Topological Characterization of Task Solvability in General Models of Computation

TL;DR

This paper extends the asynchronous computability theorem to non-compact models by establishing that protocols correspond to continuous simplicial maps, demonstrating the universality of the topological approach in distributed computing.

Contribution

It generalizes the ACT to non-compact models by proving protocols correspond to continuous simplicial maps, unifying topological methods across diverse computation models.

Findings

Generalized ACT for sub-IIS models, including non-compact ones

Protocols in general models correspond to continuous simplicial maps

Topological approach is universal for distributed task solvability

Abstract

The famous asynchronous computability theorem (ACT) relates the existence of an asynchronous wait-free shared memory protocol for solving a task with the existence of a simplicial map from a subdivision of the simplicial complex representing the inputs to the simplicial complex representing the allowable outputs. The original theorem relies on a correspondence between protocols and simplicial maps in round-structured models of computation that induce a compact topology. This correspondence, however, is far from obvious for computation models that induce a non-compact topology, and indeed previous attempts to extend the ACT have failed. This paper shows that in every non-compact model, protocols solving tasks correspond to simplicial maps that need to be continuous. It first proves a generalized ACT for sub-IIS models, some of which are non-compact, and applies it to the set agreement task. Then it proves that in general models too, protocols are simplicial maps that need to be continuous, hence showing that the topological approach is universal. Finally, it shows that the approach used in ACT that equates protocols and simplicial complexes actually works for every compact model. Our study combines, for the first time, combinatorial and point-set topological aspects of the executions admitted by the computation model.