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Consensus Impossibility

Consensus Impossibility. Dennis Shasha (following Lynch, Fischer, Patterson). Failure Model. Processes can fail-stop (but that is all; no fail traitorous). Communication is two-way. Messages can be dropped or take an arbitrarily long time.

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Consensus Impossibility

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  1. Consensus Impossibility Dennis Shasha (following Lynch, Fischer, Patterson)

  2. Failure Model • Processes can fail-stop (but that is all; no fail traitorous). • Communication is two-way. • Messages can be dropped or take an arbitrarily long time. • To show: no algorithm can guarantee to solve consensus in the face of fail-stop processes and unlimited message delays.

  3. Bald Man’s Paradox • Man with a full head of hair is not bald. • Man with no hair is bald. • Suppose you tell me that a man with x number of hairs or more is not bald. • Suppose the man has an opaque patch on his head • Outside of patch, man has less than x number of hairs. • What’s under patch determines status!

  4. Consensus Problem Simplified • Every input process has either a 0 or 1. • If all working processes have 1 initially, then all should conclude 1. • If all working processes have 0 initially, then all should conclude 0 • In any other case, all should conclude either 0 or 1 but they must all agree.

  5. Terminology • State is just set of states of individual processes. • State is univalent for 1 (respectively 0), if, regardless of failures, the conclusion will be 1 (respectively 0). • State is bivalent if could go either to 0 or to 1.

  6. There must be some bivalent initial states • Suppose that every initial state having at least x% of 1s is univalent for 1 and any less is univalent for 0. • If one process is not communicating, then maybe that would make the difference. • So others can’t decide. • Therefore that state is bivalent.

  7. From Every Bivalent State, another bivalent state is reachable • Suppose some message sent and received changes the state from bivalent to univalent. • If that message can be lost, then state is still bivalent. • Conclusion: we can be bivalent forever. • Nota Bene: might be very unlikely to remain in such a state (e.g. two phase commit).

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