Bigamy

1. Bigamy Eli Gafni UCLA GETCO 2010

2. 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

3. (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 p0 p1 0 0 1 1 p0 p1 p0 p1

4. 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

5. 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})) color output Participating set

6. 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

7. 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?

8. 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

9. Set Cons Task (n) • If a set P, |P|=k<n invoke the task each proc returns an id from P • |P|=n one of the ids is not returned by anybody.

10. 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

11. 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’’

12. 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.

13. 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.

14. Proof • The counter of the lemma implies a r/w w.f. algorithm on nprocs 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.

15. 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.:

16. 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.

17. 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 0 1 1 ? 0?

18. 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.