No abstract available.

This paper presents a weighted version of the Byzantine Agreement Problem and its solution under various conditions. In this version, each machine is assigned a weight depending on the application. Instead of assuming that at most f out of N machines fail, the algorithm assumes that the total weight of the machines that fail is at most f/N. When each machine has weight 1/N, this problem reduces to the standard Byzantine Generals Agreement Problem. By choosing weights appropriately, the weighted Byzantine Agreement Problem can be applied to situations where a subset of processes are more trusted. By using weights, the system can reach consensus in the presence of Byzantine failures, even when more than N/3 processes fail, so long as the total weight of the failed processes is less than 1/3. Also, a method to update the weights of the processes after execution of the weighted Byzantine Agreement is given. The update method guarantees that the weight of any correct process is never reduced and the weight of any faulty process, suspected by correct processes whose total weight is at least 1/4, is reduced to 0 for future instances. A short discussion of some weight assignment strategies is also given.

The Byzantine Agreement (BA) problem requires non-faulty processes to agree on a common value. In many applications, it is important that the processes agree on the correct value. In this paper, we present a problem called Accurate Byzantine Agreement with Feedback (ABAF) in which all processes receive common feedback from the environment indicating if the value they agreed upon was correct or not (accuracy). We present an algorithm that solves the ABAF problem based on a standard solution to the BA problem and a multiplicative method to maintain and update process weights indicative of how often they are correct. We make guarantees on the accuracy of the algorithm based on assumptions on the accuracy of the processes and the proportion of faulty and non-faulty processes in the system. For each iteration, if the weight of accurate processes is at least 3/4th the weight of the non-faulty processes, the algorithm always decides on the correct value. When the non-faulty processes are accurate with probability greater than 1/2, the algorithm decides on the correct value with very high probability after some initial number of mistakes. In fact, among n processes, if there exists even one process which is accurate for all iterations, the algorithm is wrong only O(log n) times for any large number of iterations of the algorithm.