Concepts and implications are two facets of the knowledge contained within a binary relation between objects and attributes. Simplification logic (SL) has proved to be valuable for the study of attribute implications in a concept lattice, a topic of interest in the more general framework of formal concept analysis (FCA). Specifically, SL has become the kernel of automated methods to remove redundancy or obtain different types of bases of implications. Although originally FCA used only the positive information contained in the dataset, negative information (explicitly stating that an attribute does not hold) has been proposed by several authors, but without an adequate set of equivalence-preserving rules for simplification. In this work, we propose a mixed simplification logic and a method to automatically remove redundancy in implications, which will serve as a foundational standpoint for the automated reasoning methods for this extended framework.
Formal concept analysis is a data analysis framework based on lattice theory. In this paper, we analyse the use, inside this framework, of positive and negative (mixed) attributes of a dataset, which has proved to represent more information on the use of just positive attributes. From a theoretical point of view, in this paper we show the structure and the relationships between minimal generators of the simple and mixed concept lattices. From a practical point of view, the obtained theoretical results allow us to ensure a greater granularity in the retrieved information. Furthermore, due to the relationship between FCA and Knowledge Space theory, on a practical level, we analyse the marks of a Mathematics course to establish the knowledge structure of the course and determine the key items providing new relevant information that is not evident without the use of the proposed tools.
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