Attribute Reduction Algorithms Determined by Invariants for Decision Tables - Liu, Guilong (Cognitive computation, 2022)
Rough set theory is a field of research pertaining to human-inspired computation. Attribute reduction is an important component of rough set theory and has been extensively studied. The reduction invariant is a key concept for an attribute reduction and finding new reduction invariants is an important task of attribute reduction. This paper explores the effect of different reduction invariants on the same attribute reduction types. The aim of this paper was to elucidate the mathematical structure of attribute reduction, thereby facilitating the use of new reduction invariants and their corresponding algorithms for positive region reduction and relative reduction in decision tables. New reduction invariants provide the opportunity to design significantly improved reduction algorithms. Two main reduction algorithms are used to identify reducts. One is a heuristic algorithm and the other is a discernibility matrix-based algorithm. Mathematically, the latter is far more complicated than the former. Although the discernibility matrix-based algorithm has a high time complexity, it remains the only approach to identify all reducts. This paper uses the discernibility matrix-based methods to study the attribute reduction problem. We focus on the mathematical structures of attribute reduction with respect to invariants and provide different algorithms to solve the same reduction problem. This research on reduction invariants provides a new perspective for attribute reduction. Positive region reduction and relative reduction are two frequently used types of reduction for decision tables. We provide three invariants for positive region reduction. Based on these invariants, we derive the corresponding discernibility matrix-based reduction algorithms that yield the same reduction results. For relative reduction, we also obtain similar results regarding invariants and algorithms. The shortcoming of this work is that we do not offer a simpler algorithm than the heuristic algorithm. However, the presented mathematical framework unifies previous work on the subject and is conducive to simplifying the associated decision tables for identifying all of the reducts.