Database Reference
In-Depth Information
, while
α
2
(
with
α
2
(r)
={
(a,b),(b,ω
1
)
}
B
)
={⊥
,α
2
(s)
}
with
α
2
(s)
={
(b,ω
0
)
}
.
0
,
ass(p
1
)
α
1
(
Hence,
ass(p
0
)
=⊥
=
A
)
={⊥
,α
1
(r)
}
with
α
1
(r)
={
(a,b)
}
,
α
2
(
α
3
(
ass(p
2
)
)
de-
fined above. It is easy to see that the final extensions of the relations
r
and
s
are
equal to those in finite canonical solution
can
F
(
I
,D)
of a global schema dis-
cussed in Sect.
4.2.4
, which was, for the query-answering purposes, equivalent
to the infinite canonical model of the global schema.
Notice that the meaning of this path-process
a
1
.a
2
where
=
B
)
={⊥
,α
2
(s)
}
with
α
2
(s)
={
(b,ω
0
)
}
, and
ass(p
3
)
=
A
a
2
=
{
⊥
∈
a
1
=
b
}
,
Act,
described after Definition
53
,is
a
1
⊗
a
2
=
Ta
1
∩
Ta
2
=
{
}∪⊥
, which is equal
to the meaning of the path-process for the Strong Data Integration semantics.
T
b
7.4
Operational Semantics for Database-Mapping Programs
In computer science, coalgebra has emerged as a convenient and suitably general
way of specifying the reactive behavior of systems, including classes in object-
oriented programming. While algebraic specification deals with functional behavior,
typically using inductive datatypes generated by constructors, coalgebraic specifi-
cation is concerned with reactive behavior modeled by coinductive process types
that are observable by selectors, much in the spirit of automata theory. An important
role is played here by final coalgebras, which are complete sets of possibly infinite
behaviors.
As we have seen, the
syntax
of a programming language with a signature
Σ
P
is formally defined by
initial Σ
P
-algebras
(and its denotational model) which in-
ductively defines the
syntax-monad
T
P
, as we have specified in Sect.
7.3.1
.The
semantics
of a programming language is dual to its
syntax
,sowewillseeinthis
section that it is based on the coinduction and
final coalgebras
and the
observation-
comonad
D
P
.
The operational semantics of a language defines
how
programs are to be executed
and what their observable effect is. Let us consider any
DB-mapping system
given by
a graph
G
=
(V
G
,E
G
)
in Definition
54
, composed of a number of database schemas
A
∈
, i.e., the edges
in
E
G
), whose current extension is determined by a mapping-interpretation
α
0
∈
Int(G)
V
G
(and inter-schema mappings between them
M
AB
:
A
−→
B
DB
Sch
(G)
.
The operational semantics is based on the process which begins with an up-
dating of the extension (instance-database) of a given database schema
⊆
A
∈
V
G
,
from the previous extension
α
0
(
α
∗
(
A
)
into this updated extension
A
=
A
)
, where
α
∗
∈
DB
Sch
(G)
, such that for any other database schema
B
∈
V
G
different from
A
,
α
∗
(
α
0
(
)
,but
α
∗
/
Mod(
Sch
(G))
(i.e., it does not satisfy all inter-schema
mappings in
E
G
). This process
P
begins by updating of the extension of the
database
B
)
=
B
∈
A
(that is the root of
P
) propagates in the graph
G
by creation of an
