Two types of explanations have received significant attention in the literature recently when analyzing the decisions made by classifiers. The first type explains why a decision was made and is known as a sufficient reason for the decision, also an abductive or PI-explanation. The second type explains why some other decision was not made and is known as a necessary reason for the decision, also a contrastive or counterfactual explanation. These explanations were defined for classifiers with binary, discrete and, in some cases, continuous features. We show that these explanations can be significantly improved in the presence of non-binary features, leading to a new class of explanations that relay more information about decisions and the underlying classifiers. Necessary and sufficient reasons were also shown to be the prime implicates and implicants of the complete reason for a decision, which can be obtained using a quantification operator. We show that our improved notions of necessary and sufficient reasons are also prime implicates and implicants but for an improved notion of complete reason obtained by a new quantification operator that we define and study in this paper.

翻译：近年来，文献中关注的两类解释在分析分类器决策时受到广泛重视。第一类解释阐明决策为何做出，被称为决策的充分理由，也称为溯因解释或PI解释。第二类解释说明为何未做出其他决策，被称为决策的必要理由，亦称为对比解释或反事实解释。这些解释最初是针对具有二元、离散以及在特定情况下连续特征的分类器定义的。研究表明，在非二元特征存在时，这些解释可得到显著改进，从而形成一类能传递更多关于决策及底层分类器信息的新解释。必要理由与充分理由被证明是决策完整理由的素蕴含项与素隐含项，可通过量化算子获得。本文证明，我们改进后的必要理由与充分理由新概念仍是素蕴含项与素隐含项，但对应的是由本文定义并研究的新型量化算子所获得的改进型完整理由概念。