Impossibility of Consensus with One Faulty Process - Papers We Love SF

Papers We LoveSan Francisco Edition

July 24th, 2014

Henry Robinson henry@cloudera.com / @henryr

• Software engineer at Cloudera since 2009

• My interests are in databases and distributed systems

• I write about them - in particular, about papers in those areas - at http://the-paper-trail.org

Papers We LoveSan Francisco Edition

July 24th, 2014

Henry Robinson henry@cloudera.com / @henryr

Papers We LoveSan Francisco Edition

July 24th, 2014

Henry Robinson henry@cloudera.com / @henryr

Papers of which we are quite fond

• Impossibility of Distributed Consensus with One Faulty Process, by Fischer, Lynch and Paterson (1985)

• Dijkstra award winner 2001

• Walk through the proof (leaving rigour for the paper itself)

• Show how this gives rise to a framework for thinking about distributed systems

or: agreeing to agree

Consensus

• Consensus is the problem of having a set of processes agree on a value proposed by one of those processes

• Validity: the value agreed upon must have been proposed by some process

• Termination: at least one non-faulty process eventually decides

• Agreement: all deciding processes agree on the same value

• Validity: the value agreed upon must have been proposed by some process - safety

• Termination: at least one non-faulty process eventually decides - liveness

• Agreement: all deciding processes agree on the same value - safety

Transactional CommitShould I commit this

transaction?

[Magic consensus protocol]

YES! No :(

Replicated State Machines

Client

Node 1

Node 2

Node 3

N-2N-3 N = SN-1

N-2N-3 N = SN-1

N-2N-3 N = SN-1

1: Client proposes !state N should !

be S2: Magic consensus !

protocol3: New state written to!

log

Strong Leader Election

1: Who’s the leader?

Strong Leader Election

A cast of millions

2: Magic consensus

protocol1: Who’s the

leader?

Strong Leader Election

A cast of millions

2: Magic consensus

protocol3: There can only

be one1: Who’s the

leader?

What does FLP actually say?

Fischer

Fischer Lynch

Fischer Lynch Paterson

Fischer Lynch Paterson

Choose at most two.

Distributed consensus is impossible when at least one process might fail

Distributed consensus is impossible when at least one process might fail

“[a] surprising result”

Distributed* consensus is impossible when at least one process might fail

*i.e. message passing

Distributed consensus is impossible when at least one process might fail

Termination Validity

Agreement

Distributed consensus is impossible when at least one process might fail

No algorithm solves consensus in every case

Distributed consensus is impossible when at least one process might fail

Crash failures

Hierarchy of Failure Modes

Crash failures!!

Fail by stopping

Omission failures!!

!

!

!

Fail by dropping messages

Hierarchy of Failure Modes

Crash failures!!

Fail by stopping

Byzantine failures!!

!

!

!

!

!

!

!

Fail by doing whatever the hell I like

Omission failures!!

!

!

!

Fail by dropping messages

Hierarchy of Failure Modes

Crash failures!!

Fail by stopping

More on the system model

• The system model is the abstraction we layer over messy computers and networks in order to actually reason about them.

• Message deliveries are the only way that nodes may communicate

• Messages are delivered in any order

• But are never lost (c.f. crash model vs. omission model), and are always delivered exactly once

• Nodes do not have access to a shared clock.

• So cannot mutually estimate the passage of time

• Messages are the only way that nodes may co-ordinate with each other

The Proof itself

Some definitions• Configuration: the state of every node in the system,

plus the set of undelivered (but sent) messages!

• Initial configuration: what each node in the system would propose as the decisions at time 0

• Univalent: a state from which only one decision is possible, no matter what messages are received (0-valent and 1-valent can only decide 0 or 1 respectively)

• Bivalent: a state from which either decision value is still possible.

Proof sketch

Initial, ‘undecided’, configuration

Proof sketch

Initial, ‘undecided’, configuration

Undecided state

Messages delivered

Proof sketch

Initial, ‘undecided’, configuration

Undecided state

Messages delivered

More messages delivered

Proof sketch

Initial, ‘undecided’, configuration

Undecided state

Messages delivered

Lemma 2: This always exists!

More messages delivered

Proof sketch

Initial, ‘undecided’, configuration

Undecided state

Messages delivered

Lemma 2: This always exists!

Lemma 3: You can always get here!

More messages delivered

Lemma 2: Communication Matters

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

(C:XY means process 0 has initial value X, process 1 has initial value Y)

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

These two configurations differ only at one node, but their

valencies are different

(C:XY means process 0 has initial value X, process 1 has initial value Y)

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

These two configurations differ only at one node, but their

valencies are different

(C:XY means process 0 has initial value X, process 1 has initial value Y)

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

(C:XY means process 0 has initial value X, process 1 has initial value Y)

I decided 1!

All executions of the protocol - i.e. set of messages delivered

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

(C:XY means process 0 has initial value X, process 1 has initial value Y)

I decided 0!

All executions of the protocol - i.e. set of messages delivered

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

(C:XY means process 0 has initial value X, process 1 has initial value Y)

I decided 0!

What if process 1 fails? Are the configurations any different?

