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.

粗糙集理论是一个与人类启发计算有关的研究领域。属性约简是粗糙集理论的重要组成部分，已被广泛研究。约简不变量是属性约简的关键概念，寻找新的约简不变量是属性约简的重要任务。本文探讨了不同归约不变量对相同属性归约类型的影响。本文的目的是阐明属性约简的数学结构，从而促进使用新的约简不变量及其相应算法在决策表中进行正区域约简和相对约简。新的归约不变量为设计显着改进的归约算法提供了机会。两种主要的归约算法用于识别归约。一种是启发式算法，另一种是基于辨别矩阵的算法。在数学上，后者远比前者复杂。尽管基于识别矩阵的算法具有很高的时间复杂度，但它仍然是识别所有约简的唯一方法。本文采用基于识别矩阵的方法来研究属性约简问题。我们专注于关于不变量的属性约简的数学结构，并提供不同的算法来解决相同的约简问题。对约简不变量的研究为属性约简提供了新的视角。正区域缩减和相对缩减是决策表的两种常用缩减类型。我们为正区域减少提供了三个不变量。基于这些不变量，我们推导出相应的基于辨别矩阵的归约算法，这些算法产生相同的归约结果。对于相对减少，我们在不变量和算法方面也获得了类似的结果。这项工作的缺点是我们没有提供比启发式算法更简单的算法。然而，所提出的数学框架统一了之前关于该主题的工作，并有助于简化用于识别所有归约的相关决策表。