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

 Cc: "'[ontolog-forum] '" Tara Athan Thu, 13 Jan 2011 13:17:04 -0800 <4D2F6BD0.9020408@xxxxxxxxxx>
 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 _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx ``` ``` ``` ```-- 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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 Current Thread Re: [ontolog-forum] Ontology of Rough Sets, Tara Athan Re: [ontolog-forum] Ontology of Rough Sets, Ronald Stamper Re: [ontolog-forum] Ontology of Rough Sets, John F. Sowa Re: [ontolog-forum] Ontology of Rough Sets, Tara Athan Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, Tara Athan <= Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, Tara Athan Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, AzamatAbdoullaev Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, Christopher Menzel Re: [ontolog-forum] Ontology of Rough Sets, Tara Athan Re: [ontolog-forum] Ontology of Rough Sets, Christopher Menzel Re: [ontolog-forum] Ontology of Rough Sets, Rich Cooper Re: [ontolog-forum] Ontology of Rough Sets, Christopher Menzel