When Is Agreement Possible? CS 188 Distributed Systems February 24, 2015 Lecture 13 Page 1 CS 188,Winter 2015
Introduction • Basics of agreement protocols • Impossibility of agreement in asynchronous system with failures • When is agreement possible? Lecture 13 Page 2 CS 188,Winter 2015
Basics of Agreement Protocols • What is agreement? • What are the necessary conditions for agreement? Lecture 13 Page 3 CS 188,Winter 2015
What Do We Mean By Agreement? • In simplest case, can n processors agree that a variable takes on value 0 or 1? – Only non-faulty processors need agree • More complex agreements can be built from this simple agreement Lecture 13 Page 4 CS 188,Winter 2015
Conditions for Agreement Protocols • Consistency – All participants agree on same value and decisions are final • Validity – Participants agree on a value at least one of them wanted • Termination – All participants choose a value in a finite number of steps Lecture 13 Page 5 CS 188,Winter 2015
Impossibility of Agreement in Async System With Failures • Assume a reliable, but asynchronous, message passing system – Any message may face arbitrary delays • Can a set of processors reach agreement if one of the processors fails? Lecture 13 Page 6 CS 188,Winter 2015
Agreement Isn’t Always Possible • In the general case for arbitrary systems • Adding some special properties to the system may change that result • But without those properties, provably impossible – A result sometimes abbreviated FLP • For Fischer, Lynch, and Patterson, who proved it Lecture 13 Page 7 CS 188,Winter 2015
Model of the System • The system consists of n processors • The goal is for all non-faulty processors to agree on value 0 or 1 • Rule out the trivial case of always agreeing on 0 (or 1) • Agreement depends on protocol, initial state, and inputs to each processor Lecture 13 Page 8 CS 188,Winter 2015
Bivalent and Univalent States • A bivalent state is a system state that could lead to either value being decided • A univalent state can only lead to one of the values being decided – 0-valent or 1-valent • Valency must take allowable failures into account! Lecture 13 Page 9 CS 188,Winter 2015
System Configuration • Processors have internal state • State of network is the set of messages sent, but not yet received • Event e is the receipt of message m by a processor – Which can lead to sending one or more new messages – Events are deterministic • A schedule is a sequence of events Lecture 13 Page 10 CS 188,Winter 2015
Proving the Result • Let’s assume the result is false – That we can reach agreement with one failure in these conditions • Use an adversarial model – Within rules of behavior, assume adversary can force any legal event • Look for contradictions Lecture 13 Page 11 CS 188,Winter 2015
What Can the Adversary Do? • Force any processor to perform an event at any moment • Choose any message to be delivered to any processor when it requests a message • Delay any message arbitrarily long • Once, it can kill one processor permanently Lecture 13 Page 12 CS 188,Winter 2015
The Necessity of Bivalency • There has to be an initial bivalent configuration for the system • Why? • If all processors started with value 1, the system would decide 1 • If all processors started with value 0, the system would decide 0 Lecture 13 Page 13 CS 188,Winter 2015
Intermediate Initial States • If some processors start with value 0 and some with value 1 – Some initial states lead to result 1 – Some initial states lead to result 0 – All initial states lead to one or the other • So there is a 1-valent initial state that differs from a 0-valent initial state by one processor’s initial value Lecture 13 Page 14 CS 188,Winter 2015
A Graphical Representation What’s in these states? State x State y Node 1:0 Node 1:0 Node 2:1 Node 2:1 Node 3: 1 Node 3: 1 . . . . . Node N: 0 . Node N: 1 They differ in only one value 0-valent initial states 1-valent initial states Lecture 13 Page 15 CS 188,Winter 2015
Why Does This Imply Bivalence? • What if that one differing processor is the processor that fails? • The system must still reach agreement from the remaining states – Which are identical, now • But on what value? Lecture 13 Page 16 CS 188,Winter 2015
