Database Reference
In-Depth Information
we will use the word “programs” for this process-algebra language, denoted by
GSOS
DB
and adequate to represent the operational semantics for the execution of
the database-mapping programs specified by a mapping graph
G
(or its “algebraic”
sketch
Sch
(G)
). Note that such processes are not specified by the users (i.e., pro-
grammers) as the original database-mapping programs specified by a graph
G
,but
are dynamically generated during the execution (forward or backward chaining) of
such programs.
Denotational model of a process-programming language, introduced in Sect.
7.1
and given by the abstract grammar (GSOS
DB
), is a
Σ
P
-algebra, where
Σ
P
=
nil, Act,
{
a.
_
}
a
∈
Act
,
is the signature of the language corresponding to the basic syntax constructs in
GSOS
DB
. The processes themselves form the initial such
Σ
P
-algebra and the cor-
responding homomorphism from this initial
Σ
P
-algebra (syntax algebra of
Σ
P
)to
the
denotational model
is called initial algebra semantics [
17
].
In what follows, we will define the denotational model
Σ
P
-algebra in our frame-
work of database mapping systems where the DB-denotational semantics is based
on the category
DB
.
In the database-mappings context, a
visible action a
∈
Act
(a nullary operator of
the signature
Σ
P
) of a database-mapping program (a graph)
G
is the
kernel
of views
that define the information-flux of a given schema mapping
M
AB
:
A
→
B
,fora
given
α
∗
∈
DB
Sch
(G)
. That is, based on point 2 of Definition
13
,
Int(G)
⊆
Δ
α, MakeOperads(
M
AB
)
a
=
∈
P
fin
(
Υ
)
⊂
Ob
DB
,
where
P
fin
is the finite powerset operator.
Consequently, the set of all possible visible actions is defined by
Act
=
P
fin
(
Υ
)
,
where
Υ
is the total object in the subcategory of only simple objects
DB
and each
visible action
a
∈
Act
is a finite set of views (i.e., relations), i.e., a simple object
in
DB
.
Note that the low-level programming “select-project-join
union” language
(SPJRU language [
1
], with the signature
Σ
R
, is encapsulated into this set of pos-
sible visible actions
Act
.
+
=
Proposition 33
(V
G
,E
G
)
,
where
V
G
is the set of vertices
(
nodes
)
in G and E
G
is the set of edges in G
,
the graph
obtained by inverting orientation only of inter-schema mappings
For any schema-database mapping program G
M
AB
:
A
→
B
by G
OP
.
There exists an LTS derived from G
(
or G
OP
)
and from a given initial mapping-
interpretation α
∗
∈
Int(G) of the sketch
Sch
(G)
,
which is a model of the root
database schema in this LTS
,
such that all states of this LTS are the models of
schema databases
(
vertices
)
in V
G
.
The set of visible atomic actions for the set of
all LTS of database-mapping systems is equal to Act
=
P
fin
(
Υ
)
,
with
Υ
introduced
in Definition
26
.
