Database Reference
In-Depth Information
3
Definition of DB Category
3.1
Why Do We Need a New Base Database Category?
For the composition of
complex
database mapping graphs, it is important to dis-
tinguish the following two basic compositions of database schemas
A
and
B
with
respect to DBMSs:
•
In the composed schema, in order to make it impossible to write a query over the
composition with relations of both databases, the two database schemas should
be mutually
separated
by two independent DBMSs: it is common case when two
databases are separated, and this binary separation-composition at the schema
level is denoted by
(disjoint union where all elements are indexed by
schema symbols (denoted by a label
A
A
†
B
, for a schema
A
=
(S
A
,Σ
A
)
), equal to
{
A
}×
{
B
}×
the complex schema
(S
A
†
B
,Σ
A
†
B
)
where
S
A
†
B
=
∪
(
S
A
)
(
S
B
)
and
{
A
}×
{
B
}×
Σ
A
†
B
=
(
Σ
A
)
∪
(
Σ
B
)
, and
×
is the Cartesian product of sets (if
, then we set
A
=
1 and
B
=
B
=
A
2).
•
In the composed schema, when the two database schemas are
connected
into the
same DBMS (without any change of the two original database schemas): in this
case, we are able to use the queries over this composed schema with relations
of
both
databases for inter database mappings. This binary federation-composition,
denoted by
, is the (simple)
union
but with
renaming
(if we have the same
relational symbol
r
A
⊕
B
S
B
, then, before the union, we rename this symbol in
S
B
by a new fresh relational symbol of the same type), so that
S
∈
S
A
∩
S
A
rn
S
B
,
A
⊕
B
=
where
rn
denotes the union with renaming.
The equality '
=
' for database schemas is naturally defined as follows: for any two
schemas
A
,
B
∈S
,
A
=
B
if
π
i
(
A
)
=
π
i
(
B
),i
=
1
,
2. Let us consider the mappings
M
:
A
†
B
→
C
and
M
:
A
⊕
B
→
C
. In the first case, in any query mapping
q(
x
)
⇒
q
C
(
x
)
in the SOtgd of
M
, all relational symbols in the query
q(
x
)
must be
of the database
A
or (mutually exclusive) of the database
B
, that is,
M
=
∃
f
∀
x
1
q
A,
1
(
x
1
)
q
C,
1
(
t
1
)
∧···∧∀
x
k
q
A,k
(
x
k
)
⇒
q
C,k
(
t
k
)
∧∀
y
1
q
B,
1
(
y
1
)
⇒
