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Time is not a Healer, but it Sure Makes Hindsight 20:20

Eli Gafni, Giuliano Losa

TL;DR

This paper addresses the classical impossibility of deterministic consensus in asynchronous or partially synchronous distributed systems. It centers on showing four models—the FLP asynchronous model, the 1-resilient shared-memory model, the fail-to-send model, and the fail-to-receive model—are mutually simulable for colorless tasks, enabling reductions between them. The authors present a new, simple constructive impossibility proof in the fail-to-send model, inspired by Völzer and Borowski-Gafni and grounded in Sperner’s lemma, and they frame the result within a colorless-task simulation framework. The work provides a unified, pedagogy-friendly view that can guide teaching and understanding of consensus impossibility by starting from model equivalence before addressing the impossibility in the easiest model.

Abstract

In the 1980s, three related impossibility results emerged in the field of distributed computing. First, Fischer, Lynch, and Paterson demonstrated that deterministic consensus is unattainable in an asynchronous message-passing system when a single process may crash-stop. Subsequently, Loui and Abu-Amara showed the infeasibility of achieving consensus in asynchronous shared-memory systems, given the possibility of one crash-stop failure. Lastly, Santoro and Widmayer established the impossibility of consensus in synchronous message-passing systems with a single process per round experiencing send-omission faults. In this paper, we revisit these seminal results. First, we observe that all these systems are equivalent in the sense of implementing each other. Then, we prove the impossibility of consensus in the synchronous system of Santoro and Widmayer, which is the easiest to reason about. Taking inspiration from Völzer's proof pearl and from the Borowski-Gafni simulation, we obtain a remarkably simple proof. We believe that a contemporary pedagogical approach to teaching these results should first address the equivalence of the systems before proving the consensus impossibility within the system where the result is most evident.

Time is not a Healer, but it Sure Makes Hindsight 20:20

TL;DR

This paper addresses the classical impossibility of deterministic consensus in asynchronous or partially synchronous distributed systems. It centers on showing four models—the FLP asynchronous model, the 1-resilient shared-memory model, the fail-to-send model, and the fail-to-receive model—are mutually simulable for colorless tasks, enabling reductions between them. The authors present a new, simple constructive impossibility proof in the fail-to-send model, inspired by Völzer and Borowski-Gafni and grounded in Sperner’s lemma, and they frame the result within a colorless-task simulation framework. The work provides a unified, pedagogy-friendly view that can guide teaching and understanding of consensus impossibility by starting from model equivalence before addressing the impossibility in the easiest model.

Abstract

In the 1980s, three related impossibility results emerged in the field of distributed computing. First, Fischer, Lynch, and Paterson demonstrated that deterministic consensus is unattainable in an asynchronous message-passing system when a single process may crash-stop. Subsequently, Loui and Abu-Amara showed the infeasibility of achieving consensus in asynchronous shared-memory systems, given the possibility of one crash-stop failure. Lastly, Santoro and Widmayer established the impossibility of consensus in synchronous message-passing systems with a single process per round experiencing send-omission faults. In this paper, we revisit these seminal results. First, we observe that all these systems are equivalent in the sense of implementing each other. Then, we prove the impossibility of consensus in the synchronous system of Santoro and Widmayer, which is the easiest to reason about. Taking inspiration from Völzer's proof pearl and from the Borowski-Gafni simulation, we obtain a remarkably simple proof. We believe that a contemporary pedagogical approach to teaching these results should first address the equivalence of the systems before proving the consensus impossibility within the system where the result is most evident.
Paper Structure (18 sections, 8 theorems, 1 equation, 2 figures)

This paper contains 18 sections, 8 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Four Equivalent Models
  3. The New Impossibility Proof
  4. The Models
  5. The FLP model
  6. The 1-resilient shared-memory model
  7. The fail-to-send model
  8. The fail-to-receive model
  9. Simulations and Colorless Tasks
  10. Model Equivalences
  11. fail-to-send = fail-to-receive
  12. fail-to-receive $\leq$ fail-to-send
  13. fail-to-send $\leq$ fail-to-receive
  14. FLP = fail-to-receive
  15. fail-to-receive $\leq$ FLP
  16. ...and 3 more sections

Key Result

lemma thmcounterlemma

For every two models $A$ and $B$ out of the four models of this section, if $B\leq A$ then $A$ solves all the colorless tasks that $B$ solves.

Figures (2)

  • Figure 1: An execution in the fail-to-send model. Each round, the messages of the process selected by the adversary are highlighted with a darker tone.
  • Figure 2: Situation in the second case of the proof of \ref{['l3']}, where $\mathcal{P}=\{p_1,p_2,p_3\}$ (so $n=3$) and $c$ is $p_2$-dependent. There must exist $q\in\mathcal{P}$ such that one of the configurations $c_1$, $c_2$, or $c_3$ is $q$-dependent.

Theorems & Definitions (20)

  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition: p-silent and 1-silent execution
  • definition thmcounterdefinition: Pseudo-consensus
  • definition thmcounterdefinition: $p$-dependent configuration
  • lemma thmcounterlemma
  • ...and 10 more