Re: category theory

Renato,

I think all the examples that you describe are covered by OWL. If you
use OWL for your ontology, you can specify that a predicate is

- transitive
- symmetric
- functional
- inverse of another predicate
- inverseFunctional

The terms injective and surjective are not used, these are equivalent (I
believe) to inverseFunctional and functonal, respectively (bijective is
simply a predicate that is both).

There are also more complicated ways in OWL to specify a group of
reseources on which a predicate has some specific behaviour (look at the
various constraints), but I believe the list above covers all your
examples below. Note that, with a very high probability, a new version
of OWL (currently named OWL1.1) will also be specified in he coming
year, which will add some more possibilities to this list.

True, none of these are defined and described in terms of category
theory, more in terms of traditional logic.

Do I still miss something? Please tell me...

Ivan

P.S. A good place to start is: http://www.w3.org/TR/owl-guide/

Renato golin wrote:
> Ivan Herman wrote:
>> However... what type of results/research would you expect to see out of
>> that? I could very vaguely see some possibilities in looking at
>> structural equivalence and analysis of RDF graphs, but I am not sure
>> that would reveal anything significant  (I may be wrong!). I am also not
>> sure that, say, the concept of injection or surjection would give
>> anything new with respect to what OWL can already describe in terms of
>> functional and inverse functional properties...
>>
>> I am interested to hear/see which way you want to go with that.
> 
> Hi Ivan,
> 
> I'm not an expert in OWL and RDF but I've been looking for a particular
> behaviour on some ontologies (dc, rdf, rdfs, etc) and couldn't find
> (what doesn't mean it's not there).
> 
> Some relationships are injective:
>     "Luke, I'm your father"
> 
> It means that Darth Vader is Luke's father but not the other way around.
> 
> Other relationships are bijective:
>     "Reuben is Stitch's cousin"
> 
> Which means also that Stitch is Reuben's cousin.
> 
> Some relation are neither exactly:
>     "I love her"
> 
> Who said that hope it to be bijective, but it might be not.
> 
> We understand all three sentences but a program will not, unless you
> specify those relationships to it. I mean, a program cannot know that
> Luke is Vader's son just reading that phrase (unless previously
> hard-coded in the program, but that's what we're trying to avoid with
> RDF I guess).
> 
> But if you say that, not only that father is injective but that there is
> an opposite and the opposite is son, voilá. If it was bijective, the
> program wouldn't look for the opposite for it'd know they're the same.
> 
> So, when I ask the program: "Who is Reuben's cousin?" it'll know that
> it's Stitch.
> 
> At last, when asked who loves me it should say: "She might love you, let
> me check" and go through the RDF jungle, searching for quotes from "her"
> saying she loves me... or not.
> 
> The third case require further enquires, on data that might not be
> available (for she's a bit shy) but the first two, by adding the
> additional category constraint, I can rely on the data itself.
> 
> Maybe there is a way already to do it using the current ontologies...
> 
> 
> thanks,
> --renato

-- 

Ivan Herman, W3C Semantic Web Activity Lead
URL: http://www.w3.org/People/Ivan/
PGP Key: http://www.cwi.nl/%7Eivan/AboutMe/pgpkey.html
FOAF: http://www.ivan-herman.net/foaf.rdf