Tight group renaming on groups of size g is equivalent to g-consensus

We address two problems, the g-tight group renaming task and what we call, safe-consensus task, and show the relations between them. We show that any g-tight group renaming task, the first problem, implements g processes consensus. We show this by introducing an intermediate task, the safe-consensus task, the second problem, and showing that g-tight group renaming implements g-safe-consensus and that the latter implements g-consensus. It is known that with g-consensus g-tight group renaming is solvable, making the two problems equivalent. The safe-consensus task, is of independent interest. In it the validity condition of consensus is weakened as follows: if the first processor to invoke the task returns before any other processor invokes, i.e., it runs in solo, then it outputs its input; Otherwise the consensus output can be arbitrary, not even the input of any process. We show the equivalence between safe-(set-)consensus and (set-)consensus.