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Re: Equality and subclass axioms

From: Ian Horrocks <horrocks@cs.man.ac.uk>
Date: Sun, 26 Nov 2000 20:43:21 +0000 (GMT)
To: Jim Hendler <jhendler@darpa.mil>
cc: www-rdf-logic@w3.org
Message-ID: <Pine.LNX.4.21.0011262035290.30772-100000@galahad.cs.man.ac.uk>
On Sat, 25 Nov 2000, Jim Hendler wrote:

> At 16:24 +0000 11/25/00, Ian Horrocks wrote:
> >I have heard some worries expressed about the effect of having
> >equality (if and only if) axioms in an ontology. For example, we could
> >state in our ontology that an object is a Triangle if and only if it
> >is in the intersection of the classes Polygon and ThreeSidedThing (in
> >this case it is sometimes said that being in the intersection of the
> >classes Polygon and ThreeSidedThing is a necessary and sufficient
> >condition for being a Triangle). The semantics of this axiom are that
> >in every satisfying "interpretation" (i.e., in every model of the
> >world that conforms to the structural constraint imposed by the
> >axiom), the set of objects that are Triangles must be equal to the
> >intersection of the set of objects that are Polygons and the set of
> >objects that are ThreeSidedThings.
> >
> >As far as I can understand it, the worry is as to what will happen if
> >another equality axiom w.r.t. Triangle is added to the ontology, e.g.,
> >that an object is a Triangle if and only if it is in the intersection
> >of the classes Polygon and ThreeAngledThing. This doesn't cause any
> >problem: the set of objects that are Triangles must be equal to the
> >intersection of the set of objects that are Polygons and the set of
> >objects that are ThreeAngledThings, and from the transitivity of
> >equality we can of course also infer that the intersection of the set
> >of objects that are Polygons and the set of objects that are
> >ThreeSidedThings is equal to the intersection of the set of objects
> >that are Polygons and the set of objects that are ThreeAngledThings.
> >
> >As with other ontological axioms, these kinds of axiom give structure
> >to the domain of discourse by restricting the set of valid models. Of
> >course it is possible to restrict the set so tightly that some (or
> >even all) classes are empty in all valid models, but this can happen
> >with or without equality axioms. If I am allowed to add a plug for
> >reasoning at this point, I would say that this is an example of how it
> >can be useful as it makes it possible for a tool to draw the users
> >attention to such an occurrence, which may indicate an error in the
> >design of the ontology.
> >
> >Regards, Ian
> Ian-
>   Here's my worry, which is controllable, but definitely worth 
> thinking about.  You, being a rational person, come about with a rule 
> that says triangles are three-sided things.  I, being an irrational 
> person, have an axiom stating triangles are 4-sided.  We both use if 
> and only if rules.  Some web crawler comes, scrapes both of our rules 
> into the same knowledge base -- and then what happens?  Do we get two 
> kinds of triangles or no kinds of triangles.  

Well, assuming that the crawler realises that both axioms are talking
about the same thing (which is a big assumption, but is a different
problem to the one we are addressing here), then there will only be one
triangle, but two axioms stating apparently contradictory things about

> Obviously, no model can 
> allow triangles that are  both  three and four-sided - so I would 
> assume by throwing in my irrational axiom I've somehow "negated" 
> yours.  

You didn't "negate" my axiom (you can never do that), you just added some
additional information (an additional constraint). Assuming it is true
that no model can allow triangles that are both three and four-sided, then
this is an example of the kind of "over-constraining" that I mentioned in
my email: our ontology now constrains allowable models to the extent that
none can ever contain an instance of triangle (i.e., we can infer that
triangle is equivalent to the class "Nothing"). If we use a reasoner to
check the ontology generated by our crawler, then it will detect this
fact, and can alert an intelligent (possibly human) agent to the fact that
there may be a problem with the axioms relating to triangle.

By the way, it isn't as obvious as it at first seems that no model can
allow triangles that are both three and four-sided: it depends on how the
axioms are stated and on what additional information is in the ontology.
Let us imagine that both the axioms use a property called "numberOfSides",
with integer values of 3 and 4 respectively. In OIL, this would lead to
triangle being "inconsistent"; this should also be the case for DAML, but
it isn't completely clear at the moment as the semantics of concrete data
types has yet to be determined (and if we treat 3 and 4 as "individuals",
then we would still need additional information, such as a unique name
assumption, to infer that triangle is inconsistent).

