Lecture 3
Introduction to Principles of
Distributed Computing
Sergio Rajsbaum
Math Institute
UNAM, Mexico
Sergio Rajsbaum 2006
Lecture 3
• Part I: synchronous uniform consensus lower
bound
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The lecture in a nutshell
• Traditionally different models were treated in
different ways
• We will see that, for consensus, this is not needed
• Consensus solvability depends on how long
connectivity preserved by a particular model
Connectivity
destroyed
Connectivity
preserved
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L2(X0)
L(X0)
X0
Initial
states after
states
states after
one round
2 rounds
CONSENSUS
A fundamental Abstraction
Each process has an input, should decide an output s.t.
Agreement: correct processes’ decisions are the same
Validity: decision is input of one process
Termination: eventually all correct processes decide
There are at least two possible input values 0 and 1
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In the rest of the course we assume
all possible vectors over the input
values V unless specified otherwise
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Basic Model
• Message passing (essentially equivalent to
read/write shared memory model)
• Channels between every pair of processes
• Crash failures
t < n potential failures out of n >1 processes
• No message loss among correct processes
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Synchronous Model
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Timing model
• Processor speeds
– All run at the same speed
• Message delays
– Constant
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Synchronous Model
• Algorithm runs in synchronous rounds:
Round
– send messages to any set of processes,
– receive messages from previous round,
– do local processing (possibly decide, halt)
• If process i crashes in a round, then any subset
of the messages i sends in this round can be lost
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Synchronous Consensus
• In a run with f failures (f<t)
– Processes can decide in f+1 rounds
– And no less !
[Lamport Fischer 82; Dolev, Reischuk, Strong 90] (early-deciding)
• 1 round with no failures
• In this talk deciding
– halting takes min(f+2,t+1) [Dolev, Reischuk, Strong 90]
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Uniform Consensus
• Uniform agreement: decision of every two
processes is the same
Recall: with consensus, only correct processes
have to agree (disagreement with the dead is
OK)
This version of consensus will be useful to extend
the lower bound argument to asynchronous
models
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Synchronous Uniform Consensus
Every algorithm has a run with f
failures (f<t-1), that takes at least
f+2 rounds to decide
• [Charron-Bost, Schiper 00; KR 01]
– as opposed to f+1 for consensus
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A Simple Proof of the Uniform
Consensus Synchronous Lower Bound
[Keidar, Rajsbaum IPL 02]
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States
• State = list of processes’ local states
• Given a fixed deterministic algorithm,
state at the end of run determined by initial
values and environment actions
– failures, message loss
– can be denoted as:
x . E1. E2. E3
x state, Ei environment actions
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Connectivity
States x, x’ are similar, x~x’, if they look the
same to all but at most one process
• Set of initial states of consensus is connected
000
~
001
~
011
~
111
n=3
• Intuition: in connected states there cannot be
different decisions
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Coloring
• Impossibility proofs color non-decided states
• Classical coloring: valency, potential
decisions state can lead to e.g. [FLP85]
• Our coloring:
val(x) = decision of correct processes in
failure-free extension of x (0 or 1)
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To Prove Lower Bounds
or impossibility results
• Sufficient to look at subset of runs, called a
system
• Simplifies proof
• A set of environment actions defines a
system
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Considered Environment Actions
• (i, [k]) - i fails,
– messages to processes {1,…,k} lost (if sent)
– [0] empty set - no loss
– applicable if i non-failed and < t failures
• (0, [0]) - no failures
– always applicable
Notice: at most one process fails in one round
– its messages lost by prefix of processes
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Layering
• Layering L = set of environment actions
– L(X) = {x.E | x  X, E  L applicable to x}
– L0(X) = X
– Lk(X) = L(Lk-1(X))
• Define system using layers
– X0 set of initial states
– System: all runs obtained from L( . )
[Moses, Rajsbaum 98; Gafni 98;
Herlihy, Rajsbaum,Tuttle 98]
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L2(X0)
L(X0)
X0
Proof Strategy
• Uniform Lemma: from connected set, under some
conditions, 2 more rounds needed for uniform
consensus (recall: 1 for consensus)
• The initial states are connected.
Connectivity lemma: for f<t+1, Lf(X0) connected
– feature of model, not of the problem
– also implies consensus f+1 lower bound
– can be proven for all Li(X0) in other models, e.g.,
mobile failure model [MosesR98], [Santoro,Widemayer89], and
asynchronous model
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Uniform Lemma
• If
– X connected
– x,x’X, s.t. val(x)= 0, val(x’)=1
– In all states in X exist at least 3 non-failed
processes and 2 can fail
• Then
– yX s.t. in y.(0,[0]) not all decide
1-round failure-free extension of y
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Uniform Lemma: Proof
• X connected, val(x)= 0, val(x’)=1
x
...
y
y’
...
x’
differ only in state of
some j
• Assume, by contradiction, in failure-free
extensions of y, y’, all decide after 1 round
• 2 cases: j either failed or non-failed
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Illustrating the Contradiction
Case 1: j is correct
y’
y
y
y’
X
y.(0,[0])
y’.(0,[0])
X
y’.(1,[2])
y.(1,[2])
the same
val(y)=0, solook
y leads
to to process 3
decision 0
in one failure-free round
X
X
X
X
y.(1,[2]).(3,[3]) y.(1,[2]).(3,[3])
look the same to process 2
A contradiction to uniform agreement!
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The uniform consensus
synchronous lower bound
•
•
•
•
n >2, t >1, f =0
X0 = {initial failure-free states} connected
x’,xX0 s.t. val(x)=0, val(x’)=1 (validity)
By Uniform Lemma, from some initial state
need 2 rounds to decide
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Connectivity Lemma:
f
L (X0) Connected for f<t+1
• Proof by induction, base immediate
• For state x, L(x) connected (next slide)
• Let x~x’X,
– x, x’ differ in state of i only, i can fail
– x.(i, [n]) = x’.(i, [n])
L(x)
L(x’)
x.(i, [n]) ~ x’.(i, [n])
x ~ x’
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L(x) is Connected
x
x
~
x
X
~
x
X
~
X
x.(0,[0])
x.(1,[0])
x.(1,[2])
x.(1,[3])
x.(0,[0])
~ x.(2,[0])
~ x.(2,[1])
~ x.(2,[3])
x.(0,[0])
~ x.(3,[0])
~ x.(3,[1])
~ x.(3,[2])
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Theorem: f+2 Lower Bound
• Assume n>t, and f < t-1
• Lf(X0) - final states of runs with  f failures
– connected
– in any state in Lf(X0) exist at least 3 non-failed
processes and 2 can fail
• Take z, z’X0 s.t. val(z)  val(z’),
– let x, x’ be failure-free extensions of z, z’:
x=z.(i,[0])f  Lf(X0)
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Exercise
1. Consider Modify the theorem and the proof
of this talk for the consensus problem
(instead of the uniform consensus problem)
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Bibliography
• Keidar and Rajsbaum, “A Simple Proof of the Uniform
Consensus Synchronous Lower Bound,” in IPL, Vol. 85, pp.
47-52, 2003.
• Keidar and Rajsbaum, “On the Cost of Fault-Tolerant
Consensus When There Are No Faults” in Keidar’s page,
including slides and papers.
• Moses, Rajsbaum, “A Layered Analysis of Consensus,”
SIAM J. Comput. 31(4): 989-1021, 2002.
• Mostéfaoui, Rajsbaum, Raynal: Conditions on
input vectors for consensus solvability in
asynchronous distributed systems. J. ACM, 2003
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Sergio Rajsbaum 2006