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Topological Characterization of Task Solvability in General Models of Computation

Authors Hagit Attiya , Armando Castañeda , Thomas Nowak

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Author Details

Hagit Attiya
  • Department of Computer Science, Technion, Haifa, Israel
Armando Castañeda
  • Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico
Thomas Nowak
  • Laboratoire Méthodes Formelles, Université Paris-Saclay, CNRS, ENS Paris-Saclay, France
  • Institut Universitaire de France, Paris, France

Funding

  • Attiya, Hagit: partially supported by the Israel Science Foundation (grants 380/18 and 22/1425).
  • Castañeda, Armando: partially supported by the DGAPA PAPIIT project IN108723.
  • Nowak, Thomas: partially supported by the ANR grant ANR-21-CE48-0003.

Cite As Get BibTex

Hagit Attiya, Armando Castañeda, and Thomas Nowak. Topological Characterization of Task Solvability in General Models of Computation. In 37th International Symposium on Distributed Computing (DISC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 281, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.DISC.2023.5

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed computing models
Keywords
  • task solvability
  • combinatorial topology
  • point-set topology

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