Database Reference
In-Depth Information
Remark
In the rest of this topic, we will use
(S
A
,Σ
A
)
also for the set of rela-
tional symbols in
S
A
and an instance-database
A
for
α
∗
(
A
=
α
∗
(S
A
)
: it will simply
be called a “database” when it is clear from the context. The functions
∂
0
and
∂
1
are
different from
dom
and
cod
functions used for the category arrows. For an
atomic
morphism or, more generally, for a
complete
morphism
f
A
)
=
B
,the
∂
0
(f )
spec-
ifies exactly the subset of relations in a database
A
used by
f
, while
∂
1
(f )
defines
the target relations in a database
B
for this mapping. Thus, for a complete mapping
∂
0
(f )
⊆
:
A
→
dom(f )
=
A
and
∂
1
(f )
⊆
cod(f )
=
B
. In the case when
f
is a simple
view-mapping,
∂
1
(f )
is a singleton.
The fact that each atomic arrow
f
:
A
→
B
is a set of functions and has the
information flux
f
B
T
(f )
is important for the composition of the arrows in
DB
:
any two simple arrows
f,g
:
A
→
B
are equal (
f
≡
g
) not only when the two sets of
functions
f
and
g
are equal sets but also when they are different, but the information
fluxes
f
and
=
g
are equal. More about this for complex arrows will be presented in
Definition
23
.
From the proof of Theorem
1
, for the composition of any two morphisms in
DB
,
α
∗
(
M
AB
)
α
∗
(
M
BC
)
f
=
:
A
→
B
and
g
=
:
B
→
C
,
α
∗
(
M
BC
)
α
∗
(
M
AB
)
α
∗
(
M
BC
◦
=
◦
=
◦
=
h
g
f
M
AB
)
=
α
q
i
α
q
i
1
×···×
α
q
i
k
i
|
≤
j
≤
k
i
1
≤
i
≤
n,
and 1
≤
i
j
≤
m
for 1
∪
α(
1
r
∅
)
=
q
B
i
·
k
i
∪{
(q
Ai
1
,...,q
Ai
k
i
)
|
1
≤
i
≤
n,
and 1
≤
i
j
≤
m
for 1
≤
j
≤
q
⊥
}
,
where
q
B
i
denotes the function
α(q
i
)
,for1
≤
i
≤
n
,
q
A
j
denotes the function
α(q
j
)
,for1
≤
≤
·
denotes the composition of functions. Graphically, such
a composition of morphisms can be represented as a composition of trees (see the
examples bellow where the 'graphical-tree' morphisms are denoted by
f
T
,g
T
,...
).
Generally, a composed morphism
h
j
m
, and
C
is not a complete morphism, that
is, it can be graphically represented by a general tree such that not all its leaves
are in
A
. Such an “
incomplete
” morphism is called a (partial)
p
-arrow. A
p
-arrow
corresponds to a morphism obtained from a schema mapping
:
A
→
M
AB
={
Φ
}:
A
→
B
where
Φ
is an SOtgd such that at least one left-hand side of its implications has
the characteristic-functional symbols (for the relations that are not in the source
schema
).
By the graphical tree-composition of two trees in Fig.
3.1
,
f
T
(incomplete) and
g
T
(complete), we obtain the tree
h
T
(of the
p
-arrow
h
=
g
◦
f
:
A
−→
C
):
h
T
=
A
(g
◦
f)
T
q
B
j
·
q
A
i
(tree)
=
α
∗
(
M
BC
)
α
∗
(
M
AB
)
&
∂
1
(q
A
i
)
q
B
j
∈
q
A
i
∈
∈
∂
0
(q
B
j
))
=
q
B
j
·
q
A
i
(tree)
|
∂
1
(q
A
i
)
∈
∂
0
(q
B
j
)
|
q
B
j
∈
α
∗
(
M
BC
)
