RE: RDF mapping of datarange complements brings problem for DL/Full relationship

Peter F. Patel-Schneider wrote:

>From: "Michael Schneider" <schneid@fzi.de>
>Subject: RE: RDF mapping of datarange complements brings problem for
>DL/Full relationship
>Date: Wed, 4 Jun 2008 09:58:12 +0200
>
>> Hi Peter!
>>
>> >> (Just to be clear: I did not suggest to extend the complements
>> >> of dataranges to the "whole" domain in OWL DL. This would
>> >> make no sense. I only stated that in *OWL Full* the complement
>> >> actually *is* relative to the whole domain, and that this
>> >> difference between DL and Full may lead to problems.
>> >> I write this mail here to show such a problem.)
>> >
>> >I don't see that the situation in OWL Full is forced.  The complement
>> >operator for dataranges could, I think, be relative to rdfs:Literal
>in
>> >OWL Full.
>> >
>> >peter
>>
>> AFAICS, this would lead to rdfs:Literal being equivalent to owl:Thing.
>>
>> Cheers,
>> Michael
>
>How so?
>
>peter

Let's regard the complement C of rdfs:Literal:

  (1) C owl:complementOf rdfs:Literal

The already existing OWL 1 Full semantics of 'owl:complementOf' are
specified for /every/ class. In particular, rdfs:Literal is a subclass of
owl:Thing:

  (2) CEXT_I(S_I(rdfs:Literal)) subset CEXT_I(S_I(owl:Thing)) 

>From the old OWL 1 spec, we thus receive:

  (3) CEXT_I(S_I(C)) = IOT - CEXT_I(S_I(rdfs:Literal))

where the set IOT is defined to be:

  (4) IOT = CEXT_I(S_I(owl:Thing))

Now, suppose we add a semantic condition, which defines the complements of
datatypes relative to rdfs:Literal. In OWL Full, datatypes are characterized
as being instances of the set IDC, defined by:

  (5) IDC = CEXT_I(S_I(rdfs:Datatype)) 

In particular, the first table in sec. 5.2 of the OWL Full spec states that

  (6) S_I(rdfs:Literal) in IDC

Hence, our new semantic condition would "capture" rdfs:Literal, leading to 

  (7) CEXT_I(S_I(C)) = CEXT_I(S_I(rdfs:Literal)) - CEXT_I(S_I(rdfs:Literal))
                     = {}

So from (3), (4) and (7) we get:

  (8) CEXT_I(S_I(owl:Thing)) - CEXT_I(S_I(rdfs:Literal)) = {}

This allows us to conclude:

  (9) CEXT_I(S_I(owl:Thing)) subset CEXT_I(S_I(rdfs:Literal))

And from (2) and (9) we get:

  (10) CEXT_I(S_I(rdfs:Literal)) = CEXT_I(S_I(owl:Thing))

Finally, the IFF semantics of owl:equivalentClass lead from (10) to:

  (11) rdfs:Literal owl:equivalentClaas owl:Thing

Cheers,
Michael