Consider a point-to-point network in which nodes are connected by directed links. This paper proves tight necessary and sufficient conditions on the underlying communication graphs for solving the following fault-tolerant consensus problems: Exact crash-tolerant consensus in synchronous systems, Approximate crash-tolerant consensus in asynchronous systems, and Exact Byzantine consensus in synchronous systems. The problem of asynchronous Byzantine consensus in directed graphs remains open. Prior work has developed analogous necessary and sufficient conditions for undirected graphs [11,3,9]. However, the conditions for undirected graphs are not adequate to completely characterize these consensus problems in directed graphs. Moreover, the algorithms for directed graphs presented in this paper are substantially different from those previously developed for undirected graphs. Other prior work on directed graphs has explored somewhat different problems than those solved in this paper.

The CAP theorem is a fundamental result that applies to distributed storage systems. In this article, we first present and prove two CAP-like impossibility theorems. To state these theorems, we present probabilistic models to characterize the three important elements of the CAP theorem: consistency (C), availability or latency (A), and partition tolerance (P). The theorems show the un-achievable envelope, that is, which combinations of the parameters of the three models make them impossible to achieve together. Next, we present the design of a class of systems called Probabilistic CAP (PCAP) that perform close to the envelope described by our theorems. In addition, these systems allow applications running on a single data center to specify either a latency Service Level Agreement (SLA) or a consistency SLA. The PCAP systems automatically adapt, in real time and under changing network conditions, to meet the SLA while optimizing the other C/A metric. We incorporate PCAP into two popular key-value stores: Apache Cassandra and Riak. Our experiments with these two deployments, under realistic workloads, reveal that the PCAP systems satisfactorily meets SLAs and perform close to the achievable envelope. We also extend PCAP from a single data center to multiple geo-distributed data centers.

Industry 5.0 is emerging as a result of the advancement in networking and communication technologies, artificial intelligence, distributed computing, and beyond 5G. Among the important enabling technologies, federated learning, industrial edge computing, and Byzantine-tolerant machine learning (ML) are key accelerators in Industry 5.0. We propose a framework to integrate these key components. Recent works have designed various Byzantine-tolerant ML algorithms for a datacenter or a cluster. However, these algorithms are difficult to be applied to industrial edge computing paradigms. In this article, a novel Byzantine-tolerant federated learning algorithm, CRACAU, is designed for the popular three-level edge computing architecture. In this algorithm, edge devices jointly learn an ML model using the data collected at each device, and their private data are never shared with others. Under standard assumptions, we formally prove that CRACAU converges to the optimal point, i.e., CRACAU finds the optimal parameters of the ML model. We also implement CRACAU in the MXNet framework and evaluate it on the popular benchmark MNIST and CIFAR-10 image classification datasets. Experimental results show that CRACAU achieves satisfying accuracy.