Boolean functions and their representation through logics, circuits, machine learning classifiers, or binary decision diagrams (BDDs) play a central role in the design and analysis of computing systems. Quantifying the relative impact of variables on the truth value by means of importance values can provide useful insights to steer system design and debugging. In this paper, we introduce a uniform framework for reasoning about such values, relying on a generic notion of importance value functions (IVFs). The class of IVFs is defined by axioms motivated from several notions of importance values introduced in the literature, including Ben-Or and Linial's influence and Chockler, Halpern, and Kupferman's notion of responsibility and blame. We establish a connection between IVFs and game-theoretic concepts such as Shapley and Banzhaf values, both of which measure the impact of players on outcomes in cooperative games. Exploiting BDD-based symbolic methods and projected model counting, we devise and evaluate practical computation schemes for IVFs.

翻译：布尔函数及其通过逻辑、电路、机器学习分类器或二元决策图（BDDs）的表达在计算系统的设计与分析中扮演着核心角色。通过重要性度量量化变量对真值的影响程度，能为系统设计导向与调试提供重要洞见。本文提出一种关于此类度量的统一推理框架，该框架基于泛化的重要性值函数（IVFs）概念。IVF类由文献中提出的多种重要性度量概念（包括Ben-Or与Linial的影响力，以及Chockler、Halpern和Kupferman的责任与归因概念）所激发的公理定义。我们建立了IVFs与博弈论概念（如Shapley值和Banzhaf值）之间的关联，后者衡量合作博弈中玩家对结果的贡献度。借助基于BDD的符号化方法与投影模型计数，我们设计并评估了IVFs的实用计算方案。