Timely Common Knowledge Strategy Timely Characterising Asymmetric - - PowerPoint PPT Presentation

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Timely Common Knowledge Strategy Timely Characterising Asymmetric - - PowerPoint PPT Presentation

Aug 14, 2022 120 likes •1.27k views

Introduction Background Problem & Timely Common Knowledge Strategy Timely Characterising Asymmetric Distributed Coordination via Common Knowledge Vectorial Fixed Points Timely- Coordinated Response Nested- Yannai A. Gonczarowski

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge

Characterising Asymmetric Distributed Coordination via Vectorial Fixed Points Yannai A. Gonczarowski

Einstein Institute of Mathematics and Center for the Study of Rationality The Hebrew University of Jerusalem

January 8, 2013 Joint work with Yoram Moses

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 1 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.
  • Drying should not begin more than 5 minutes after the

first washing robot finished.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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SLIDE 7

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.
  • Drying should not begin more than 5 minutes after the

first washing robot finished.

  • A few immediate consequences:

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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SLIDE 8

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.
  • Drying should not begin more than 5 minutes after the

first washing robot finished.

  • A few immediate consequences:
  • L and R may not finish more than 5 minutes apart.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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SLIDE 9

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.
  • Drying should not begin more than 5 minutes after the

first washing robot finished.

  • A few immediate consequences:
  • L and R may not finish more than 5 minutes apart.
  • R must start washing no more than 3 minutes after, and

no more than 7 minutes before L starts.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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SLIDE 10

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Motivational Example

  • Consider a robotic Car Wash.
  • Two washing robots: L and R.
  • A drying robot D.
  • At some point after a car enters, it must be soaped &

rinsed from both sides, and then dried.

  • L is of a new model — takes 4 minutes to wash.
  • R is older — takes 6 minutes.
  • Drying should not begin more than 5 minutes after the

first washing robot finished.

  • A few immediate consequences:
  • L and R may not finish more than 5 minutes apart.
  • R must start washing no more than 3 minutes after, and

no more than 7 minutes before L starts.

  • ...

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 2 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be coordinated?y

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be coordinated?y
  • What if the robots cannot see each other / the car?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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SLIDE 13

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be coordinated?y
  • What if the robots cannot see each other / the car?
  • What if messages may take a while to arrive?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be coordinated?y
  • What if the robots cannot see each other / the car?
  • What if messages may take a while to arrive?
  • What if messages may never arrive?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be optimally coordinated?y
  • What if the robots cannot see each other / the car?
  • What if messages may take a while to arrive?
  • What if messages may never arrive?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Primary Goal

Constraints

−7 ≤ startr − startl ≤ 3 6 ≤ startd − startr ≤ 11 −9 ≤ startl − startd ≤ −4

  • How can the tasks be optimally coordinated?y
  • What if the robots cannot see each other / the car?
  • What if messages may take a while to arrive?
  • What if messages may never arrive?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 3 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

  • When L acts, it must know that the car has arrived.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

  • When L acts, it must know that the car has arrived.
  • It must also know that between 7 time units earlier and 3

time units later, R knows that the car has already arrived.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

  • When L acts, it must know that the car has arrived.
  • It must also know that between 7 time units earlier and 3

time units later, R knows that the car has already arrived.

  • It must also know that between 7 time units earlier and 3

time units later, R knows that between 3 time units earlier and 7 time units later, L knows that the car has already arrived.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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SLIDE 22

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

  • When L acts, it must know that the car has arrived.
  • It must also know that between 7 time units earlier and 3

time units later, R knows that the car has already arrived.

  • It must also know that between 7 time units earlier and 3

time units later, R knows that between 3 time units earlier and 7 time units later, L knows that the car has already arrived.

  • etc.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Consequences

Knowledge of Precondition

If a protocol P guaranteeing that L starts to wash only if a given event ψ holds, then whenever L starts to wash (according to P), L knows that ψ holds.

  • When L acts, it must know that the car has arrived.
  • It must also know that between 7 time units earlier and 3

time units later, R knows that the car has already arrived.

  • It must also know that between 7 time units earlier and 3

time units later, R knows that between 3 time units earlier and 7 time units later, L knows that the car has already arrived.

