Chap 15. Agreement

Problem

Processes need to agree on a single bit No link failures A process can fail by crashing (no malicious

behavior) Messages take finite (though unbounded)

time Looks easy, can this be solved ?

Consensus in Asynchronous systems Impossible even if just one process can fail !

(Fischer, Lynch, Peterson – FLP result)

N (N ¸ 2) processes Each process starts with an initial value {0,1}

that is modeled as the input register x Making a decision is modeled by writing to

the output register y Output registers are write once

Assumptions

Initial independence Processes can choose their input independently

Commute property : If events e and f are

on different processes

they commute

Assumptions (contd.)

Asynchrony of events: Any receive event can be arbitrarily delayed Every message is eventually delivered If e is a receive event

and e is enabled at G then

se is also enabled at G

Requirements

Agreement Two non-faulty processes cannot commit on

different values Non-triviality

Both 0 and 1 should be possible outcomes Termination

A non-faulty process terminates in finite time

Informal proof of the impossibility result We show that no protocol can satisfy

agreement, non-triviality and termination in the presence of even 1 failure

We show that : There is an initial global state in which the system

is non-decisive There exists a way to keep the system non-

decisive

Indecision

Lat G.V be the set of decision values reachable from a global state G

Since a non-faulty process terminates, G.V is non-empty

G is : Bivalent: G.V = { 0 ,1 } – indecisive 0-Valent: G.V = { 0 } – always leads to deciding 0 1-Valent: G.V = { 1 } – always leads to deciding 1

We show that there exists a bivalent initial state

Claim: Every consensus protocol has a bivalent initial state Assume claim is false Non-triviality : The initial set of global states must

contain 0-valent and 1-valent states Adjacent global states: If they differ in the state of

exactly one process There must be adjacent 0-valent and 1-valent states

which differ in the state of, say, p Apply a sequence where p does not take any steps Contradiction

Claim: There exists a method to keep the system indecisive Event e (on process p) is applicable to G G is the set of global states reachable from

G without applying e H = e(G )

Claim : H contains a bivalent global state

Assume that H contains no bivalent states Claim 1: H contains both 0-valent and 1-

valent states

Neighbors : 2 global states are neighbors if one results from the other in a single step

Claim 2: There exist neighbors G0, G1 such that H0 = e(G0) is 0-valent and

H1 = e(G1) is 1-valent

Claim 2:There exist neighbors G0, G1 :

H0 = e(G0) is 0-valent andH1 = e(G1) is 1-valent Let the the smallest sequence of events

applied to G without applying e such that et(G) has a different valency from e(G) Such a sequence exists The last two global states in the sequence give us

the required neighbors

w.l.o.g. let G1 = f(G0) where f is an event on process q.

Case 1 : p is different from q F is applicable to H0 resulting in H1

But H0 is 0-valent and H1 is 1-valent

Case 2: p=q Commute property

Application: Terminating Reliable Broadcast (TRB) There are N processes in the system and P0 wants

to broadcast a message to all processes. Termination: Every correct process eventually delivers

some message Validity: If the sender is correct and broadcasts m then all

correct processes deliver m Agreement: If a correct process delivers m then all correct

processes deliver m Integrity: Every correct process delivers at most one

message, and if it delivers m ( and m ‘sender faulty’) then the sender must have broadcasted m

TRB is impossible in asynchronous systems Can use TRB to solve consensus If a process receives ‘sender faulty’ it decides

on 0 Else it decides on the value of the message

received

Faults in a distributed system Crash: Processor halts, does not perform any

other action and does not recover Crash+Link: Either processor crashes or the

link fails and remains inactive. The network may get partitioned

Omission: Process sends or receives only a proper subset of messages required for correct operation

Byzantine: Process can exhibit arbitrary behavior

Consensus in synchronous systems There is an upper bound on the on the

message delay and the durations of actions performed by the processes

Consensus under crash failures

Consensus under Byzantine faults

Consensus under crash failures Requirements :

Agreement: Non faulty processes cannot decide on different values

Validity: If all processes propose the same value, v, then the decided value should be v

Termination: A non-faulty process decides in a finite time

Algorithm

f denotes the maximum number of failures Each process maintains V the set of values

proposed by other processes (initially it contains only its own value)

In every round a process: Sends to all other processes the values from V

that it has not sent before After f+1 rounds each process decides on the

minimum value in V

Algorithm

Proof: Agreement

If value x is in Vi at correct process i then belongs to the V of all correct processes

If x was added to Vi in round k<f+1, all correct process will receive that value in round k+1

If x was added to Vi in the last round (f+1) then there exists a chain of f+1 processes that have x in their V. At least one of them is non-faulty and will broadcast the value to other correct processes

Complexity

Message complexity: O((f+1)N2) If each value needs b bits then the total bits

communicated per round is O(bN3) Time:

Needs f+1 rounds

Consensus under Byzantine faults Story:

N Byzantine generals out to repel an attack by a Turkish Sultan

Each general has a preference – attack or retreat Coordinated attack or retreat by loyal generals

necessary for victory Treacherous Byzantine generals could conspire

together and send conflicting messages to mislead loyal generals

Byzantine General Agreement (BGA) Reliable messages Possible to show that no protocol can tolerate

f failures if N · 3f

Lets assume N > 4f

BGA Algorithm

Takes f+1 rounds Rotating coordinator processes (kings) Pi is the king in round i Phase 1:

Exchange V with other processes Based on V decide myvalue (majority value)

Phase 2: Receive value from king- kingvalue If V has more than N/2 + f copies of myvalue then

V[i]=myvalue else V[i]= kingvalue

After f+1 rounds decide on V[i]

BGA Algorithm

Informal proof argument

If correct processes agree on a value at the beginning of a round they continue to do so at the end

N>4f N-N/2 > 2f N-f > N/2 +f

Each process will receive > N/2+f identical messages

At least one non-faulty process becomes the king (f+1 rounds) In the correct round if any process chooses myvalue then it

received more than N/2+f myvalue messages) Therefore king received more than N/2 myvalue messages, i.e.,

kingvalue = myvalue

Knowledge

Knowledge about the system can be increased by communicating with other processes

Can use notion of knowledge to prove fundamental results, e.g. Agreement is impossible in asynchronous unreliable systems

Notations and definitions

Ki(b) : process i in group G of processors knows b

Someone knows b:

Everyone knows b:

Everyone knows E(b): E(E(b)) Ek(b) : k ¸ 0

E0(b) = b and Ek+1(b) = E(Ek(b))

Notations and definitions

Common knowledge C(b):

Hence for any k

C(b) )Ek(b)

Application: Two generals problem The situation:

Enemy camped in valley Two generals hills separated by enemy Communication by messengers who have to pass through

enemy territory … may be delayed or caught Generals need to agree whether to attack or retreat

Protocol which always solves problem impossible

Can we design a protocol that can lead to agreement in some run?

Application: Two generals problem Solution: Don’t start a war if your enemy controls the

valley Agreement not possible Let r be the run corresponding to the least number

of messages that lead to common knowledge Let m be the last message, say it was sent from P

to Q Since channel is unreliable P does not know if m

was received, hence P can assert C(b) before m was sent

Contradiction – r is the minimal run