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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

  1. Introduction Epistemic Consequences Background Problem & Knowledge of Precondition Strategy Timely If a protocol P guaranteeing that L starts to wash only if a Common Knowledge given event ψ holds, then whenever L starts to wash (according Timely- to P ), L knows that ψ holds. Coordinated Response Nested- • When L acts, it must know that the car has arrived. Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

  2. Introduction Epistemic Consequences Background Problem & Knowledge of Precondition Strategy Timely If a protocol P guaranteeing that L starts to wash only if a Common Knowledge given event ψ holds, then whenever L starts to wash (according Timely- to P ), L knows that ψ holds. Coordinated Response Nested- • When L acts, it must know that the car has arrived. Knowledge Formulation • It must also know that between 7 time units earlier and 3 Summary time units later, R knows that the car has already arrived. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 4 / 21

  3. Introduction Epistemic Consequences Background Problem & Knowledge of Precondition Strategy Timely If a protocol P guaranteeing that L starts to wash only if a Common Knowledge given event ψ holds, then whenever L starts to wash (according Timely- to P ), L knows that ψ holds. Coordinated Response Nested- • When L acts, it must know that the car has arrived. Knowledge Formulation • It must also know that between 7 time units earlier and 3 Summary 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

  4. Introduction Epistemic Consequences Background Problem & Knowledge of Precondition Strategy Timely If a protocol P guaranteeing that L starts to wash only if a Common Knowledge given event ψ holds, then whenever L starts to wash (according Timely- to P ), L knows that ψ holds. Coordinated Response Nested- • When L acts, it must know that the car has arrived. Knowledge Formulation • It must also know that between 7 time units earlier and 3 Summary 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

  5. Introduction Epistemic Consequences Background Problem & Knowledge of Precondition Strategy Timely If a protocol P guaranteeing that L starts to wash only if a Common Knowledge given event ψ holds, then whenever L starts to wash (according Timely- to P ), L knows that ψ holds. Coordinated Response Nested- • When L acts, it must know that the car has arrived. Knowledge Formulation • It must also know that between 7 time units earlier and 3 Summary 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

  6. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

  7. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Halpern and Moses (1990): Common knowledge is Knowledge Formulation necessary and sufficient for simultaneous actions. Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

  8. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Halpern and Moses (1990): Common knowledge is Knowledge Formulation necessary and sufficient for simultaneous actions. Summary • Necessary: A simultaneous action implies common knowledge of the action. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

  9. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Halpern and Moses (1990): Common knowledge is Knowledge Formulation necessary and sufficient for simultaneous actions. Summary • 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

  10. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Ben-Zvi and Moses (2010): Nested knowledge is necessary Knowledge Formulation and sufficient for linear ordering of actions. Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

  11. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Ben-Zvi and Moses (2010): Nested knowledge is necessary Knowledge Formulation and sufficient for linear ordering of actions. Summary • Necessary: act 2 ⇒ K 2 (previously act 1 ), act 3 ⇒ K 3 K 2 (previously act 1 ), etc. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 5 / 21

  12. Introduction Review: Knowledge Gain and Coordination Background Problem & Strategy Many forms of coordination have a necessary and sufficient Timely Common characterising epistemic state. Knowledge Timely- Coordinated Response Nested- Ben-Zvi and Moses (2010): Nested knowledge is necessary Knowledge Formulation and sufficient for linear ordering of actions. Summary • Necessary: act 2 ⇒ K 2 (previously act 1 ), act 3 ⇒ K 3 K 2 (previously act 1 ), 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

  13. Introduction Reminder: The Runs and Systems Model Background Based upon Fagin et al. (1995) Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

  14. Introduction Reminder: The Runs and Systems Model Background Based upon Fagin et al. (1995) Problem & Strategy Timely Common Knowledge Timely- • Agents: I . Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

  15. Introduction Reminder: The Runs and Systems Model Background Based upon Fagin et al. (1995) Problem & Strategy Timely Common Knowledge Timely- • Agents: I . Coordinated Response Nested- • Time: T = N ∪ { 0 } . (For ease of presentation.) Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

