Database Reference
In-Depth Information
Notice that in the transition system of
M
U
of the host program (Program 1) we
consider the case when
M
U
receives the answer from
M
R
just in time and without
error.
The significant transitions in the computation system of
M
U
, for the synchro-
nization of
M
U
, are denoted by the symbol
in Fig.
6.3
, while the two horizontal
red lines are the time-synchronization instances.
6.2.1 The Categorial Approach to SQL Embedding
The synchrony conversations, for a given source host program
P
, between
M
U
and
M
R
, described by the transition diagram in Fig.
6.3
, represent the execution of the
embedding (of the compiled SQL statements). The states of the machines are pro-
gressively enumerated from their initial states
st(
0
)
S
U
, st
(
0
)
∈
∈
S
R
, w.r.t. a de-
terministic transition system
TrS
→
P
)
(in Definition
37
). We can abstract
this deterministic transition system by forgetting the instruction addresses, that is,
by substituting the transitions
((j, st(i)),(k,f (st(i))))
=
(S
P
,
∈→
P
by the simple state-
transitions
(st(i),f (st(i)))
∈→
P
.
Let
ATrS
U
=
(π
2
(S
PU
),
→
P
)
be an abstracted deterministic transition sys-
tem obtained from an execution of the program
P
on a host machine
M
U
, with
π
2
(S
PU
)
S
U
the set of its internal states progressively enu-
merated from 0 (initial state) to the last
N
th state (end of this program) and
ATrS
R
=
={
st(i)
|
0
≤
i
≤
N
}⊆
→
P
)
be the deterministic abstracted transitions system ob-
tained from the execution of the program
P
on an RDB machine
M
R
, represented
in Fig.
6.3
, Sect.
6.2
. The most significant parts of the
TrS
(in what follows, we will
use this simplified abstracted transition system) of the
M
R
, underlined in Fig.
6.3
by delimited intervals from diagram on the left (signed by
(π
2
(S
PR
),
) and right sides) are
the following:
1.
Variable binding.
It transforms an Application Plan into an arrow in
RA
.
The transition
st
(k)
λIn
∗
ER
(Out
U
(st(j)))
st
(k
ATrS
R
that corresponds to
the binding of bounded variables in the variable-binding operators of the called
Application Plan.
The Bind module during the compilation of the original source host program
P
generates the Application Plans for each SQL statement in
P
, which are the
“variable-binding”
RA
arrows: it means that instead of the concrete operators of
Σ
RA
algebra, the constants (i.e., values) used for such operators are here substi-
tuted by the bounded variables. Only when the
M
U
calls an Application Plan,
by passing the triple
(n,
0
,L)
with the list
L
of values for this
n
th plan, then
the bounded variables of these value-binding operators are substituted (variable
binding) with the values in
L
, and hence these operators become the operators of
Σ
RA
algebra, and the Application Plan becomes exactly an arrow in
RA
.
2.
Execution.
By an application of the evaluation functor
Eval
+
1
)
∈
:
RA
→
DB
.
f
n
st
(m)
The transition
st
(k
+
1
)
∈
ATrS
R
,
m
≥
k
+
3
,n
≥
1, which cor-
responds to the composition of transactions
f
n
=
RETURN
◦
Out
R
◦
Ap
n
, where
DEC
)
n
, and
Ap
n
=
P
n
◦
(t
4
◦
