On different ways to be (dis)similar to elements in a set. Boolean analysis and graded extension

Prade, H; Richard, G

On different ways to be (dis)similar to elements in a set. Boolean analysis and graded extension

Prade, H

Richard, G

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Publication Type:: Conference Proceeding
Citation:: Communications in Computer and Information Science, 2016, 611 pp. 605 - 618
Issue Date:: 2016-01-01

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Field	Value	Language
dc.contributor.author	Prade, H https://orcid.org/0000-0003-4586-8527	en_US
dc.contributor.author	Richard, G	en_US
dc.date.issued	2016-01-01	en_US
dc.identifier.citation	Communications in Computer and Information Science, 2016, 611 pp. 605 - 618	en_US
dc.identifier.isbn	9783319405803	en_US
dc.identifier.issn	1865-0929	en_US
dc.identifier.uri	http://hdl.handle.net/10453/122120
dc.description.abstract	© Springer International Publishing Switzerland 2016. We investigate here two questions, first in a Boolean setting and then in a gradual setting: Can we give a formal meaning to “being at odds” (in the sense of being an outlayer) with regard to a subset and, as a dual problem, can we give a meaning to “being even” (in the sense of conforming to a given set of values). Is there a relation between oddness and evenness? Such questions emerge from recent proposals for using oddness or evenness measures in classification problems. This paper is dedicated to a formal study of the oddness and evenness indices in the case of subsets with three or four elements, which are at the basis of the associated measures. Triples are indeed the only subsets such that adding an item that conforms to the triple minority, if any, destroys the majority. It appears that the notions of oddness and evenness are not simple dual of each other; a third notion of being “balanced” interplays with the two others. This is discussed in the setting of squares and hexagons of opposition. The notions of oddness and evenness are related to the study of homogeneous and heterogeneous logical proportions that link four Boolean variables through the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to pairs of these variables. Although elementary, the analysis provides an organized view of new notions that appear to be meaningful when revisiting the old ideas of similarity and dissimilarity in a new perspective. As a side result, it is also mentioned that the logical proportion underlying the idea of being balanced corresponds to the logical encoding of Bongard problems.	en_US
dc.relation.ispartof	Communications in Computer and Information Science	en_US
dc.relation.isbasedon	10.1007/978-3-319-40581-0_49	en_US
dc.title	On different ways to be (dis)similar to elements in a set. Boolean analysis and graded extension	en_US
dc.type	Conference Proceeding
utslib.citation.volume	611	en_US
utslib.for	0804 Data Format	en_US
utslib.for	0803 Computer Software	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Software
pubs.organisational-group	/University of Technology Sydney/Strength - CAI - Centre for Artificial Intelligence
utslib.copyright.status	closed_access
pubs.publication-status	Published	en_US
pubs.volume	611	en_US

Abstract:

© Springer International Publishing Switzerland 2016. We investigate here two questions, first in a Boolean setting and then in a gradual setting: Can we give a formal meaning to “being at odds” (in the sense of being an outlayer) with regard to a subset and, as a dual problem, can we give a meaning to “being even” (in the sense of conforming to a given set of values). Is there a relation between oddness and evenness? Such questions emerge from recent proposals for using oddness or evenness measures in classification problems. This paper is dedicated to a formal study of the oddness and evenness indices in the case of subsets with three or four elements, which are at the basis of the associated measures. Triples are indeed the only subsets such that adding an item that conforms to the triple minority, if any, destroys the majority. It appears that the notions of oddness and evenness are not simple dual of each other; a third notion of being “balanced” interplays with the two others. This is discussed in the setting of squares and hexagons of opposition. The notions of oddness and evenness are related to the study of homogeneous and heterogeneous logical proportions that link four Boolean variables through the conjunction of two equivalences between similarity or dissimilarity indicators pertaining to pairs of these variables. Although elementary, the analysis provides an organized view of new notions that appear to be meaningful when revisiting the old ideas of similarity and dissimilarity in a new perspective. As a side result, it is also mentioned that the logical proportion underlying the idea of being balanced corresponds to the logical encoding of Bongard problems.

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http://hdl.handle.net/10453/122120