1. Consensus with Partial Synchrony Kevin Schaffer Chapter 25 from “Distributed Algorithms” by Nancy A. Lynch

2. Consensus  System inputs init(v) i stop i  System outputs decide(v) i

3. Constraints  Well-formedness: Interactions between U i and A are well-formed for i  Agreement: All decision values are identical  Validity: If all init actions specify the same value v, then v is the only possible decision value  f-failure termination, 0  f  n: If init occurs on all ports and stop occurs on at most f ports, then a decide occurs on all non-failing ports.

4. PSynchFD Failure Detector  Each process continually sends messages to all other processes  If P i performs m steps without receiving a message from P j, then it outputs inform-stopped(j)  m must be strictly greater than (d + 2 )/ 1 + 1

5. Correctness of PSynchFD  All failures are eventually detected by all other non-failed processes  After performing (d + 2 )/ 1 + 1 steps, at least d + 2 time has passed; it takes at most d + 2 for a message sent from P j to reach P i ; hence, P j must have stopped  Once a stop j occurs, it takes at most Ld + d + O(L 2 ) for either an inform-stopped(j) i or a stop i to occur

6. Transforming Synchronous to Partially Synchronous  In A each process P i is the composition of two MMT automata: Q i and R i Q i is a PSynchFD process R i is simulates the synchronous algorithm  Each R i keeps track of the current round r and moves to the next round once it has received all round r messages from all non-failed processes  B is A with the input and output corrected

7. Process P in B

8. Upper Bound

9. Synchronous Algorithm  Synchronous algorithm adapted to partially synchronous model  Uses PSynchFD to detect process failures  Complexity: T(0) + f(Ld + d) + d + O(fL 2 )

10. PSynchAgreement  Uses PSynchFD to detect process failures  Proceeds in rounds  A process can decide 0 in even-numbered rounds and 1 in odd-numbered rounds  Processes start round 0 after receiving input  Each process has a variable decided to keep track of which processes have decided

11. PSA Algorithm  Round 0 If input is 1 then  Send goto(1) to all processes  Go to round 1 If input is 0 then  Send goto(2) to all processes  Output decide(0)  Send decided to all processes

12. PSA Algorithm (2)  Round r > 0 If process received goto(r + 1) from anyone  Send goto(r + 1) to all processes  Go to round r + 1 If process received goto(r) from everyone not in stopped  decided  Send goto(r + 2) to all processes  Output decide(r mod 2)  Send decided to all processes

13. Lemma 25.7  If any process sends goto(r + 2), then some process tries to decide at round r  If any process reaches round r + 2, then some process tries to decide at round r

14. Lemma 25.8  If a process i decides at round r then: R i sends no goto(r + 1) messages R i sends a goto(r + 2) message to every process No process tries to decide at round r + 1

15. Validity  If all processes start with 0, then no process can over leave round 0. No process can decide 1 in round 0.  If all processes start with 1, then no process tries to decide in round 0. Lemma 25.7 implies that no process reaches round 2.

16. Agreement  Suppose that Ri decides at round r and no process decides at any earlier round  By Lemma 25.8, no process tries to decide at round r + 1. By Lemma 25.7, no process can reach round r + 3  All processes must decide in r or r + 2, which result in the same output value

17. Liveness  Suppose R i gets stuck at round r > 0  All non-stopped, non-decided processes must eventually reach r  Since r > 0, each process must send goto(r) to R i and R i must eventually receive it  Therefore R i ’s condition for deciding it satisfied; a contradiction

18. Wait-Free Termination  A quiet round r is one in which some process never receives goto(r + 1) If no process tries to decide at round r, then round r + 1 is quiet If some process decides at round r, then r + 2 is quiet  There must eventually be a quiet round, hence wait-free termination is guaranteed  Complexity: Ld + (2f + 2)d + O(f 2 + L 2 )

19. Lower Bound  Suppose that n  f + 1  There is no n-process agreement algorithm for the partially synchronous model that guarantees f-failure termination, in which all nonfaulty processes always decide strictly before time Ld + (f – 1)d

20. Other Results  Synchronous processes, asynchronous channels: agreement is not solvable with even 1 failure  Asynchronous processes, synchronous channels: same  Eventual time bounds: solvable, but only if n > 2f