Database Reference
In-Depth Information
diagram (I1) commutative): for each unary (or binary) operator
o
i
∈
Σ
P
and the
relations
f
#
(t),f
#
(t
1
),f
#
(t
2
)
∈
Z
(obtained from terms
t,t
1
,t
2
∈
T
P
X
)
1.
h(f
#
(t),i)
=
o
i
(t)
#
, where
o
i
(t)
∈
T
P
X
is a term, and
_
#
:
T
P
R→
Υ
in
Definition
31
is the evaluation of terms (R-algebra
f
is a restriction of
_
to
⊆R
).
2.
h((f
#
(t
1
),f
#
(t
2
)),
X
−
1
)
=
t
1
TIMES
t
2
#
,h
A
((f
#
(t
1
),f
#
(t
2
)),
0
)
=
t
1
UNION
t
2
#
and
h
A
((f
#
(t
1
),f
#
(t
2
)),
1
)
=
t
1
MINUS
t
2
#
if
f
#
(t
1
)
and
f
#
(t
2
)
are union compatible;
⊥
otherwise.
Let
Σ
P
=
(S
Z
,Σ
Z
)
be a schema of the instance
database
Z
, with the set of relational symbols in
S
Z
. Let us show that
Z
has to be a
closed object in
DB
. We have two possible cases:
1. When
S
Z
⊆
Σ
R
be SPRJU signature and
Z
=
X
, the assignment
f
:
X
→
Z
must satisfy the condition for each
r
∈
X
⊆R
,
f(r)
=⊥
if
r
∈
S
Z
. (In each object (instance database) in
DB
the
empty relation
is an element of it.)
In this case, from the fact that the elements (i.e., relations) in
Z
areinthe
image of the mapping
f
#
(which is an evaluation of terms in
⊥
T
P
X
), this database
instance has to be a closed object, that is,
Z
=
TD
of an database instance
D
⊆
Z
.
2. When
S
Z
⊃
. So, the
image of
f
#
is the set of all views that can be obtained with SPRJU algebra from
this instance database
C
and, consequently,
TC
⊆
Z
. However,
Z
is the image
of the
h
X
, let us define the database instance
C
={
f(r)
|
r
∈
X
}
Z
which computes all views over
Z
as well, so that, again
Z
has to be a closed object in
DB
,
Z
:
Σ
R
Z
→
=
TD
, with
TC
⊆
TD
. The maximal case
for
Z
is when
Z
T
Υ
.
In what follows, we will consider a particular case when
X
=
Υ
=
S
Z
,asinthe
case which defines the power-view operator
T
for the instance databases, based
on the initial
X
=
Σ
R
-algebra semantics.
Consequently, we obtain the
syntax
monad
(
T
P
,η,μ)
with
η
X
=
inl
X
, while
μ
X
=
(id
T
P
X
)
#
is the unique inductive extension of
inr
X
:
Σ
P
(
T
P
X)
−→
T
P
X
along the identity on
T
P
X
, as represented in the following commutative diagram:
This syntax monad framework permits us to form the terms from any operator deriv-
able from the signature
Σ
P
:
