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Re: [ontolog-forum] borrowing terminology

To: <conrad.bock@xxxxxxxx>
Cc: "'[ontolog-forum] '" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Pat Hayes <phayes@xxxxxxx>
Date: Fri, 9 Mar 2007 11:12:55 -0600
Message-id: <p06230906c2174206a1ee@[]>
>  > >So OWL and as the rest of computer science are using "range" a
>  > >nonmathematical way:
>  > >
>  > >   http://en.wikipedia.org/wiki/Range_%28computer_science%29
>  >
>  > No, wait. There is range-math and range-CS, let
>  > us agree (although in fact 'range' is used in
>  > both disciplines in both senses). But the
>  > owl/RDFS sense of 'range' is closer to the first,
>  > mathematical, sense than the second. 
>Not following, the definitions in the math links say all elements in the
>range have mappings from some element in the domain, while the co-domain
>can have elements that aren't mapping to by any element in the domain.    (01)

Yes. But at least the issue is about sets and 
relations. The definitions for 'range' in the 
wikipedia CS entry (pointer above) are to do with 
boundary values in arrays and limits on values of 
variables, which are a COMPLETELY different 
topic. I thought this was your point.    (02)

>In OWL, if the range of a property pet is Animal, it isn't required that
>all animals are pets.    (03)

Right, so the owl:range is a subset of the 
co-domain, as I said. There is no explicit OWL 
construction for codomain as such, but if the 
codomain of P is C, it would not be incorrect to 
assert that C is (a) range of P.   In fact, if S 
is any set which is a subset of the codomain and 
which has the math-range as a subset, then it 
would be correct to assert that S is an owl-range 
for the property. In OWL there can be many ranges 
asserted of a property, and 'the' range is 
somewhere inside their intersection. One can 
bound it from above, as it were, but one cannot 
bound it from below with an owl:range assertion.    (04)

>  Or at least if you tell me it is (and since it's
>you, I'd pretty much hafto believe it!), I'd need to ask what the OWL
>construct is for co-domain.
>  > >Idempotence is another example:
>  > >
>  > >   http://en.wikipedia.org/wiki/Idempotent
>  > >   http://en.wikipedia.org/wiki/Idempotence_(computer_science)
>  > >
>  >
>  > This isn't a very good example, as these really
>  > are the same concept, almost word-for-word the
>  > same definition in fact, but they are applied to
>  > rather different kinds of application area. But I
>  > don't think there would be any risk of a
>  > mathematician seriously misunderstanding what was
>  > meant here.
>Not the way I've heard idempotence used by software people.  They're
>referring to calling the same function twice on the same arguments (the
>wikipedia ambiguously calls this "used multiple times"), where the
>second call has no effect.    (05)

Ah, I see your point, sorry. Hmm. I think I see 
where this happened.  If you assume that every 
application involves a 'state', then multiple 
applications become nested function applications 
to a starting state, which maps the computer 
science sense into the mathematical sense. But I 
agree the CS folk have kind of diluted the 
mathematical meaning.    (06)

>  This is important in cases where you need to
>make sure a function has been called on an argument, but aren't sure if
>it has already, and don't have a way of finding out.  The mathematical
>definition "calls" the function once, and calls it again with the result
>of the first call, and gets the same result back it did the first time
>(the composition of the function with itself is the function).
>Bummer about the coopting of completeness.
>  > Another example which came up on the OWL-1.1 discussion recently was
>  > the notion of 'disjoint union', which has been given a non-standard
>  > (and genuinely faulty) interpretation by formal ontologists, which is
>  > not in line with its established mathematical
>I thought it equivalent to a set of disjoint classes that are also a
>union for another.  What's the mathematical definition?    (07)

It unions distinct 'copies' of the sets, so even 
if they are identical or share elements, one gets 
distinct copies in the disjoint union (aka 
'separated union'). This can be described 
formally in various ways, but the key property is 
that the cardinality of the disjoint union is 
always the sum of the cardinalities of the 
unioned sets. So the n-fold disjoint union of A 
with itself is meaningful and isomorphic to A x 
n.  If the sets are disjoint, then the disjoint 
union is the ordinary union. But it is defined 
even if they aren't disjoint; and this is 
important for the category-theory treatment of 
set theory.    (08)

Pat    (09)

>Conrad    (010)

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