The set agreement power of a shared object O describes O ’s ability to solve set agreement problems: it is the sequence $$(n_1, n_2, {\ldots }, n_k, {\ldots })$$ ( n 1 , n 2 , … , n k , … ) such that, for every $$k\ge 1$$ k ≥ 1 , using O and registers one can solve the k -set agreement problem among at most $$n_k$$ n k processes. It has been shown that the ability of an object O to implement other objects is not fully characterized by its consensus number (the first component of its set agreement power). This raises the following natural question: is the ability of an object O to implement other objects fully characterized by its set agreement power ? We prove that the answer is no: every level $$n \ge 2$$ n ≥ 2 of Herlihy’s consensus hierarchy has two linearizable objects that have the same set agreement power but are not equivalent, i.e., at least one cannot implement the other. We also show that every level $$n \ge 2$$ n ≥ 2 of the consensus hierarchy contains a deterministic linearizable object $$O_n$$ O n with some set agreement power $$(n_1,n_2,\ldots ,n_k,\ldots )$$ ( n 1 , n 2 , … , n k , … ) such that being able to solve the k -set agreement problems among $$n_k$$ n k processes, for all $$k\ge 1$$ k ≥ 1 , is not enough to implement $$O_n$$ O n .

中文翻译：

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超出约定的生活

共享对象 O 的集合一致性能力描述了 O 解决集合一致性问题的能力：它是序列 $$(n_1, n_2, {\ldots }, n_k, {\ldots })$$ ( n 1 , n 2 , ... , nk , ... ) 使得，对于每 $$k\ge 1$$ k ≥ 1 ，使用 O 和寄存器可以解决最多 $$n_k$$ nk 个进程之间的 k 集一致性问题。已经表明，对象 O 实现其他对象的能力并不完全由其共识数（其设定的协议权的第一个组成部分）表征。这提出了以下自然问题：对象 O 实现其他对象的能力是否完全以其设定的协议能力为特征？我们证明答案是否定的：Herlihy 共识层次结构的每一层 $$n \ge 2$$ n ≥ 2 都有两个线性化对象，它们具有相同的集合一致能力但不等价，即，至少一个不能实现另一个。我们还表明，共识层次结构的每个级别 $$n \ge 2$$ n ≥ 2 都包含一个确定性可线性化对象 $$O_n$$ O n 具有一些设定的协议能力 $$(n_1,n_2,\ldots,n_k, \ldots )$$ ( n 1 , n 2 , … , nk , … ) 使得能够解决 $$n_k$$ nk 个进程之间的 k 集一致性问题，对于所有 $$k\ge 1$$ k ≥ 1 ，不足以实现 $$O_n$$ O n 。