Re: Comments on Norbert's topological extension of relational algebra

Op zondag 20 december 2015 10:27:17 UTC+1 schreef vadi..._at_gmail.com:
> On Tuesday, December 15, 2015 at 8:14:17 AM UTC-8, Jan Hidders wrote:
> > What *I* am interested in is the connection with this work:
> >
> > http://alpha.uhasselt.be/~lucp1080/queries_reals.pdf
>
> Do you prefer to work with algebraic or semi-algebraic constraints?
>
> Here is how Heath's theorem from database dependency theory is made obvious in algebraic settings. Basically we have:
> 1. A system of constraints in 3 variables x,y,z
> 2. A function x->y
> We need to reorganize the system's constraints into two parts: the ones expressed in variables x and y only, on one hand, and the other in x and z. For the first system we take the equation that defines the active domain x together with the equation that explicitly defines FD x->y. For the second system, we take the whole original system, and eliminate y by substitution its formula in terms of x.
> https://vadimtropashko.wordpress.com/2014/01/03/analytic-view-of-functional-dependency/
>
> I struggle to prove Heath's theorem for semialgebraic sets. For example, is it possible to have functional dependency which is a function but not expressible polynomially?

In principle yes, but that was already the case in the algebraic setting where it is also not a priori the case that all functional dependencies are expressible by a polynomial, unless that is how you define them, as you apparently do. You are anyway looking here at restricted classes of relations and dependencies, so it's up to you to say how you want to restrict them.

• Jan Hidders