On the Power of Weaker Pairwise Interaction: Fault-Tolerant - PowerPoint PPT Presentation

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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