Simplifying Implications with Positive and Negative Attributes: A Logic-Based Approach

Francisco Pérez-Gámez,Pablo Cordero,Manuel Ojeda-Aciego,Domingo López-Rodríguez,Ángel Mora

doi:10.3390/math10040607

Francisco Pérez-Gámez, Pablo Cordero + Show 3 more

Open Access

https://doi.org/10.3390/math10040607

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Journal: Mathematics	Publication Date: Feb 16, 2022
Citations: 4	License type: CC BY 4.0

Affiliation: Universidad de Málaga

Abstract

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.

Highlights

Since the 1980s, formal concept analysis (FCA) has been a solid framework to analyze data and extract hidden knowledge, comparable to other well-known techniques in terms of cost
We focus on exact association rules; we resort to the framework of FCA, where the first occurrences of negative information appear in Missaoui et al [10,11], in which the authors computed mixed implications from a double context formed by the initial context together with its opposite
Formal concept analysis (FCA) [20,21] is a mathematical theory based on lattice theory which analyses the information given in a formal context, i.e., a relationship between a set of objects and a set of attributes stored in a table

Summary

Introduction

Since the 1980s, formal concept analysis (FCA) has been a solid framework to analyze data and extract hidden knowledge, comparable to other well-known techniques in terms of cost. In the same process, FCA returns sets of implications and/or association rules (well-known in other areas such as data mining, machine learning, and rough set theories) with a rich algebraic framework in which we can compute closed sets and their minimal generators, pseudointents, different types of bases, etc. This knowledge can reveal interesting patterns and solve significant problems in modern areas as social network analysis [2,3] or recommender systems [4,5]. For “ostrich”, the positive information is relevant (“large”, “heavy”, “fast”); the negative information (“does not fly”) is relevant

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