W3C home > Mailing lists > Public > www-webont-wg@w3.org > July 2002

comments on issue 5.19 (classes as instances) and 4.6 (equivalentTo)

From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
Date: Fri, 26 Jul 2002 18:43:46 -0400
To: www-webont-wg@w3.org
Message-Id: <20020726184346O.pfps@research.bell-labs.com>

[I found this note surprisingly difficult to write.  I may end up
significantly revising it due to comments from the group.]


Some preliminary discussion on classes as instances must preceed discussion
of equivalentTo, because equivalence and identity for classes and
individuals depends on how they are handled which is impacted by the stance
on classes as instances.

References:

[RDFMT] RDF Model Theory, http://www.w3.org/TR/rdf-mt/
[D+OMT] A Model-Theoretic Semantics for DAML+OIL (March 2001)


Issue 5.19: Classes as instances

I see four stances that can be taken with respect to classes as instances.
By the way, the same four stances exist with respect to properties as
instances.  I will assume that any reasonable system takes the same stance
for properties as instances that it takes for classes as instances.

1/ Super-strong: A super-strong stance on classes as instances says that a
class is just something attached to an individual.  This is the stance
taken in the new RDF model theory, where the extension of a class is
defined from an individual, not from the name of the class.  This is shown
in the definition of CEXT(x), which is defined on resources (individuals),
not on names.  (Contrast this to IS, which gives meanings for URI refs as
individual names by mapping them into resources.)

2/ Strong: A strong stance on classes as instances says that if two names
denote the same individual then their meaning as classes must be the same.
This stance is, of course, compatible with the super-strong stance, and, I
think, has no observable difference from the super-strong stance.

Consequence of the super-strong and strong stances:

  E1 - if a and b are names that denote the same individual
       then the class extensions of a and b are the same

(Strangely enough, having class and instance names disjoint does not
totally preclude the strong stance.  It is possible to have a mapping from
class names to individual names that serves to relate classes with
individuals.  Of course, this approach has some differences from the
regular strong stance, as the name mapping has to be used in appropriate
places. )

3/ Weak:  The weak stance on classes as individuals only says that every
class has an associated individual.  There is no commitment that if two
classes are associated with the same individual then their meaning as
classes are the same.  One way of performing this association is to use the
name of the class as the name of the individual.  This is the stance taken
in the DAML+OIL model theory.

In the weak stance E1 is not valid.

4/ None:  This stance denys any relationship between classes and instances
besides the instance relationship.  Many DL-based representation systems
have this stance.  This stance often comes along with a partitioning of the
space names into separate subspaces for instances, classes, and
properties.  Even if this is not the case, there is no relationship between
a class and an instance that share the same name.


Issue 4.6: equivalentTo

There are several semantic relationships that have to be considered when
investigating how to treat sameIndividualAs, sameClassAs, samePropertyAs,
and equivalentTo.  The first three are rather easy to describe and do not
depend on any of the stances taken on classes as individuals.

sameIndividualAs/ two names mean the same individual
R2/ two names have the same class extension
R3/ two names have the same property extension

However, there are other semantic relationships that are harder to define. 

sameClassAs/ two names mean the same class, which implies that they have
    the same class extension, and maybe something more
samePropertyAs/ two names mean the same property, which implies that they
    have the same property extension, and maybe something more 
equivalentTo/ two names are equivalent, 
    which implies at least that they have the same denotations as
    individuals and the same class and property extensions

(In RDF and DAML+OIL individual denotation, class extension, and property
extension is all there is, but other logics may have more to
interpretations, which is why I'm being pedantic here.)

So
	sameClassAs(x,y) implies R2(x,y)
	samePropertyAs(x,y) implies R3(x,y)
	equivalentTo(x,y) implies sameIndividualAs(x,y)
	equivalentTo(x,y) implies R2(x,y)
	equivalentTo(x,y) implies R3(x,y)


Strong and super-strong stances:

In the strong and super-strong stances, we have the following implications:

	sameIndividualAs(x,y) implies R2(x,y)
	sameIndividualAs(x,y) implies R3(x,y)

Further, there is only one reasonable choice for ``the same class'' and
``the same property'', namely having the names denote the same individual
as well as having the same class or property extension, so:

	sameClassAs(x,y) implies sameIndividualAs(x,y) and R2(x,y)
	samePropertyAs(x,y) implies sameIndividualAs(x,y) and R3(x,y)

However, because of the previous two implications this reduces to:

	sameClassAs(x,y) implies sameIndividualAs(x,y)
	samePropertyAs(x,y) implies sameIndividualAs(x,y)

Now two names are equivalent here exactly when they denote the same
individual, so:

	equivalentTo(x,y) iff sameIndividualAs(x,y)

So in the strong and super-strong cases equivalentTo is just another name
for sameIndividualAs, certainly nothing more.  


Weak stance:

Because in the weak stance E1 is not valid, 

	sameIndividualAs(x,y) does not imply R2(x,y)
	sameIndividualAs(x,y) does not imply R3(x,y)
so
	sameIndividualAs(x,y) does not imply sameClassAs(x,y)
	sameIndividualAs(x,y) does not imply samePropertyAs(x,y)

It would not make sense to have sameClassAs imply sameIndividualAs, so:

	sameClassAs(x,y) does not imply sameIndividualAs(x,y)
	samePropertyAs(x,y) does not imply sameIndividualAs(x,y)

Now two names are equivalent exactly when they are the same individual,
class, and property

	equivalentTo(x,y) iff 
	sameIndividualAs(x,y) & sameClassAs(x,y) & samePropertyAs(x,y) 

so equivalentTo is the conjunction of the three sameXxxAs relationships.


None stance:

In the none stance, classes and properties really have nothing more than
their extension, so

	R2(x,y) iff sameClassAs(x,y)
	R3(x,y) iff samePropertyAs(x,y)

In the none stance with a separated name space, equivalentTo is an
ill-typed relationship.  

In the none stance with a single name space
	
	equivalentTo(x,y) iff 
	sameIndividualAs(x,y) & sameClassAs(x,y) & samePropertyAs(x,y) 

but this doesn't make much sense.




Peter F. Patel-Schneider
Bell Labs Research
Received on Friday, 26 July 2002 18:43:55 GMT

This archive was generated by hypermail 2.2.0+W3C-0.50 : Monday, 7 December 2009 10:57:51 GMT