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On the Power of Weaker Pairwise Interaction: Fault-Tolerant Simulation of Population Protocols ICDCS 2017 Giuseppe Antonio Di Luna, Paola Flocchini, Taisuke Izumi, Tomoko Izumi, Nicola Santoro, Giovanni Viglietta Atlanta June 8, 2017


  1. On the Power of Weaker Pairwise Interaction: Fault-Tolerant Simulation of Population Protocols ICDCS 2017 Giuseppe Antonio Di Luna, Paola Flocchini, Taisuke Izumi, Tomoko Izumi, Nicola Santoro, Giovanni Viglietta Atlanta – June 8, 2017 Fault-Tolerant Simulation of Population Protocols

  2. Population Protocols a d a b c b c b / Setting: a set of finite-state agents./ Fault-Tolerant Simulation of Population Protocols

  3. Population Protocols a d a b δ c b c b /Pairs of agents interact in a non-deterministic order.../ Fault-Tolerant Simulation of Population Protocols

  4. Population Protocols c d a b c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  5. Population Protocols c d a b c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  6. Population Protocols c d a b δ c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  7. Population Protocols c d c b c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  8. Population Protocols c d c b c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  9. Population Protocols c d c b δ c e c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  10. Population Protocols c b c b c a c b /...and change states according to a transition function./ Fault-Tolerant Simulation of Population Protocols

  11. One-way models and omission faults TW T 3 IT f ( a, b ) f ( a, b ) s r a I 4 b I 3 T 2 I 1 I 2 T 1 IO /The traditional interaction model is called Two-Way ./ Fault-Tolerant Simulation of Population Protocols

  12. One-way models and omission faults TW T 3 IT f ( a, b ) a I 4 b I 3 T 2 I 1 I 2 T 1 IO / Immediate Observation: only the second agent transitions./ Fault-Tolerant Simulation of Population Protocols

  13. One-way models and omission faults TW T 3 IT g ( a ) f ( a, b ) a I 4 b I 3 T 2 I 1 I 2 T 1 IO / Immediate Transmission: the first agent detects proximity ./ Fault-Tolerant Simulation of Population Protocols

  14. One-way models and omission faults TW ( ) g ( a ) f a, b a T 3 IT b I 4 I 3 T 2 g ( a ) I 1 I 2 a T 1 b IO / I 1 : IT with omission faults, no detection./ Fault-Tolerant Simulation of Population Protocols

  15. One-way models and omission faults TW ( ) g ( a ) f a, b a T 3 IT b I 4 I 3 T 2 g ( a ) g ( b ) I 1 I 2 a T 1 b IO / I 2 : IT with omission faults, proximity detection./ Fault-Tolerant Simulation of Population Protocols

  16. One-way models and omission faults TW ( ) g ( a ) f a, b a T 3 IT b I 4 I 3 T 2 g ( a ) h ( b ) I 1 I 2 a T 1 b IO / I 3 : IT with omission faults, reactor-side omission detection./ Fault-Tolerant Simulation of Population Protocols

  17. One-way models and omission faults TW ( ) g ( a ) f a, b a T 3 IT b I 4 I 3 T 2 o ( a ) g ( b ) I 1 I 2 a T 1 b IO / I 4 : IT with omission faults, starter-side omission detection./ Fault-Tolerant Simulation of Population Protocols

  18. One-way models and omission faults f ( a, b ) f ( a, b ) s r a TW b T 3 IT f ( a, b ) r I 4 I 3 a T 2 b I 1 I 2 T 1 ( ) f a, b s IO a b / T 1 : TW with omission faults, no detection./ Fault-Tolerant Simulation of Population Protocols

  19. One-way models and omission faults ( ) ( ) f a, b f a, b s r a b TW ( ) f ( a, b ) o a r T 3 IT a b I 4 I 3 T 2 f ( a, b ) s I 1 I 2 a b T 1 IO o ( a ) a b / T 2 : TW with omission faults, starter-side omission detection./ Fault-Tolerant Simulation of Population Protocols

  20. One-way models and omission faults ( ) ( ) f a, b f a, b s r a b TW ( ) f ( a, b ) o a r T 3 IT a b I 4 I 3 T 2 f ( a, b ) ( ) h b s I 1 I 2 a b T 1 IO o ( a ) h ( b ) a b / T 3 : TW with omission faults, omission detection by both sides./ Fault-Tolerant Simulation of Population Protocols

