But the
big challenge of ontology O is that its rather a relation than a
set, to be specified by two mathematical objects known in the theory of
relations as the ground/set G (the domain, a sequence of classes/sets)
and the figure/graph F (the Cartesian
product),
Sets relating names
to values? That is, if G is the set of domains and F is the set of
objects ranging over G?s domains, then that is equivalent to a two column
table of names and values. Maybe I am missing something in your
statement ? please explain if you don?t mind spending the time to do
so.
Not exactly... This is a
standardEntity-Attribute-Value Model, or Object-Parameter-Value Schema, a
relational table, like an RDF triple comprising an object, a property, a
value. And It's all about an information system/spreadsheet/flat
data/information table/classification system/object-predicate table/relation
table, etc, organized as rows/tuples//entities/instances/dependent variables
and
columns/attributes//features/dimensions/characteristics/predicates/indvariables.
Formally, to be expressed as W = (U, P, V), the universe of things, the set
of properties, and the set of values. Now introducing as an equivalence
relation a P-indiscernibility, you start dealing with the rough-set
representations of the original/crisp set.
Seemingly modifying the
equivalence relation, there must be different extensions to the
RS, with the different equivalence classes like probabilistic, inconsistency
or fuzzy. And the application areas are everywhere you have a need in
approximate patterns/rules for data streams/flows.
The ontological schema
describes the world of things, W, in terms of general
classes/kinds/categories, as Entity, State, Change, and Relationship,
which could be considered as Metadata Table for the standard
information tables as above.
described
as a couple of the mathematical things, O = (G(O), F(O)). Further the
context is to be reified as the world states, W, with its model, M and
language, L, as it was done in the Reality book, http://www.igi-global.com/bookstore/TitleDetails.aspx?TitleId=859&DetailsType=Chapters
Sent: Friday,
January 14, 2011 5:50 AM
Subject: Re:
[ontolog-forum] Ontology of Rough Sets
Tara,
I don?t recall
any specific reference on rough sets ? I think I first read something in
the IEEE suite of pubs, maybe ten years ago.
Rough sets should
be fairly easy to implement in SQL, so it would be interesting to know
what components you were considering, and whether they met your needs.
If any reference
info comes back to me, I?ll post it.
-Rich
Sincerely,
Rich Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT
com
9 4 9 \ 5 2 5 - 5 7 1
2
From:
ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Tara Athan
Sent: Thursday, January 13, 2011 3:31
PM
To: Rich
Cooper
Cc:
'[ontolog-forum] '
Subject: Re: [ontolog-forum] Ontology
of Rough Sets
Rich Cooper wrote:
Hi
Tara,
Are you are
visualizing the slice of an orange while I?m visualizing the remainder of
the orange with the slice removed?
Must be, because I don't catch how the application
you outline below fits the mathematical theory of rough sets that I have
read about.
Can you provide a reference?
Without that
visualization shift, I can?t fit my imaging into your response
?a rough set would be a sub-lattice with top node of the
upper approximation and bottom node being the lower approximation?.
My view of a
rough sets application is that we have many overlapping ways of sensing
objects with instruments (blood tests, MRI ?) that can detect and report
on what was found. The trajectory, or process, (e.g., life cycle of
a disease) of the said object shows up in different instruments at
different phases of its existence. Before computers, statisticians
could only dismiss outliers, but with computers in everything, we can look
at the outliers as valid data points. There must be subtleties yet
unmined in those signatures.
Rough sets
provide a way of modeling that kind of dynamic signal and differencing.
Potentially they can help explain nuances of the data that weren?t
detectable with less analysis.
Sensors are often
modeled as probabilistically detecting or not detecting objects, but
remember that sensors only detect COLUMNS, of object properties -
attributes. So the probabilistic view may not make practical
analysis of data outliers.
So I would still
like to see an ontology of rough sets, if there is one available.
Thanks,
-Rich
Sincerely,
Rich
Cooper
EnglishLogicKernel.com
Rich AT EnglishLogicKernel DOT
com
9 4 9 \ 5 2 5 - 5 7 1
2
I'm more inclined to think of a lattice of crisp
sets, and a rough set would be a sub-lattice 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: ontolog-forum-bounces@xxxxxxxxxxxxxxxx
[mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Tara Athan
Sent: Thursday, January 13, 2011 10:53 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] 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.341-356.
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.3-27.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets and Boolean reasoning.
Information Sciences, 177(1), pp.41-73.
PAWLAK, Z. and A. SKOWRON. 2007. Rough sets: Some extensions.
Information Sciences, 177(1), pp.28-40.
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 3-valued rough logic accepting decision
rules. Fundam. Inf., 61(1), pp.37-45.
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 3-valued 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
Owner, Athan Ecological Reconciliation Services
tara_athan at alt2is.com
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--
Tara Athan
Owner, Athan Ecological Reconciliation Services
tara_athan at alt2is.com
707-272-2115 (cell, preferred)
707-485-1198 (office)
249 W. Gobbi St. #A
Ukiah, CA 95482
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