  16. Introduction Reminder: The Runs and Systems Model Background Based upon Fagin et al. (1995) Problem & Strategy Timely Common Knowledge Timely- • Agents: I . Coordinated Response Nested- • Time: T = N ∪ { 0 } . (For ease of presentation.) Knowledge Formulation • Runs: R . Each run is a function from time to: Summary • A state of the environment — and — • A state for each agent. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 6 / 21

  17. Introduction Reminder: The Runs and Systems Model Background Based upon Fagin et al. (1995) Problem & Strategy Timely Common Knowledge Timely- • Agents: I . Coordinated Response Nested- • Time: T = N ∪ { 0 } . (For ease of presentation.) Knowledge Formulation • Runs: R . Each run is a function from time to: Summary • 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

  18. Introduction Set-Theoretic Reasoning Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  19. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  20. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  21. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response • ψ ⊆ φ means “ ψ validly implies φ ”. Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  22. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response • ψ ⊆ φ means “ ψ validly implies φ ”. Nested- Knowledge Formulation Summary For any event ψ , Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  23. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response • ψ ⊆ φ means “ ψ validly implies φ ”. Nested- Knowledge Formulation Summary For any event ψ , ◯ d ψ � � � ( r , t ) | ( r , t + d ) ∈ ψ Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  24. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response • ψ ⊆ φ means “ ψ validly implies φ ”. Nested- Knowledge Formulation Summary For any event ψ , ◯ d ψ � � � ( r , t ) | ( r , t + d ) ∈ ψ ( r , t ) | ∃ t ′ ≤ t + d : ( r , t ′ ) ∈ ψ ◯ ≤ d ψ � � � Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  25. Introduction Set-Theoretic Reasoning Background Problem & Strategy • A point ( r , t ) ∈ Ω R � R × T defines a snapshot of the Timely system. Common Knowledge • An event is associated with the set of points Timely- ψ ∈ F R � 2 Ω R at which it holds. Coordinated Response • ψ ⊆ φ means “ ψ validly implies φ ”. Nested- Knowledge Formulation Summary For any event ψ , ◯ d ψ � � � ( r , t ) | ( r , t + d ) ∈ ψ ( r , t ) | ∃ t ′ ≤ t + d : ( r , t ′ ) ∈ ψ ◯ ≤ d ψ � � � K j ψ � � � ( r , t ) | [( r , t )] j ⊆ ψ [( r , t )] j is the event “the local state of j is r j ( t )”. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 7 / 21

  26. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

  27. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Halpern and Moses (1990) Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

  28. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Halpern and Moses (1990) Common Knowledge • Common knowledge of ψ is the greatest fixed point of Timely- Coordinated x �→ E I ( ψ ∩ x ). Response • In the lattice F R . Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

  29. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Halpern and Moses (1990) Common Knowledge • Common knowledge of ψ is the greatest fixed point of Timely- Coordinated x �→ E I ( ψ ∩ x ). Response • In the lattice F R . Nested- Knowledge Formulation • A few variants: Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

  30. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Halpern and Moses (1990) Common Knowledge • Common knowledge of ψ is the greatest fixed point of Timely- Coordinated x �→ E I ( ψ ∩ x ). Response • In the lattice F R . Nested- Knowledge Formulation • A few variants: Summary • Define Eventual Common Knowledge of ψ as the greatest fixed point of x �→ ∩ j ∈ I ◇ K j ( ψ ∩ x ). • Necessary and sufficient for eventual coordination. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 8 / 21

  31. Introduction Epistemic Fixed-Point Analysis Background Problem & Strategy Timely Halpern and Moses (1990) Common Knowledge • Common knowledge of ψ is the greatest fixed point of Timely- Coordinated x �→ E I ( ψ ∩ x ). Response • In the lattice F R . Nested- Knowledge Formulation • A few variants: Summary • Define Eventual Common Knowledge of ψ as the greatest fixed point of x �→ ∩ j ∈ I ◇ K j ( 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

  32. Introduction Reasoning about Coordination Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 9 / 21

  33. Introduction Reasoning about Coordination Background Problem & Strategy Timely Fagin et al. (1995) Common Knowledge Timely- An I -ensemble, where I ⊆ I is a set of agents, is an I -tuple of Coordinated e = ( e j ) j ∈ I ∈ F RI , in which for every agent j ∈ I , the Response events ¯ Nested- event e j is a j -local event. Knowledge Formulation • An event is j -local if it is known to j whenever it occurs. Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 9 / 21

