Re: Proposal: 6NF

"J M Davitt" <jdavitt_at_aeneas.net> wrote in message news:OLnUg.7759$OE1.6886_at_tornado.ohiordc.rr.com...

> Brian Selzer wrote:

>
> So, let me get this straight: we round up all the NULLs,
> pack 'em together in Guantanamo Bay, and the world will
> be a better place?
>

The moon might be better than Camp Delta--or maybe Bikini island. The problem is that not all NULLs want to kill us and are willing to die in the process.

>>
>>> In addition, when used correctly, no subjectivity is
>>>
>>>>introduced. For every database schema with NULLable attributes, there
>>>>is always an equivalent schema without.
>>>
>>>Well, being the god* you seem to be, I won't doubt
>>>that solely because you said so.
>>>
>>>* Genius of Design
>>>
>>
>>
>> I'm not sure I understand your point.

>
> How are two schemas - one "with NULLable attributes" and
> one "without" - equivalent?
>

Two database schemata are equivalent if all of the information contained in an instance of one can be derived from the information contained in an instance of the other. Database schemata A and B are equivalent if and only if a pair of inverse mappings exist between A and B: one mapping from A to B such that for every legal instance of A, there exists one and only one legal instance of B, and one mapping from B to A such that for every legal instance of B, there exists one and only one legal instance of A. In other words, given any instance of A, only one instance of B can be derived using the mapping from A to B such that when the mapping from B to A is applied to that derived instance, the result is the original instance for A, and given any instance of B, only one instance of A can be derived using the mapping from B to A such that when the mapping from A to B is applied to that derived instance, the result is the original instance for B. Note that this applies to database schemata with and without nullable attributes. For example, a database schema D consisting of a single relation,

R {A, B, C}

that satisfies the functional dependency A --> B is not equivalent to a database schema D' consisting of two relations,

S {A, B}, T {A, C}

without the circular inclusion dependency S[A] <--> T[A] because for any given instance of D, there can be a huge number (infinite if any domain for A, B or C is infinite) of instances of D' where the natural join of the extensions of S and T is equal to the extension of R. The circular inclusion dependency ensures that the projections of S JOIN T over {A, B} and {A, C} are equal to the extensions of S and T respectively.

I hope that the above description and example illustrate what I mean by equivalent database schemata. For a database schema without nullable attributes, the mappings involve sets of projections and natural joins. For a database schema with nullable attributes, the operations are different. A simple projection would still include nullable attributes. To eliminate them, the decomposition operation must also restrict the projection to eliminate any tuple with a null. As a consequence, the inverse operation must be an outer join which restores the eliminated tuples. In addition, instead of a circular inclusion dependency, all that is needed is a single foreign key constraint. There is no need to maintain a 1:1 correspondence between tuples from each projection because since the original attribute is nullable, the 1:1 correspondence didn't exist to begin with.

>   Are you asking for proof that a

>
> What does SQL's OUTER JOIN have to do with what you described?
>
> How does a description of COMPOSE make your point?
>