Set agreement power is not a precise characterization for oblivious deterministic anonymous objects

Anonymous shared memory systems, recently introduced in [36], are composed of objects for which there is no a priori agreement between processes on their names. We resolve the following foundational open problems in theoretical distributed computing, for a model which includes both non-anonymous and anonymous shared objects: (1) Are non-trivial oblivious deterministic objects with the same set agreement power have the same computational power? (2) Is there a non-trivial oblivious deterministic object which is strictly weaker than an atomic read/write register? We prove that the answer to the first problem is negative, while the answer to the second problem is positive. The positive answer to the second problem implies that the common belief that every non-trivial deterministic object of consensus number one is at least as strong as atomic read/write registers is false. A noteworthy property of the proofs of our results lies in their simplicity.