An Axiomatic System for Conditional Attribute Implications in Triadic Concept Analysis |
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| Authors: | Estrella Rodríguez‐Lorenzo Pablo Cordero Manuel Enciso Rokia Missaoui Ángel Mora |
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| Affiliation: | 1. Departamento de Matemática Aplicada, Andalucía Tech, Universidad de Málaga, Málaga, Spain;2. Departamento de Lenguajes y Ciencias de la Computación, Andalucía Tech, Universidad de Málaga, Málaga, Spain;3. Department of Computer Science and Engineering, University of Quebec in Outaouais, Gatineau, Canada |
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| Abstract: | In this paper, we define a sound and complete inference system for triadic implications generated from a formal triadic context  , where G, M, and B are object, attribute, and condition sets, respectively, and I is a ternary relation  . The inference system is expressed as a set of axioms “à la Armstrong.” The type of triadic implications we are considering in this paper is called conditional attribute implication (CAI) and has the following form:  , where X and Y are subsets of M, and  is a subset of B. Such implication states that Ximplies Y under all conditions in  and any subset of it. Moreover, we propose a method to compute CAIs from Biedermann's implications. We also introduce an algorithm to compute the closure of an attribute set X w.r.t. a set Σ of CAIs given a set  of conditions. |
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, where G, M, and B are object, attribute, and condition sets, respectively, and I is a ternary relation 
. The inference system is expressed as a set of axioms “à la Armstrong.” The type of triadic implications we are considering in this paper is called conditional attribute implication (CAI) and has the following form: 
, where X and Y are subsets of M, and 
is a subset of B. Such implication states that Ximplies Y under all conditions in 
and any subset of it. Moreover, we propose a method to compute CAIs from Biedermann's implications. We also introduce an algorithm to compute the closure of an attribute set X w.r.t. a set Σ of CAIs given a set 
of conditions.