Re: Why are [Database] Mathematicians Crippled ?

On Tuesday, February 3, 2015 at 10:10:20 AM UTC-8, Norbert_Paul wrote:
> Tegiri Nenashi wrote:
> > On Monday, February 2, 2015 at 9:45:02 AM UTC-8, Norbert_Paul wrote:
> >>> .... How do you represent polynomials in database?
> > My question is how useful this is;
>
> No. Your question was as cited above.
>
> > ..., but how about "give me all systems which roots are zero dimensional
> > varieties"?
>
> If this requires to find the root, you could stumble into undamental problems:
> en.wikipedia.org/wiki/Galois_theory#Solvable_groups_and_solution_by_radicals
> But maybe there is a way do decide "p has roots of dimension zero" without
> actually solving the polynomial p.

Depends on the field.

Berlekamp's algorithm works in univariate case.

Groubner basis is used in multivariate case.

However, this is starting to get off the topic (if databases are used in science). Sure they are, take for example relatively recent discovery of Higgs signal in a huge pile of CERN data (ref Dr. Maaike Limper). What I'm trying to convey here is the lack of impression with the tools that database technology can offer to a scientist in the field.

> > Yes, transitive closure is required. It can also go the opposite way: given
> > adjacency matrix of a graph, if we know it's inverse, then the adjacency
> > matrix of transitive closure is the following matrix inverse
> >
> > M^* = 1 + M + M^2 +... = (1-M)^(-1)
>
> Is this supposed to be a proof that transotove closure is required?
> Then I don't get it.

You said that, if we know how to calculate determinant, then we can compute matrix inverse. The two expressions for transitive closure M^* are there to demonstrate the opposite, that if we know the determinant (and, therefore matrix inverse), then we can compute transitive closure. But, again, I doubt there is a single scientist in the field who is seriously considering matrix operations done via SQL queries.  

> > However, I don't consider transitive closure as a part of relational algebra.
>
> This is more a personal opinion you have.
> I prefer to distinguish between "relational algebra with transitive closure"
> and "relational algebra without transitive closure".
>
> See Page 527 (or Page 5)
> 14.3 LIMITATIONS OF RELATIONAL ALGEBRA
> in
> http://web.cecs.pdx.edu/~maier/TheoryBook/MAIER/C14.pdf .
>
> > It is something that belongs to algebra of binary relations, which as been
> > married against its will. She is unhappy in her marriage because she is
> > allowed to make love only on second day of each month (i.e. transitive
> > closure operator is not total in relational algebra).
>
> How sad! Poor binary! But why are operaitons on binary relations not part of
> relational algebra?
>
> given
> R(a,b) and S(b,c) .
>
> Then
>
> R NATJOIN S
> PROJECT[a](R)
> SELECT[b=7](S)
>
> look like Relational Algebra expressions to me.

You are mathematician. Therefore, you know many examples of algebras. Does relational algebra looks like genuine algebra to you?

Classic Relational Algebra employs six basic operations: projection, restriction, join, union, set difference, and renaming. First, the set operators - union, and difference - admit only "attribute"-compatible operands. Second, several other operations - namely, restriction,projection, and renaming - are parametrized by relation attributes.

Compare this to algebra of binary relations.

However, this appearance of classic Relational Algebra is misleading. The operations can be amended so that resulting system don't suffer the above deficiencies. What can't be fixed are the two operations which are foreign to relational algebra: - composition
- transitive closure
Sure composition can be expressed via join followed by projection. Then, transitive closure is some repetitive composition and union. However, while doing so, you would have to rename attributes, and explicit referencing attributes is something that makes your algebra deficient.   Received on Tue Feb 03 2015 - 20:31:46 CET