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Brief Announcement: Stone Duality Proofs for Colorless Distributed Computability Theorems

Authors Cameron Calk, Emmanuel Godard

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LIPIcs.DISC.2026.61.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed computing models
  • Theory of computation → Distributed algorithms
  • Theory of computation → Categorical semantics
  • Mathematics of computing → Point-set topology
  • Theory of computation → Computability
Keywords
  • simplicial complex
  • partial order
  • spectral spaces
  • Stone duality
  • categorical semantics

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Abstract

Herlihy and Shavit [Herlihy and Shavit, 1999], Saks and Zaharoglou [M. Saks and F. Zaharoglou, 2000], building on the work of Borowsky and Gafni [Borowsky and Gafni, 1997], have shown that questions of distributed computability were amenable to a topological approach by encoding distributed systems as simplicial complexes. We present an extension of the simplicial semantics for distributed computing by identifying spectral topology as a natural generalization thereof. In particular, we extend the topological approach of the Asynchronous Computability Theorem to any round-based, content-neutral, full-information protocol, a family of models containing the Iterated Immediate Snapshot protocol. The main insight of this work is in considering global states obtained after finite executions of a distributed protocol Π not as abstract simplicial complexes as was previously done, but as finite posets.
This extension enables us to prove a new general distributed computability theorem : a protocol Π solves a colorless task (ℐ,𝒪,Δ) if and only if there exists a spectral map f:Π^∞(ℐ) → 𝒪 compatible with Δ, where Π^∞(ℐ) is a spectral space associated to the protocol Π. This space is introduced in this work. From this general characterization, many known colorless computability theorems can be directly derived. In particular, we provide new insights into the previously established connection between colorless task-solvability and continuous maps between geometric realizations. Our proofs rely on Stone duality, a well established tool in semantics theory which links spectral spaces with distributive lattices. In this semantics, spaces correspond to program behaviours and lattices to program specifications, related by a crucial "ingredient" (see [Mai Gehrke, 2016]), namely topology.

Cite As Get BibTex

Cameron Calk and Emmanuel Godard. Brief Announcement: Stone Duality Proofs for Colorless Distributed Computability Theorems. In 40th International Symposium on Distributed Computing (DISC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 397, pp. 61:1-61:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.DISC.2026.61

Author Details

Cameron Calk
  • Aix-Marseille University & CNRS UMR7020, Marseille, France
Emmanuel Godard
  • Aix Marseille University & CNRS UMR7020, Marseille, France

Funding

This work is supported by grant DisQC ANR-22-CE47-0002-01 from the French National Research Agency, the Amidex fondation and project ANR-23-PECL-0009 TRUSTINCloudS (PEPR Cloud).

References

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