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Bigamy Eli Gafni UCLA GETCO 2010

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Outline Models, tasks, and solvability What is SM? r/w w.f. asynch computability Sperner Lemma as a consequence of the impossibility of set consensus Impossibility of 0-1 coloring 1111… Simple algorithmic reduction of t-resiliency to wait-free (characterizing t-resiliency) – Consensus contingent on a resolver t-resiliency by r/w reduction Algorithmic Asynch Computability

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(Inputless) Tasks Point to set map from a chromatic simplex to a chromatic labeled complex that preserve dimension and colors Eg two proc cons between p0 and p1 p0p1 p0p1p0 0011

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Tasks cont’d A task is ``invoked’’ by a proc and returns a value to the invoker. If just p0 invoked, the task will return 0 If just p1 invoked, the task will return 1 If both invoked, it will either return 0 to both or 1 to both

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A model A task whose labels also identify a single consistent participating set for each label To make 2 cons a task we have 4 outputs – ((P0,0,p0),(p1,0,{p0,p1})) – ((P1,1,p1),(p0,1,{p0,p1})) – ((P0,0,{p0,p1}),(p1,0,{p0,p1})) – ((P1,1,{p0,p1}),(p0,1,{p0,p1})) coloroutput Participating set

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A model cont’ed A task/model solves a task if when iterating the model/task enough times to get a task A then there exists a chromatic simplicial map ``boundary preserving’’ from A to the task. SM wait-free model/task is the immediate snaps task

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What is SM? The task implemented operationally as follows: All interleaving of w(i), followed by any permutation of of r(i,j) j=1,…,n An r(I,j) ``reads’’ j for pi iff w(i) precedes it. pi returns all the ids it read What is the (succinct) ``spec’’ of the possible simplexes returned?

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SM spec There must be at least one proc who returns all (the proc to write last). Take all the procs that return the set n and eliminate their ids from smaller returned sets, ie move their write up in the sequence… Continue inductively. The returned sets left is an ``immediate snapshot’’ simplex SM is a ``fat’’ immediate snapshot n iterations of SM/task maps to immediate snapshot

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Set Cons Task (n) If a set P, |P|=k

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Sperner Lemma Sperner coloring of a subdivided simplex: Each 0-face distinct color. A vertex gets the color of one of the 0-faces carrying it Lemma: Any Sperner coloring forces a fully colored n-simplex

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Asynch Computability Any r/w w.f. algorithm for set cons implies a Sperner coloring of a chromatic subdivided simplex Any Sperner coloring of a chromaic subdivided simplex without fully colored simplex implies a r/w w.f. algorithm for set cons Conclusion: Sperner holds for chromatic subdivided simplexes Will later show a property that holds only for chromatic, so general Sperner is not ``automatic’’

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General Sperner from Chromatic Sperner Let A be a subdivided simplex and let A be Sperner colored, then there exists a refinement r(A) of A which is chromatic and moreover, If A has no fully colored simplex so does r(A). This is not a ``disguised derivation’’ of Sperner as no ``combinatorics’’ were relied on.

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Impossibility of 0-1 1111…task Want to prove the following Lemma: A 0-1 coloring of a chromatic subdivided simplex such that each boundary simplex has at least one vertex colored 0, forces a mono-chromatic full simplex Want to prove the Lemma from the impossibility of set cons.

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Proof The counter of the lemma implies a r/w w.f. algorithm on n procs such that of n processors at least one returns 0 and at least one returns 1. A proc that return 0 now chooses it name, at proc that returns 1 writes it in SM and reads and return a participating id that did not write it returned 1. The first to write 1 will not be returned by anybody = set cons.

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R/W Reduction What’s a researcher in Distributed Algorithms to do after the discovery of the connection to topology? – Take topo 101 in order to remain ``viable’’? Analogue thing occurred with NP-completeness. Some do ``algorithms’’ some do ``complexity’’ and suddenly a big implication from complexity on algorithm: – NP-completeness reduction Here: Do r/w reductions Characterization of t-resilient by reduction to w.f.:

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t-resiliency Model/Task implementation: – w(i) followed by any permutation of r(i,j) j=1,…,n – Interleave on I – T-resiliency: The first read is preceded by at least n-t w’s No ``fat’’ ``t-resilient immediate snapshot’’! Can be shown: 2 iterations of immediate snaps with ``views’’ that ``see’’ less than n-t removed Will show equivalent to: Wait once until see n-t then proceed w.f.

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t-res by r/w reduction Consensus contingent on a ``resolver’’ – If all propose same all output same – Else if resolver is alive all output its value – Epsilon ¼ agreement – If not ? Then output else wait for resolver 01 0?0? 1?1?

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t-res cont For simplicity t=1 Wait until n-1, or n In Round Robin service the ``reads’’ by resolver consensus, with the resolver for he code of pi is pi If stuck with no output then from there on al agree on the ``reads’’ until the stuck is resolved and jumps to another code.

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