Lecture 35: Consensus I

COSC 273: Parallel and Distributed Computing

Spring 2023

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Binary Consensus

Setup:

  • $n$ processes/threads, $P_1, P_2, \ldots, P_n$
  • processes have unique IDs, $1, 2, \ldots, n$ (like ThreadId.get())
  • each process $i$ holds input $x_i = 0$ or $1$

Output:

  • each process $i$ outputs $b_i = 0$ or $1$

Failure:

  • Some process(es) may crash
    • failing process may perform some steps before failing
    • cannot distinguish slow from crash

Conditions for Consensus

  • Agreement: all processes output the same value

  • Validity: if all systems have the same input, they all output that value

  • Termination: all (non-faulty) processes decide on an output and terminate after a finite number of steps

Exercise

Devise an algorithm for consensus assuming:

  1. Extremely fair scheduler: all processes take a step before anyone takes a next step
    • execution in synchronous rounds
  2. No faulty processes
  3. Access to shared array of size $n$

Consensus with Faults

Suppose some process(es) may crash at any time during an execution…

  • Other processes can’t tell that a process crashes
    • e.g., cannot distinguish slow process from crashed

Not Consensus 1

How can we achieve…

  • Agreement: all processes output the same value
  • Validity: if all systems have the same input, they all output that value
  • Termination: all (non-faulty) processes decide on an output and terminate after a finite number of steps

Not Consensus 2

How can we achieve…

  • Agreement: all processes output the same value
  • Validity: if all systems have the same input, they all output that value
  • Termination: all (non-faulty) processes decide on an output and terminate after a finite number of steps

Not Consensus 3

How can we achieve…

  • Agreement: all processes output the same value
  • Validity: if all systems have the same input, they all output that value
  • Termination: all (non-faulty) processes decide on an output and terminate after a finite number of steps

So

Without too much trouble, we can achieve…

  • …consensus with synchronous rounds, no faults
  • …“consensus” without agreement
  • …“consensus” without validity
  • …“consensus” without termination

What about consensus with faults?

A Remarkable Fact

Theorem (FLP, 1985). There is no algorithm that achieves consensus in the presence of even a single faulty process.

  • Proven by Fischer, Lynch, and Paterson in 1985
    • their version of result for “message passing” model
    • ours is for shared memory with atomic read/writes
  • Surprise to the parallel/distributed computing community
  • Among most influential results in CS

Our Plan

Prove version of FLP result:

  • There is no wait-free protocol for consensus with read/write registers for any $n > 1$
  • wait-free means all processes terminate after a finite number of steps independently of other processors’ behavior
  • wait-free is a stronger assumption than termination
  • termination just means that all processors halt eventually
    • our result is weaker than the FLP result

Computational Model

  • Processes have shared memory (registers)
  • Atomic read/write access
    • behavior like volatile variables in Java
  • Scheduler decides which process makes a step
    • assume each step is read/write
  • Some processes may crash
    • such a process is never scheduled again
  • Scheduler is otherwise fair: non-crashed processes are scheduled eventually

Algorithms

An algorithm $A$ specifies next operation:

  • read value from shared register
  • write value to shared register
  • terminate

as a function of

  • input ($x_i$)
  • outcomes of previous (read) operations

Next step is uniquely determined by local input and values read in previous steps

Example: Default to 0

Idea: output 0 unless all processes have input 1

int in = getLocalInput();
int i = ThreadId.get();
write(i, in); // write my value to register i
if (in == 0) return 0;
for (int j = 0; j < nProcesses; j++) {
    // wait until register j has been written	
    while (read(j) != 0 && read(j) != 1) { };	
    if (read(j) == 0) return 0;
}
// all processors have in == 1
return 1;

Executions

An execution $E$ of algorithm $A$ specifies

  • Inputs of all processes
  • Sequence of steps taken by processes
    • read
    • write
    • terminate
    • crash

Executions may be incomplete

  • Not all nodes have terminated/crashed yet
    • encodes current state/history of execution

Executions may be extended by scheduling more steps

Example of Execution $E$

$E$ Step 01

$E$ Step 02

$E$ Step 03

$E$ Step 04

$E$ Step 05

$E$ Step 06

Extending Executions

In $E$, no process has terminated yet

  • We can consider extensions of a given execution
  • Start with $E$, and perform more steps

$E’$ Step 06

$E’$ Step 07

$E’$ Step 08

$E’$ Step 09

$E’$ Step 10

Note

We can consider many different extensions of $E$

Extension $E’$ of $E$

Alternate extension $E’’$

Indistinguishable Executions

  • $E$ and $E’$ are executions
  • they are indistinguishable at process $P_i$ if in $E$ and $E’$:
    1. $P_i$ has same input
    2. sequence of read/write operations performed by $P_i$ are same
    3. the sequence of values read and written by $P_i$ are the same

$E’$ for P1

$E’’$ for P1

First Important Observation

Lemma 1. If executions $E$ and $E’$ are indistinguishable to process $P_i$ then:

  1. If $P_i$ has not yet terminated, then $P_i$’s next step will be the same in any extension
  2. If $P_i$ has terminated, then $P_i$’s output is the same in $E$ and $E’$

Bivalent Executions

  • Consider a (hypothetical) wait-free consensus protocol $A$
  • Let $E$ be an execution of $A$

We say that $E$ is…

  1. $0$-valent if in every extension of $E$, all processes output $0$
  2. $1$-valent if in every extension of $E$, all processes output $1$
  3. bivalent if there exist
    • an extension $E’$ of $E$ in which all processes output $0$
    • an extension $E’’$ of $E$ in which all processes output $1$

Second Important Observation

Lemma 2. Suppose $A$ solves consensus. Then there is a bivalent initial state.

  • Here an initial state is an execution in which no process has yet taken a step
    • the execution consists of only inputs for each process

Proof of Lemma 2

Must show: there is a bivalent initial state

Argument:

  • by contradiction: suppose no bivalent initial state
  • consider sequence of initial states
  • show some are $0$-valent, some are $1$-valent
  • show that some must be bivalent

$E_1$ is $0$-valent (Why?)

$E_5$ is $1$-valent

More Initial States

Assume: All Univalent

Adjacent Pair, Different Valency

All Extensions of $E_2$ Return $0$

All Extensions of $E_3$ Return $1$

$E_2’$ and $E_3’$ Indistinguishable

$E_2$ and $E_3$ Bivalent

Note

Don’t need to assume $P_2$ crashes

  • just assume first step of $P_2$ is scheduled after some other thread outputs
  • this is possible because we assume $A$ is wait-free
    • some process guaranteed to terminate even if one is not scheduled

Mere possibility of a crash together with wait-free assumption implies existence a bivalent initial state

  • same holds if we require only termination with one fault

Next Time

  • Bivalent initial conditions have critical executions
  • Wait-free consensus is impossible!
    • assuming only read/write registers…