  • etc.
  • Codependence ⇒ infinitely many epistemic requirements.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Halpern and Moses (1990): Common knowledge is necessary and sufficient for simultaneous actions.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Halpern and Moses (1990): Common knowledge is necessary and sufficient for simultaneous actions.

  • Necessary: A simultaneous action implies common

knowledge of the action.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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SLIDE 27

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Halpern and Moses (1990): Common knowledge is necessary and sufficient for simultaneous actions.

  • Necessary: A simultaneous action implies common

knowledge of the action.

  • Sufficient: Common knowledge of any fact arises

simultaneously among all agents, and thus can be used to coordinate a simultaneous action.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Ben-Zvi and Moses (2010): Nested knowledge is necessary and sufficient for linear ordering of actions.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Ben-Zvi and Moses (2010): Nested knowledge is necessary and sufficient for linear ordering of actions.

  • Necessary:

act2 ⇒ K2(previously act1), act3 ⇒ K3K2(previously act1), etc.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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SLIDE 30

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Review: Knowledge Gain and Coordination

Many forms of coordination have a necessary and sufficient characterising epistemic state.

Ben-Zvi and Moses (2010): Nested knowledge is necessary and sufficient for linear ordering of actions.

  • Necessary:

act2 ⇒ K2(previously act1), act3 ⇒ K3K2(previously act1), etc.

  • Sufficient: Nested knowledge of any fact arises linearly, and

thus can be used to coordinate linearly-ordered actions.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reminder: The Runs and Systems Model

Based upon Fagin et al. (1995)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reminder: The Runs and Systems Model

Based upon Fagin et al. (1995)

  • Agents: I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reminder: The Runs and Systems Model

Based upon Fagin et al. (1995)

  • Agents: I.
  • Time: T = N ∪ {0}. (For ease of presentation.)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reminder: The Runs and Systems Model

Based upon Fagin et al. (1995)

  • Agents: I.
  • Time: T = N ∪ {0}. (For ease of presentation.)
  • Runs: R. Each run is a function from time to:
  • A state of the environment — and —
  • A state for each agent.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reminder: The Runs and Systems Model

Based upon Fagin et al. (1995)

  • Agents: I.
  • Time: T = N ∪ {0}. (For ease of presentation.)
  • Runs: R. Each run is a function from time to:
  • A state of the environment — and —
  • A state for each agent.
  • A protocol governs the agents’ behaviour.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

  • ψ ⊆ φ means “ψ validly implies φ”.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

  • ψ ⊆ φ means “ψ validly implies φ”.

For any event ψ,

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

  • ψ ⊆ φ means “ψ validly implies φ”.

For any event ψ, ◯dψ

  • (r, t) | (r, t + d) ∈ ψ
  • Yannai A. Gonczarowski (HUJI)

Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

  • ψ ⊆ φ means “ψ validly implies φ”.

For any event ψ, ◯dψ

  • (r, t) | (r, t + d) ∈ ψ
  • ◯≤dψ
  • (r, t) | ∃t′ ≤ t + d : (r, t′) ∈ ψ
  • Yannai A. Gonczarowski (HUJI)

Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Set-Theoretic Reasoning

  • A point (r, t) ∈ ΩR R × T defines a snapshot of the

system.

  • An event is associated with the set of points

ψ ∈ FR 2ΩR at which it holds.

  • ψ ⊆ φ means “ψ validly implies φ”.

For any event ψ, ◯dψ

  • (r, t) | (r, t + d) ∈ ψ
  • ◯≤dψ
  • (r, t) | ∃t′ ≤ t + d : (r, t′) ∈ ψ
  • Kjψ
  • (r, t) | [(r, t)]j ⊆ ψ
  • [(r, t)]j is the event “the local state of j is rj(t)”.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Halpern and Moses (1990)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Halpern and Moses (1990)

  • Common knowledge of ψ is the greatest fixed point of

x → EI(ψ ∩ x).

  • In the lattice FR.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Halpern and Moses (1990)

  • Common knowledge of ψ is the greatest fixed point of

x → EI(ψ ∩ x).

  • In the lattice FR.
  • A few variants:

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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SLIDE 48

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Halpern and Moses (1990)

  • Common knowledge of ψ is the greatest fixed point of

x → EI(ψ ∩ x).