  21. One-way models and omission faults TW δ ( a , a ) = ( f ( a , a ) , f ( a , a )) s r s s r r s r T 3 IT δ ( a , a ) = { ( f ( a , a ) , f ( a , a )) , s r s s r r s r ( ) = ( ( ) ( )) δ a , a g a , f a , a ( o ( a ) , f ( a , a )) , ( f ( a , a ) , h ( a )) , ( o ( a ) , h ( a )) } s r s s r s r s r s s r r s r I I 3 4 T 2 δ ( a , a ) = { ( g ( a ) , f ( a , a )) , ( g ( a ) , h ( a )) } δ ( a , a ) = { ( g ( a ) , f ( a , a )) , ( o ( a ) , g ( a )) } ( ) = { ( ( ) ( )) s r s s r s r s r s s r s r δ a , a f a , a , f a , a , s r s s r r s r ( o ( a ) , f ( a , a )) , ( f ( a , a ) , a ) , ( o ( a ) , a ) } s r s r s s r r s r I I 2 1 ( ) = { ( ( ) ( )) ( ( ) ( )) } ( ) = { ( ( ) ( )) ( ( ) ) } δ a , a g a , f a , a , g a , g a δ a , a g a , f a , a , g a , a s r s s r s r T s r s s r s r 1 δ ( a , a ) = { ( f ( a , a ) , f ( a , a )) , s r s s r r s r ( ( )) ( ( ) ) } a , f a , a , f a , a , a s r s r s s r r IO δ ( a , a ) = ( a , f ( a , a )) s r s s r Theorem: all possible models obtained by combining one-way and two-way interactions with omission detection and proximity detection, starter-side or reactor-side, fall into one of these classes. Fault-Tolerant Simulation of Population Protocols

  22. Simulating TW protocols with weaker ones f ( a, b ) f ( a, b ) s r a b /We seek to simulate two-way interactions in weaker models./ Fault-Tolerant Simulation of Population Protocols

  23. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b a ′ w b w 1 1 /The simulating agents have a simulated state and a work state ./ Fault-Tolerant Simulation of Population Protocols

  24. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b a ′ w b w 2 2 /Typically, an interaction determines a change in the work state./ Fault-Tolerant Simulation of Population Protocols

  25. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b a ′ w b w 3 2 /Typically, an interaction determines a change in the work state./ Fault-Tolerant Simulation of Population Protocols

  26. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b a ′ w b w 3 3 /Typically, an interaction determines a change in the work state./ Fault-Tolerant Simulation of Population Protocols

  27. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b c ′ w b w 4 3 /Occasionally, changes in the simulated state may occur./ Fault-Tolerant Simulation of Population Protocols

  28. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b c ′ w b w 4 4 /Occasionally, changes in the simulated state may occur./ Fault-Tolerant Simulation of Population Protocols

  29. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b c ′ w b w 4 5 /Occasionally, changes in the simulated state may occur./ Fault-Tolerant Simulation of Population Protocols

  30. Simulating TW protocols with weaker ones f ( a, b ) = c f ( a, b ) = d s r a b c ′ w d w 4 6 /These have to mimic transitions in the simulated TW protocol./ Fault-Tolerant Simulation of Population Protocols

  31. Simulating TW protocols with weaker ones c d c ′ w d w 4 6 /Globally, we want to pair up simulated states transitions.../ Fault-Tolerant Simulation of Population Protocols

  32. Simulating TW protocols with weaker ones c d c ′ w d w 4 6 /...in a way that is compatible with the simulated TW protocol./ Fault-Tolerant Simulation of Population Protocols

  33. Results: infinite memory TW T 3 IT I 4 I 3 T 2 I 1 I 2 T 1 IO /Suppose the simulating agents have infinite memory : what models can simulate all TW population protocols?/ Fault-Tolerant Simulation of Population Protocols

  34. Results: infinite memory TW T 3 IT I 4 I 3 T 2 I 1 I 2 T 1 IO /In IT , we can implement a token-passing technique that can be used to simulate two-way interactions./ Fault-Tolerant Simulation of Population Protocols

  35. Results: infinite memory TW T 3 IT I 4 I 3 T 2 I 1 I 2 T 1 IO /In T 3 , it is impossible to simulate a two-way protocol for the pairing problem ./ Fault-Tolerant Simulation of Population Protocols

  36. Results: infinite memory TW T 3 IT I 4 I 3 T 2 I 1 I 2 T 1 IO /As a consequence, simulation is impossible also in the weaker interaction models./ Fault-Tolerant Simulation of Population Protocols

  37. Results: unique IDs TW T 3 IT I 4 I 3 T 2 I 1 I 2 T 1 IO /Suppose the simulating agents have unique IDs as part of their initial state./ Fault-Tolerant Simulation of Population Protocols

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