We present a new abstraction to replace the t of n assumption used in designing fault-tolerant algorithms. This abstraction models dependent process failures yet it is as simple to use as the t of n assumption. To illustrate this abstraction, we look at the Consensus problem for synchronous systems with both crash and arbitrary process failures. By considering failure correlations, we are able to reduce latency and enable the solution of Consensus for system configurations in which it is not possible when forced to use protocols designed under the t of n assumption. We give lower bounds for the number of rounds and replication requirements that are sufficient to solve Consensus. We show that, in general, the lower bound for number of rounds in the worst case assuming crash failures is different from the lower bound assuming arbitrary failures given the same system configuration. This is in contrast with the traditional result under the t of n assumption.
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