Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A General Input-Dependent Colorless Computability Theorem and Applications to Core-Dependent Adversaries

Yannis Coutouly, Emmanuel Godard

TL;DR

The paper extends the general colorless computability framework to input-dependent adversaries within the $IIS$ model, proving that a colorless task is solvable against an adapted adversary $\mathcal{A}$ iff there exists a continuous map $f: geo(\mathcal{A})\to|\mathcal{O}|$ carried by $\Delta$. It then shows that core-resilient adversaries of $IIS_n$ have the same computability power as core-resilient adversaries with crashes only at the start, via the construction $\mathcal{U}(\mathcal{H})$ and related geometry. Building on these, the authors provide a necessary and sufficient characterization of solvability for condition-based, core-dependent adversaries solving $k$-Set Agreement, and identify four task-representation settings and structural properties of the carrier map $\Delta$ that simplify proofs without altering computability. Overall, the work broadens the reach of topological methods in distributed computing to dynamic and input-dependent adversaries, and yields a computable framework for deciding solvability in $k$-Set Agreement scenarios, with potential implications for round complexity and reductions across adversarial models.

Abstract

Distributed computing tasks can be presented with a triple $(\I,\Ou,Δ)$. The solvability of a colorless task on the Iterated Immediate Snapshot model (IIS) has been characterized by the Colorless Computability Theorem \cite[Th.4.3.1]{HKRbook}. A recent paper~\cite{CG-24} generalizes this theorem for any message adversaries $\ma \subseteq IIS$ by geometric methods. In 2001, Mostéfaoui, Rajsbaum, Raynal, and Roy \cite{condbased} introduced \emph{condition-based adversaries}. This setting considers a particular adversary that will be applied only to a subset of input configurations. In this setting, they studied the $k$-set agreement task with condition-based $t$-resilient adversaries and obtained a sufficient condition on the conditions that make $k$-Set Agreement solvable. In this paper we have three contributions: -We generalize the characterization of~\cite{CG-24} to \emph{input-dependent} adversaries, which means that the adversaries can change depending on the input configuration. - We show that core-resilient adversaries of $IIS_n$ have the same computability power as the core-resilient adversaries of $IIS_n$ where crashes only happen at the start. - Using the two previous contributions, we provide a necessary and sufficient characterization of the condition-based, core-dependent adversaries that can solve $k$-Set Agreement. We also distinguish four settings that may appear when presenting a distributed task as $(\I,\Ou,Δ)$. Finally, in a later section, we present structural properties on the carrier map $Δ$. Such properties allow simpler proof, without changing the computability power of the task. Most of the proofs in this article leverage the topological framework used in distributed computing by using simple geometric constructions.

A General Input-Dependent Colorless Computability Theorem and Applications to Core-Dependent Adversaries

TL;DR

The paper extends the general colorless computability framework to input-dependent adversaries within the model, proving that a colorless task is solvable against an adapted adversary iff there exists a continuous map carried by . It then shows that core-resilient adversaries of have the same computability power as core-resilient adversaries with crashes only at the start, via the construction and related geometry. Building on these, the authors provide a necessary and sufficient characterization of solvability for condition-based, core-dependent adversaries solving -Set Agreement, and identify four task-representation settings and structural properties of the carrier map that simplify proofs without altering computability. Overall, the work broadens the reach of topological methods in distributed computing to dynamic and input-dependent adversaries, and yields a computable framework for deciding solvability in -Set Agreement scenarios, with potential implications for round complexity and reductions across adversarial models.

Abstract

Distributed computing tasks can be presented with a triple . The solvability of a colorless task on the Iterated Immediate Snapshot model (IIS) has been characterized by the Colorless Computability Theorem \cite[Th.4.3.1]{HKRbook}. A recent paper~\cite{CG-24} generalizes this theorem for any message adversaries by geometric methods. In 2001, Mostéfaoui, Rajsbaum, Raynal, and Roy \cite{condbased} introduced \emph{condition-based adversaries}. This setting considers a particular adversary that will be applied only to a subset of input configurations. In this setting, they studied the -set agreement task with condition-based -resilient adversaries and obtained a sufficient condition on the conditions that make -Set Agreement solvable. In this paper we have three contributions: -We generalize the characterization of~\cite{CG-24} to \emph{input-dependent} adversaries, which means that the adversaries can change depending on the input configuration. - We show that core-resilient adversaries of have the same computability power as the core-resilient adversaries of where crashes only happen at the start. - Using the two previous contributions, we provide a necessary and sufficient characterization of the condition-based, core-dependent adversaries that can solve -Set Agreement. We also distinguish four settings that may appear when presenting a distributed task as . Finally, in a later section, we present structural properties on the carrier map . Such properties allow simpler proof, without changing the computability power of the task. Most of the proofs in this article leverage the topological framework used in distributed computing by using simple geometric constructions.

Paper Structure

This paper contains 29 sections, 18 theorems, 2 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Topological Methods and Computability Theorems
  3. The general message adversary line of work
  4. Application to Open Problems
  5. Models of Computation and Definitions
  6. The message adversaries framework
  7. Execution of a Distributed Algorithm
  8. Relevant communication model
  9. Abstract simplicial complexes and colorless tasks
  10. Colorless tasks into a colored world
  11. Encoding colorless tasks with processes
  12. Problem Statements
  13. Geometric Definition of Simplicial Complexes
  14. Standard Definitions
  15. Geometric Encoding of Iterated Immediate Snapshots
  16. ...and 14 more sections

Key Result

Proposition 10

There is an algorithm to solve $({{\mathcal{I}}}\xspace,\mathcal{O}\xspace,\Delta)$ on $IIS_n$ with condition $\mathcal{V}({{\mathcal{I}}}\xspace)$ if and only if there is an algorithm to solve $({{\mathcal{I}}}\xspace,\mathcal{O}\xspace,\Delta)$ on $IIS_n$ with condition $\mathcal{P}_n({{\mathcal{I

Figures (8)

  • Figure 1: The Binary Consensus task.
  • Figure 2:
  • Figure 3: Examples of the different possible settings.
  • Figure 4: Construction of $Chr(\mathbb S^2)$ as a geometric encoding for $IIS_2.$
  • Figure 5: Example of $\mathcal{K}in({{\mathcal{I}}}\xspace,1,2)$ and $\mathcal{U}_1({{\mathcal{I}}}\xspace)$
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 1: Dynamic Graph and message adversary
  • Definition 2: IIS model HKRbook
  • Definition 3: Iterated $t$-resilient adversary
  • Definition 4: Core-resilient adversary
  • Definition 5: Abstract simplicial complex
  • Definition 6: Simplicial map
  • Definition 7: Carrier map
  • Definition 8: Colorless Task
  • Definition 9: Solvability of a Colorless Task
  • Proposition 10
  • ...and 28 more