Re: A Topological Relational Algebra in Lisp

Tegiri Nenashi wrote:
> On Saturday, January 17, 2015 at 3:07:12 PM UTC-8, Norbert_Paul wrote:
>> Tegiri Nenashi wrote:
>>> Can we lower discussion level a little? Instead of topologies lets focus
>>> on metric spaces. Now consider multivariate relation:
>>>
>>> x y z
>>> -----
>>> 1 1 1
>>> 2 1 1
>>> 3 2 1
>>> 3 2 2
>>>
>>> What metric space do you expend this relation to?
>>
>> I will call that relation "X".
>
> X capital is fine (as long as reader is not confusing it with lower x which
> is the first attribute).

OK. It is commonplace among topologists to denote the point set by X and the topology by T (or, Tau). I overlooked the possible confusion.
>
>> Then, of course, the metric space will be the discrete space (X.P(X))
>> where P(X) is the power set of X. It will be modelled as (X,\emptyset)
>> because the empty set creates the discrete topology.
>
> What is the power set of a relation? Most of the time, when people refer to
> set structure of relation they mean set of tuples. Therefore, I interpret
> your answer as
>
> {{},{(1,1,1)},{(1,1,1),(2,1,1)},...,{(1,1,1),(2,1,1),(3,2,1),(3,2,2)}}
>
> Why this is an interesting object? It seems that further along you argue it
> isn't?

Yes, the power set P(X) of a set is the set of all subsets of X.

It is interesting for formal reasons: All topologies for a set form al lattice where the discrete topology is the maximal element and its counterpart, the indiscrete topology {{}, X}, is the minimal element. It is also often used in proofs but it has not many practical application. One might be its being a suitable default topology (IMHO) if no other topology is specified.

>> Actually, when considering databases, I am not (yet) interested very much
>> in metric topologies for databases.
>
> Metric space is something that you can explain to undergraduate student from
> STEM field. I doubt every math major undergraduate can coherently explain
> what abstract topology is.
>
> This is one of the reasons why databases enjoyed such a success. The basic
> object is a predicate -- a concept easily grasped by anybody taking
> introductory course in mathematical logic.

Maybe future introductory courses in mathematics for STEM also cover some topology. For example, Calculus uses mucht topology (using the metric topology of the reals).

>> The reason is that a database relation is always finite (one might dispute
>> that but it is a assumption a make). My model so far only considers finite
>> database relations, and takes them as topological point sets. But a finite
>> metric space is always discrete (where every subset of the point set is
>> open), and, hence, quite uninteresting.
>
> Forget about metric spaces then. Can you elaborate a toy example with one or
> two relations and what topological objects they correspond to?

Take a set

   X = {bathroom, bdoor, hallway}.

Each element can be considered a subset of the 3D real space R^3 (its "geometry"). The set could be coded as a database table

   roomid name floor department

   bathroom    "Bathroom 007"    2nd        007
   hallway     "Hall 007"        2nd        007
   bdoor       "Bathroom door"   2nd        007.

A topology could be

   T = { {}, {bathroom}, {hallway}, X}.

Note that it does not contain the set {bdoor}. This topology tells us, that bdoor is close to the bathroom and also close to hallway.
So it represents the fact, that bdoor connects the bathroom to the hallway. (Here also the connection to the metric topology of the space in R^3 where these three entities live, could be discussed.)

The above topology can be coded as a binary relation on X:

   R = bounded boundary

       bathroom bdoor
       hallway  bdoor

If you transpose the relation you get

   RT = bounded boundary

        bdoor    bathroom
        bdoor    hallway

This gives a topology

   { {}, {bdoor}, X}

the so-called "dual topology".
In the example case this topology is a graph where bdoor is an edge connecting bathroom and hallway. These graphs are often called "room adjacency graphs" by planners.
Example:

   http://www.arch.virginia.edu/insightlab/student.php?postid=68

>> However, when spatial entities that live in some R^n (R^2 - the plane, R^3
>> - the Euclidean space, R^4 - the space-time, ...) are modelled and
>> (according to my assumption) are partitioned into a a finite set of
>> spatial entities (say, faces, edges, volumes, hyper-volumes, ...) then the
>> set of entities immediately fits into my model (for being finite).
>> Interestingly the entities get their topology by the same operations that
>> produce topological result spaces from input spaces (so-called !quotient
>> space"). Anyhow, the resulting topology for these entities (like an edge
>> being bounded by vertices) usually does not constitute a metric sapce any
>> more.
>
> Again, in order not to loose your reader (me), can you please exhibit an
> example? For example, take a concrete simplicial complex, are you saying you
> have innovative way to represent it "relationally"? (Please keep the example
> concrete).

See the above example.

The relational representaion of a topology is an old and well-established fact in mathematics. The innovation is (IMHO) to combine it with the relational model.

>> Maybe, however, your post might be the start of developing a concept to
>> introduce concepts for metric topologies. Note, however, that, in order to
>> get interesting topologies (not only the dioscrete one), the point set must
>> be infinite.
>
> No, if metric space correspondence is vacuous, there is no point elaborating
> this -- I understand that metric space view can be not interesting, while
> topology offering some nontrivial insight.

The bathroom door example topology is derived from the metric space that door and rooms live in. Received on Sun Jan 18 2015 - 10:03:25 CET