Formal Concept Analysis from the Standpoint of Possibility Theory

Abstract

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15496The contribution was presented at : http://www.matap.uma.es/icfca2015/files/a61b70b0124cd72ef5b1442c0223db4a-0.html Abstract. Formal concept analysis (FCA) and possibility theory (PoTh) have been developed independently. They address different concerns in information processing: while FCA exploits relations linking objects and properties, and has applications in data mining and clustering, PoTh deals with the modeling of (graded) epistemic uncertainty. However, making a formal parallel between FCA and PoTh is fruitful. The four set-functions at work in PoTh have meaningful counterparts in FCA; this leads to consider operators neglected in FCA, and thus new fixed point equations. One of these pairs of equations, paralleling the one defining formal concepts in FCA, defines independent sub-contexts of objects and properties that have nothing in common. The similarity of the structures underlying FCA and PoTh is still more striking, using a cube of opposition (a device extending the traditional square of opposition in logic). Beyond the parallel between FCA and PoTh, this invited contribution, which largely relies on several past publications by the authors, also addresses issues pertaining to the possible meanings, degree of satisfaction vs. degree of certainty, of graded object-property links, which calls for distinct manners of handling the degrees. Other lines of interest for further research are briefly mentioned.

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…Within the latter field, Aristotelian diagrams have been used to study various logic-based approaches to knowledge representation, including fuzzy logic [16][17][18][19][20], modal-epistemic logic [21][22][23][24][25] and probabilistic logic [26][27][28]. Furthermore, Aristotelian diagrams are also used extensively to study (the connections between) other types of knowledge representation formalisms, such as formal argumentation theory [29][30][31][32], fuzzy set theory [33][34][35][36], formal concept analysis and possibility theory [37][38][39], rough set theory [37,40,41], multiple-criteria decision-making [42][43][44] and the theory of logical and analogical proportions [45][46][47][48][49]. In sum, then, Aristotelian diagrams have come to serve as visual tools that greatly facilitate communication, research and teaching in a wide variety of disciplines that deal with logical reasoning in all its facets.…”

Asymmetric Composition of Possibilistic Operators in Formal Concept Analysis: Application to the Extraction of Attribute Implications from Incomplete Contexts

“…are changed into (x, a, +) give birth to lower and upper completions, respectively. 23,24 In this way, two classical (Boolean) formal contexts, denoted by K * and K * are obtained as extreme results of the two replacements. More formally: There exist exactly 2 n formal contexts obtained by arbitrarily replacing each "?"…”

Logic-Sensitivity of Aristotelian Diagrams in Non-Normal Modal Logics

“…Aristotelian diagrams, such as the so-called square of opposition, visualize a number of formulas from some logical system, as well as certain logical relations holding between them. These diagrams have a rich history in philosophy and logic [1][2][3], and today they are also widely used in artificial intelligence, to study and compare knowledge representation formalisms such as the rough set theory [4][5][6], formal concept analysis and possibility theory [7][8][9], formal argumentation theory [10][11][12][13], fuzzy set theory [14][15][16][17], logical theories of analogical and proportional reasoning [18][19][20][21][22][23], probabilistic logic [24][25][26] and multiple-criterion decision-making [27][28][29]. Without a doubt, the oldest and most widely used Aristotelian diagram is the square of opposition for the categorical statements from syllogistics, such as 'all Greeks are mortal' and 'some Greeks are mortal'.…”