Re: Extending my question. Was: The relational model and relational algebra - why did SQL become the industry standard?

"Paul Vernon" <paul.vernon_at_ukk.ibmm.comm> wrote in message news:<b3de03$nfa$2_at_sp15at20.hursley.ibm.com>...
> > As long as the polar representation has a
> > representable value in that area, I see no problems. Even if the polar
> > representation has no value in that area, but a point near that area I'm
> > still okay with it.
>
> But what happens if many polar points are 'near' that area?. If you can define
> a way of having exactly one polar point 'near' every Cartesian point (and vis
> versa), then ok.

When we have domains of "rationals" or "reals" in a database, really we're just talking about domains of integers with the scale shifted. So for theoretical purposes we can ignore any basic number domain except integers.

So the domain that is the subset of the plane of (cartesian) points (n,m) where n,m are integers between some set of values is surely a *different* domain than that of (polar) points (r,t) where r,t are integers. Here I'm assuming we can specify the angle t to an arbitrary scale. In other words they aren't just different representations of the same domain (unless you have a logical model that can deal with uncountable infinities).

If the function f maps one represention of a domain to another, then surely we want f(a*b) = f(a)*f(b) for any operation *? Can anyone give an example of such a mapping between the polar and cartesian domains defined above that works for standard addition and multiplication?

For example the polar domain isn't closed under conventional addition. Though neither I suppose is the cartesian domain if you consider that standard (non modulo) addition is undefined for large enough values.

So the question is, should the logical model be allowed to include things (like infinities) that can't be realised in the physical implementation? Do we need two logical models: one that is purer and corresponds closer to "reality" and another slightly modified version that corresponds to the limitations of finite machines?

For example to our set of integers between -1000 and +1000 we could add two more elements +INF and -INF say. Although then things like "a + b - b = a" wouldn't necessarily be true but at least we'd have closed operations. Are there any other ways to neatly deal with finite subsets of integers apart from using modulo arithmetic?

Maybe we need an implementation of the integer domain in DBMSs that will cover ALL integers i.e. given enough machine space any integer can be stored. So this would deal with the countably infinite in some logical sense, but we'd still have problems with the polar-cartesian mapping (which requires uncountable infinities).

Paul. Received on Wed Feb 26 2003 - 15:14:35 CET