Note that sketch proofs in the paper are meant to give the reader a general idea
of why the (finite) RDFS rules are APS. As Peter Patel-Schneider pointed out,
rule rdfs3 is not purely APF.
(rdf:type rdfs:range rdfs:Class) [axiomatic and ontological]
^ (?x rdf:type :foo) [assertional]
--> (:foo rdf:type rdfs:Class) [ontological]
While this is true, in this particular case, rule rdfs3 behaves as an APT rule,
and thus, it is still APS (which is what's important for parallelization). To
give an idea as to why it is APT in this case, consider what inferences can be
derived from (:foo rdf:type rdfs:Class).
rdfs8: (:foo rdf:type rdfs:Class)
--> (:foo rdfs:subClassOf rdfs:Resource)
rdfs10: (:foo rdf:type rdfs:Class)
--> (:foo rdfs:subClassOf :foo)
Clearly, triples of the form (:foo rdfs:subClassOf :foo) do not help derive more
inferences (similar to why rule rdfs6 is APT). Rule rdfs8 will only result in
deriving more triples of the form (?a rdf:type rdfs:Resource) via rules rdfs11
and rdfs9. However, in the context of rules rdfs4a and rdfs4b, such triples
will be generated anyway. Therefore, it stands that rdfs3 is APS (in the
context of the other finite RDFS rules).
Then some might say "Then couldn't triples of the form (?x rdf:type rdfs:Class)
be considered assertional?" The answer is yes, they could be, making proofs of
APS a little more difficult. As I started this note, the proofs in the paper
are meant to give the reader a general idea as to why the rules are APS (within
a paper of limited length).
Peter also pointed out that since we excluded the possibility of extending the
RDF/S terms (e.g., triples of the form (:mySubPropertyOf rdfs:subPropertyOf
rdfs:subPropertyOf)), we technically produce the closure for the RDFS fragment
of OWL 1 DL. We point out, though, that if we change the definition of
"ontological triples" to include such extensions (e.g., considering all
:mySubPropertyOf triples as ontological), then we can handle extensions of the
RDF/S terms (although multiple passes with some rules would then be required).