Database Reference
In-Depth Information
Notice that the bottom square commutes (it exists only if the set of integrity con-
straints
Σ
A
is not empty) because the information flux of
f
Σ
A
is always empty
(independently if
α
satisfies the integrity-constraint mapping
T
AA
or
not
, from
Corollary
6
, Sect.
2.4.3
), i.e.,
f
Σ
A
=⊥
0
, and hence the composition with another
simple arrows will have empty flux as well.
2. For each object (a functor)
α
∗
in
Int(G(
DB
Sch
(G(
A
))
, its identity arrow
A
))
⊆
α
∗
α
∗
, that is, a transition
is a natural transformation,
id
α
∗
=
η
P
NOP
:
M
P
NOP
⇒
α
∗
.
3. The associativity of the compositions of the arrows (i.e., natural transformations)
in
Int(G(
α
∗
====
DB
Sch
(G(
A
))
is based on the property of composition of the ar-
rows, and the fact that for the given three arrows,
α
∗
===
A
⊆
))
M
P
k
⇒
M
P
n
⇒
α
1
,
α
1
===
α
2
M
P
m
◦
P
k
◦
P
n
⇒
α
3
.But
we have the associativity for the sequential composition of the programs, that is,
P
m
(
◦
P
k
◦
P
n
)
=
(P
m
◦
P
k
)
◦
P
n
=
P
m
◦
P
k
◦
P
n
. Consequently, the associativity
for the composition of the arrows in
Int(G(
A
))
⊆
M
P
m
⇒
α
3
, their composition is equal to the arrow
α
∗
======
and
α
2
===
DB
Sch
(G(
A
))
holds as well.
A system (a machine
M
R
) that supports
transaction processing
guarantees that
if the transaction executes some updates and then a failure occurs (for whatever
reason) before the transaction reaches its normal termination, then those updates will
be undone. Thus, the transaction either executes in its entirety or is totally canceled.
In this way, a sequence of operations that is fundamentally not atomic can be made
to look as if it really were atomic from an external point of view (and hence
M
R
encapsulates transactional atomicity).
The COMMIT and ROLLBACK operations are the key to the way it works:
•
The COMMIT operation signals successful end-of-transaction, if its execution
(of
P
1
)in
M
R
returns without errors to
M
U
. It tells to transaction manager that a
logical unit of work has been successfully completed, the database is in consistent
state again, and all updates made by that unit of work can now be “committed”
