Computing the Mixed Concept Lattice

Abstract

The classical approach on Formal Concept Analysis (FCA) extracts knowledge from a binary table $\mathbb {K}=(G,M,I)$ taking into account the existing relationships (given by the binary relation I) between objects G and attributes M. Thus, this classical setting accounts only for positive information. Particularly, FCA allows to define and compute the concept lattice $\underline{\mathbb {B}}(\mathbb {K})$ from this positive information. As an extension of this framework, some works consider not only this positive information, but also the negative information that is explicit when objects have no relation to specific attributes (denoted by $\overline{\mathbb {K}}$). These works, therefore, use the apposition of positive and negative information to compute the mixed concept lattice $\underline{\mathbb {B}}^{\#}(\mathbb {K})$. In this paper, we propose to establish the relationships between extents and intents of concepts in $\underline{\mathbb {B}}(\mathbb {K})$, ${\underline{\mathbb {B}}(\overline{\mathbb {K}})}$ and $\underline{\mathbb {B}}^{\#}(\mathbb {K})$ and how to address an incremental algorithm to compute $\underline{\mathbb {B}}^{\#}(\mathbb {K})$ merging the knowledge on $\underline{\mathbb {B}}(\mathbb {K})$, $\underline{\mathbb {B}}(\overline{\mathbb {K}})$ previously obtained with classical methods.

Partially supported by the Spanish Ministry of Science, Innovation, and Universities (MCIU), State Agency of Research (AEI), Junta de Andalucía (JA), Universidad de Málaga (UMA) and European Regional Development Fund (FEDER) through the projects PGC2018-095869-B-I00 (MCIU/AEI/FEDER), TIN2017-89023-P (MCIU/AEI/FEDER), PRE2018-085199 and UMA2018-FEDERJA-001 (JA/UMA/FEDER).