  • In the lattice FR.
  • A few variants:
  • Define Eventual Common Knowledge of ψ as the greatest

fixed point of x → ∩j∈I ◇ Kj(ψ ∩ x).

  • Necessary and sufficient for eventual coordination.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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SLIDE 49

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Epistemic Fixed-Point Analysis

Halpern and Moses (1990)

  • Common knowledge of ψ is the greatest fixed point of

x → EI(ψ ∩ x).

  • In the lattice FR.
  • A few variants:
  • Define Eventual Common Knowledge of ψ as the greatest

fixed point of x → ∩j∈I ◇ Kj(ψ ∩ x).

  • Necessary and sufficient for eventual coordination.
  • Define ε-Common Knowledge of ψ as the greatest fixed

point of x → E ε

I (ψ ∩ x).

  • Necessary and sufficient for up-to-ε coordination.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reasoning about Coordination

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 9 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reasoning about Coordination

Fagin et al. (1995)

An I-ensemble, where I ⊆ I is a set of agents, is an I-tuple of events ¯ e = (ej)j∈I ∈ FRI, in which for every agent j ∈ I, the event ej is a j-local event.

  • An event is j-local if it is known to j whenever it occurs.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 9 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Reasoning about Coordination

Fagin et al. (1995)

An I-ensemble, where I ⊆ I is a set of agents, is an I-tuple of events ¯ e = (ej)j∈I ∈ FRI, in which for every agent j ∈ I, the event ej is a j-local event.

  • An event is j-local if it is known to j whenever it occurs.

Central example: the tuple ¯ e = {“j is starting to clean right now”}j∈I is an ensemble.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 9 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Can these be Generalized to deal with Arbitrary Timely Constraints?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Can these be Generalized to deal with Arbitrary Timely Constraints?

Definition (Timely Coordination)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Can these be Generalized to deal with Arbitrary Timely Constraints?

Definition (Timely Coordination)

1 A timely-coordination spec is a pair (I, δ), where I ⊆ I is a

set of agents and δ : I 2 → Z ∪ {∞}.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Can these be Generalized to deal with Arbitrary Timely Constraints?

Definition (Timely Coordination)

1 A timely-coordination spec is a pair (I, δ), where I ⊆ I is a

set of agents and δ : I 2 → Z ∪ {∞}. δ(r, l) = −7 δ(l, r) = 3 δ(d, r) = 6 δ(r, d) = 11 δ(l, d) = −9 δ(d, l) = −4

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Can these be Generalized to deal with Arbitrary Timely Constraints?

Definition (Timely Coordination)

1 A timely-coordination spec is a pair (I, δ), where I ⊆ I is a

set of agents and δ : I 2 → Z ∪ {∞}.

2 Given a timely-coordination spec (I, δ) and a system

R ⊆ R, we say that an I-ensemble ¯ e ∈ FRI is δ-coordinated (in R) if for every (j, k) ∈ I 2 and for every (r, t) ∈ ej, there exists t′ ≤ t + δ(j, k) s.t. (r, t′) ∈ ek. δ(r, l) = −7 δ(l, r) = 3 δ(d, r) = 6 δ(r, d) = 11 δ(l, d) = −9 δ(d, l) = −4

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Can these be Generalized to deal with Arbitrary Timely Constraints?

Definition (Timely Coordination)

1 A timely-coordination spec is a pair (I, δ), where I ⊆ I is a

set of agents and δ : I 2 → Z ∪ {∞}.

2 Given a timely-coordination spec (I, δ) and a system

R ⊆ R, we say that an I-ensemble ¯ e ∈ FRI is δ-coordinated (in R) if for every (j, k) ∈ I 2 and for every (r, t) ∈ ej, there exists t′ ≤ t + δ(j, k) s.t. (r, t′) ∈ ek. δ(r, l) = −7 δ(l, r) = 3 δ(d, r) = 6 δ(r, d) = 11 δ(l, d) = −9 δ(d, l) = −4 Captures Car-Wash, Simultaneous/Ordered Coordination, etc.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Main Analysis Difficulties

  • Actions are not necessarily ordered.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Main Analysis Difficulties

  • Actions are not necessarily ordered.
  • Dependencies may be circular.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Main Analysis Difficulties

  • Actions are not necessarily ordered.
  • Dependencies may be circular.
  • Infinitely many epistemic consequences.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Main Analysis Difficulties

