We present an 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 consider the Consensus problem for synchronous and asynchronous systems with both crash and arbitrary process failures. We give process replication requirements for our model and describe algorithms for system configurations satisfying these requirements. By considering failure correlations, we enable the solution of Consensus for system configurations with arbitrary failures in which it is not possible when forced to use protocols designed under the t of n assumption. Additionally, we are able to solve Consensus with fewer rounds in synchronous systems with crash failures.
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