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Lecture 3 Introduction to Principles of Distributed Computing

Lecture 3 Introduction to Principles of Distributed Computing. Sergio Rajsbaum Math Institute UNAM, Mexico. Lecture 3. Part I : synchronous uniform consensus lower bound. L 2 (X 0 ). L(X 0 ). X 0. The lecture in a nutshell. Traditionally different models were treated in different ways

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Lecture 3 Introduction to Principles of Distributed Computing

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  1. Lecture 3Introduction to Principles of Distributed Computing Sergio Rajsbaum Math Institute UNAM, Mexico

  2. Lecture 3 • Part I: synchronous uniform consensus lower bound

  3. L2(X0) L(X0) X0 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 Connectivity preserved states after 2 rounds

  4. 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

  5. In the rest of the course we assume all possible vectors over the input values V unless specified otherwise

  6. 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

  7. Synchronous Model

  8. Timing model • Processor speeds • All run at the same speed • Message delays • Constant

  9. 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

  10. 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]

  11. 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

  12. 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

  13. A Simple Proof of the Uniform Consensus Synchronous Lower Bound[Keidar, Rajsbaum IPL 02]

  14. 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

  15. n = 3 000 ~ 001 ~ 011 ~ 111 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 • Intuition: in connected states there cannot be different decisions

  16. 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)

  17. 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

  18. 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

  19. L2(X0) L(X0) X0 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]

  20. 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 allLi(X0) in other models, e.g., mobile failure model [MosesR98], [Santoro,Widemayer89], and asynchronous model

  21. 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

  22. ... ... x x’ y y’ Uniform Lemma: Proof • X connected, val(x)= 0, val(x’)=1 • Assume, by contradiction, in failure-free extensions of y, y’, all decide after 1 round • 2 cases: j either failed or non-failed differ only in state of some j

  23. y’ y y y’ X X y.(0,[0]) y’.(0,[0]) y.(1,[2]) y’.(1,[2]) look the same to process 3 X X X X y.(1,[2]).(3,[3]) y.(1,[2]).(3,[3]) 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!

  24. The uniform consensus synchronous lower bound • n >2, t >1, f =0 • X0 = {initial failure-free states} connected • x’,xX0s.t.val(x)=0, val(x’)=1 (validity) • By Uniform Lemma, from some initial state need 2 rounds to decide

  25. L(x) L(x’) Connectivity Lemma: Lf(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]) x.(i, [n]) ~ x’.(i, [n]) x ~ x’

  26. 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]) L(x) is Connected

  27. 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)

  28. Exercise • Consider Modify the theorem and the proof of this talk for the consensus problem (instead of the uniform consensus problem)

  29. 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, “Onthe 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

  30. End of Lecture 3

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