Re: Proposal: 6NF

From: Bob Badour <bbadour_at_pei.sympatico.ca>
Date: Sat, 14 Oct 2006 13:14:49 GMT
Message-ID: <dp5Yg.5239$cz.75184@ursa-nb00s0.nbnet.nb.ca>

Jan Hidders wrote:

> vc wrote:
>

>>Jan Hidders wrote:
>>
>>>vc wrote:
>>>
>>>>Jan Hidders wrote:
>>>>
>>>>>vc wrote:
>>>>>
>>>>>>More interestingly,  why bother with a trivial fact that Z may be
>>>>>>[considered] a subset of R in whatever sense ?
>>>>>
>>>>>Because you want a simple formalization of subtyping a la Cardelli and
>>>>>Wegner where subtyping coincides with being a subset. It means that the
>>>>>meanig of the + operation in (a + b) can be considered independent of
>>>>>the type of a and b.
>>>>
>>>>RIght,  but my understanding of the OOP stuff leads me to believe that
>>>>a subtype should '"inherit" its type operations.  If so,  assuming R is
>>>>our base type,  what do we do about division in Z (regarded as an R
>>>>subtype)  ;) ?
>>>
>>>Call it integer division to distinguish it from the real division? :-)
>>
>>Whatever you define  as "integer division"  cannot be claimed to be the
>>division as understood in elementary arithmetic, as well as in abstract
>>algebra, and therefore to be "inheritable" from R-division as I imagine
>>it should be in the OOP sense of the word.  In simpler terms,  Z ain't
>>a field while R is.

>
>
> Ah, right, I see your problem. Your reasoning is: (1) there is a
> division for R and (2) Z is a subtype of R so (3) there must be some
> inherited division for Z, but (4) Z is not closed under division. What
> gives?
>
> Well, it's (3) that you should take a closer look at. Is that really
> true? The answer is: no, if you look at the algebras that define the
> domains it is not necessarily true that the algebra of the subtype has
> more operations.
>
> Why is that so? In order to allow Z to be a subtype of R in a database
> universe we only need to be able to injectively map the elements of Z
> to elements of R such that it is a homomorphism for presumed identical
> operations. The latter means for example that if f : Z -> R is the
> mapping and +_R the real addition and +_Z the integer addtion then it
> should hold that f(a) +_R f(b) = f(a +Z b). Why is that sufficient? If
> this holds then we can define a two-sorted algebra that combines the
> two algebras and their operations such that Z is subset of R. You see
> that the division for R can then also be used for Z since it has type
> RxR->R and Z is a subset of R. So, in that sense, it is then
> "inherited" by Z.
>
> So, summarizing, you see that the requirement is not that the algebra
> for the subtype should contain all the operations of the the algebra of
> the supertype.

I don't see that at all, Jan. The Integer inherits the division operation that returns a Real. It's just not part of the integer algebra. The integer type does indeed introduce additional operations, which is why many languages will distinguish between / and div for instance. Received on Sat Oct 14 2006 - 08:14:49 CDT