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Towards a Theory of Peer-to-Peer Computability Joachim Giesen Roger Wattenhofer Aaron Zollinger

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Presentation Overview Introduction Peer-to-peer framework Protocol graph Peer-to-peer computability Summary

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Introduction Peer-to-Peer: –Sharing of computation resources directly between clients Current research: fault-tolerant distributed content location services Our focus: fundamental coordination primitives

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Peer-to-Peer Model Agents, registers: Atomic access to powerful read-modify-write registers Wait-free implementations Asynchronicity Ordering decision tasks

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Ordering Decision Tasks Position Predecessor Leader Election 312 <<

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Strategy Strategy: one agent‘s register access order Example: Agent 1: (1,2), (1,3), (1,2) Agent 2: (2,1), (2,3), (2,1) Agent 3: (3,2), (3,1), (3,2) Protocol: all agents‘ strategies

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Execution Execution: one access sequence of all strategies Example: Three Executions (1,2), (1,3), (1,2), (2,1), (2,3), (2,1), (3,2), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (2,3), (3,2), (2,1), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (3,2), (2,3), (2,1), (3,1), (3,2) Agent 1: (1,2), (1,3), (1,2) Agent 2: (2,1), (2,3), (2,1) Agent 3: (3,2), (3,1), (3,2)

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Execution DAG Execution DAG: –Vertices: execution accesses –Edge a l ! a l’ :(1) a l, a l’ of the form (i,¢) (2) a l, a l’ of the form (i,j), (j,i) Example: Three Executions (1,2), (1,3), (1,2), (2,1), (2,3), (2,1), (3,2), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (2,3), (3,2), (2,1), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (3,2), (2,3), (2,1), (3,1), (3,2)

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Indistinguishability Relation Two executions A, B can be indistinguishable for an agent i: A ~ i B Simply indistinguishable if so for all agents: A ~ B ~ i and ~ equivalence relations

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Pruned Execution DAG Indistinguishability relation defined via pruned execution DAG Prune execution DAG w.r.t. agent i: drop all vertices “irrelevant” to agent i Example: Three Executions (1,2), (1,3), (1,2), (2,1), (2,3), (2,1), (3,2), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (2,3), (3,2), (2,1), (3,1), (3,2) (1,2), (1,3), (1,2), (2,1), (3,2), (2,3), (2,1), (3,1), (3,2)

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Protocol Graph Protocol Graph defined on the executions of a protocol: –Vertices: equivalence classes of all executions w.r.t. to indistinguishability relation ~ –Edges E i : {u,v} with label i iff A 2 u, B 2 v s.t. A ~ i B

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Example: Protocol Graph II

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Invisibility Relation Basis for serializability in ordering decision tasks Invisibility relation i 8 A j: in the execution A, j is invisible for i Formally: in the execution DAG of A there is no oriented path from any (j,¢) to any (i,¢)

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Ordering Decision Tasks II Agents compute total order Task result compliant with invisibility Leader Election (1) 8 i,j: lead(i) = lead(j) (2) i 8 A j ! lead(i) j Position(1) i j ! pos(i) pos(j) (2) i 8 A j ! pos(i) < pos(j) Predecessor i if pos(i) = 1 j if pos(j) = pos(i) - 1 pred(i) =

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Computability I Theorem: Leader Election is impossible for n > 2 agents. Proof by connectivity of protocol graph. Theorem: Position is possible for n = 3. Proof:

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Computability II Theorem: Predecessor is impossible for n = 3. Proof by reduction to Leader Election. Corollary: Predecessor is impossible for n > 3. Proof by reduction to n = 3. Corollary: Position is possible for n > 3. Proof by simulation of counting networks.

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Summary Proposed peer-to-peer model Defined structures based on execution in this model Results: –Leader Election and Predecessor are impossible –Position is possible

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