
(x
1
,y
3
)
.
1
⇒
Enrolment
(y
3
,x
2
)
).
∃
x
1
∀
x
2
∀
=
By application of the algorithm
VarTransform
to the implications in
Φ,Φ
E
and
Φ
M
, we obtain the following respective impli
ca
tio
n
s:
2.1.
(
Takes
(x
1
,x
2
)
∧
Student
(y
1
,y
3
)
∧
(
1
.
=
1
)
∧
(y
1
=
x
1
)
∧
r
(
1
,
1
))
⇒
Enrolment
(y
3
,x
2
)
.
2.2.
(
Takes
(x
1
,x
2
)
(
1
.
∧
Student
(y
1
,y
3
)
∧
=
1
)
∧
(y
1
=
x
1
)
∧
r
(
1
,
1
))
⇒
r
q
2
(y
3
,x
2
)
.
2.3.
(r
q
2
(y
3
,x
2
)
∧
Student
(x
1
,y
4
)
∧
(
1
.
=
1
)
∧
(y
4
=
y
3
)
∧
r
(
1
,
1
))
⇒
Enrolment
(y
3
,x
2
)
.
3. The mapping
M
AC
={
Φ
}:
A
→
D
in Example
9
, with
Φ
equal to SOtgd with
the two implications
