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

TL;DR

The paper establishes a tight boundary between the asynchronous message-passing model with crashes ($\mathrm{AMP}_f$) and the Heard-Of model with omissions ($\mathrm{HO}_f$): they are equivalent for colorless tasks when $n>2f$, and equivalent for colored tasks only when $f=1$ (with $n>2$); for $f>1$ a colored task like renaming separates the models. A novel intermediate model, Silenced-Faulty Heard-Of ($\mathrm{SFHO}_f$), captures silenced processes and enables bidirectional simulations that underpin the equivalence/separation results. The analysis extends to randomized protocols with non-adaptive adversaries, showing the boundary is structural (information flow) rather than purely probabilistic. Together, the results delineate precisely when canonical rounds suffice to model asynchronous computation and when they fail to capture its expressive power.

Abstract

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.

Figures (3)

• Figure 1: Simulating $\mathcal{M}_2$ in $\mathcal{M}_1$
• Figure 2: Constructing $q_i$ from $p_i$ and $P_i$
• Figure 3: Strategy for equivalence results; $\mathcal{M}_1 \hookrightarrow \mathcal{M}_2$ indicates a simulation of $\mathcal{M}_1$ in $\mathcal{M}_2$

Theorems & Definitions (42)

• Lemma 1
• proof
• Definition 1
• Definition 2
• Definition 3
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof