Lecture 3 Introduction to Principles of Distributed Computing

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Lecture 3Introduction to Principles of
Distributed Computing
Sergio Rajsbaum Math Institute UNAM, Mexico
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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
Initial states
states after one round
states after 2 rounds
Connectivity preserved
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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 lt n potential failures out of n gt1 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
• send messages to any set of processes,
• receive messages from previous round,
• do local processing (possibly decide, halt)

Round

• 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 (fltt)
• Processes can decide in f1 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(f2,t1) 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
  (fltt-1), that takes at least f2 rounds to decide
• Charron-Bost, Schiper 00 KR 01
• as opposed to f1 for consensus

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A Simple Proof of the Uniform Consensus
Synchronous Lower BoundKeidar, Rajsbaum IPL
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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, xx, if they look the
  same to all but at most one process
• Set of initial states of consensus is connected
• 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 Boundsor 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 lt 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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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 fltt1, Lf(X0) connected
• feature of model, not of the problem
• also implies consensus f1 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

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
val(y)0, so y leads to decision 0 in one
failure-free round
look the same to process 2
A contradiction to uniform agreement!
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The uniform consensus synchronous lower bound

• n gt2, t gt1, 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 Lf(X0) Connected for fltt1

• Proof by induction, base immediate
• For state x, L(x) connected (next slide)
• Let xx?X,
• x, x differ in state of i only, i can fail
• x.(i, n) x.(i, n)

x.(i, n) x.(i, n)
x x
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L(x) is Connected
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Theorem f2 Lower Bound

• Assume ngtt, and f lt 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
  xz.(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 Keidars 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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End of Lecture 3