1. Impossibility of Distributed Consensus with One Faulty Process
Michael J. Fischer
Nancy A. Lynch
Michael S. Paterson

2. Introduction
Problem
Reach agreement among remote processes
Example: transaction commit problem
Easy if the participating processes and network are completely reliable
Real systems are subject to possible faults
Process Crash
Unreliable Communication
Byzantine failure

3. Result
No completely asynchronous consensus protocol can tolerate even a single unannounced process death

4. Assumption Don’t consider Byzantine failure Reliable message system:
messages are delivered correctly and exactly once
Asynchronous:
No assumption the relative speeds of processes or the delay time in delivering a message
No synchronized clock
Algorithms based on time-out can’t be used
No ability to detect the death of a process

5. The weak consensus problem
Initial state: 0 or 1
Decision state:
Nonfaulty process decides on a value in {0, 1}
Requirement:
All nonfaulty processes that make a decision must choose the same value.
Some processes eventually make a decision(for proof)
Trivial solution is ruled out

6. System model
Processes are modeled as automata and communicate by messages
One atomic step
receive a message, perform local computation, send arbitrary but finite messages
Atomic broadcast
send a message to all other processes in one step
all nonfaulty processes or none will receive it

7. Consensus Protocols
A consensus protocol P is an asynchronous system of N processes
Each process p has a one-bit input register xp, an output register yp, with values in {b, 0, 1}
Internal state:
input/output register, pc, internal storage
Initial state:
all fixed starting values except the input register
output register starts with b

8. p acts deterministically according to a transition function
Decision state
output register has the value 0 or 1
Once a decision is made, the value can’t be changed
p acts deterministically according to a transition function
Processes communicate by messages
Message system
send(p, m)
receive(p)
nondeterministically

9. A configuration consists of
All internal state of each process, the contents of message buffer
A step is a transition of one configuration C to another e(C), including 2 phases:
First, receive(p) to get a message m
Based on p’s internal state and m, p enters a new internal state and sends finite messages to other
e = (p, m) is called an event and said e can be applied to C

10. Schedule, run, reachable and accessible
A schedule from C
a finite or infinite sequence σ of events that can be applied, in turn, starting from C
The associated sequence of steps is called a run
σ(C) denotes the reulting configuration and is said to be reachable from C
An accessible configuration C
If C is reachable from some initial configuration

11. Lemma 1
Suppose that from some configuration C, the schedules σ1 and σ2 lead to configuration C1 ,C2 , respectively. If the sets of Processes taking steps in σ1 and σ2 respectively, are disjoint, then σ2 can be applied to C1 and σ1 can be applied to C2 , and both lead to the same configuration.

13. Decision value, partially correct, nonfaulty
A configuration C has decision value v if some process p is in a decision state with yp =v.
P is partially correct if
No accessible configuration has more than one decision value
For each v {0, 1}, some accessible configuration has decision value v.
A process is nonfaulty
If it takes infinitely many steps

14. Admissible and deciding run, totally correct
Admissible run
At most one process is faulty and all messages sent to nonfaulty processes are eventually received
Deciding run
Some process reaches a decision state
A consensus protocol P is totally correct in spite of one fault if it is partially correct and every admissible run is a deciding run

15. Theorem 1
No consensus protocol is totally correct in spite of one fault.

16. Bivalent, 0-valent/1-valent
Let C be a configuration, V the set of decision values of configurations reachable from C. C is bivalent if |V| = 2. C is univalent if |V| = 1.
0-valent or 1-valent according to the corresponding decision value.

17. Lemma 2
P has a bivalent initial configuration

18. Lemma 3
Let C be a bivalent configuration of P, and let e = (p, m) be an event that is applicable to C. Let C be the set of configurations reachable from C without applying e, and let D= e(C) ={e(E) | E C and e is applicable to E}. Then, D contains a bivalent configuration.

21. Theorem 2
There is a partially correct consensus protocol in which all nonfaulty processes always reach a decision, provided no processes die during its execution and a strict majority of the processes are alive initially

22. Conclusion
The problem of fault-tolerant cooperative computing cannot be solved in a totally asynchronous model of computation..
To solve this problem in practice, more refined models of distributed computing is needed