This paper employs switching-algebraic techniques for the calculation of a fundamental index of voting powers, namely, the total Banzhaf power. This calculation involves two distinct operations: (a) Boolean differencing or differentiation, and (b) computation of the weight (the number of true vectors or minterms) of a switching function. Both operations can be considerably simplified and facilitated if the pertinent switching function is symmetric or it is expressed in a disjoint sum-of-products form. We provide a tutorial exposition on how to implement these two operations, with a stress on situations in which partial symmetry is observed among certain subsets of a set of arguments. We introduce novel Boolean-based symmetry-aware techniques for computing the Banzhaf index by way of two prominent voting systems. These are scalar systems involving six variables and nine variables, respectively. The paper is a part of our ongoing effort for transforming the methodologies and concepts of voting systems to the switching-algebraic domain, and subsequently utilizing switching-algebraic tools in the calculation of pertinent quantities in voting theory.

翻译：本文采用开关代数技术计算投票权力的基本指数，即总Banzhaf权力。该计算涉及两个不同操作：(a)布尔差分或微分，以及(b)开关函数权重（即真向量或最小项数量）的计算。若相关开关函数具有对称性，或以不相交积之和形式表示，这两个操作可被显著简化和便利化。我们提供了这两种操作的教程式阐述，重点讨论论点集合某子集存在部分对称性的情形。通过两个著名的投票系统，我们引入了新颖的基于布尔理论的对称感知方法用于计算Banzhaf指数，这两个系统分别为涉及六个变量和九个变量的标量系统。本文是我们持续将投票系统方法论与概念转化至开关代数领域，并随后利用开关代数工具计算投票理论中相关量工作的组成部分。