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

Nicola wrote:
> On 2015-12-15 16:14:14 +0000, Jan Hidders said:
>
>> What *I* am interested in is the connection with this work:
>>
>> http://alpha.uhasselt.be/~lucp1080/queries_reals.pdf
>
> Do you mean, with the constraint database model (by Kanellakis et al.)
> in general or with the results in that specific paper?
>
> I think that constraint databases might be a nice complement to Norbert's
> topological model. They might provide an alternative way to represent the
> geometry of cw-complexes, for example. Then, you would encode both
> geometric and topological information in a purely relational way.

I is funny that you mention complexes.

Actually our (Patrick's and my) initial concept was (after a generalization of Jan's PLA structure) a data model for complexes:

To model an n-complex we collected i-cells in i-tables:

   Cells_n(idn,...)
   Cells_[n-1](idm,...)
...

   Cells_1(id1,...)
   Cells_0(id0,...)

Then we modelled the boundary operator by i-boundary tables (foreign keys from Bd_i to Cells_i and Cells_j are straightforward):

   Bd_n(idn, idm, alpha : integer)
   Bd_[n-1](idm, id[m-1], alpha : integer) ...

   Bd_1(id1, id0, alpha : integer)

with the restriction that for two consecutive boundary tables Bd_i, Bd_j with k + 2 = j + 1 = i the predicate

   not exists(

     select Bd_i.idi, Bd_j.idk
     from   Bd_i, Bd_j
     where  Bd_i.idj = Bd_j.idy
     group by Bd_i.idi, Bd_j.idk
     having sum(Bd_i.alpha * Bd_j.alpha) != 0

must be true. Note that the inner select without the "having" clause carries out matrix multiplication and hence is the composition of two consecutive (linear) boundary operators. The having clause only returns non-zero entries. As the boundary of the boundary must be the zero map, such non-ero entries must not occur in this composition.

We then realized that you can collect all cells into one table X and the boundary into one another table R and thereby get a point-set topological space (X, T) associated to the complex and modelled by (X,R). You can then store the dimension of the cells by a "dim" attribute for X and, eventually, if "dim" is a key attribute, by two dimension attributes, say, "dim" and "bddim", in R.

Conversely it is still possible to extend a topological data type (X,R) by an attribute R.alpha to get a complex. Then you can consider R an square matrix with index set X and multiply it to itself. When R^2 = 0 (a partial null matrix) the space is a complex. You can also add dimension attributes if you wish so.

Norbert

>
> Nicola
Received on Fri Dec 18 2015 - 20:00:18 CET