Database Reference
In-Depth Information
The most interesting concrete syntax is an interpretation of the abstract
T
P
0
, any unary operation '
a
i
.
_' by the unary
⊥
syntax, where '
nil
' is interpreted by
database operation
a
i
⊗
_ and the binary operation '_
_' by the
DB
coproduct _
+
_
n
by the corresponding
DB
n
-ary coproducts,
(and other finitary
n
-ary operations
n
). Thus:
denoted by
+
Definition 53
T
P
syntax is defined by
the following algebra isomorphism (which is identity for elements in
Act
):
T
:
T
P
X, Act, nil,
The concrete interpretation of the abstract
}
a
i
∈
Act
,
n
n
≥
2
→
T
P
X, Act,
{
a
i
.
_
}
a
i
∈
Act
,
+
n
n
≥
2
0
,
⊥
{
a
i
⊗
_
between the abstract algebra and its concrete DB-interpretation.
This concrete DB-syntax will be denoted by
T
P
, so that for any
X
there is a bi-
jection between the set of abstract and these concrete terms,
→
T
P
(X)
,
T
:
T
P
(X)
0
,
such that
T
(nil)
=⊥
T
(a
i
.t)
=
a
i
⊗
(
T
(t))
and
T
(t
1
t
2
)
=
(
T
(t
1
))
+
n
',
n
(
T
(t
2
))
(and analogously for all derived finitary operators '
3).
In the case of the database-mapping DB-denotational semantics where
Z
≥
⊆
Ob
DB
, given by Definition
52
, with this concrete interpretation of the operators
in Definition
53
, the mapping
h
Z
represents the composition of objects
in
DB
as ground terms of the grammar
(
G
r
DB
)
in Definition
52
. In this case, each
path
a
1
⊗
a
2
⊗···⊗
a
n
of a given process
p
∈
X
(which is an tree) is just the inter-
section of all information fluxes of the corresponding composition of arrows in
DB
,
f
n
◦···◦
:
Σ
P
Z
→
B
, where
A
is the root state (a database) of this process and
B
is its final state (a database): thus, the
meaning of each path of such a process
is
represented by the information transferred from the initial state
A
into the final state
B
of this process-path.
Note that, when the
Σ
P
-algebra
h
f
2
◦
f
1
:
A
→
:
→
Σ
P
(Z)
Z
is the
denotational semantics
for the initial (free) syntax algebra
Ob
DB
as well),
then
f
:
X
→
Z
is an assignment function which assigns to process variables in
X
the database-instances in
Ob
DB
, and
f
#
:
T
P
X
T
P
X
(when
Z
⊆
Ob
DB
(
Act
⊂
Z
is its unique homomorphic
extension to all terms with variables, called
initial algebra semantics
[
17
].
In what follows, we will investigate which assignment
f
→
Z
will be
ade-
quate
in order to infer the operational behavior of database-mapping programs from
the DB-denotational semantics.
:
X
→
7.3.2 Database-Mapping Processes and DB-Denotational
Semantics
In order to define the database-mapping processes, for a given database-mapping
program specified by a graph
G
, when the extension of a particular database schema
is changed (by Insertion, Deletion or Updating (considered as deleting of old up-
dated tuples and then by Insertion of new updated tuples)), we will consider the
