Impossibility results in the presence of multiple faulty processes

We investigate the possibility of solving certain problems in an unreliable asynchronous distributed system where multiple processes may fail. We assume undetectable crash failures, which means that a process may become faulty at any time during execution and that no event can occur on a process after it fails. A sufficient condition is provided for the unsolvability of problems in the presence of multiple faulty processes. Families of problems are shown to be solvable in the presence of t - 1 faulty processes but unsolvable in the presence of t faulty processes for any t. These are variants of problems unsolvable in the presence of a single faulty process such as consensus, choosing a leader, or renaming. Consequently, we exhibit a strict solvability hierarchy of classes of problems, depending on the number of failures. In order to prove the impossibility result a contradiction is shown among a set of axioms that characterize any fault-tolerant protocol solving the problems we treat. Six axioms are presented that define the essential properties of asynchronous computation. An additional axiom defines a protocol to be nontrivial if in some execution n - t processes have their input values read, and yet the output value for one of the processes is still undetermined. In the course of the proof, we present two results of independent interest. We show that any nontrivial asynchronous protocol must have a splitter process, regardless of any faults. Intuitively, if left to run on its own at some point, a splitter -nay force choosing either of two distinct output values for some (possibly other) process. Then we show that in any nontrivial protocol that tolerates up to t ≥ 1 crash failures, such a splitter must be a decider, where a decider is a splitter for its own values.