  • Actions are not necessarily ordered.
  • Dependencies may be circular.
  • Infinitely many epistemic consequences.
  • δ is not necessarily symmetric w.r.t. the agents in I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Symmetric Forms of Coordination are Easier

In the simultaneous / up-to-ε / eventual coordination scenarios, the constraints are symmetric (invariant under permutations on I):

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Symmetric Forms of Coordination are Easier

In the simultaneous / up-to-ε / eventual coordination scenarios, the constraints are symmetric (invariant under permutations on I):

  • All agents have to know that within a certain (global)

temporal constraint all agents will act.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Symmetric Forms of Coordination are Easier

In the simultaneous / up-to-ε / eventual coordination scenarios, the constraints are symmetric (invariant under permutations on I):

  • All agents have to know that within a certain (global)

temporal constraint all agents will act.

  • All agents have to know that within a certain (global)

temporal constraint all agents will have to know that within these temporal constraints all agents will act.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Symmetric Forms of Coordination are Easier

In the simultaneous / up-to-ε / eventual coordination scenarios, the constraints are symmetric (invariant under permutations on I):

  • All agents have to know that within a certain (global)

temporal constraint all agents will act.

  • All agents have to know that within a certain (global)

temporal constraint all agents will have to know that within these temporal constraints all agents will act.

  • etc.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Symmetric Forms of Coordination are Easier

In the simultaneous / up-to-ε / eventual coordination scenarios, the constraints are symmetric (invariant under permutations on I):

  • All agents have to know that within a certain (global)

temporal constraint all agents will act.

  • All agents have to know that within a certain (global)

temporal constraint all agents will have to know that within these temporal constraints all agents will act.

  • etc.
  • Solutions are greatest fixed point of functions of the form

x → E ∗(ψ ∩ x), for some “triggering” ψ, and for E ∗ along the lines of “within a certain temporal constraint, everyone will know that . . . ”.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

δ-Coordination is Generally Asymmetric

For general δ, such a fixed point is elusive:

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

δ-Coordination is Generally Asymmetric

For general δ, such a fixed point is elusive:

  • In the car-wash scenario,

startl ⇒ Kl : within -7 to 3: startr.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

δ-Coordination is Generally Asymmetric

For general δ, such a fixed point is elusive:

  • In the car-wash scenario,

startl ⇒ Kl : within -7 to 3: startr.

  • Similarly yet asymmetrically,

startd ⇒ Kd : within -11 to -6: startr.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

δ-Coordination is Generally Asymmetric

For general δ, such a fixed point is elusive:

  • In the car-wash scenario,

startl ⇒ Kl : within -7 to 3: startr.

  • Similarly yet asymmetrically,

startd ⇒ Kd : within -11 to -6: startr.

  • What should E ∗ encompass? −7 to 3 or −11 to −6?

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. x → K?

  • ψc ∩

“in ? to ? time units: x”

  • Yannai A. Gonczarowski (HUJI)

Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. startr → Kr

  • ψc ∩ “in −3 to 7 time units: startl” ∩

“in 6 to 11 time units: startd”

  • Yannai A. Gonczarowski (HUJI)

Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. startl → Kl

  • ψc ∩ “in −7 to 3 time units : startr” ∩

“in 4 to 9 time units: startd”

  • startr

→ Kr

  • ψc ∩ “in −3 to 7 time units: startl” ∩

“in 6 to 11 time units: startd”

  • startd

→ Kd

  • ψc ∩ “in −9 to −4 time units: startl” ∩

“in −11 to −6 time units: startr”

  • Yannai A. Gonczarowski (HUJI)

Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. startl → Kl

  • ψc ∩ “in −7 to 3 time units : startr” ∩

“in 4 to 9 time units: startd”

  • startr

→ Kr

  • ψc ∩ “in −3 to 7 time units: startl” ∩

“in 6 to 11 time units: startd”

  • startd

→ Kd

  • ψc ∩ “in −9 to −4 time units: startl” ∩

“in −11 to −6 time units: startr”

  • A vectorial E ∗!

