Topological Characterization of Task Solvability in General Models of Computation

Authors Hagit Attiya , Armando Castañeda , Thomas Nowak

Thumbnail PDF


  • Filesize: 0.8 MB
  • 21 pages

Document Identifiers

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

Cite AsGet 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)


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
  • task solvability
  • combinatorial topology
  • point-set topology


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Dan Alistarh, James Aspnes, Faith Ellen, Rati Gelashvili, and Leqi Zhu. Why extension-based proofs fail. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM Symposium on Theory of Computing (STOC 2019), pages 986-996. ACM, New York, 2019. Google Scholar
  2. Bowen Alpern and Fred B. Schneider. Defining liveness. Information Processing Letters, 21(4):181-185, 1985. Google Scholar
  3. Hagit Attiya, Armando Castañeda, and Sergio Rajsbaum. Locally solvable tasks and the limitations of valency arguments. Journal of Parallel and Distributed Computing, 176:28-40, 2023. URL:
  4. Hagit Attiya and Sergio Rajsbaum. The combinatorial structure of wait-free solvable tasks. SIAM Journal on Computing, 31(4):1286-1313, 2002. URL:
  5. Nicolas Bourbaki. General Topology. Chapters 1-4. Springer, Heidelberg, 1989. Google Scholar
  6. Armando Castañeda, Pierre Fraigniaud, Ami Paz, Sergio Rajsbaum, Matthieu Roy, and Corentin Travers. A topological perspective on distributed network algorithms. Theoretical Computer Science, 849:121-137, 2021. Google Scholar
  7. Étienne Coulouma, Emmanuel Godard, and Joseph Peters. A characterization of oblivious message adversaries for which consensus is solvable. Theoretical Computer Science, 584:80-90, June 2015. URL:
  8. Yannis Coutouly and Emmanuel Godard. A topology by geometrization for sub-iterated immediate snapshot message adversaries and applications to set-agreement. In Rotem Oshman, editor, Proceedings of the 37th International Symposium on Distributed Computing (DISC 2023). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, 2023. To appear. Google Scholar
  9. Tristan Fevat and Emmanuel Godard. Minimal obstructions for the coordinated attack problem and beyond. In Proceedings of the 25th IEEE International Parallel & Distributed Processing Symposium (IPDPS 2011), pages 1001-1011. IEEE, New York, 2011. Google Scholar
  10. Pierre Fraigniaud, Ran Gelles, and Zvi Lotker. The topology of randomized symmetry-breaking distributed computing. In Avery Miller, Keren Censor-Hillel, and Janne H. Korhonen, editors, Proceedings of the 40th ACM Symposium on Principles of Distributed Computing (PODC 2021), pages 415-425. ACM, New York, 2021. URL:
  11. Eli Gafni, Petr Kuznetsov, and Ciprian Manolescu. A generalized asynchronous computability theorem. In Shlomi Dolev, editor, Proceedings of the 33rd ACM Symposium on Principles of Distributed Computing (PODC 2014), pages 222-231. ACM, New York, 2014. Google Scholar
  12. Emmanuel Godard and Eloi Perdereau. Back to the coordinated attack problem. Mathematical Structures in Computer Science, 30(10):1089-1113, 2020. URL:
  13. Maurice Herlihy, Dmitry N. Kozlov, and Sergio Rajsbaum. Distributed Computing Through Combinatorial Topology. Morgan Kaufmann, Waltham, 2014. URL:
  14. Maurice Herlihy and Sergio Rajsbaum. Simulations and reductions for colorless tasks. In Darek Kowalski and Alessandro Panconesi, editors, Proceedings of the 31st ACM Symposium on Principles of Distributed Computing (PODC 2012), pages 253-260. ACM, New York, 2012. URL:
  15. Maurice Herlihy and Nir Shavit. The topological structure of asynchronous computability. Journal of the ACM, 46(6):858-923, 1999. Google Scholar
  16. Gunnar Hoest and Nir Shavit. Toward a topological characterization of asynchronous complexity. SIAM Journal on Computing, 36(2):457-497, 2006. URL:
  17. Petr Kuznetsov, Thibault Rieutord, and Yuan He. An asynchronous computability theorem for fair adversaries. In Calvin Newport and Idit Keidar, editors, Proceedings of the 37th ACM Symposium on Principles of Distributed Computing (PODC 2018), pages 387-396. ACM, New York, 2018. URL:
  18. Ronit Lubitch and Shlomo Moran. Closed schedulers: a novel technique for analyzing asynchronous protocols. Distributed Computing, 8:203-210, 1995. Google Scholar
  19. Saunders Mac Lane. Categories for the Working Mathematician. Springer, Heidelberg, 2nd edition, 1987. Google Scholar
  20. Thomas Nowak, Ulrich Schmid, and Kyrill Winkler. Topological characterization of consensus under general message adversaries. In Proceedings of the 38th ACM Symposium on Principles of Distributed Computing (PODC 2019), pages 218-227. ACM, New York, 2019. URL:
  21. Nicola Santoro and Peter Widmayer. Time is not a healer. In B. Monien and R. Cori, editors, Proceedings of the 6th Annual Symposium on Theoretical Aspects of Computer Science (STACS 1989), pages 304-313. Springer, Heidelberg, 1989. URL:
  22. Vikram Saraph, Maurice Herlihy, and Eli Gafni. Asynchronous computability theorems for t-resilient systems. In Cyril Gavoille and David Ilcinkas, editors, Proceedings of the 30th International Symposium on Distributed Computing (DISC 2016), pages 428-441. Springer, Heidelberg, 2016. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail