Proof : Let

The asynchronous computability theorem for t-resilient tasks. A simple constructive computability theorem for wait-free computation. a spanning tree of G, starting (and ending) at a leaf, and then omitting the last two edges of the tour). Let V be an arbitrary set of n-dimensional vectors such that the adjacency graph G(V) is connected. We now complete the proof of Theorem 2.1 by presenting a wait-free protocol for n processes, P V , such that P V () = V. Let be a traversal of length L of G (L 2jV j ? 4). The protocol for p i is given below: 1. Execute the L-multi-consensus protocol MC L. 2. Let y i be the output of phase 1. p i decides on the i ? th component of the vector (y i). P V is clearly wait-free. We now prove that P V () V. By the Adjacency property of MC L , in each execution e of the protocol, there are two integers k and k+1 such that in phase 1 of the protocol every process decides either on k or on k + 1. Hence there are two adjacent vectors in G, (k) = ~ u = (u 1 ; ; u n) and (k+1) = ~ v = (v 1 ; ; v n), such that every process p i decides either on u i or on v i , and hence the output vector of that execution is either ~ u or ~ v. It remains to prove that every vector ~ v 2 V is the output of some execution of P V. Let ~ u = (u 1 ; ; u n) be any vertex in G. Then there is an integer k, 0 k L, such that (k) = ~ u. By the Multi Consensus property of MC L , there is an execution of MC L during phase 1 of the protocol, in which all the processes decide on k. By extending this execution to any execution in which all the processes are non-faulty, we get that each process p i decides on u i , and hence the output vector is ~ u. This completes the proof of Theorem 2.1. We have characterized the output sets deened by input-free, fault tolerant protocols, and showed that these sets need to preserve connectivity of the output graphs, but otherwise are arbitrary. When considering …