Equivalence and Separation between Heard-Of and Asynchronous Message-Passing Models

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.