feat(FLP): develop the theory of pseudo-consensus algorithms that can tolerate up to 1 fault - #755
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feat(FLP): develop the theory of pseudo-consensus algorithms that can tolerate up to 1 fault#755ctchou wants to merge 2 commits into
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fmontesi
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Jul 30, 2026
| /-- `a.OneSilentDecided s p b` means that the boolean value `b` is decided on in a state that | ||
| is reachable from `s` without the participation of `p`. In the notation of [Volzer2004], this | ||
| is equivalent to `b ∈ val(p,s)`. -/ | ||
| def Algorithm.OneSilentDecided (a : Algorithm P M S) |
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I'd call this CanDecideWithout or CanDecideWithoutOne.
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| /-- `a.Uniform s b` means that for every process `p`, `a.OneSilentDecided s p b` but | ||
| not `a.OneSilentDecided s p !b`. -/ | ||
| def Algorithm.Uniform (a : Algorithm P M S) (s : State P M S) (b : Bool) : Prop := |
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I find the naming a bit confusing here:
- For a given boolean b,
Uniformmeans that, (1) there's a state reachable without involving p where some process has decided b and (2) no state reachable without involving p has a process that's decided !b. NonUniformmeans that for any boolean b, there's a process p such that we can reach a state without involving p where some process has decided b.
Do these come from the paper?
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| /-- Assuming `a.PseudoConsensus 1` and there are at least 2 processes, there must exist an input | ||
| that gives rise to a non-uniform initial state. This theorem formalizes Lemma 1 of [Volzer2004]. -/ | ||
| theorem nonUniform_inp (hpc1 : a.PseudoConsensus 1) (hc : card P ≥ 2) : |
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Very nice result!
The name and content makes me wonder a bit if this lemma (and perhaps also some of the defs above, like NonUniform) should be extended in name or namespaced so that it's clear they're about 1-pseudo consensus.
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This PR develops a theory of pseudo-consensus algorithms that can tolerate up to 1 fault, which is the core of Völzer's argument. The central notion is "non-uniformity": a state
sis non-uniform iff for each boolean valueb, there exists a processpsuch that there is an execution of the algorithm fromsthat ends up deciding onbwithout the participation ofp. Then any algorithm that can tolerate up to 1 fault is proved to have two properties:(1)
nonUniform_inp, which says that there must an initial non-uniform state.(2)
nonUniform_step, which will be shown in the next PR to guarantee that any execution ending in a non-uniform state can be extended to a longer execution ending in a non-uniform state.