# An Algorithmic Approach to the Asynchronous Computability Theorem

## TL;DR

This paper provides the first complete description and proof of correctness for the convergence algorithm, an alternative, algorithmic proof of the asynchronous computability theorem in distributed computing.

## Contribution

It offers the first comprehensive description and validation of the convergence algorithm, complementing the combinatorial proof of the ACT.

## Key findings

- Complete description of the convergence algorithm
- Proof of correctness for the convergence algorithm
- Bridges combinatorial and algorithmic proofs of the ACT

## Abstract

The asynchronous computability theorem (ACT) uses concepts from combinatorial topology to characterize which tasks have wait-free solutions in read-write memory. A task can be expressed as a relation between two chromatic simplicial complexes. The theorem states that a task has a protocol (algorithm) if and only if there is a certain chromatic simplicial map compatible with that relation. While the original proof of the ACT relied on an involved combinatorial argument, Borowsky and Gafni later proposed an alternative proof that relied on a algorithmic construction, termed the "convergence algorithm". The description of this algorithm was incomplete, and presented without proof. In this paper, we give the first complete description, along with a proof of correctness.