Re: Another attempt...

Andrew Newman wrote:
> On 21/03/2008, Arjohn Kampman <arjohn.kampman@aduna-software.com> wrote:
>> Hi Andrew, others,
[...]
>>  U and 0 are both typed relations of some type T. U contains all possible
>>  tuples of this type, whereas 0 contains zero tuples of this type. Note
>>  that this means that these relations actually (can) have attributes. DEE
>>  and DUM are specific types of U and 0, namely the ones with zero
>>  attributes.
> 
> I think the term date uses is isomorphism - I'm sayings there is one
> between sets, bags, an untyped and typed and relational algebra (page
> 246).
> 
> The reason I think there is an isomorphism for U and 0 for DEE and DUM
> is taken from page 261:
> "As I showed earlier, the identity element with respect to
> intersection is the universal relation of the pertinent type.  But
> join is a generalized intersection; in particular, it doesn't require
> its operands to be of the same type, and indeed they usually aren't.
> As direct consequence of this fact, join has was might be called a
> general identity element namely, TABLE_DEE, which is the unique
> relation with no attributes and exactly one tuple (necessarily the
> empty tuple).  To elaborate: If A is a relation of type T and U is the
> corresponding universal relation, then it's certainly true that the
> join of A and U is equal to A.  But the join of A and TABLE_DEE is
> also equal to A, and this latter equality is guaranteed to hold no
> matter what the type of A happens to be.  This, we might resonably say
> that join (i.e., <AND>) has both (a) a specific identity element for
> each specific relation type and (b) a generic identity element,
> TABLE_DEE, that's independent of relation type"
> 
> Hopefully, you can see why I interpreted this to mean that DEE and U
> and the next paragraph likewise make DUM and 0 to be isomorphic.  Now
> he doesn't actually say that DEE UNION A = DEE but I don't think he
> needs to (given the previous definitions).

I read the same paragraph and already guessed that this is where your
interpretation came from. However, I still think that this
interpretation is incorrect. Isomorphisms exists between the univeral
and empty relations in various algebras, but I doubt that they exist
between DEE/DUM and U/0, respectively.

[...]
>>  As an empty graph pattern "{}" corresponds to DEE and a false graph
>>  pattern "{filter(false)}" corresponds to DUM, the SPARQL algebra seems
>>  to be in line with Date's definitions.
>>
> 
> I'd disagree - even if I'm wrong about the untyped relational
> identities - he's goes on further in chapter 12 to define the *same*
> identities for bags.

I'm sorry, but I can only find definitions for U and 0, I can't find any
equivalences of DEE and DUM. Can you point me to a specific page and/or
paragraph?

--
Arjohn

Received on Tuesday, 25 March 2008 15:21:03 UTC