  34. Introduction Reasoning about Coordination Background Problem & Strategy Timely Fagin et al. (1995) Common Knowledge Timely- An I -ensemble, where I ⊆ I is a set of agents, is an I -tuple of Coordinated e = ( e j ) j ∈ I ∈ F RI , in which for every agent j ∈ I , the Response events ¯ Nested- event e j is a j -local event. Knowledge Formulation • An event is j -local if it is known to j whenever it occurs. Summary 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

  35. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

  36. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Definition (Timely Coordination) Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

  37. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Definition (Timely Coordination) Timely Common Knowledge 1 A timely-coordination spec is a pair ( I , δ ), where I ⊆ I is a Timely- set of agents and δ : I 2 → Z ∪ {∞} . Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 10 / 21

  38. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Definition (Timely Coordination) Timely Common Knowledge 1 A timely-coordination spec is a pair ( I , δ ), where I ⊆ I is a Timely- set of agents and δ : I 2 → Z ∪ {∞} . Coordinated Response Nested- Knowledge Formulation Summary 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

  39. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Definition (Timely Coordination) Timely Common Knowledge 1 A timely-coordination spec is a pair ( I , δ ), where I ⊆ I is a Timely- set of agents and δ : I 2 → Z ∪ {∞} . Coordinated Response 2 Given a timely-coordination spec ( I , δ ) and a system Nested- Knowledge e ∈ F RI is R ⊆ R , we say that an I -ensemble ¯ Formulation δ -coordinated (in R ) if for every ( j , k ) ∈ I 2 and for every Summary ( r , t ) ∈ e j , there exists t ′ ≤ t + δ ( j , k ) s.t. ( r , t ′ ) ∈ e k . 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

  40. Introduction Can these be Generalized to deal with Arbitrary Background Timely Constraints? Problem & Strategy Definition (Timely Coordination) Timely Common Knowledge 1 A timely-coordination spec is a pair ( I , δ ), where I ⊆ I is a Timely- set of agents and δ : I 2 → Z ∪ {∞} . Coordinated Response 2 Given a timely-coordination spec ( I , δ ) and a system Nested- Knowledge e ∈ F RI is R ⊆ R , we say that an I -ensemble ¯ Formulation δ -coordinated (in R ) if for every ( j , k ) ∈ I 2 and for every Summary ( r , t ) ∈ e j , there exists t ′ ≤ t + δ ( j , k ) s.t. ( r , t ′ ) ∈ e k . 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

  41. Introduction Main Analysis Difficulties Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response • Actions are not necessarily ordered. Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

  42. Introduction Main Analysis Difficulties Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response • Actions are not necessarily ordered. Nested- • Dependencies may be circular. Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

  43. Introduction Main Analysis Difficulties Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response • Actions are not necessarily ordered. Nested- • Dependencies may be circular. Knowledge Formulation • Infinitely many epistemic consequences. Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

  44. Introduction Main Analysis Difficulties Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response • Actions are not necessarily ordered. Nested- • Dependencies may be circular. Knowledge Formulation • Infinitely many epistemic consequences. Summary • δ is not necessarily symmetric w.r.t. the agents in I . Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 11 / 21

  45. Introduction Symmetric Forms of Coordination are Easier Background Problem & Strategy In the simultaneous / up-to- ε / eventual coordination Timely scenarios, the constraints are symmetric (invariant under Common Knowledge permutations on I ): Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

  46. Introduction Symmetric Forms of Coordination are Easier Background Problem & Strategy In the simultaneous / up-to- ε / eventual coordination Timely scenarios, the constraints are symmetric (invariant under Common Knowledge permutations on I ): Timely- Coordinated • All agents have to know that within a certain (global) Response temporal constraint all agents will act. Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

  47. Introduction Symmetric Forms of Coordination are Easier Background Problem & Strategy In the simultaneous / up-to- ε / eventual coordination Timely scenarios, the constraints are symmetric (invariant under Common Knowledge permutations on I ): Timely- Coordinated • All agents have to know that within a certain (global) Response temporal constraint all agents will act. Nested- Knowledge • All agents have to know that within a certain (global) Formulation temporal constraint all agents will have to know that Summary within these temporal constraints all agents will act. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

