Locally Solvable Tasks and the Limitations of Valency Arguments

TL;DR

This paper explains why impossibility proofs for set agreement and renaming tasks in distributed computing are inherently global, showing that protocols can always solve these tasks locally given certain conditions.

Contribution

It demonstrates that protocols for set agreement and renaming can always be solved locally, explaining the absence of local impossibility proofs for these tasks.

Findings

Local protocols can solve set agreement and renaming tasks.

Impossibility proofs for these tasks are necessarily global.

No local impossibility proofs exist for set agreement and renaming.

Abstract

An elegant strategy for proving impossibility results in distributed computing was introduced in the celebrated FLP consensus impossibility proof. This strategy is local in nature as at each stage, one configuration of a hypothetical protocol for consensus is considered, together with future valencies of possible extensions. This proof strategy has been used in numerous situations related to consensus, leading one to wonder why it has not been used in impossibility results of two other well-known tasks: set agreement and renaming. This paper provides an explanation of why impossibility proofs of these tasks have been of a global nature. It shows that a protocol can always solve such tasks locally, in the following sense. Given a configuration and all its future valencies, if a single successor configuration is selected, then the protocol can reveal all decisions in this branch of executions, satisfying the task specification. This result is shown for both set agreement and renaming, implying that there are no local impossibility proofs for these tasks.