Is This Possible? Looks like x Does the Does the and y must be system decide system decide bivalent on 1? on 0? State x State y Node 1:0 Node 1:0 Node 2:1 Node 2:1 Node 3: 1 Node 3: 1 . . . . . Node N: 0 . Node N: 1 Then State x Then State y wasn’t 1-valent, wasn’t 0-valent, after all after all 0-valent initial states 1-valent initial states Lecture 13 Page 17 CS 188,Winter 2015
So What? • So there has to be at least one bivalent initial state • Why’s that so bad? • If the system never leaves a bivalent state, it never makes a decision • We must show our adversary can’t perpetually force bivalency Lecture 13 Page 18 CS 188,Winter 2015
The Persistence of Bivalency • Let’s assume bivalency doesn’t persist • At some point, some bivalent state must transition to a univalent state – Implying at least two events • One to go to 0-valent • One to go to 1-valent • With no events leading to bivalent states Lecture 13 Page 19 CS 188,Winter 2015
A Graphical Representation C e e’ D D’ Remember, these events are each delivery of a message So m and m’ must have been in the message delivery system state simultaneously Lecture 13 Page 20 CS 188,Winter 2015
Looking Closely at Events e and e’ • What would happen if we executed e first, then e’ ? • What would happen if we executed them in the opposite order? • Well, why should I care? • Would executing them in either order lead to the same state? • If so, there’s a contradiction Lecture 13 Page 21 CS 188,Winter 2015
Order of Events e and e’ C e e’ D D’ e’ e Lecture 13 Page 22 CS 188,Winter 2015
Why Should They Lead to the Same State? • What if e and e’ occur on different processors? • Then they’re independent events • So they should produce the same result if executed in either order • So e and e’ could not have occurred on different processors Lecture 13 Page 23 CS 188,Winter 2015
Could the Events Occur on the Same Processor P ? • If e was first, the state became 0-valent • If e’ was first, the state became 1- valent • But what if P then fails? • Since the event happened only at P , only P sees the effects • So we’re still in a bivalent state Lecture 13 Page 24 CS 188,Winter 2015
Recapitulating the Argument • It’s possible to start in a bivalent state • There must be some point at some processor P at which the bivalent state changes to univalent • If P fails before anyone knows the valency, the system becomes bivalent – And can never settle to univalency • Perpetual bivalency implies no agreement Lecture 13 Page 25 CS 188,Winter 2015
When Is Agreement Possible? • Didn’t we show in the last class that we can reach agreement if less than 1/3 of our processors are faulty? • Yes, but only if the message passing system is synchronous • Whether agreement is possible in a system depends on certain parameters Lecture 13 Page 26 CS 188,Winter 2015
Parameters for Agreement In Distributed Systems • Synchronous vs. asynchronous processors • Bounded vs. unbounded communications delay • Ordered vs. unordered messages • Point-to-point vs. broadcast communications Lecture 13 Page 27 CS 188,Winter 2015
Synchronous vs. Asynchronous Processors • Synchronous processors imply that all processors make progress predictably • More precisely, there is a constant s such that – for every s+1 steps taken by P i – all P j will take at least one step Lecture 13 Page 28 CS 188,Winter 2015
Bounded vs. Unbounded Communications Delay • Delay is bounded if and only if all messages arrive at their destination within t steps – Implies no lost messages • Doesn’t imply messages arrive in the order sent Lecture 13 Page 29 CS 188,Winter 2015
Ordered vs. Unordered Messages • Messages are ordered if they are received in the same real time order as their sending – Using true real time • In some cases, merely receiving all messages in same order at all processors is enough Lecture 13 Page 30 CS 188,Winter 2015
Point-to-Point vs. Broadcast Communications • Point-to-point communications means a given message sent by P i is seen only by its destination P j • Broadcast communications mean that P i can send a message to all other processors in a single atomic step • Most typically by hardware broadcast Lecture 13 Page 31 CS 188,Winter 2015
So, When Can We Reach Agreement? • Case 1: Processors are synchronous and communications is bounded • Case 2: Messages are ordered and the transmission medium is broadcast • Case 3: Processors are synchronous and messages are ordered • And that’s it – (Case 1 covers Byzantine agreement) Lecture 13 Page 32 CS 188,Winter 2015
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