Database Reference
In-Depth Information
each relational symbol
r
i
j
,j
by the place symbol
(
_
)
j
(while replacing
¬
r
i
j
,j
by
(
_
)
j
) and by replacing
r
by unlabeled
(
_
)
.
We will represent each operation
q
i
by a composition of two operations
v
i
·
q
A,i
, where
q
A,i
∈
O(r
i
1
,
1
,...,r
i
k
,k
,r
q
)
is the same expression as
q
i
above
where
r
q
is a relational symbol of the same type as
r
, and
v
i
∈
O(r
q
,r
)
is the
expression
(
_
)
1
(
y
i
)
¬
the place symbol
⇒
(
_
)(
y
i
)
where
y
i
is the tuple of variables of the atoms
r
(
y
i
)
and
r
q
(
y
i
)
.
In the simplest case when
χ
i
is a tautology
r(
x
i
)
⇒
r(
x
i
)
,
χ
i
is replaced in
S
by
q
i
=
q
A,i
=
v
i
=
1
r
∈
O(r,r)
(that is, by the identity mapping expression
(
_
)
1
(
x
i
)
⇒
(
_
)(
x
i
)
if
x
i
is not empty; and by the expression
(
_
)
1
⇒
(
_
)
for the
O(r
∅
,r
∅
)
, otherwise).
4. (
Construct operad's operations
)
Add the 'empty operation' operad 1
r
∅
∈
identity operation 1
r
∅
∈
O(r
∅
,r
∅
)
in
S
, represented by the map-
ping expression
(
_
)
1
⇒
(
_
)
.
Return
The set of abstract operad's operations denoted by an (mapping) arrow
M
AB
=
S
:
A
→
B
.
In what follows, we will use the term 'mapping-operad' from such a set of ab-
stract operad operations. The reason why we insert the identity 'empty operad's
operation' 1
r
∅
∈
S
is that, as we will see, we can have the
Tarski's interpretations of database mappings where we do not transfer any infor-
mation from the source to the target database (i.e., with the empty information flux
of such a mapping). Consequently, 1
r
∅
is always an element of the set of operad's
operations obtained from a given SOtgd. As we will see in the algorithm of de-
composition of SOtgds, we will have cases where the mappings are generated with
SOtgd equal to the tautology
r
O(r
∅
,r
∅
)
in
M
AB
=
∅
⇒
r
.
∅
Example 7
In the most trivial case when
M
AB
={
r
∅
⇒
r
∅
}
(that is, when there is
no effective mapping between the schemas
) where
r
∅
is a nullary predicate
symbol (i.e., the
truth
propositional letter in FOL, Definition
1
), so that SOtgd is a
banal tautology
r
∅
⇒
A
and
B
M
AB
is always satisfied, we obtain that
r
∅
and, consequently,
MakeOperads(
M
AB
)
={
1
r
∅
}
. The particular cases for such trivial mappings are the
following:
M
AA
∅
={
r
∅
⇒
r
∅
}:
A
→
A
∅
,
M
A
∅
A
={
r
∅
⇒
r
∅
}:
A
∅
→
A
,
and
Id
A
∅
={
r
∅
⇒
r
∅
}:
A
∅
→
A
∅
,
the identity mapping
,
where
A
∅
=
(
{
r
∅
}
,
∅
)
is the empty database schema. It is easy to verify that
MakeOperads
{
r
∅
}
={
r
∅
⇒
1
r
∅
}
.
Obviously, we cannot have the Tarski's interpretations for the empty schema
,
and we will need to define its interpretation in an ad hoc way in Sect.
2.4.1
dedicated
to R-algebras for the operads.
A
∅