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. startl → Kl

  • ψc ∩ “in −7 to 3 time units : startr” ∩

“in 4 to 9 time units: startd”

  • startr

→ Kr

  • ψc ∩ “in −3 to 7 time units: startl” ∩

“in 6 to 11 time units: startd”

  • startd

→ Kd

  • ψc ∩ “in −9 to −4 time units: startl” ∩

“in −11 to −6 time units: startr”

  • A vectorial E ∗!

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Searching for a Suitable Fixed Point

Let ψc be the event “the car c is in the car-wash facility”. startl → Kl

  • ψc ∩ “in −7 to 3 time units : startr” ∩

“in 4 to 9 time units: startd”

  • startr

→ Kr

  • ψc ∩ “in −3 to 7 time units: startl” ∩

“in 6 to 11 time units: startd”

  • startd

→ Kd

  • ψc ∩ “in −9 to −4 time units: startl” ∩

“in −11 to −6 time units: startr”

  • A vectorial E ∗!

(xj)j∈I →  Kj

  • ψ ∩
  • k∈I\{j}

◯≤δ(j,k)xk

j∈I

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge

Definition (Timely Common Knowledge)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 15 / 21

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Timely Common Knowledge

We work in the lattice FRI.

Definition (Timely Common Knowledge)

Let R ⊆ R and let (I, δ) be a timely-coordination spec. For each ψ ∈ FR, we define δ-common knowledge of ψ by I, denoted by C δ

I ψ, to be the greatest fixed point of the function

f δ

ψ :

FRI → FRI (xi)i∈I →  Ki

  • ψ ∩
  • j∈I\{i}

◯≤δ(i,j)xj

i∈I

Note

C δ

I ψ is a tuple.

Lemma

C δ

I ψ is well-defined.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 15 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Timely Common Knowledge Properties

Many Desirable Properties — E.g. Induction Rule

Every ¯ ξ ∈ FRI satisfying ¯ ξ ≤ f δ

ψ(¯

ξ) also satisfies ¯ ξ ≤ C δ

I ψ.

  • Recall that ≤ means “coordinate-wise validly implies”.

Theorem

C δ

I ψ is the largest δ-coordinated ensemble s.t. (C δ I )j ⊆ ψ for

every agent j ∈ I.

Observation

The ensemble in which the j’th event is “(C δ

I )j holds for the

first time” is δ-coordinated.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

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Applying Timely Common Knowledge

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

  • The protocol in which each robot j starts washing/drying

at the first instant at which (C δ

I ψc)j holds (during each

run) is an optimal solution for the car-wash problem.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

  • The protocol in which each robot j starts washing/drying

at the first instant at which (C δ

I ψc)j holds (during each

run) is an optimal solution for the car-wash problem.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

  • The protocol in which each robot j starts washing/drying

at the first instant at which (C δ

I ψc)j holds (during each

run) is an optimal solution for the car-wash problem.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

  • The protocol in which each robot j starts washing/drying

at the first instant at which (C δ

I ψc)j holds (during each

run) is an optimal solution for the car-wash problem.

  • The car-wash problem is solvable in a run r iff there exists

t s.t. (r, t) ∈ (C δ

I ψc)j for some/all j.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Applying Timely Common Knowledge

(ψc is the event “the car c is in the car-wash facility”.)

Observation

Every protocol solving the car-wash problem corresponds to a δ-coordinated ensemble ¯ e s.t. ej ⊆ ψc for every agent j ∈ I.

Corollary (Necessary and Sufficient for Solving Car-Wash)

  • The protocol in which each robot j starts washing/drying

at the first instant at which (C δ

I ψc)j holds (during each

run) is an optimal solution for the car-wash problem.

  • The car-wash problem is solvable in a run r iff there exists

t s.t. (r, t) ∈ (C δ

I ψc)j for some/all j.

