Re: Notions of Type

Bob Badour wrote:
Marshall wrote:

paul c wrote:

Marshall wrote:

erk wrote:

Sorry if this is obvious to everyone else, but does an algebra include
only operations defined on values of the type in question?

Yes.


I ask
because in relational algebra, at least the rename operator involves a
different type ("attribute name") than the "core type" (relation).

Very true. Of the various relational operators that have been
identified over the years, only a few, like union, are really
algebraic. RESTRICT, PROJECT, etc. aren't. ...

Just a quick question - is your reason for making that statement that
restrict, project, etc. result in a relation that doesn't necessarily
have all the attributes of the operand relations?


No; that's actually not a problem.

The problem is the operands. Strictly speaking, an algebraic operator
would be one which had the type

op: 'a, 'a -> 'a

("The thing named "op" has the type: function taking operands of
type some-a and some-a and returning a value of type some-a.")

So the algebra of the integers has operators like:

+: int, int -> int

Consider PROJECT:

PROJECT: Relation, Set-of-attributes -> Relation

So for PROJECT of an x,y point over x, we pass it two
things:

1) A relation defined over attributes x and y
2) ???
and it returns
3) A relation defined over attribute x (aka "profit")

Whoops! Doesn't fit the template. The second argument isn't
a relation. So, strictly speaking, this is not an algebraic operator,
because it isn't closed over the type Relation. Exercise for the
reader: what *is* the type of the other argument? This should
make your head hurt a least a little bit.

It is a relation of degree 1 and cardinality N representing a set of N
attribute names.

Must it be a relation, given that its tuples aren't used at all? And if
one uses a relation, isn't the degree N and the cardinality 0? (I'm
basing this on TABLE_DEE (true) being degree zero, cardinality one and
TABLE_DUM (false) being degree zero, cardinality zero).

.

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