Quorum systems are a key mathematical abstraction in distributed fault-tolerant computing for capturing trust assumptions. A quorum system is a collection of subsets of all processes, called quorums, with the property that each pair of quorums have a non-empty intersection. They can be found at the core of many reliable distributed systems, such as cloud computing platforms, distributed storage systems and blockchains. In this paper we give a new interpretation of quorum systems, starting with classical majority-based quorum systems and extending this to Byzantine quorum systems. We propose an algebraic representation of the theory underlying quorum systems making use of multivariate polynomial ideals, incorporating properties of these systems, and studying their algebraic varieties. To achieve this goal we will exploit properties of Boolean Groebner bases. The nice nature of Boolean Groebner bases allows us to avoid part of the combinatorial computations required to check consistency and availability of quorum systems. Our results provide a novel approach to test quorum systems properties from both algebraic and algorithmic perspectives.

中文翻译：

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群体系统的代数模型

仲裁系统是分布式容错计算中用于捕获信任假设的关键数学抽象。仲裁系统是所有进程的子集的集合，称为仲裁，其特性是每对仲裁都有一个非空交集。它们位于许多可靠的分布式系统的核心，例如云计算平台、分布式存储系统和区块链。在本文中，我们对仲裁系统进行了新的解释，从经典的基于多数的仲裁系统开始，并将其扩展到拜占庭仲裁系统。我们提出了群体系统理论的代数表示，利用多元多项式理想，结合这些系统的特性，并研究它们的代数变体。为了实现这个目标，我们将利用布尔 Groebner 基的特性。Boolean Groebner 基的良好性质使我们能够避免检查群体系统的一致性和可用性所需的部分组合计算。我们的结果提供了一种从代数和算法角度测试群体系统属性的新方法。