I'm more inclined to think of a lattice of crisp sets, and a rough set
would be a sublattice with top node of the upper approximation and
bottom node being the lower approximation.
One of the interesting things, to me, of this set theory is two
orthogonal partial orders:
1. the usual containment (U1 (upper approx of set 1) contained in U2,
L1 contained in L2) > RSet1 contained in RSet2
2. roughness (U1 contained in U3, L3 contained in L1) > RSet1 is
rougher than RSet3
And actually this was why I started to look into the theory in the
first place because I was considering a domain with more than one
partial order  graphs.
Tara
Rich Cooper wrote:
Tara, John,
Also, the lower and upper bounds on a rough set parallels some of the
lattice properties John has suggested for ontologies. Would it be accurate
to characterize rough sets in that way  as a specification of the GLB and
LUB in a lattice of rough sets, based on data collected in the rough sets so
far?
Musingly,
Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT com
9 4 9 \ 5 2 5  5 7 1 2
Original Message
From: ontologforumbounces@xxxxxxxxxxxxxxxx
[mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of Tara Athan
Sent: Thursday, January 13, 2011 10:53 AM
To: [ontologforum]
Subject: Re: [ontologforum] Ontology of Rough Sets
OK, let's be careful.
There is a mathematical set theory called "rough set theory", that was
introduced in the 1980's
PAWLAK, Z. 1982. ROUGH SETS. International Journal of Computer &
Information Sciences, 11(5), pp.341356.
and has been studied quite a bit since. Here are some more recent articles:
PAWLAK, Z. and A. SKOWRON. 2007. Rudiments of rough sets. Information
Sciences, 177(1), pp.327.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets and Boolean reasoning.
Information Sciences, 177(1), pp.4173.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets: Some extensions.
Information Sciences, 177(1), pp.2840.
It might be a good idea to emphasize at this point that sets and
concepts are not the same thing. A set is determined by its membership,
a concept is determined by its definition. The extension of a concept is
a set, but the same concept can have extensions with different
membership in different contexts, such as at different times (this may
be an abuse of the term "context", I'm not sure.) However, if we,
implicitly or explicitly, specify the context so that a concept has a
unique extension, then we can work with sets through their defining
concepts.
Rough set theory uses the notion of "indiscernability", where there is
an attribute which can be measured to some precision or "granularity"
and this precision does not fully describe reality. The universe of
discourse is clustered into groups that indiscernible with respect to a
set of attributes.
"Crisp" sets consist of unions of these clusters. But it is also
possible to have sets, as determined by listing their membership, which
cannot be defined precisely by specifying regions in the quality spaces
of the available attributes because there are indiscernible clusters
that fall partly within and partly outside of the set.
Such sets are rough sets, and they have two approximations:
an lower approximation, which consists of the greatest crisp subset,
a upper approximation, which consists of the smallest crisp superset.
Given only the available attributes of an individual x, and the upper
and lower approximations of a rough set, then the question "Is x a
member of rough set B" has three possible answers, definitely yes,
definitely no, or maybe. Thus rough sets intuitively carry a 3 valued
logic, although the mathematics is more nuanced  see
POLKOWSKI, L. 2003. A note on 3valued rough logic accepting decision
rules. Fundam. Inf., 61(1), pp.3745.
So we can also wonder about the extension of the "rough" notion to
ontologies.
That approach has been explored:
INUKAI, Y., A. GEHRMANN, Y. NAGAI and S. ISHIZU. 2007. ROUGH SET THEORY
USING SIMILARITY OF OBJECTS DESCRIBED BY ONTOLOGY [online]. [Accessed
Jan 13, 2011]. Available from:
http://journals.isss.org/index.php/proceedings51st/article/viewFile/504/294.
Or we may ask a different but related question: Instead of an atomic
concept being a Boolean unary predicate, what if it is a 3valued unary
predicate? I'm not sure if there are published papers addressing this
question.
If there are other perspectives on this issue, it would be interesting
to hear about it from other list members.
Tara
John F. Sowa wrote:
Folks,
We have to make a clear distinction between ontology and the tools,
languages, logics, and reasoning methods used with any ontology.
The subject line of this thread could be very misleading.
Ever since Aristotle, categories and hierarchies of categories
have been useful for ontology  primarily because the study
of existence leads to a study of what kinds of things exist.
A's syllogisms and his method of definition by genus and
differentiae have also been useful. But many people (starting
with Aristotle himself) have noted that prototypes rather
than strict definitions are better for some applications.
In general, there is *no* specific logic or reasoning method
that is either essential or irrelevant to ontology. The
choice depends entirely on specific applications  or even
on very narrow questions or problems about an application.
Re rough sets: This is an important topic, but I would be
very cautious about any way of thinking that combines the
word 'ontology' with any particular reasoning method.
That is in fact why I have been unhappy with the phrase
"Web Ontology Language" used as a scrambled acronym for OWL.
It suggests to many novices, that the word 'the' belongs in
front of that phrase  but that idea is hopelessly misguided.
Ontology is *orthogonal* to any and all versions of logic,
reasoning methods, and implementation tools. Any specific
language or tool tends to channel thinking into certain paths
that may be useful for some applications. But ways of thinking
that are specialized for one kind of application can often be
inappropriate or even misguided for other applications.
John
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Tara Athan
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