Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tight Conditions for Binary-Output Tasks under Crashes

Timothé Albouy, Antonio Fernández Anta, Chryssis Georgiou, Nicolas Nicolaou, Junlang Wang

TL;DR

The paper introduces a unifying framework for binary-output tasks in crash-prone distributed systems by abstracting tasks to sets of possible output sets $OS$ that executions may realize. It delivers a complete tight solvability characterization for all 16 binary-output classes under both asynchronous and synchronous timing models, providing both impossibility proofs and constructive algorithms, including a novel disagreement symmetry-breaking problem. By decoupling from inputs and focusing on output sets, the results generalize classical problems such as binary consensus and symmetry breaking, and yield lower bounds applicable to stronger task formulations with validity or multi-valued outputs. The framework paves the way for broad applicability and extensions, such as handling partial synchrony, non-binary outputs, or multiplicity and validity constraints, while offering deep insights into the fundamental limits of distributed coordination under crashes.

Abstract

This paper explores necessary and sufficient system conditions to solve distributed tasks with binary outputs (\textit{i.e.}, tasks with output values in $\{0,1\}$). We focus on the distinct output sets of values a task can produce (intentionally disregarding validity and value multiplicity), considering that some processes may output no value. In a distributed system with $n$ processes, of which up to $t \leq n$ can crash, we provide a complete characterization of the tight conditions on $n$ and $t$ under which every class of tasks with binary outputs is solvable, for both synchronous and asynchronous systems. This output-set approach yields highly general results: it unifies multiple distributed computing problems, such as binary consensus and symmetry breaking, and it produces impossibility proofs that hold for stronger task formulations, including those that consider validity, account for value multiplicity, or move beyond binary outputs.

Tight Conditions for Binary-Output Tasks under Crashes

TL;DR

The paper introduces a unifying framework for binary-output tasks in crash-prone distributed systems by abstracting tasks to sets of possible output sets that executions may realize. It delivers a complete tight solvability characterization for all 16 binary-output classes under both asynchronous and synchronous timing models, providing both impossibility proofs and constructive algorithms, including a novel disagreement symmetry-breaking problem. By decoupling from inputs and focusing on output sets, the results generalize classical problems such as binary consensus and symmetry breaking, and yield lower bounds applicable to stronger task formulations with validity or multi-valued outputs. The framework paves the way for broad applicability and extensions, such as handling partial synchrony, non-binary outputs, or multiplicity and validity constraints, while offering deep insights into the fundamental limits of distributed coordination under crashes.

Abstract

This paper explores necessary and sufficient system conditions to solve distributed tasks with binary outputs (\textit{i.e.}, tasks with output values in ). We focus on the distinct output sets of values a task can produce (intentionally disregarding validity and value multiplicity), considering that some processes may output no value. In a distributed system with processes, of which up to can crash, we provide a complete characterization of the tight conditions on and under which every class of tasks with binary outputs is solvable, for both synchronous and asynchronous systems. This output-set approach yields highly general results: it unifies multiple distributed computing problems, such as binary consensus and symmetry breaking, and it produces impossibility proofs that hold for stronger task formulations, including those that consider validity, account for value multiplicity, or move beyond binary outputs.
Paper Structure (32 sections, 15 theorems, 6 equations, 2 tables, 4 algorithms)

This paper contains 32 sections, 15 theorems, 6 equations, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Towards a unifying framework for task solvability.
  3. Our approach.
  4. Contributions and roadmap.
  5. Related Work
  6. Agreement.
  7. Symmetry breaking.
  8. Unification attempts.
  9. Computing Model and Problem Formalization
  10. Computing Model
  11. Process model.
  12. Communication model.
  13. Timing models: $\text{Async}\xspace$ and $\text{Sync}\xspace$.
  14. Pseudo-randomness.
  15. Problem Formalization
  16. ...and 17 more sections

Key Result

Theorem 4

Under any timing model ($\text{Async}\xspace$ or $\text{Sync}\xspace$), the condition $n-t \geq 2$ is necessary for implementing the set of output sets $O = \{\varnothing,\{0,1\}\}$.

Theorems & Definitions (17)

  • Definition 1: Implementation of a set of output sets
  • Definition 2: Binary-output task solvability
  • Theorem 4
  • Theorem 5
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Theorem 10
  • Lemma 11
  • Lemma 12
  • ...and 7 more