A Hierarchy of Unrestricted Deterministic Objects with Consensus Number 1

The consensus number of a shared object is the maximum number of processes that can solve consensus in a wait-free manner using copies of the object and registers. In 2016, to prove that an object is not fully characterized by its consensus number, Afek, Ellen and Gafni showed that for all integers n ≥ 2, there exists an infinite sequence of deterministic objects of consensus number n with strictly increasing computational power. In 2018, Daian, Losa, Afek, and Gafni constructed an infinite sequence of deterministic objects of consensus number 1 with strictly decreasing computational power, but the single operation that each of these objects supports is restricted in how it can be used during an execution. As restrictions can have an effect on an object’s consensus number, it was left as an open question whether the same result holds without this restriction. In this paper, we construct an infinite sequence of unrestricted deterministic objects with strictly decreasing computational power. All objects in this sequence have consensus number 1.