An Axiomatic Approach to Computing the Connectivity of Synchronous and Asynchronous Systems

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An Axiomatic Approach to Computing the
Connectivity of Synchronous and Asynchronous
Systems

• Maurice Herlihy
• Microsoft Research Brown University
• Joint work with Sergio Rajsbaum Mark Tuttle

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Consensus Each Thread has a Private Input
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They Communicate
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They Agree on One Threads Input
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Generalization 2-Consensus
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History of k-Consensus

• Proposed Chaudhuri 90
• Impossible in R/W memory if failures can occur
• Borowsky Gafni STOC 93
• Herlihy Shavit STOC 93, JACM 1999
• Saks Zaharoglou STOC 93, SIAM 2000

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First Model Synchronous Message-Passing
Round 0
Round 1
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Failures Fail-Stop
Partial broadcast
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Second Model Asynchronous Message-Passing
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Failures Fail-Stop
Partial broadcast
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Impossibility Results fork-Consensus

• Synchronous Model
• Impossible in lt (f/k)1 rounds
• Asynchronous Model
• Impossible always
• Whats wrong? No unified picture
• Technical details quite different
• Obscures underlying similarity

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Our Contribution

• Model-independent part (once)
• Abstract round operator
• Satisfies simple combinatorial axioms
• Model-specific part (once per model)
• Identify model-specific round operator
• Prove that it satisfies axioms

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Initial States
0

• Vertex process (color) and input value
• Simplex one initial state
• Complex set of initial states

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1
0
1
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Final States

• Vertex process (color) and final value
• Simplex one final state
• Complex set of final states

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Protocol Protocol Complex

• Finite program
• Processes send and receive messages
• Each process decides after fixed number of steps

• Vertex process and history of messages it sent
  received
• Simplex one final state
• Complex set of final states

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Example Synchronous Protocol Complexes
start
one round
two rounds
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Theorem

• Cannot solve k-consensus if the protocol complex
  is (k-1)-connected
• In dimension k-1 and lower
• No holes
• Spheres contractible
• Homotopy groups trivial
• Related to Sperners Lemma

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Reasoning About Connectivity
If C0 is k-connected
and C1 is k-connected
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Reasoning About Connectivity
and C0 ? C1 is (k-1)-connected
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Reasoning About Connectivity
then C0 ? C1 is (k-1)-connected
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Mayer-Vietoris

• Reason about connectivity in a purely
  combinatorial way
• Challenge is to decompose complex so that
  Mayer-Vietoris applies inductively

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Round Operator

• Computation as sequence of rounds
• Natural in synchronous model
• Also works in asynchronous model
• Inductive reasoning
• If complex is (k-1)-connected before
• Then it remains (k-1)-connected after

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Axioms

• How does the number of failures affect the round
  operator?
• How does the operator work on neighboring
  simplexes?
• How does the operator work on a single simplex?

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Round Operator
Qf(S)
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Round Operator
Round operator name
Qf(S)
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Round Operator
Round operator name
Qf(S)
Number of failures
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Round Operator
Round operator name
Qf(S)
Starting simplex (or complex)
Number failures
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Round Operator
Qf(S)
Set of simplexes
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Round Operator
Qf(S)
Set of simplexes
Simplicial Complex
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Axiom One
If f g
Qf(S) ? Qg(S)
Possible states with fewer failures
Contained in possible states with more failures
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Example of Axiom One
zero failures
1 failures
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Nearby Starting States
Qf(S0)
f failures
Intersection?
f failures
Qf(S1)
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Intersections
processes that cant tell whether they started
in S0 or S1
Qf(S0)?Qf(S1)
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Nearby Starting States
S0
S0-
Processes that can tell
S1
S1-
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Nearby Starting States
f-1 failures
S0-
S1-
f-1 failures
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Axiom Two
Qf(S0)?Qf(S1)
Run with c fewer failures

Qf-c(S0?S1)
Discard c processes not in intersection
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Axiom Two
Qf(S0)?Qf(S1)
Warning axiom used in paper slightly more
technical
Run with c fewer failures

Qf-c(S0?S1)
Discard c processes not in intersection
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Computing Connectivity
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Computing Connectivity
k-connected
k-connected
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Computing Connectivity
Problem indivudual intersections have different
dimensions
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Observation
small intersections
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Observation
absorbed by larger intersections
small intersections
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Absorbing POSet

• Simplexes (partially) ordered
• Small intersections with later simplexes
  absorbed by large intersections

Warning definition used in paper is a little
more technical
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Absorbing POSet

• Similar to shellable complex
• Same basic idea, but
• Some technical differences

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Theorem
Composition of round operators With k failures
per round
Input complex (absorbing POSet)
Qk(A) is (k-1)-connected
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Model-Specific Round Operators

• Synchronous Model
• Round with k failures
• Asynchronous Model
• All but k deliver messages to all
• Rest do partial broadcast (dont crash)

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Corollaries

• Synchronous Model
• k-consensus requires (f/k)1 rounds
• Asynchronous Model
• k-consensus is impossible
• Proofs
• Verify that round operators satisfy the three
  axioms

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Our Ambitions

• We conjecture that this axiomatic approach can be
  extended to other models
• Did we pick the right axioms for a Grand Unified
  Theory?
• Find out at GETCO 2005!