Tight Conditions for Binary-Output Tasks under Crashes

2025-10-15

AI Summary

This work investigates the solvability boundary of binary-output tasks (output ∈ {0,1}) in an n-process distributed system tolerating up to t crash failures. Addressing both synchronous and asynchronous models, it introduces a unified abstraction—“output sets”—that subsumes canonical problems such as consensus and symmetry breaking within a single framework, overcoming limitations of task-specific analyses. Integrating crash-fault models, combinatorial topology, and distributed computability theory, the paper formally characterizes necessary and sufficient conditions for solvability of all binary tasks and proves their tightness. For the first time, it precisely identifies solvability phase-transition thresholds across varying n/t ratios under a unified framework. Moreover, the derived impossibility results extend to broader task classes, establishing foundational solvability criteria for crash-tolerant distributed computing.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Multiagent Systems: Distributed Problem Solving
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Systems and Infrastructure for Web, Mobile and WoT: Federated Web and WoT systems, including distributed, federated and edge-based data processing

Abstract

This paper explores necessary and sufficient system conditions to solve distributed tasks with binary outputs ( extit{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.

Problem

• Characterizes tight system conditions for binary-output tasks
• Unifies distributed computing problems like consensus and symmetry breaking
• Provides impossibility proofs for stronger task formulations beyond binary outputs

Innovation

• Characterizes tight conditions for binary-output tasks
• Unifies binary consensus and symmetry breaking problems
• Provides impossibility proofs for stronger task formulations