Two types of explanations have been receiving increased attention in the literature 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 explanation or a 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 also define and study.

翻译：在分析分类器决策时，文献中日益关注两类解释。第一类解释阐明为何做出某个决策，被称为决策的充分理由，亦可理解为溯因解释或PI-解释。第二类解释阐明为何未做出其他决策，被称为决策的必要理由，亦可理解为对比解释或反事实解释。这些解释最初针对具备二元、离散特征（某些情况下包括连续特征）的分类器而定义。研究表明，当特征为非二元时，这些解释可得到显著改进，从而形成一类能够传递更多决策信息及底层分类器信息的新型解释。必要理由与充分理由也被证实为决策完整理由的素蕴含项与素蕴含式，可通过量化算子获取。本文证明，改进后的必要理由与充分理由概念同样是素蕴含项与素蕴含式，但对应的是通过一种新型量化算子（本文亦对其进行了定义与研究）获得的改进型完整理由概念。