Database Reference
In-Depth Information
where
τ
E
represents the embedding of the SQL statements from the host univer-
sal machine
M
U
into the categorial RDB machine
M
R
(by “forgetting” the parts
of the program which are not the embedded SQL statements), while
τ
T
maps the
embedded transitions in
M
R
into the RDBMS transactions.
Let, for a given source program
P
, the arrow in
U
P
=
→
st(j
2
)
, be the sequence of atomic transitions of the compiled host program
P
U
be-
tween two consecutive operations 'COMMIT'. That is, with
i
1
, st(j
1
−
U
M
(P )
,
f
:
st(j
1
)
1
)
,
i
2
,
COMMIT
st(j
1
−
1
)
∈→
P
COMMIT
st(j
1
−
1
)
,
with
st(j
1
)
=
(6.6)
and
i
3
, st(j
2
−
1
)
,
i
3
,
COMMIT
st(j
2
−
1
)
∈→
P
COMMIT
st(j
1
−
1
)
,
with
st(j
1
)
=
(6.7)
in the computation system of
M
U
. Then the states
st
(m)
F
P
st(j
1
)
=
τ
E
(P )
st(j
1
)
and
=
F
P
st(j
2
)
=
τ
E
(P )
st(j
2
)
st
(k)
=
are the two consecutive synchpoints in the RDB categorial machine
M
R
(because
t
2
(st
(m))
(t
3
(st
(m))
1 and
t
2
(st
(k))
(t
3
(st
(k))
1, with
st
(m)
∗
+
1
)
=
∗
+
1
)
=
=
st
(k
μy(t
2
(st
(k
(t
3
(st
(k
0
))
and hence,
from Definition
42
, the transition from
st
(m)
to
st
(k)
is an
atomic transaction
.
Hence, the vertical composition of natural transformations above, for this source
program
P
, is represented by the following diagram of functors in
Cat
on the left,
and with the result of these functors on the right:
−
1
−
−
1
−
y))
∗
−
1
−
y))
+
1
)
−
1
=
