1. Sergio Rajsbaum 2006 Lecture 3 Introduction to Principles of Distributed Computing Sergio Rajsbaum Math Institute UNAM, Mexico

2. Sergio Rajsbaum 2006 Lecture 3 Part I: synchronous uniform consensus lower bound

3. Sergio Rajsbaum 2006 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 X0X0 L(X 0 ) L 2 (X 0 ) Initial states states after one round states after 2 rounds Connectivity preserved Connectivity destroyed

4. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 In the rest of the course we assume all possible vectors over the input values V unless specified otherwise

6. Sergio Rajsbaum 2006 Basic Model Message passing (essentially equivalent to read/write shared memory model) Channels between every pair of processes Crash failures t 1 processes No message loss among correct processes

7. Sergio Rajsbaum 2006 Synchronous Model

8. Sergio Rajsbaum 2006 Timing model Processor speeds –All run at the same speed Message delays –Constant

9. Sergio Rajsbaum 2006 Round 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) If process i crashes in a round, then any subset of the messages i sends in this round can be lost

10. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 A Simple Proof of the Uniform Consensus Synchronous Lower Bound [Keidar, Rajsbaum IPL 02]

14. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 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 000001111011 ~~~ n = 3

16. Sergio Rajsbaum 2006 ColoringColoring 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. Sergio Rajsbaum 2006 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

18. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 Layering Layering L = set of environment actions –L(X) = {x.E | x  X, E  L applicable to x} –L 0 (X) = X –L k (X) = L(L k-1 (X)) Define system using layers –X 0 set of initial states –System: all runs obtained from L(. ) [Moses, Rajsbaum 98; Gafni 98; Herlihy, Rajsbaum,Tuttle 98] X0X0 L(X 0 ) L 2 (X 0 )

20. Sergio Rajsbaum 2006 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, L f (X 0 ) connected –feature of model, not of the problem –also implies consensus f+1 lower bound –can be proven for all L i (X 0 ) in other models, e.g., mobile failure model [MosesR98], [Santoro,Widemayer89], and asynchronous model

21. Sergio Rajsbaum 2006 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. Sergio Rajsbaum 2006 Uniform Lemma: Proof Assume, by contradiction, in failure-free extensions of y, y’, all decide after 1 round 2 cases: j either failed or non-failed y’y xx’... X connected, val(x)= 0, val(x’)=1 differ only in state of some j

23. Sergio Rajsbaum 2006 Illustrating the Contradiction C ase 1: j is correct y y’ y.(0,[0])y’.(0,[0]) X y y’ X y.(1,[2])y’.(1,[2]) XXXX y.(1,[2]).(3,[3]) A contradiction to uniform agreement! val(y)=0, so y leads to decision 0 in one failure-free round look the same to process 2 look the same to process 3

24. Sergio Rajsbaum 2006 The uniform consensus synchronous lower bound n >2, t >1, f =0 X 0 = {initial failure-free states} connected  x’,x  X 0 s.t. val(x)=0, val(x’)=1 (validity) By Uniform Lemma, from some initial state need 2 rounds to decide

25. Sergio Rajsbaum 2006 Connectivity Lemma: L f (X 0 ) 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 ~ x’ L(x)L(x’) x.(i, [n]) ~ x’.(i, [n])

26. Sergio Rajsbaum 2006 L(x) is Connected xx x.(0,[0]) ~ x.(1,[0]) X x.(0,[0]) ~ x.(2,[0]) ~ x.(2,[1]) ~ x.(2,[3]) x.(0,[0]) ~ x.(3,[0]) ~ x.(3,[1]) ~ x.(3,[2]) X x x.(1,[2]) X x x.(1,[3]) ~ ~

27. Sergio Rajsbaum 2006 Theorem: f+2 Lower Bound Assume n>t, and f < t-1 L f (X 0 ) - final states of runs with  f failures –connected –in any state in L f (X 0 ) exist at least 3 non-failed processes and 2 can fail Take z, z’  X 0 s.t. val(z)  val(z’), –let x, x’ be failure-free extensions of z, z’: x=z.(i,[0]) f  L f (X 0 )

28. Sergio Rajsbaum 2006 Exercise 1.Consider Modify the theorem and the proof of this talk for the consensus problem (instead of the uniform consensus problem)

29. Sergio Rajsbaum 2006 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

30. Sergio Rajsbaum 2006