1. DISTRIBUTED ALGORITHMS AND SYSTEMS Spring 2014 Prof. Jennifer Welch Set 11: Asynchronous Consensus 1

2. Impossibility of Asynchronous Consensus Set 11: Asynchronous Consensus 2  Show impossible in read/write shared memory with n processors and n - 1 faults  prove directly: not hard since so many faults  implies there is no 2-proc algorithm for 1 fault  Show impossible in r/w shared memory with n processors and 1 fault. Two approaches:  Reduction: use a hypothetical n-proc algorithm for 1 fault as a subroutine to design a 2-proc algorithm for 1 fault  Direct proof: Use similar ideas to n - 1 failures case

3. Impossibility of Asynchronous Consensus Set 11: Asynchronous Consensus 3  Show impossible in message passing with n processors and 1 fault. Two approaches:  Reduction: Use a hypothetical message passing algorithm for n procs and 1 fault as a subroutine to design a shared memory algorithm for n procs and 1 fault. This would contradict previous result.  Direct approach: Use similar ideas to shared memory case, augmented to handle messages. (Historically, this was the first version that was proven.)

4. Modeling Asynchronous Systems with Crash Failures Set 11: Asynchronous Consensus 4  Let f be the maximum number of faulty processors.  For both SM and MP: All but f of the processors must take an infinite number of steps in an admissible execution.  For MP: Also require that all messages sent to a nonfaulty processor must eventually be delivered, except for those sent by a faulty processor in its last step, which might or might not be delivered.

5. Wait-Free Algorithms Set 11: Asynchronous Consensus 5  An algorithm for n processors is wait-free if it can tolerate n - 1 failures.  Intuition is that a nonfaulty processor does not wait for other processors to do something: it cannot, because it might be the only processor left alive.  First result is to show that there is no wait-free consensus algorithm in the asynchronous r/w shared memory model.

6. Impossibility of Wait-Free Consensus Set 11: Asynchronous Consensus 6  Suppose in contradiction there is an n-processor algorithm for n - 1 faults in the asynchronous read/write shared memory model.  Proof is similar to that showing f + 1 rounds are necessary in the synchronous message passing model. bivalent initial config bivalent config bivalent config bivalent config bivalent config …

7. Modified Notion of Bivalence Set 11: Asynchronous Consensus 7  In the synchronous round lower bound proof, valency referred to which decisions are reachable in failure- sparse admissible executions.  For this proof, we are concerned with which decisions are reachable in any execution, as long as it is admissible (for the asynchronous shared memory model with up to n - 1 failures).

8. Univalent Similarity Set 11: Asynchronous Consensus 8 Lemma (5.15): If C 1 and C 2 are both univalent and they are similar w.r.t. p i (shared memory state is same, p i ’s local state is same), then they have the same valency. Proof: C 1 v-valent p i -only  p i decides v C 2 w-valent  p i decides v

9. Bivalent Initial Configuration Set 11: Asynchronous Consensus 9 Lemma (5.16): There exists a bivalent initial configuration. Proof is similar to what we did for the synchronous f + 1 round lower bound proof.

10. Critical Processors Set 11: Asynchronous Consensus 10 Def: If C is bivalent and i(C) (result of p i taking one step) is univalent, then p i is critical in C. Lemma (5.17): If C is bivalent, then at least one processor is not critical in C, i.e., there is a bivalent extension. Proof: Suppose in contradiction all processors are critical. C bival. j(C) 1-val. i(C) 0-val. pipi pjpj Rest of proof is case analysis of what p i and p j do in their two steps

11. Critical Processors Set 11: Asynchronous Consensus 11 Case 1: p i and p j access different registers. j(C) 1-val. pjpj C bival. i(C) 0-val. pipi pjpj pipi Case 2: p i and p j read same register. Same proof.

12. Critical Processors Set 11: Asynchronous Consensus 12 Case 3: p i writes to a register R and p j reads from R. C bival. p j reads from R p i writes to R i(C) 0-val j(C) 1-val i(j(C)) 1-val similar w.r.t. p i p i writes to R

13. Critical Processors Set 11: Asynchronous Consensus 13 Case 4: What if p i and p j both write to the same shared variable?  Can "assume away" the problem by assuming we only have single-writer shared variables.  Or, can do a similar proof for this case.

14. Finishing the Impossibility Proof Set 11: Asynchronous Consensus 14  Create an admissible execution C 0,i 1,C 1,i 2,C 2,… in which all configurations are bivalent.  contradicts termination requirement  Start with bivalent initial configuration.  Suppose we have bivalent C k. To get bivalent C k+1 :  Let p i_k+1 be a processor that is not critical in C k.  Let C k+1 be i k+1 (C k ).

15. Impossibility of 1-Resilient Consensus: Reduction Idea Set 11: Asynchronous Consensus 15 Even if the ratio of nonfaulty processors becomes overwhelming, consensus still cannot be solved in asynchronous SM (with read/write registers). 1. Assume there exists an algorithm A for n processors and 1 failure. 2. Use A as a subroutine to design an algorithm A' for 2 processors and 1 failure. 3. We just showed such an A' cannot exist. 4. Thus A cannot exist.

16. Impossibility of 1-Resilient Consensus: Direct Proof Idea 16  Suppose in contradiction there is such an algorithm.  Strategy: Construct an admissible execution (at most 1 fault) that never terminates:  show there is a bivalent initial configuration  show how to go from one bivalent configuration to another, forever (so can never terminate)  Technically more involved because in constructing this execution, we cannot kill more than one processor.

17. Impossibility of Consensus in Message Passing: Reduction 17 Strategy: 1. Assume there exists an n-processor 1-resilient consensus algorithm A for the asynchronous message passing model. 2. Use A as a subroutine to design an n-processor 1-resilient consensus algorithm A' for asynchronous shared memory (with read/write variables). 3. Previous result shows A' cannot exist. 4. Thus A cannot exist.

18. Impossibility of Consensus in MP 18 Idea of A':  Simulate message channels with read/write registers.  Then run algorithm A on top of these simulated channels. To simulate channel from p i to p j :  Use one register to hold the sequence of messages sent over the channel  p i "sends" a message m by writing the old value of the register with m appended  p j "receives" a message by reading the register and checking for new values at the end