An Algebraic Model For Quorum Systems

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.Comment: 15 pages, 3 algorithm

Brief Announcement: Asymmetric Distributed Trust

Quorum systems are a key abstraction in distributed fault-tolerant computing for capturing trust assumptions. They can be found at the core of many algorithms for implementing reliable broadcasts, shared memory, consensus and other problems. This paper introduces asymmetric Byzantine quorum systems that model subjective trust. Every process is free to choose which combinations of other processes it trusts and which ones it considers faulty. Asymmetric quorum systems strictly generalize standard Byzantine quorum systems, which have only one global trust assumption for all processes. This work also presents protocols that implement abstractions of shared memory and broadcast primitives with processes prone to Byzantine faults and asymmetric trust. The model and protocols pave the way for realizing more elaborate algorithms with asymmetric trust

Asymmetric Distributed Trust

Quorum systems are a key abstraction in distributed fault-tolerant computing for capturing trust assumptions. They can be found at the core of many algorithms for implementing reliable broadcasts, shared memory, consensus and other problems. This paper introduces asymmetric Byzantine quorum systems that model subjective trust. Every process is free to choose which combinations of other processes it trusts and which ones it considers faulty. Asymmetric quorum systems strictly generalize standard Byzantine quorum systems, which have only one global trust assumption for all processes. This work also presents protocols that implement abstractions of shared memory and broadcast primitives with processes prone to Byzantine faults and asymmetric trust. The model and protocols pave the way for realizing more elaborate algorithms with asymmetric trust

Stellar Consensus by Instantiation

Stellar introduced a new type of quorum system called a Federated Byzantine Agreement System. A major difference between this novel type of quorum system and a threshold quorum system is that each participant has its own, personal notion of a quorum. Thus, unlike in a traditional BFT system, designed for a uniform notion of quorum, even in a time of synchrony one well-behaved participant may observe a quorum of well-behaved participants, while others may not. To tackle this new problem in a more general setting, we abstract the Stellar Network as an instance of what we call Personal Byzantine Quorum Systems. Using this notion, we streamline the theory behind the Stellar Network, removing the clutter of unnecessary details, and refute the conjecture that Stellar\u27s notion of intact set is optimally fault-tolerant. Most importantly, we develop a new consensus algorithm for the new setting

The Evolution of Quorum Sensing as a Mechanism to Infer Kinship.

Bacteria regulate many phenotypes via quorum sensing systems. Quorum sensing is typically thought to evolve because the regulated cooperative phenotypes are only beneficial at certain cell densities. However, quorum sensing systems are also threatened by non-cooperative "cheaters" that may exploit quorum-sensing regulated cooperation, which begs the question of how quorum sensing systems are maintained in nature. Here we study the evolution of quorum sensing using an individual-based model that captures the natural ecology and population structuring of microbial communities. We first recapitulate the two existing observations on quorum sensing evolution: density-dependent benefits favor quorum sensing but competition and cheating will destabilize it. We then model quorum sensing in a dense community like a biofilm, which reveals a novel benefit to quorum sensing that is intrinsically evolutionarily stable. In these communities, competing microbial genotypes gradually segregate over time leading to positive correlation between density and genetic similarity between neighboring cells (relatedness). This enables quorum sensing to track genetic relatedness and ensures that costly cooperative traits are only activated once a cell is safely surrounded by clonemates. We hypothesize that under similar natural conditions, the benefits of quorum sensing will not result from an assessment of density but from the ability to infer kinship