Database Reference
In-Depth Information
1. The function
α(q
A,
1
)
:
α(
Local
)
×
α(
Emp
)
→
α(r
q
1
)
such that for any tuple
∈
∈
a
α(
Local
)
and
b
α(
Emp
)
,
α(q
A,
1
)
=
a,I
T
f
1
(a)
if
a
a
,
b
=
b
;
otherwise.
And for any
a,b
∈
α(r
q
1
)
,
α(v
1
)(
a,b
)
=
a,b
if
a,b
∈
α(
Office
)
;
otherwise.
2. The function
α(q
A,
2
)
:
×
→
α(
Emp
)
α(
Over65
)
α(r
q
2
)
such that for any tuple
a
∈
α(
Emp
)
and
b
∈
α(
Over65
)
α(q
A,
2
)
=
a
,
b
a
if
a
=
b
;
otherwise.
∈
=
∈
And for any
a
α(r
q
2
)
,
α(v
2
)(
a
)
a
if
a
α(
CanRetire
)
;
other-
wise.
From the fact that mapping-interpretation satisfies the schema mappings, based on
Corollary
4
, all functions
α(v
i
)
,for1
≤
i
≤
2, are injections.
2.4.2 Query-Answering Abstract Data-Object Types and Operads
We consider the views as a universal property for the databases: they are the possible
observations of the information contained in an instance-database, and we can use
them in order to establish an equivalence relation between databases.
In the theory of
algebraic specifications
, an Abstract Data Type (ADT) is speci-
fied by a set of operations (constructors) that determine how the values of the carrier
set are built up and by a set of formulae (in the simplest case, the equations) stat-
ing which values should be identified. In the standard initial algebra semantics, the
defining equations impose a congruence on the initial algebra. Dually, a
coagebraic
specification
of a class of systems, i.e., Abstract Object Types (AOT), is character-
ized by a set of operations (destructors) that specify what can be
observed
out of a
system-
state
(i.e., an element of the carrier) and how a state can be transformed to a
successor-state.
We start by introducing the class of coalgebras for database query-answering
systems for a given instance-database (a set of relations)
A
. They are presented
in an algebraic style by providing a co-signature. In particular, the sorts include one
single “hidden sort” corresponding to the carrier of the coalgebra and other “visible”
sorts, for the inputs and outputs, that have a given fixed interpretation. Visible sorts
will be interpreted as sets without any algebraic structure defined on them. For us,
the coalgebraic terms built over destructors are interpreted as the basic
observations
that one can make on the states of a coalgebra. Input sorts are considered as a set
L
A
of the finite unions of conjunctive finite-length queries
q(
x
)
for a given instance-
database
A
, as specified in Sect.
1.4.1
.
Based on the theory of database observations and its power-view operator
T
,
defined in Sect.
1.4.1
, the output sort of this database AOT is the set
TA
of all
resulting views (i.e., resulting
n
-ary relations) obtained by computation of queries
q(
x
)
∈
L
A
. It is considered as the carrier of a coalgebra as well.
