Agreement via Symmetry Breaking: On the Structure of Weak Subconsensus Tasks

This paper is on the relative power and the relations linking two important synchronization problems in n-process wait-free shared memory models, namely, set agreement and renaming, which are two of the most studied subconsensus tasks. Since the 2006 seminal paper of Gafni, Rajsbaum and Herlihy, it is known that some renaming instances are strictly weaker than set agreement. Indeed, it was later on shown that not even (n + 1)-renaming (the strongest task in the renaming family, after perfect n-renaming) can implement (n - 1)-set agreement (the weakest non-trivial task in the set agreement family). These and other results seem to imply that renaming and, more generally, the tasks called generalized symmetry breaking tasks (GSB) are weaker than agreement tasks. This paper shows that this is not the case, namely, it shows that there is a large family of GSB tasks that are more powerful than (n - 1)-set agreement. Some of these tasks are equivalent to n-renaming, while others lie strictly between n-renaming and (n+1)-renaming. Moreover, none of these GSB tasks can solve (n - 2)-set agreement. Hence, these subconsensus tasks have a rich structure and are interesting in their own. The proofs of these results are based on algebraic topology techniques and new ideas about different notions of nondeterminism that can be associated with shared objects. Interestingly, this paper sheds a new light on the relations linking set agreement and renaming.