  48. Introduction Symmetric Forms of Coordination are Easier Background Problem & Strategy In the simultaneous / up-to- ε / eventual coordination Timely scenarios, the constraints are symmetric (invariant under Common Knowledge permutations on I ): Timely- Coordinated • All agents have to know that within a certain (global) Response temporal constraint all agents will act. Nested- Knowledge • All agents have to know that within a certain (global) Formulation temporal constraint all agents will have to know that Summary within these temporal constraints all agents will act. • etc. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 12 / 21

  49. Introduction Symmetric Forms of Coordination are Easier Background Problem & Strategy In the simultaneous / up-to- ε / eventual coordination Timely scenarios, the constraints are symmetric (invariant under Common Knowledge permutations on I ): Timely- Coordinated • All agents have to know that within a certain (global) Response temporal constraint all agents will act. Nested- Knowledge • All agents have to know that within a certain (global) Formulation temporal constraint all agents will have to know that Summary 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

  50. Introduction δ -Coordination is Generally Asymmetric Background Problem & Strategy Timely Common Knowledge For general δ , such a fixed point is elusive: Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

  51. Introduction δ -Coordination is Generally Asymmetric Background Problem & Strategy Timely Common Knowledge For general δ , such a fixed point is elusive: Timely- Coordinated Response • In the car-wash scenario, Nested- start l ⇒ K l : within -7 to 3: start r . Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

  52. Introduction δ -Coordination is Generally Asymmetric Background Problem & Strategy Timely Common Knowledge For general δ , such a fixed point is elusive: Timely- Coordinated Response • In the car-wash scenario, Nested- start l ⇒ K l : within -7 to 3: start r . Knowledge Formulation Summary • Similarly yet asymmetrically, start d ⇒ K d : within -11 to -6: start r . Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

  53. Introduction δ -Coordination is Generally Asymmetric Background Problem & Strategy Timely Common Knowledge For general δ , such a fixed point is elusive: Timely- Coordinated Response • In the car-wash scenario, Nested- start l ⇒ K l : within -7 to 3: start r . Knowledge Formulation Summary • Similarly yet asymmetrically, start d ⇒ K d : within -11 to -6: start r . • What should E ∗ encompass? − 7 to 3 or − 11 to − 6? Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 13 / 21

  54. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely Common Knowledge Timely- Coordinated � � Response x �→ K ? ψ c ∩ “in ? to ? time units: x ” Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  55. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely Common Knowledge Timely- Coordinated � � ψ c ∩ “in − 3 to 7 time units: start l ” ∩ Response start r �→ K r “in 6 to 11 time units: start d ” Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  56. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely � � Common ψ c ∩ “in − 7 to 3 time units : start r ” ∩ Knowledge start l �→ K l “in 4 to 9 time units: start d ” Timely- Coordinated � � ψ c ∩ “in − 3 to 7 time units: start l ” ∩ Response start r �→ K r “in 6 to 11 time units: start d ” Nested- Knowledge � � ψ c ∩ “in − 9 to − 4 time units: start l ” ∩ Formulation start d �→ K d “in − 11 to − 6 time units: start r ” Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  57. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely � � Common ψ c ∩ “in − 7 to 3 time units : start r ” ∩ Knowledge start l �→ K l “in 4 to 9 time units: start d ” Timely- Coordinated � � ψ c ∩ “in − 3 to 7 time units: start l ” ∩ Response start r �→ K r “in 6 to 11 time units: start d ” Nested- Knowledge � � ψ c ∩ “in − 9 to − 4 time units: start l ” ∩ Formulation start d �→ K d “in − 11 to − 6 time units: start r ” Summary A vectorial E ∗ ! Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  58. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely � � Common ψ c ∩ “in − 7 to 3 time units : start r ” ∩ Knowledge start l �→ K l “in 4 to 9 time units: start d ” Timely- Coordinated � � ψ c ∩ “in − 3 to 7 time units: start l ” ∩ Response start r �→ K r “in 6 to 11 time units: start d ” Nested- Knowledge � � ψ c ∩ “in − 9 to − 4 time units: start l ” ∩ Formulation start d �→ K d “in − 11 to − 6 time units: start r ” Summary A vectorial E ∗ ! Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  59. Introduction Searching for a Suitable Fixed Point Background Let ψ c be the event “the car c is in the car-wash facility”. Problem & Strategy Timely � � Common ψ c ∩ “in − 7 to 3 time units : start r ” ∩ Knowledge start l �→ K l “in 4 to 9 time units: start d ” Timely- Coordinated � � ψ c ∩ “in − 3 to 7 time units: start l ” ∩ Response start r �→ K r “in 6 to 11 time units: start d ” Nested- Knowledge � � ψ c ∩ “in − 9 to − 4 time units: start l ” ∩ Formulation start d �→ K d “in − 11 to − 6 time units: start r ” Summary A vectorial E ∗ !   � � � ◯ ≤ δ ( j , k ) x k ( x j ) j ∈ I �→  K j ψ ∩  k ∈ I \{ j } j ∈ I Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 14 / 21

