Recent advances in nanotechnology have led to the emergence of energy efficient circuit technologies like spintronics and domain wall magnets that use Majority gates as their primary logic elements. For logic synthesis methods targeted to such technologies to be effective and efficient, they need to be able to use, manipulate, and exploit large Majority terms in their synthesis flow. One of the problems that turn up in such an endeavor is the determination of equivalence of two n-input Majority terms. As Majority gates implement more complex Boolean functions than traditional AND/OR gates, this is a non-trivial problem—one that demands to be solved before proceeding to harder problems dealing with networks of Majority gates. We provide an algorithm based on prime implicants as a solution to this problem. In addition, we provide an algorithm that compacts an n-input Majority term to an equivalent n-input Majority term that has the fewest number of inputs. In this quest, we extend the concept of implicants to two cases — 1-implicants and prime 1-implicants that imply that a function evaluates to ‘1’, and 0-implicants and prime 0-implicants that imply that it evaluates to ‘0’. We exploit the properties of Majority to create an efficient algorithm to generate the sums of all prime 1-implicants and all prime 0-implicants of an n-input Majority term. As Majority and Threshold functions have been shown to be logically equivalent, our algorithms are applicable to Threshold functions as well. Being based on implicants of Majority, our algorithms improve on the known naive algorithms for equivalence checking and compaction for threshold logic terms.

中文翻译：

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使用多数蕴涵的 n 输入多数项的等价检查和压缩

纳米技术的最新进展导致了节能电路技术的出现，例如使用多数门作为其主要逻辑元件的自旋电子学和畴壁磁铁。为了使针对此类技术的逻辑综合方法有效且高效，它们需要能够在其综合流程中使用、操纵和利用大众数项。在这种努力中出现的问题之一是确定两个 n 输入多数项的等价性。由于多数门实现了比传统 AND/OR 门更复杂的布尔函数，因此这是一个非常重要的问题——在处理多数门网络的更难问题之前需要先解决这个问题。我们提供了一种基于素蕴涵的算法来解决这个问题。此外，我们提供了一种算法，将 n 输入多数项压缩为具有最少输入数量的等效 n 输入多数项。在这个探索中，我们将蕴涵的概念扩展到两种情况——1-蕴涵和素数 1-蕴涵意味着函数的计算结果为“1”，以及 0-蕴涵和素数 0-蕴涵意味着它的计算结果为“0” '。我们利用 Majority 的属性来创建一种有效的算法，以生成 n 输入多数项的所有质数 1-蕴涵项和所有质数 0-蕴涵项的总和。由于已证明多数函数和阈值函数在逻辑上是等效的，因此我们的算法也适用于阈值函数。基于多数的蕴涵，我们的算法改进了已知的朴素算法，用于阈值逻辑项的等价检查和压缩。