A well-known generalization of the consensus problem, namely, set agreement (SA), limits the number of distinct decision values that processes decide. In some settings, it may be more important to limit the number of “disagreers”. Thus, we introduce another natural generalization of the consensus problem, namely, bounded disagreement (BD), which limits the number of processes that decide differently from the plurality. More precisely, in a system with n processes, the $(n,\ell )\text{-BD}$ task has the following requirement: there is a value v such that at most ℓ processes (the disagreers) decide a value other than v. Despite their apparent similarities, the results described below show that bounded disagreement, consensus, and set agreement are in fact fundamentally different problems.

We investigate the relationship between bounded disagreement, consensus, and set agreement. In particular, we determine the consensus number [15] for every instance of the BD task. We also determine values of n, ℓ, m, and k such that the $(n,\ell )\text{-BD}$ task can solve the $(m,k)\text{-SA}$ task (where m processes can decide at most k distinct values). Using our results and a previously-known impossibility result for set agreement [7], we prove that for all $n\ge 2$, there is a BD task (and a corresponding BD object) that has consensus number n but cannot be solved using n-consensus and registers. Prior to our paper, the only objects known to have this unusual characteristic for $n\ge 2$ (which shows that the consensus number of an object is not sufficient to fully capture its power) were artificial objects crafted solely for the purpose of exhibiting this behavior [1], [17].

共识问题的众所周知的概括（即集合协议（SA））限制了流程决定的不同决定值的数量。在某些情况下，限制“不同意者”的数量可能更为重要。因此，我们引入了共识问题的另一种自然概括，即有界分歧（BD），它限制了从多个决策中脱颖而出的流程数量。更确切地说，在具有n个进程的系统中，$（ñ，\ell ）\text{-BD}$任务具有以下要求：存在一个值v，使得最多ℓ个进程（不同意者）决定v以外的其他值。尽管它们之间存在明显的相似之处，但以下描述的结果表明，有限的分歧，共识和约定一致实际上是根本不同的问题。

我们调查有限分歧，共识和设定一致之间的关系。特别是，我们确定BD任务的每个实例的共识数[15]。我们还确定n，ℓ，m和k的值，使得$（ñ，\ell ）\text{-BD}$ 任务可以解决 $（米，ķ）\text{-SA}$任务（其中m个进程最多可以决定k个不同的值）。使用我们的结果和先前已知的不可能达成设定协议的结果[7]，我们证明了$ñ\ge 2$，有一个BD任务（和相应的BD对象），其共识号为n，但无法使用n -consensus和寄存器解决。在我们撰写本文之前，已知唯一具有这种非同寻常特征的物体$ñ\ge 2$ （这表明一个对象的共识数不足以完全捕获其力量）是仅为了展示这种行为而制造的人造对象[1]，[17]。