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：我们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 $f$ 个进程崩溃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。对于更大的 $f$ 值出现的分离性，源于 $\operatorname{HO}_f$ 中存在被静默的进程，这可能导致不相容的决策。证明通过 $\operatorname{AMP}_f$ 与 $\operatorname{HO}_f$ 之间经由一个捕捉这种静默概念的中间模型进行双向模拟来完成。这些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。综上，这些结果精确地界定了基于轮次的抽象在何处能够刻画异步计算，以及在何处不能。