We prove this for a general class of Timely-Coordinated Response (TCR) problems that we define. (General I, δ, actions, etc.)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

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The Iterative Definition of Common Knowledge

Definition (Common Knowledge - Popular Definiton)

Let R ⊆ R and let I ⊆ I. For every ψ ∈ FR, CIψ

∞

  • n=1

EI nψ, where EI 0ψ ψ and EI nψ EIEI n−1ψ for every n ∈ N.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 18 / 21

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The Iterative Definition of Common Knowledge

Definition (Common Knowledge - Popular Definiton)

Let R ⊆ R and let I ⊆ I. For every ψ ∈ FR, CIψ

∞

  • n=1

EI nψ, where EI 0ψ ψ and EI nψ EIEI n−1ψ for every n ∈ N. Equivalently, CIψ =

  • (i1,...,in)∈I ∗

Ki1 · · · Kinψ where I ∗ is the set of all finite sequences of elements of I.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 18 / 21

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A Generalization for Timely-Common Knowledge

Theorem (Nested-Knowledge Characterisation)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 19 / 21

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A Generalization for Timely-Common Knowledge

Theorem (Nested-Knowledge Characterisation)

Under natural assumptions (mainly perfect recall), we have: (C δ

I ψ)i

=

  • (i,i2,...,in)∈I ∗

Ki ◯δ(i,i2) Ki2 ◯δ(i2,i3) Ki3 · · · ◯δ(in−1,in) Kinψ,

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 19 / 21

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A Generalization for Timely-Common Knowledge

Theorem (Nested-Knowledge Characterisation)

Under natural assumptions (mainly perfect recall), we have: (C δ

I ψ)i

=

  • (i,i2,...,in)∈I ∗

Ki ◯δ(i,i2) Ki2 ◯δ(i2,i3) Ki3 · · · ◯δ(in−1,in) Kinψ,

  • In the general case,

the r.h.s. above is not sufficient for solving TCR, while the fixed-point definition always is.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 19 / 21

slide-105
SLIDE 105

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Generalization for Timely-Common Knowledge

Theorem (Nested-Knowledge Characterisation)

Under natural assumptions (mainly perfect recall), we have: (C δ

I ψ)i

=

  • (i,i2,...,in)∈I ∗

Ki ◯δ(i,i2) Ki2 ◯δ(i2,i3) Ki3 · · · ◯δ(in−1,in) Kinψ,

  • In the general case,

the r.h.s. above is not sufficient for solving TCR, while the fixed-point definition always is.

  • An iterative definition that does not require any

assumptions is uglier. (Involves a definition by transfinite induction.)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 19 / 21

slide-106
SLIDE 106

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

A Generalization for Timely-Common Knowledge

Theorem (Nested-Knowledge Characterisation)

Under natural assumptions (mainly perfect recall), we have: (C δ

I ψ)i

=

  • (i,i2,...,in)∈I ∗

Ki ◯δ(i,i2) Ki2 ◯δ(i2,i3) Ki3 · · · ◯δ(in−1,in) Kinψ,

  • In the general case,

the r.h.s. above is not sufficient for solving TCR, while the fixed-point definition always is.

  • An iterative definition that does not require any

assumptions is uglier. (Involves a definition by transfinite induction.)

  • In many naturally-occurring models, timely common

knowledge is attainable (and thus TCR is solvable) even when common knowledge is not. (E.g. the two generals.)

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 19 / 21

slide-107
SLIDE 107

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Summary

  • The connection between coordination and epistemology

extends deeper than realized so far.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 20 / 21

slide-108
SLIDE 108

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Summary

  • The connection between coordination and epistemology

extends deeper than realized so far.

  • Loose circular codependence between actions introduces

significant complexity.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 20 / 21

slide-109
SLIDE 109

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Summary

  • The connection between coordination and epistemology

extends deeper than realized so far.

  • Loose circular codependence between actions introduces

significant complexity.

  • Asymmetric coordination necessitates a leap to vectorial

fixed points.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 20 / 21

slide-110
SLIDE 110

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Summary

  • The connection between coordination and epistemology

extends deeper than realized so far.

  • Loose circular codependence between actions introduces

significant complexity.

  • Asymmetric coordination necessitates a leap to vectorial

fixed points.

  • Timely Common Knowledge strictly generalizes common

knowledge and previous variants.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 20 / 21

slide-111
SLIDE 111

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Summary

  • The connection between coordination and epistemology

extends deeper than realized so far.

  • Loose circular codependence between actions introduces

significant complexity.

  • Asymmetric coordination necessitates a leap to vectorial

fixed points.

  • Timely Common Knowledge strictly generalizes common

knowledge and previous variants.

  • Vectorial fixed points are a natural, powerful tool for

protocol analysis.

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 20 / 21

slide-112
SLIDE 112

Introduction Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary

Questions?

Thank you!

Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 21 / 21