  60. Introduction Timely Common Knowledge Background Problem & Strategy Definition (Timely Common Knowledge) Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 15 / 21

  61. Introduction Timely Common Knowledge Background We work in the lattice F RI . Problem & Strategy Definition (Timely Common Knowledge) Timely Common Knowledge Let R ⊆ R and let ( I , δ ) be a timely-coordination spec. For Timely- each ψ ∈ F R , we define δ -common knowledge of ψ by I , Coordinated Response denoted by C δ I ψ , to be the greatest fixed point of the function Nested- Knowledge Formulation Summary f δ F RI F RI ψ : →   � � � ◯ ≤ δ ( i , j ) x j ( x i ) i ∈ I �→  K i ψ ∩  j ∈ I \{ i } i ∈ I Note Lemma C δ C δ I ψ is a tuple. I ψ is well-defined. Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 15 / 21

  62. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

  63. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 16 / 21

  64. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  65. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  66. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  67. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  68. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  69. Introduction Timely Common Knowledge Properties Background Problem & Strategy Timely Many Desirable Properties — E.g. Induction Rule Common Knowledge ξ ∈ F RI satisfying ¯ Every ¯ ψ (¯ ξ ) also satisfies ¯ ξ ≤ f δ ξ ≤ C δ I ψ . Timely- Coordinated • Recall that ≤ means “coordinate-wise validly implies”. Response Nested- Knowledge Theorem Formulation Summary 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

  70. Introduction Applying Timely Common Knowledge Background Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  71. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Timely Common Knowledge Timely- Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  72. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  73. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  74. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  75. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  76. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  77. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary • The protocol in which each robot j starts washing/drying at the first instant at which ( C δ I ψ c ) 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

  78. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary • The protocol in which each robot j starts washing/drying at the first instant at which ( C δ I ψ c ) 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

  79. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary • The protocol in which each robot j starts washing/drying at the first instant at which ( C δ I ψ c ) 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

  80. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary • The protocol in which each robot j starts washing/drying at the first instant at which ( C δ I ψ c ) 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 ψ c ) j for some/all j . Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 17 / 21

  81. Introduction Applying Timely Common Knowledge Background ( ψ c is the event “the car c is in the car-wash facility”.) Problem & Strategy Observation Timely Common Every protocol solving the car-wash problem corresponds to a Knowledge Timely- δ -coordinated ensemble ¯ e s.t. e j ⊆ ψ c for every agent j ∈ I. Coordinated Response Nested- Corollary (Necessary and Sufficient for Solving Car-Wash) Knowledge Formulation Summary • The protocol in which each robot j starts washing/drying at the first instant at which ( C δ I ψ c ) 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 ψ c ) 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

  82. Introduction The Iterative Definition of Common Knowledge Background Problem & Strategy Definition (Common Knowledge - Popular Definiton) Timely Common Knowledge Let R ⊆ R and let I ⊆ I . For every ψ ∈ F R , Timely- Coordinated ∞ Response � E I n ψ, C I ψ � Nested- Knowledge n =1 Formulation Summary where E I 0 ψ � ψ and E I n ψ � E I E I n − 1 ψ for every n ∈ N . Yannai A. Gonczarowski (HUJI) Timely Common Knowledge January 8, 2013 18 / 21

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