I decided 1!

2-node systemC: 00!V: 1

C: 01!V: 0

C: 11!V: 0

C: 10!V: 1

(C:XY means process 0 has initial value X, process 1 has initial value Y)

I decided 0!

For the remaining processes: no difference in initial state, but

different outcome ?!

I decided 1!

Same execution

Every protocol has an undecided (‘bivalent’) initial state

Lemma 3: Indecisiveness is Sticky

Configuration C (bivalent)

e-not-delivered

Configuration

Configuration

Configuration

Configuration

Configuration

e-arrived-last

Configuration

Configuration

Configuration

Configuration

Configuration

Some message e is sent in C

Configuration C (bivalent)

e-not-delivered

Configuration

Configuration

Configuration

Configuration

Configuration

e-arrived-last

Configuration

Configuration

Configuration

Configuration

Configuration

One of these must be

bivalent

Some message e is sent in C

Configuration set D

• Consider the possibilities:

• If one of those configurations in D is bivalent, we’re done

• Otherwise show that lack of bivalent state leads to contradiction

• Do this by first showing that there must be both 0-valent and 1-valent configurations in D

• and that this leads to a contradiction

D

Configuration C (bivalent)

0-valent!e not received

0-valent!e received

Either the protocol goes through D before it reaches the 0-valent

configuration…

2. e is received

1. C moves to 0-valent configuration

before receiving e

D

Configuration C (bivalent)

0-valent!e not received

0-valent!e received

Or the protocol gets to the 0-valent configuration after

receiving e in which case this state also must be 0-valent and in D

1. e is received

2. 0-valent state is arrived at

Now for the contradiction

• There must be two configurations C0 and C1 that are separated by a single message m where receiving e in Ci moves the configuration to Di

• We will write that as Ci + e = Di

• So C0 + m = C1

• and C0 + m + e = C1 + e = D1

• and C0 + e = D0

• Now consider the destinations of m and e. If they go to different processes, their receipt is commutative

• C0 + m + e = D1

• C0 + e + m = D0 + m = D1

• Contradiction: D0 is 0-valent!

• Instead, e and m might go to the same process p.

• Consider a deciding computation R from the original bivalent state C, where p does nothing (i.e. looks like it failed)

• Since to get to D0 and D1, only e and m have been received, only p took any steps to get there.

• So R can apply to both D0 and D1.

• Since D0 and D1 are both univalent, so the configurations D0 + R and D1 + R are both univalent.

• Now remember:

• A = C + R

• D1 = C + m + e

• D0 = C + e

• But what about

• C + R + m + e = A + m + e = D1 + R => 1-valent

• C + R + e = A + e = D0 + R => 0-valent

• Let e be some event that might be sent in configuration C. Then let D be the set of all configurations where e is received last and let C be the set of configurations where e has not been received.

• D either contains a bivalent configuration, or both 0- and 1-valent configurations. If it contains a bivalent configuration, we’re done. So assume it does not.

• Now there must be some C0 and C1 in C where C0 + e is 0-valent, but C1 + e is 1-valent, and C1 = C0 + e’

• Consider two possibilities for the destination of e’ and e. If they are not the same, then we can say C0 + e + e’ == C0 + e’ + e = C1 + e = D1 -> 1-valent. But C0 + e -> 0-valent.

• If they are the same, then let A be the configuration reached by a deciding run from C0 when p does nothing (looks like it failed). We can also apply that run from D0 and D1 to get to E0 and E1. But we can get from A to either E0 or E1 by applying e or e’ + e. This is a contradiction.

What are the consequences?

!

“These results do not show that such problems cannot be “solved” in practice; rather, they

point up the need for more refined models of distributed computing that better reflect realistic

assumptions about processor and communication timings, and for less stringent

requirements on the solution to such problems. (For example, termination might be required

only with probability 1.) “

Paxos

• Paxos cleverly defers to its leader election scheme

• If leader election is perfect, so is Paxos!

• But perfect leader election is solvable iff consensus is.

• Impossibilities all the way down…

Randomized Consensus

• Nice way to circumvent technical impossibilities: make their probability vanishingly small

• Ben-Or gave an algorithm that terminates with probability 1

• (But the rate of convergence might be high)

Failure Detectors

• Deep connection between the ability to tell if a machine has failed, and consensus.

• Lots of research into ‘weak’ failure detectors, and how weak they can be and still solve consensus

FLP vs CAP

• FLP and CAP are not the same thing (see http://the-paper-trail.org/blog/flp-and-cap-arent-the-same-thing/)

• FLP is a stronger result, because the system model has fewer restrictions (crash stop vs omission)

• Theorem: CAP is actually really boring

Further reading

• 100 Impossibility Proofs for Distributed Computing (Lynch, 1989)

• The Weakest Failure Detector for Solving Consensus (Chandra and Toueg, 1996)

• Sharing Memory Robustly in Message-Passing Systems (Attiya et. al., 1995)

• Wait-Free Synchronization (Herlihy, 1991)

• Another Advantage of Free Choice: Completely Asynchronous Agreement Protocols (Ben-Or, 1983)