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How Exhaustive Does an Extension-Based Proof Need to Be?

Authors Faith Ellen , Shihao Liu , Leqi Zhu , Eli Gafni , Rati Gelashvili

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LIPIcs.OPODIS.2025.29.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computability
  • Theory of computation → Interactive proof systems
  • Theory of computation → Distributed algorithms
  • Theory of computation → Distributed computing models
Keywords
  • Extension-based proof
  • set agreement
  • valency argument
  • zero-one exclusion

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Abstract

The class of extension-based proofs encompasses traditional valency arguments. It has been shown that they are insufficient to establish the impossibility of (n-1)-set agreement among n ≥ 3 processes in an asynchronous system with crash failures. We generalize this definition to k-exhaustive extension-based proofs, in which a prover can learn the maximum length of all executions involving a set of at most k processes from a specified configuration (which may be infinite). An upper bound on the length of these executions enables the prover to determine the outputs of all these executions. When k = n, this enables the prover to perform an exhaustive search of all reachable configurations, so it knows everything about the protocol. On the other hand, extension based proofs are as powerful as 1-exhaustive extension-based proofs. For any task with no deterministic, wait-free solution among n ≥ 2 processes, we show that there is an (n-1)-exhaustive extension-based proof of its impossibility. This is done using a new characterization of such tasks. In contrast, we prove that for 1 ≤ k ≤ n-2, there is no k-exhaustive extension-based proof of the impossibility of (n-1)-set agreement.

Cite As Get BibTex

Faith Ellen, Shihao Liu, Leqi Zhu, Eli Gafni, and Rati Gelashvili. How Exhaustive Does an Extension-Based Proof Need to Be?. In 29th International Conference on Principles of Distributed Systems (OPODIS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 361, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.OPODIS.2025.29

Author Details

Faith Ellen
  • University of Toronto, Canada
Shihao Liu
  • University of Toronto, Canada
Leqi Zhu
  • University of Manitoba, Winnipeg, Canada
Eli Gafni
  • University of California, Los Angeles, CA, USA
Rati Gelashvili
  • New Hampshire, USA

Funding

Faith Ellen and Shihao Liu were supported by the Natural Science and Engineering Research Council of Canada (NSERC) grant RGPIN-2020-04178 and Leqi Zhu was supported by the Natural Science and Engineering Research Council of Canada (NSERC) grant RGPIN-2025-07125.

References

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