Database Reference
In-Depth Information
where
τ
E
represents the embedding of the SQL statements from the host universal
concurrent machine
M
UC
into the concurrent categorial RDB machine
M
RC
(by
“forgetting” the parts of the program which are not the embedded SQL statements),
while
τ
T
maps the embedded transitions in
M
RC
into the RDBMS transactions.
The vertical composition of natural transformations above, for a given concur-
rent program
P
1, is represented as in non-concurrent case
in Sect.
6.2.2
by the following diagram of functors in
Cat
on the left and with the
result of these functors on the right:
=
P(
1
)
|···|
P(N),N
≥
where
P
i
n
◦···◦
P
i
1
is the sequential composition of
RA
-arrows (contained in
one atomic transaction) from
st
(m)
to
st
(k)
(as specified by point 2 of Theo-
M
P
in
◦···◦
P
i
1
⇒
α
1
)
=
rem
11
) and
α
∗
=
T
P
(st
(m))
and
α
1
=
T
P
(st
(k))
, and hence
(α
∗
======
T
P
(F
P
(f ))
.
The arrow
F
P
(f )
in
R
P
is an
atomic transaction
, while the bottom arrow
g
:
A
→
A
1
is a morphism in the
DB
category from the instance-database
A
=
Out
DB
(st
(m))
into the updated database
A
1
=
Out
DB
(st
(m))
after this atomic
transaction.
This morphism
g
=
g
n
◦···◦
g
1
, where each
g
j
,1
≤
j
≤
n
, is a morphism ob-
tained by Corollary
17
for a given schema mapping
M
i
={
Φ
i
}:
A
→
A
and ob-
tained by Proposition
26
for a given
RA
-algebra arrow
P
i
j
(in Sect.
5.3
).
As a result, we obtain that each atomic transaction produce one morphism in
DB
category, between the (consistent) database instances (that are models) of a schema
A
, before and after such an atomic transaction:
Corollary 22
For a given database schema
A
and a concurrent program
P
=
P(
1
)
|···|
P(n), n
≥
1
,
