RDF Semantics: two issues, connected to OWL

Two issues need to be mentioned involving the RDF Semantics
document, both connected to OWL, one smaller, one larger.
(This adds to another issue, dealt with in [1], and not yet 
handled in the document: the central completeness claim 
does not hold, but could be made true by addition of one 
RDFS entailment rule.)


Issue 1- It seems that two changes made to RDF Semantics
during LC2 have not yet been incorporated completely
in the definition of D-interpretations.
Issue 2- There remains a problem with the combination of the two 
documents  OWL S&AS (Semantics and Abstract Syntax) 
and  RDF Semantics, for which the optimal solution 
(in fact, any complete solution) seems to require changes to 
RDF Semantics [2].

The problem I refer to in issue 2 is that the OWL design 
includes the possibility to do without the semantic conditions
of XMLLiteral.
This aspect of the OWL semantics design is not reflected in S&AS, 
and came to my attention only a few days ago [3] [4].
This possibility is not realized by S&AS because OWL is a 
semantic extension of RDFS, and each rdfs-interpretation 
satisfies the XMLLiteral conditions.

I discuss a possibility to solve both issues.
This possibility leads in addition, without much additional
cost, to a possible, interesting extension of the RDF Semantics 
document, namely a sound and complete proof theory for reasoning
with datatypes: an "RDFS-D-entailment lemma", generalizing
the RDFS-entailment lemma to a large class of datatype maps.



The two changes I refer to in issue 1 above are the omission of
the terminology "from V" and the change not to require all
possible literal values in LV.  In line 1 of the definition
of D-interpretations the conclusion aaa in V would need
to be added before I(aaa)=x, and line 2 of this definition
could be generalized to include not all values in LV.
The point of such a generalization would be - in line with
the spirit of the document - that RDF aims at generality,
so that semantic extensions have most freedom, 
and RDF is a flexible basis for the Semantic Web.
(It was noted earlier that line 3 of this definition has also
not been completely corrected, see [5].)

The following definition of D-interpretations is formulated
as a strict extension from the XMLLiteral conditions:

Given a datatype map D and a vocabulary V, a D-interpretation
of V is an rdfs-interpretation of V such that for all
<a,d> in D we have:
- a in V and I(a) = d
- I satisfies the triples
    a type Datatype
    a subClassOf Literal
- if l=s^^a is a typed literal in V and s in L(d),
  then IL(l)=L2V(d)(s) in LV  and  <IL(l),d> in IEXT(I(type))
- if l=s^^a is a typed literal in V and s not in L(d),
  then IL(l) not in LV  and  <IL(l),d> not in IEXT(I(type)

This definition of D-interpretations would solve issue 1 above.

If this change would be made, then the XMLLiteral conditions
could be removed from Section 3 of RDF Semantics, and
incorporated by including XMLLiteral in a datatype map.
When the RDF Semantics document would be changed so that XMLLiteral is 
not necessarily present in every datatype map, then issue 2 would
be solved as well, and the inconsistency between the documents 
OWL S&AS and RDF Semantics that was noted would be gone.

If these changes would be made, then the RDF and RDFS entailment 
lemmas would need change as well, since they include XMLLiteral.
Below I show that an attractive generalization of the
RDFS entailment lemma could be obtained, extending the result
to include a large class of datatype maps.


Given a datatype map  D, consider the following "D-entailment rules":
for each pair <a,d> in D, in order to replace rule rdf2: 
rule "rdf-a":
  if E contains  uuu bbb lll, where lll = sss^^a is a well-typed literal,
  then add  _:nnn type a, where _:nnn identifies a blank node
  allocated to lll by rule lg.

Define a "D-clash" to be a triple
   b type Literal
where b is a blank node allocated by rule lg to a literal s^^a, where
<a,d> is in the datatype map D for some d.

The definitions suggested above lead to the following generalization 
of the RDFS entailment lemma:
Suppose that S is a set of RDF graphs, G an RDF graph, and D
a datatype map with disjoint value spaces and injective
lexical-to-value mappings.
Then S D-entails G iff there is a graph H that can be derived
from S combined with axiomatic triples, by application of the
literal generalization rule, rdf1, and RDFS and D-entailment
rules, and that either simply entails G or contains a D-clash.

The injectivity assumption can often be handled in practice with
a suitable canonicalization operation.
Assuming here the presence of rule rdfs14 :-) [1], this result
holds: the details of the proof can be worked out.
The assumptions on D are used in connection with the definition 
of the surrogate function sur.

Herman ter Horst

[1] http://lists.w3.org/Archives/Public/www-rdf-comments/2003OctDec/0205.html
[2] http://lists.w3.org/Archives/Public/www-webont-wg/2003Dec/0042.html
[3] http://lists.w3.org/Archives/Public/www-webont-wg/2003Dec/0034.html
[4] http://lists.w3.org/Archives/Public/www-webont-wg/2003Dec/0035.html
[5] http://lists.w3.org/Archives/Public/www-rdf-comments/2003OctDec/0218.html

Received on Tuesday, 9 December 2003 09:30:18 UTC