# Re: Some questions about the exact meanings

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Wed, 01 Oct 2003 09:36:32 -0400 (EDT)
Message-Id: <20031001.093632.04558381.pfps@research.bell-labs.com>

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From: Tanel Tammet <tammet@staff.ttu.ee>
Subject: Some questions about the exact meanings
Date: Wed, 01 Oct 2003 15:20:37 +0300

> Hi,
>
> I have been teaching a group of students some RDF and OWL.
>
> Concerning semantics, the approach has been transforming
> everything to classical FOL, using "holds(predicate, subject, object)"
> to encode triples.

> As expected, mostly it is straight[fo]rward, but in some places
> I have not been able to understand what would be the
> right translation (ie what would be the exact semantics)
> of RDF and OWL constructions. Right now I have two
> main questions.
>
> First, suppose we want to say that "P" is a symmetric property.
>
> We can axiomatize what "symmetric" means by:
>
> forall X,Y,Z.  holds(symmetric,X,Y) <=>
>                          (holds(X,Y,Z) => holds(X,Z,Y)).

I think that you meant to say

forall X,Y,Z.  holds(symmetric,X) <=>
(holds(X,Y,Z) => holds(X,Z,Y)).

or maybe even

forall X,Y,Z.  holds(rdf:type,X,owl:SymmetricProperty) <=>
(holds(X,Y,Z) => holds(X,Z,Y)).

> The question is whether in OWL the equivalence <=>
> in this definition should really be an implication =>
> or it should be an equivalence <=>
>
> What is the _right_ axiom schema for OWL: with implication
> or with equivalence?

The answers to these questions can be found in ``OWL Web Ontology Language:
Semantics and Abstract Synax'' (http://www.w3.org/TR/owl-semantics).
However, actually determining the answer is generally not trivial because
this document gives a model theory for OWL, and one has to investigate the
consequences of this model theory.  I invite you to do the investigation.

However, if you want to peek, the answer is <=>.  OWL generally takes an
extensional stance on such questions.  If the conditions for some
characteristic hold, then the characteristic holds.  (RDF generally takes a
different stance here.)

Note that it is possible to non-trivially infer that a property is
symmetric in OWL, as the inverse of a symmetric property is also
symmetric.

> The second question stems from collections in RDF.
^^^containers

> We have bag, seq and alt collections. The meanings
^^^containers
> of these seem to be a bit vague in the sense that
> I am not sure how to axiomatize them.
>
> I'd start the axiomatisation by using function terms
> (we can later convert the function terms to
> predicates if we wish).
>
> So, let us have a bag of three objects: (car, plane, train).
>
> We can
>
> a) construct a list in FOF, say like this:
>
>     bagcons(car, bagcons(plane, bagcons(train,nil)))
>
>     (where we could later replace "bagcons" terms
>      by "applies(bagcons, ...." etc) if we want)
>
> b) axiomatising "bagcons".
>
>      For example, when a collection is a bag, it
^^^container
>      means that the order of elements is unimportant.
>      Hence we should axiomatise:
>
>      forall X,Y,Z. bagcons(X,bagcons(Y,Z))=bagcons(Y,bagcons(X,Z))
>
>       in FOL.

I don't think that this is getting you at all close to RDF containers.

> The question now arises: which are the right axioms
> for bag, seq and alt?

The answer is, honestly, that there are none.  RDF collections have no (or
very, very little) semantics.

> While "seq" seems fine, since it looks like being a
> plain list (ie no axioms), bag and alt pose problems.

Even seq poses severe problems.  For example, how many elements are there
in a sequence?

> I am not sure that the "bagcons" axiom above is really
> what is meant by RDF "bag". It is even more unclear
> with "alt": how should we really axiomatise "alt"?

Alt is the worst of the three.

> Most RDF documents leave this to be understood
> "intuitively", which is not OK in RDF context.

Agreed.  That is why RDF Semantics has clarified the issue.

> And I could not understand the corresponding parts
> of RDF semantics paper.

Section 3.2.2 of RDF Semantics is actually quite clear.  It specifically
states that ``[t]here are no special semantic conditions on the container
vocabulary''.

Collections are a slightly different matter.

> Regards,
>                Tanel Tammet

Peter F. Patel-Schneider
```
Received on Wednesday, 1 October 2003 09:38:04 GMT

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