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Re: Possible semantic bugs concerning domain and range

From: pat hayes <phayes@ai.uwf.edu>
Date: Tue, 24 Sep 2002 19:51:23 -0500
Message-Id: <p05111b57b9b6a167b6a1@[65.217.30.172]>
To: Ian Horrocks <horrocks@cs.man.ac.uk>
Cc: www-webont-wg@w3.org

>Pat,
>
>DAML+OIL, and I hope OWL, can be viewed a fragment of FOL, with atomic
>classes and properties corresponding to unary and binary predicates
>respectively. According to this correspondence, subClassOf axioms
>become implications, e.g., A subClassOf B corresponds to:
>
>forall x . A(x) -> B(x)
>
>Similarly, a property range axiom P range A corresponds to:
>
>forall x,y P(x,y) -> A(y).
>
>What could be simpler and clearer than that?

Nothing, I agree. Both of those are parts of RDFS, I note. But the 
translation of an OWL restriction is a bit hairier. The OWL 
restriction semantics (ignoring the object/datatype distinction for 
now) says for example that for any class (ie any unary relation) A 
and property (ie binary relation) B, a class (unary relation) C 
exists such that for example (this is minCardinality n, the others 
are similar)

(forall x, C(x)) <-> (exists y1 y2,....yn, (and B(x,y1) B(x, 
y2),...B(x,yn) (not (= x1 x2)) ... (not (= xn-1 xn))))

Now, that is FO, if not quite so simple and clear; but that isn't a 
statement of the restriction semantics, as it stands, because you 
also have to say that this applies for *any* A and B, which comes out 
as:

(forall A, B, (exists C, ((forall x, C(x)) <-> (exists y1 y2,....yn, 
(and B(x,y1) B(x, y2),...B(x,yn) (not (= x1 x2)) ... (not (= xn-1 
xn)))) ))

which isn't FO , on your view, I gather (though it is in KIF/CL, 
interestingly enough).

If you are going to reply that we don't need the outer quantifiers, 
then please stop grousing about the need for domain closure 
conditions and agree to put up with weak OWL entailment, because that 
is then all you get. Then I will agree that OWL is a subset of FOL; 
but then, there is no deep problem embedding OWL into RDF. Or, if you 
want to say that outer-quantifier form really is FOL, then I can go 
with that; but then you shouldn't have any trouble allowing classes 
of classes and things like that into OWL. Make up your mind, and 
maybe we can agree.

>
>The combination of these two sentences entails
>forall x,y P(x,y) -> B(y).
>
>What could be simpler and clearer than that?
>
>If you want some alternative semantics, could you please explain in
>similar terms what it is?

Im just following what it says in the OWL semantics. I didn't write 
it; better ask the author why he wrote it this way.

Pat

PS the above examples are written in a barbaric mixture of prefix and 
infix notation, I hope they are readable.
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Received on Tuesday, 24 September 2002 20:51:13 GMT

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