Database Reference
In-Depth Information
EXTEND
...
EXTEND
(r
WHERE
C)
ADD
att
r
(
1
), name
1
AS
e
1
...
ADD
att
r
(n), name
n
AS
e
n
[
S
]
,
such that for each 1
≤
m
≤
n
,if
m/
∈{
i
1
,...,i
k
}
then
e
m
=
nr
r
(m)
, and
S
=
.
Consequently, all update operators of the relational algebra can be obtained by ad-
dition of these 'EXTEND _ ADD
a,name
AS
e
' operations.
name
1
,...,name
n
Definition 31
We denote the algebra of the set of operations, introduced previ-
ously in this section (points from 1 to 6 and EXTEND _ ADD
a,name
AS
e
) with
additional nullary operator (empty-relation constant)
,by
Σ
RE
. Its subalgebra
without _ MINUS _ operator is denoted by
Σ
R
, and without
⊥
and unary operators
EXTEND _ ADD
a,name
AS
e
is denoted by
Σ
R
(it is the “select-project-join-
rename+union” (SPJRU) subalgebra). We define the set of terms
⊥
T
P
X
with vari-
T
P
X
of
Σ
R
-algebra),
ables in
X
of this
Σ
R
-algebra (and analogously for the terms
inductively as follows:
1. Each relational symbol (a variable)
r
∈
X
⊆ R
and a constant (i.e., a nullary
T
P
X
;
2. Given any term
t
R
∈
T
P
X
and an unary operation
o
i
∈
operation) is a term in
Σ
R
,
o
i
(t
R
)
∈
T
P
X
;
3. Given any two terms
t
R
,t
R
∈
T
P
X
and a binary operation
o
i
∈
Σ
R
,
o
i
(t
R
,t
R
)
∈
T
P
X
.
We define the evaluation of terms in
T
P
X
,for
X
=R
, by extending the assignment
(i.e., R-algebra)
Υ
which assigns a relation to each relational symbol (a
variable) to all terms by the function
_
:R→
), where
Υ
is the universal database instance (set of all relations in Definition
26
). For a
given term
t
R
with relational symbols
r
1
,...,r
k
∈R
_
#
:
T
P
R→
Υ
(with
r
#
=
r
t
R
#
is the relational table
obtained from this expression for the given set of relations
,
r
1
,...,
r
k
∈
Υ
, with
t
R
UNION
t
R
#
=
t
R
#
if the relations
the constraint that
t
R
#
∪
t
R
#
and
t
R
#
are union compatible;
⊥
otherwise.
.
We say that two terms
t
R
,t
R
∈
T
P
X
are equivalent (or equal), denoted by
t
R
≈
Each R-algebra
α
:
X
→
Υ
is a restriction of an assignment
_
to
X
⊆R
t
R
, if for all assignments
t
R
#
.
t
R
#
=
Let us consider an example for terms of the
Σ
RE
algebra of this Definition
31
:
Example 28
Consider the term
t
R
of the
Σ
RE
-algebra, equal to the algebraic ex-
pression
r
1
[
S
1
]
WHERE
C
1
UNION
(r
2
WHERE
C
2
)
S
2
]
WHERE
C
5
[
S
5
]
[
MINUS
EXTEND
r
3
[
S
3
]
UNION
(r
4
WHERE
C
4
)
S
4
]
[
ADD
a,name
AS
e
[
S
6
]
,
