Database Reference
In-Depth Information
3.
f
R
C
j
,t
j
:
R
C
j
−
1
→
R
C
j
, where
t
j
=
π
j
(f (
x
))
,1
≤
j
≤
n
;
4.
π
]
:
R
C
n
→
R
q
is a projection function;
[
N
+
1
,...,N
+
n
5.
in
:
R
q
→
R
is a inclusion (injective function).
Consequently, for any R-operad
q
i
∈
O(r
1
,...,r
k
,r)
, the function
f
=
α(q
i
)
:
α(r
1
)
×···×
α(r
k
)
→
α(r)
is composed of the operations of the
Σ
α
algebra. Thus, this signature
Σ
α
represents
all compositions of R-operads.
From this Corollary and from Theorem
9
, we conclude that every R-algebra op-
erad has an equivalent
Σ
RE
algebra term and hence
Σ
α
relational algebra is compu-
tationally equivalent to the
Σ
RE
(full extension of Codd's algebra). The significant
difference is that the
Σ
α
-algebra terms are the functions
between
the relations, and
can be used for a composition of the arrows (morphisms) in the
DB
category, while
the evaluation of the
Σ
RE
-algebra terms are only the single relations. Consequently,
the
Σ
α
relational algebra (of R-operads) is more appropriate for a specification of
the database mappings.
From the fact that the
DB
category is computationally equivalent to the full exten-
sion of Codd's (SPRJU) algebra, exactly to
Σ
RE
algebra with all update statements
for relational tables (as in SQL), based on this equivalence, in the next chapter we
will develop a kind of the categorial RDB machines able to support not only all
standard computations for the embedded SQL updates of RDBS but also all compu-
tations for the Big Data Integration systems based on the SOtgds database mappings.
5.5
Review Questions
1. In his SPJRU relational algebra, Codd restricted this algebra by eliminating the
negation operator. In our case, the universe
SK
is the union of a finite
(but large enough) set of values of a domain
dom
and an infinite enumerable
set of Skolem constants (i.e., marked null values), so that each choice of
dom
defines a different version of the
DB
category. Consequently, the negation ap-
plied to a query formula may return with an infinite set of possible tuples. In our
framework, where we deal with the infinite database with the relations that can
have an infinite number of tuples, can we accept the negation operation as well?
Have we already used the negation for the relations and where?
2. Prepare an example and explain how we are able to use the unary relational alge-
bra operators 'EXTEND _ ADD
a,name
AS
e
' denoted shortly as _
U
=
dom
∪
a,name,e
in order to represent the existentially quantified Skolem functions in the mapping
SOtgds.
3. In order to be able do define any relation, or to extract any information (a view)
from a given database instance by using only the
finite
length expressions of rela-
tional algebras, it is important to have only the finitary relations in our theoretical
