We revisit the relationship between two fundamental models of distributed computation: the asynchronous message-passing model with up to $f$ crash failures ($\operatorname{AMP}_f$) and the Heard-Of model with up to $f$ message omissions ($\operatorname{HO}_f$). We show that for $n > 2f$, the two models are equivalent with respect to the solvability of colorless tasks, and that for colored tasks the equivalence holds only when $f = 1$ (and $n > 2$). The separation for larger $f$ arises from the presence of silenced processes in $\operatorname{HO}_f$, which may lead to incompatible decisions. The proofs proceed through bidirectional simulations between $\operatorname{AMP}_f$ and $\operatorname{HO}_f$ via an intermediate model that captures this notion of silencing. The results extend to randomized protocols against a non-adaptive adversary, indicating that the expressive limits of canonical rounds are structural rather than probabilistic. Together, these results delineate precisely where round-based abstractions capture asynchronous computation, and where they do not.

翻译：本文重新审视了分布式计算中两个基本模型之间的关系：最多允许$f$个崩溃故障的异步消息传递模型（$\operatorname{AMP}_f$）与最多允许$f$条消息遗漏的Heard-Of模型（$\operatorname{HO}_f$）。我们证明当$n > 2f$时，这两个模型在无色任务的可解性方面是等价的；而对于有色任务，仅当$f = 1$（且$n > 2$）时等价性成立。当$f$较大时出现的分离性源于$\operatorname{HO}_f$中可能存在的静默进程，这可能导致决策冲突。证明通过$\operatorname{AMP}_f$与$\operatorname{HO}_f$之间的双向模拟实现，并引入了一个捕捉静默概念的中间模型。该结果可推广至针对非自适应敌手的随机化协议，表明规范轮次的表达能力限制是结构性的而非概率性的。这些结果共同精确刻画了基于轮次的抽象在何处能够捕捉异步计算特征，以及在何处无法实现。