Traitorous Failures and Consensus Dennis Shasha (following Lynch, Fischer, Merritt)
Problem Statement 1 • Three generals A, B, and C. • At most one is a traitor. • The traitor knows the protocol and the inputs of the others. • The generals must decide to attack or not to attack (analogous to commit or abort).
Problem Statement 2 • Each day each general wakes up with an inclination to attack or not to attack. • The generals then talk to one another by two way phone. • So, A cannot overhear the conversation of B with C and symmetrically.
Problem Statement 3 • If all generals wake up with an attack inclination, the non-traitors should attack (liveness 1) • If all generals wake up with a non-attack inclination, the non-traitors should not attack (liveness 2). • If some wake up wanting to attack and others not, then either both non-traitors should attack or both should not attack (safety).
What Makes this Hard • If all generals could get together in a single room, they could simply vote. • If two vote to attack then at least one non-traitor wanted to attack, so attacking accords with the rule. • If two vote not to attack then at least one non-traitor didn’t want to attack, so that’s ok.
Communication Is Only Two Way • If A is a traitor, then A could say one thing to B and another thing to C. • B and C could later communicate, but they wouldn’t know whether the inconsistency came from A or from one another.
Scenario 1 • All generals have an inclination to attack. • However C is a traitor. What each general says is in parentheses next to it. • C(no) (yes)A(yes) (yes)B(yes) (yes)C • A and B should both say attack.
Scenario 2 • A has an inclination to attack. C does not and B is the traitor. • C(no) (yes)A(yes) (yes)B(no) (no)C • To A, the situation is as it was for Scenario 1, so A attacks. C, in order to preserve safety, must also attack.
Scenario 3 • Nobody has an inclination to attack. A is the traitor • C(no) (yes)A(no) (no)B(no) (no)C • To C, the situation is as it was in scenario 2, so C attacks. But this violates our second liveness condition.
Summary • Even if you have only one traitor, two way communication cannot guarantee a correct decision among three generals. • Three way communication could have done so and so could signed communication, because then no general could have told inconsistent stories.
Postscript • “If I were two-faced, would I be wearing this one?” Abraham Lincoln