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Re: [ontolog-forum] Ontology of Rough Sets

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "AzamatAbdoullaev" <abdoul@xxxxxxxxxxxxxx>
Date: Wed, 19 Jan 2011 17:44:52 +0200
Message-id: <B5DE5F750381441FB61B3D506D2ED8F0@personalpc>
How we talk about sets, classes, types, kinds, or categories depends on how we see things. Things possess properties of triple kinds: law characteristics, general or specific, as intrinsic features; idiosyncratic properties as individual characters; and conventional, artificial, or constructed distinctions.
Characterized by the universal law, Thing is the universal class of all classes, the highest category/metakind/ultimate type...
Characterized by the specific law, thing is a natural kind, a set of entities distinguished by natural laws. 
Characterized by the singular properties, thing is an individual, instance, examplar, instance, case, occurrance, a single entity. 
So the ladder of reality is individuals, sets, classes, types, kinds, categories, Thing.
What all science is doing: grouping/dividing its objects/particulars/instances/tokens/occurrences into natural kinds to analyze them further, as you also noticed in your last.
All the achievement of humand mind consists in discoverying the natural kinds as natural groupings of entities, natural classifications, the manifestation of real sets or ontological kinds/classes.
Now we emjoy all sorts of natural kinds as fundamental particles, chemical elements, and human beings, possibly technological, social and cultural kinds.   
My advise is just follow one simple but wise rule to escape any confusion: "Intension always determines Extension".
Azamat Abdoullaev
 
----- Original Message -----
Sent: Tuesday, January 18, 2011 1:00 AM
Subject: Re: [ontolog-forum] Ontology of Rough Sets

Hi Azamat,

 

Only the individuals and instances are real ? they present themselves in data.  Perhaps you prefer that I use the term ?classes? instead of the term ?sets? in that no class can contain itself as an element.  

 

That constraint leaves us free to use the individuals, instances of data types, as the model of ground truth.  In that representation, classes are groups of individuals, instances, et cetera.  Using that representation, the equivalence function could be composed on any or all of the four class types ? rough, fuzzy, probabilistic and crisp.  

 

In my opinion, it is a mistake to construe sets as containing themselves, though mathematicians have invested heavily into that interpretation.  

 

HTH,

-Rich

 

Sincerely,

Rich Cooper

EnglishLogicKernel.com

Rich AT EnglishLogicKernel DOT com

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From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of AzamatAbdoullaev
Sent: Monday, January 17, 2011 2:15 PM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

It's a common practice/confusion in the computing/information science ontologies to view individuals, instances, or objects as the ground level. In reality, the basic level consists of classes (fundamental kinds of things), properties and relations, all depending on the extensive or intensive approaches.

----- Original Message -----

From: Rich Cooper

Sent: Monday, January 17, 2011 11:28 PM

Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

Hi Azamat,

 

In your explanation below, you use the term ?equivalence relation? as though you agree that the various views of sets (fuzzy, rough, probabilistic, crisp) are EACH equivalence relations.  That is, each one implies a different classification of the individuals.  

 

In identifying INDIVIDUALS, only the columns of information (properties, relations, functions, ..) through which EACH INDIVIDUAL is viewed can be used to discern individuals, so what happens when multiple  individuals have the same value for all observable columns?  My view is to call that value a class.  So there may be classes resulting from individual views which would not even be CONSISTENT  with other individual views of exactly the same data.  Otherwise, why would there have been the four different views of sets invented?

 

Or have I missed the point of your posting?

 

-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 AzamatAbdoullaev
Sent: Monday, January 17, 2011 11:24 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

The issue of issues is how to classify the universe (world) of things. Or, which equivalence relation is most fundamental, and which equivalent classes constitute the fundamental set of ontological classes, mutually exclusive and collectively exhaustive. Living aside ontological dichotomies as abstracta//concreta, universals//particulars, being//existence, and substance/accident, the major partitions widely practiced might be: 

  • individuals and events,
  • objects and processes,
  • entities and relations,
  • entities, properties, relations 
  • entities, states, changes and relations.  

The rought set concepts could be useful tools in such a formal ontological division.

Azamat Abdoullaev

----- Original Message -----

From: Rich Cooper

Sent: Sunday, January 16, 2011 12:56 AM

Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

Hi Azamat,

 

But there is more.  We have definitions for four kinds of sets - fuzzy, probabilistic, rough, and crisp sets.  How many more KINDS of definitions (i.e., aspects like fuzzy, probabilistic, ..) are there in principal?  

 

I.e., with those four examples, is there a way to describe the ?signature? of how various aspects of reality are chosen to model sets?  If so, can the number four (4) be increased?  What limit could be placed on the increase, if any?

 

Your book looks great, but its too expensive for my budget!  Do you have an editor?s copy you can send me?

 

Thanks,

-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 AzamatAbdoullaev
Sent: Saturday, January 15, 2011 11:48 AM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

Rich,

Some comments to your stimulating questions.

Thanks.

Azamat

----- Original Message -----

From: Rich Cooper

Sent: Saturday, January 15, 2011 12:16 AM

Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

Azamat,

 

Agreed, rough set theory is a different view of sets, not probabilistic, not fuzzy, not ambiguous, but bounded and multivalent.  (more=>>

 

Sincerely,

Rich Cooper

EnglishLogicKernel.com

Rich AT EnglishLogicKernel DOT com

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From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of AzamatAbdoullaev
Sent: Friday, January 14, 2011 1:32 PM
To: [ontolog-forum]
Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

That's indeed an original idea. The rough sets as well as fuzzy sets, extending the classical sets as new mathematical tools,  are designed to cope with the things/concepts which are too ambiguous, vague and uncertain, like ontology is.

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

 

- nice book!

 

-Rich

 

 

Azamat Abdoullaev

 

 ----- Original Message -----

From: Rich Cooper

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

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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


From: ontolog-forum-bounces@xxxxxxxxxxxxxxxx [mailto:ontolog-forum-bounces@xxxxxxxxxxxxxxxx] On Behalf Of Tara Athan
Sent: Thursday, January 13, 2011 1:17 PM
Cc: '[ontolog-forum] '
Subject: Re: [ontolog-forum] Ontology of Rough Sets

 

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
707-272-2115 (cell, preferred)
707-485-1198 (office)
249 W. Gobbi St. #A
Ukiah, CA 95482

 

-- 
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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