N-Consensus is the Second Strongest Object for N+1

Processes

Eli Gafni UCLA

Petr Kuznetsov Max Planck Institute for

Software Systems

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Distributed shared memory

Processes communicate via objects Each object is an instance of a type

Type N-consensus: first N operations propose(v), v∈{0,1} return the argument of the first operation, N+1-th fails

Object types can be classified by their consensus power

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Consensus power

Cons(T)=N iff objects of T and RW registers can implement N-consensus but not N+1-consensus

N-consensus is universal in a system of N processes : can implement an object of any type [Her91]

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What about N or more processes?

Can objects of consensus power N “do more” than N-consensus objects?

No: if an N+1-process task can be solved by deterministic objects of consensus power N, then it can also be solved by N-consensus objects

⇨ There are no objects between N- and N+1-consensus

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Main results

Theorem 1. Any protocol over N+1 processes using deterministic objects of consensus power N is connected.

Theorem 2. Any connected protocol over N+1 processes can be emulated using N-consensus

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The rest of it

Definitions and tools Inseparability Theorem 1: connectivity of protocols Theorem 2: emulation Conclusions and future

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Processes and protocols

N+1 processes communicate via objects of consensus power N, RW registers

Protocol P: a collection of deterministic automata P1,…PN+1, assume a single initial state

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Objects and synchrony

Objects are atomic and deterministic: an execution of P can be seen as a sequence of process idsa state of P is a finite execution a view of Pi in x is the local state of Pi in x

System is asynchronous: we consider all possible executions2-consensus cannot be implemented using RW

registers N+1-consensus cannot be implemented using N-

consensus

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Some tools

2-process ε-agreement task for q0 and q1 If qi goes alone, qi outputs i outputs are within ε in [0,1](can be implemented using RW registers)

N-consensus can be used to split processes in two teams, each running as a single process in a 2-process RW protocol (e.g., ε-agreement)

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InseparabilityStates x0 and x1 of P are inseparable (x0~x1)

iff some process Pi cannot distinguish x0 and x1 even in an infinite execution

Pi goes through the same sequence of views

in e0 and e1

x1x0

e1e0

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Inseparably connected protocols

P is inseparably connected iff for each state x of P, the set G(x) of N+1 immediate descendants of x is transitively inseparable

x1

x2

xN xN+1

~ ~ ~ ~…

There is a path of inseparable states between any two

states in G(x)

…

x

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Connectivity and consensus power

Theorem 1. Any protocol over N+1 processes using deterministic objects of consensus power N is connected

Proof sketch: suppose some G(x) is ~ disconnected each execution extending x reveals the

component of G(x) we can solve consensus using x and P

(use determinism here)

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Emulation of connected protocols

Let P be connected Then we can emulate it using N-

consensus!

Emulation: Processes periodically output views of P

Safety: All output views are compatible with an execution of P

Liveness: at least one active process makes progress (outputs infinitely many views of P)

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Emulation invariants Each process maintains a local estimate

of the “current” state of P Asynchronous phases, at the end of

each phase:There are at most two different state

estimates, and they are inseparable At least one participant outputs a new view All views output so far are compatible with

both state estimates

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Phase description

Processes split in two teams q0 and q1

Each qi elects a member to promote: proposes a new “next” state xi and a new view vi

q0 and q1 runs ε-agreement and map the results to vertices of the non-separating path connecting x0 and x1

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Look in the future: x0~x1

Go along inseparable executions until a confusing view v

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Look in the future: x0 and x1 are not inseparable

But their parents x0’ and x1’ are – concatenate the paths

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Choosing the next state: safety

ε=Ω(1/N): the decided vertices are neighbors

Neighboring states are inseparable

Each view is compatible with neighboring states: safety is guaranteed!

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Joining a team: liveness

In every phase, a process:

Try to agree on the system state using N-consensus

If succeeded, join team q0 If failed (only if N came earlier)– join q1

≤N processes participate ⇨ q1 is empty Only a participant obtains a new view:

liveness is guaranteed!

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A corollary that matters

Let P be any protocol that solves a task M

A non-blocking emulation of P is also wait-free: every active process makes progress

Every N+1-process task that can be solved by objects of consensus power N can also be solved by N-consensus objects

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Summary

Inseparability (a general connectivity) Emulation of any connected protocol

In a system of N+1 processes, there is no object between N- and N+1-consensus

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Related questions Can objects of consensus power N be

implemented by N-consensus objects?Common2 class: is queue in it? [Afek et al. 93,

…]

How to emulate a protocol P that uses objects of two types, each of consensus power NRobustness [Jay93,…]

Non-determinism?

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Thank you!

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Deterministic objects

Q, set of states O, set of operations R, set of responses σ, Q x O → Q x R, sequential specification

RW-register: a read returns the most recent written value

N-consensus: first N agree on a proposed value, N+1-th fails