An Algebraic Model For Quorum Systems

TL;DR

This paper introduces an algebraic framework using polynomial ideals and Boolean Groebner bases to analyze and verify properties of quorum systems, enhancing understanding and testing in distributed computing.

Contribution

It presents a novel algebraic interpretation of quorum systems, extending classical models to Byzantine systems and enabling algebraic and algorithmic property verification.

Findings

Algebraic representation of quorum systems using polynomial ideals.

Application of Boolean Groebner bases to simplify consistency checks.

New methods for testing quorum system properties algebraically.

Abstract

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.