Extension-Based Proofs for Synchronous Message Passing
There is no wait-free algorithm that solves k-set agreement among n ≥ k+1 processes in asynchronous systems where processes communicate using only registers. However, proofs of this result for k ≥ 2 are complicated and involve topological reasoning. To explain why such sophisticated arguments are necessary, Alistarh, Aspnes, Ellen, Gelashvili, and Zhu recently introduced extension-based proofs, which generalize valency arguments, and proved that there are no extension-based proofs of this result.
In the synchronous message passing model, k-set agreement is solvable, but there is a lower bound of t rounds for any k-set agreement algorithm among n > kt processes when at most k processes can crash each round. The proof of this result for k ≥ 2 is also a complicated topological argument. We define a notion of extension-based proofs for this model and we show there are no extension-based proofs that t rounds are necessary for any k-set agreement algorithm among n = kt+1 processes, for k ≥ 2 and t > 2, when at most k processes can crash each round. In particular, our result shows that no valency argument can prove this lower bound.
Set agreement
lower bounds
valency arguments
Theory of computation~Interactive proof systems
Theory of computation~Complexity theory and logic
Theory of computation~Distributed algorithms
Theory of computation~Distributed computing models
36:1-36:17
Regular Paper
Faith Ellen would like to thank Sergio Rajsbaum and Leqi Zhu for some preliminary discussions about this problem. Support is gratefully acknowledged from the Natural Science and Engineering Research Council of Canada under grant RGPIN-2020-04178. Part of this work was done while Yilun Sheng was virtually visiting University of Toronto.
Yilun
Sheng
Yilun Sheng
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Faith
Ellen
Faith Ellen
Department of Computer Science, University of Toronto, Canada
https://orcid.org/0000-0003-4473-931X
10.4230/LIPIcs.DISC.2021.36
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