Counting-based impossibility proofs for set agreement and renaming

Set agreement and renaming are two tasks that allow processes to coordinate, even when agreement is impossible. In k-set agreement, n processes must decide on at most k of their input values. While n-set agreement is trivially wait-free solvable by each process deciding on its input, (n-1)-set agreement is not wait-free solvable. In M-renaming, processes must decide on distinct names in a range of size M. For any number n of processes, (2n-1)-renaming is wait-free solvable, but surprisingly, (2n-2)-renaming is wait-free solvable if and only if n is not a prime power; the only previous lower bound on the number of names necessary for renaming, when n is not a prime power, is n+1. In adaptive renaming, M decreases when the number p of participants in the execution decreases. It is known that (2p-1)-adaptive renaming is wait-free solvable, while (2p-pn-1-adaptive renaming is not. This paper presents counting-based proofs for the above mentioned impossibility results: n processes can wait-free solve neither (n-1)-set agreement nor (2p-pn-1-adaptive renaming; if n is a prime power, n processes cannot wait-free solve (2n-2)-renaming. For an arbitrary number of processes, we give a lower bound for renaming, by reduction from renaming for a different number of processes, and relying on the distribution of prime numbers. Our proofs combine simple operational properties of a restricted set of executions with elementary counting arguments to show the existence of an execution violating the task's conditions. This makes the proofs easier to understand, verify, and, we hope, extend.