However, even if we assume that this representation of the axioms would
lead to triangle being inconsistent, there is also the possibility that we
chose to represent 3 and 4-sidedness using classes, e.g., I stated that
triangle is equivalent to the intersection of "Polygon" and
"ThreeSidedThing" and you stated that it is equivalent to the intersection
of "Polygon" and "FourSidedThing". This will be perfectly consistent
unless the ontology contains additional axioms implying that
"ThreeSidedThing" and "FourSidedThing" are disjoint. Note that even in the
case that "ThreeSidedThing" and "FourSidedThing" are not disjoint, a
reasoning engine may still be able to spot the potential problem. E.g., if
I have also stated that square is equivalent to the intersection of
"Polygon" and "FourSidedThing", then it can generate a warning about the
implicit equivalence of square and triangle.

Finally, it is worth pointing out that none of these scenarios is likely
to arise as it is much more probable that we have used different names,
spellings, capitalisations and/or representations, so that the connection
between my triangle and your triangle will not be noticed, or will not
lead to any conflict. This is the real problem with ontology integration,
whether done by humans or crawlers/scrapers, and it is one for which there
is as yet no easy solution.

> Notice if we're not using If and only if, this problem 
> doesn't come up (i.e. IF three sides and IF 4 sided can concurrently 
> occur).

It depends on what you mean. If we stated that triangle is necessarily
three/four-sided respectively (i.e., that triangle is a subClassOf
three/four-sided), then the problem would be much the same (depending on
the representation scheme, triangle would be equivalent to "Nothing"). If
we only stated that a three/four-sided polygon is necessarily a triangle,
then the problem does not arise. However, I would see this as a bad thing
rather than a good thing - by making a weaker statement than we were
justified in making, we allow a real problem in our "integrated" ontology
to go unnoticed.

I should also point out that the suggested revision to daml-ont does not
force us to use if-and-only-if axioms, it simply allows us to do so.
Moreover, it is open to an application to ignore the semantics of such
axioms if it deems this to be appropriate.

>   I think in general we who are worrying about web semantics will 
> either need something that tags axioms to their ontologies or 
> something that allows very permissive semantics.  Notice that I'm not 
> bothered by having something that uses the term triangle for two 
> different things - because both intended extensions are in there (so, 
> for a "search engine" we'd just be getting extra answers and we'd use 
> some rules to eliminate the ones we don't like).  I'm more worried 
> about something that let's someone whose beliefs differ from mine do 
> something that causes my inferences to fail.

They don't cause your inferences to fail, they allow you to make
additional inferences. E.g., in the above case we can make the additional
inference that, not only are both three and four-sided polygons triangles,
but that there is actually no such thing as a triangle, so don't waste
time searching for one. In general, this is a very useful inference
(particularly if the search engine can inform the user not simply that it
failed to find any relevant information, but that according to the
ontology they are using their query is logically inconsistent).

Of course if a user chooses to employ an ontology literally scraped
together from unreliable sources, and without the benefit of any (possibly
reasoning supported) integration procedure, then (s)he cannot be surprised
at getting unreliable results.  My own view is that due to the difficulty
of integrating multiple ontologies (and indeed of building a single
ontology), there is almost no chance of getting anything useful out of
throwing several ontologies into a large pot and giving them a brisk stir.
Instead, I expect that web users will vote with their feet and choose to
employ well designed ontologies that they have found to give good results,
for example in improving search. Even if it is possible to use crudely
integrated ontologies, users would soon discover that your ontology was a
bad one and stop using it - the best that can happen is that it will cause
their search to return lots of information about squares when they asked
about triangles. Of course this "natural selection" would be facilitated
by a kind of "tagging" mechanism that could inform a user as to which
axioms/ontologies led to undesirable results, i.e., an explanation

>   Again, these situations are sure to come up on something as 
> permissive as the net (imagine the pro-choice and anti-abortion folks 
> trying to defeat each others' theorem provers) - but we need to 
> decide at some point how to tag or mark or distinguish or something 
> where axioms come from, or something like that.  This is clearly 
> beyond RDF, but crucial to any really practical web semantics.
>   -Jim H.
> Prof. James Hendler		Program Manager
> DARPA/ISO			703-696-2238 (phone)
> 3701 N. Fairfax Dr.		703-696-2201 (Fax)
> Arlington, VA 22203		jhendler@darpa.mil

Regards, Ian
Ian Horrocks, Department of Computer Science,
University of Manchester, Oxford Road, Manchester, M13 9PL, UK.
Tel: +44 161 275 6133  Fax: +44 161 275 6204  Email: horrocks@cs.man.ac.uk
WWW: http://www.cs.man.ac.uk/~horrocks
Received on Sunday, 26 November 2000 15:42:26 UTC

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