The Structure of Oppositions in Rough Set Theory and Formal Concept Analysis - Toward a New Bridge between the Two Settings

Ciucci, Davide; Dubois, Didier; Prade, Henri

doi:10.1007/978-3-319-04939-7_7

Cited by 26 publications

(24 citation statements)

References 39 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Demey

Smessaert

2017

Symmetry

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Aristotelian diagrams visualize the logical relations among a finite set of objects. These diagrams originated in philosophy, but recently, they have also been used extensively in artificial intelligence, in order to study (connections between) various knowledge representation formalisms. In this paper, we develop the idea that Aristotelian diagrams can be fruitfully studied as geometrical entities. In particular, we focus on four polyhedral Aristotelian diagrams for the Boolean algebra B 4 , viz. the rhombic dodecahedron, the tetrakis hexahedron, the tetraicosahedron and the nested tetrahedron. After an in-depth investigation of the geometrical properties and interrelationships of these polyhedral diagrams, we analyze the correlation (or lack thereof) between logical (Hamming) and geometrical (Euclidean) distance in each of these diagrams. The outcome of this analysis is that the Aristotelian rhombic dodecahedron and tetrakis hexahedron exhibit the strongest degree of correlation between logical and geometrical distance; the tetraicosahedron performs worse; and the nested tetrahedron has the lowest degree of correlation. Finally, these results are used to shed new light on the relative strengths and weaknesses of these polyhedral Aristotelian diagrams, by appealing to the congruence principle from cognitive research on diagram design.

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Section: Introductionmentioning

confidence: 99%

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Demey

Smessaert

2017

Symmetry

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“…Recently, it has been noticed that such a square can be generated by a binary relation and a subset that can be composed together [12]. Indeed, let R be a binary relation on a Cartesian product X × Y (nothing forbids Y = X in the construction we are going to describe).…”

Section: The Cube Of Opposition In Fcamentioning

confidence: 99%

“…Lastly, it can be checked that we also have R The cube of oppositions not only underlie FCA (and PoTh) [25], but also is a setting of interest for building bridges with rough set theory [40] (see [12]), or even formal argumentation [1]!…”

Section: The Cube Of Opposition In Fcamentioning

confidence: 99%

Formal Concept Analysis from the Standpoint of Possibility Theory

Dubois

Prade

2015

Formal Concept Analysis

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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.

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“…In addition, a large amount of computing time and resources are consumed for the input data of an evaluation system when the data is directly analyzed using a WNN. Therefore, a model for production performance evaluation was proposed based on the WNN theory, which combines data preprocessed by rough sets (RS) [2] in order to improve the speed of solving problems using the WNN, simplify the structure of the neural network, reduce the input variables, training steps, and time, and speed up network learning, as well as enhance the accuracy of the judgments. Figure 1 shows the production performance evaluation model.…”

Section: Production Performance Evaluation Modelmentioning

confidence: 99%

“…The influence degree of each index was transformed into the weight, and then the weight of each index was normalization processed by Equation (2).…”

Section: The Reduction Of the Production Performance Evaluation Indexmentioning

confidence: 99%

Production performance evaluation based on rough set theory and wavelet neural network

Song

Jin

2015

IFS

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Abstract. Aimed at overcoming subjectivity and improving the accuracy of traditional production performance evaluation methods for manufacturing enterprises, a new model of performance evaluation was proposed based on rough sets and a wavelet neural network (RS -WNN). Firstly, an evaluation index system considering innovation performance was constructed. Secondly, a theory of rough sets and fuzzy mathematics was utilized to preprocess and simplify the index system, and then, the input dimensionality of wavelet neural network was reduced. Finally, algorithms of stepwise checkout and iterative descending grads were employed to decide the parameters of WNN and to obtain the synthetic evaluation value of production performance. A case study showed that the proposed model was effective and feasible in measuring production performance.

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The Structure of Oppositions in Rough Set Theory and Formal Concept Analysis - Toward a New Bridge between the Two Settings

Cited by 26 publications

References 39 publications

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

Formal Concept Analysis from the Standpoint of Possibility Theory

Production performance evaluation based on rough set theory and wavelet neural network

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