Continuous Tasks and the Chromatic Simplicial Approximation Theorem

TL;DR

This paper introduces continuous tasks and a chromatic simplicial approximation theorem, offering a new perspective on the 1999 Asynchronous Computability Theorem and expanding the expressive power of task specifications in distributed computing.

Contribution

It provides a novel interpretation of the ACT through continuous functions and introduces a chromatic simplicial approximation theorem, offering new proof techniques and expressive capabilities.

Findings

Provides a new proof of the ACT using continuous tasks

Introduces a chromatic version of the simplicial approximation theorem

Shows continuous tasks can specify output densities for input combinations

Abstract

The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized the distributed tasks that are wait-free solvable, and thus uncovered a deep connection with algebraic topology. We present a novel interpretation of this theorem, through the notion of continuous task, defined by an input/output specification that is a continuous function. To do so, we introduce a chromatic version of a foundational result for algebraic topology: the simplicial approximation theorem. In addition to providing a different proof of the ACT, the notion of continuous task seems interesting in itself. Indeed, besides the fact that certain distributed problems are naturally specified by continuous functions, continuous tasks have an expressive power that also allows to specify the density of desired outputs for each combination of possible inputs,for example.