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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

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SLIDE 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

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SLIDE 2

Population Protocols

a b b b c c d a

/Setting: a set of finite-state agents./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 3

Population Protocols

a b b b c c d a

δ

/Pairs of agents interact in a non-deterministic order.../

Fault-Tolerant Simulation of Population Protocols

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SLIDE 4

Population Protocols

b b c c d a c e

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 5

Population Protocols

b b c c d a c e

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 6

Population Protocols

b b c c d a

δ

c e

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 7

Population Protocols

b b c c d c c e

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 8

Population Protocols

b b c c d c c e

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 9

Population Protocols

b b c c d c c e

δ

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 10

Population Protocols

b b c c c c a b

/...and change states according to a transition function./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 11

One-way models and omission faults

) a, b (

s

f ) a, b (

r

f

a b

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/The traditional interaction model is called Two-Way./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 12

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1 IO

) a, b ( f /Immediate Observation: only the second agent transitions./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 13

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1

) a ( g

IO

) a, b ( f /Immediate Transmission: the first agent detects proximity./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 14

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1

) a ( g

IO

) a, b ( f

a b

) a ( g /I1: IT with omission faults, no detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 15

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1

) a ( g

IO

) a, b ( f

a b

) a ( g ) b ( g /I2: IT with omission faults, proximity detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 16

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1

) a ( g

IO

) a, b ( f

a b

) a ( g ) b ( h /I3: IT with omission faults, reactor-side omission detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 17

One-way models and omission faults a b

I2 TW T3 IT I4 I3 T2 I1 T1

) a ( g

IO

) a, b ( f

a b

) b ( g ) a (

  • /I4: IT with omission faults, starter-side omission detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 18

One-way models and omission faults

I2 TW T3 IT

) a, b (

s

f ) a, b (

r

f

a b

I4 I3 T2 I1 T1 IO

a b

) a, b (

r

f

a b

) a, b (

s

f /T1: TW with omission faults, no detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 19

One-way models and omission faults

I2 TW T3 IT

) a, b (

s

f ) a, b (

r

f

a b

I4 I3 T2 I1 T1 IO

) a, b (

r

f

a b a b a b

) a, b (

s

f ) a (

  • )

a (

  • /T2: TW with omission faults, starter-side omission detection./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 20

One-way models and omission faults

I2 TW T3 IT

) a, b (

s

f ) a, b (

r

f

a b

I4 I3 T2 I1 T1 IO

) a, b (

r

f

a b a b a b

) a, b (

s

f ) a (

  • )

a (

  • )

b ( h ) b ( h /T3: TW with omission faults, omission detection by both sides./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 21

One-way models and omission faults

} ))

r

a ( , h )

s

a (

  • (

, ))

r

a ( , h )

r

, a

s

a (

s

f ( , ))

r

, a

s

a (

r

, f )

s

a (

  • (

, ))

r

, a

s

a (

r

, f )

r

, a

s

a (

s

f ( { ) =

r

, a

s

a ( δ

3

T ))

r

, a

s

a (

r

, f )

r

, a

s

a (

s

f ) = (

r

, a

s

a ( δ TW } )

r

, a )

s

a (

  • (

, )

r

, a )

r

, a

s

a (

s

f ( , ))

r

, a

s

a (

r

, f )

s

a (

  • (

, ))

r

, a

s

a (

r

, f )

r

, a

s

a (

s

f ( { ) =

r

, a

s

a ( δ

2

T } )

r

, a )

r

, a

s

a (

s

f ( , ))

r

, a

s

a (

r

, f

s

a ( , ))

r

, a

s

a (

r

, f )

r

, a

s

a (

s

f ( { ) =

r

, a

s

a ( δ

1

T } ))

r

a ( , h )

s

a ( g ( , ))

r

, a

s

a ( , f )

s

a ( g ( { ) =

r

, a

s

a ( δ

3

I } ))

r

a ( , g )

s

a (

  • (

, ))

r

, a

s

a ( , f )

s

a ( g ( { ) =

r

, a

s

a ( δ

4

I } ))

r

a ( , g )

s

a ( g ( , ))

r

, a

s

a ( , f )

s

a ( g ( { ) =

r

, a

s

a ( δ

2

I } )

r

, a )

s

a ( g ( , ))

r

, a

s

a ( , f )

s

a ( g ( { ) =

r

, a

s

a ( δ

1

I ))

r

, a

s

a ( , f )

s

a ( g ) = (

r

, a

s

a ( δ IT ))

r

, a

s

a ( , f

s

a ) = (

r

, a

s

a ( δ IO

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

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SLIDE 22

Simulating TW protocols with weaker ones

) a, b (

s

f ) a, b (

r

f

a b

/We seek to simulate two-way interactions in weaker models./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 23

Simulating TW protocols with weaker ones a b a

1

w

b

1 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /The simulating agents have a simulated state and a work state./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 24

Simulating TW protocols with weaker ones a b a

2

w

b

2 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Typically, an interaction determines a change in the work state./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 25

Simulating TW protocols with weaker ones a b a

3

w

b

2 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Typically, an interaction determines a change in the work state./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 26

Simulating TW protocols with weaker ones a b a

3

w

b

3 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Typically, an interaction determines a change in the work state./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 27

Simulating TW protocols with weaker ones a b c

4

w

b

3 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Occasionally, changes in the simulated state may occur./

Fault-Tolerant Simulation of Population Protocols

slide-28
SLIDE 28

Simulating TW protocols with weaker ones a b c

4

w

b

4 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Occasionally, changes in the simulated state may occur./

Fault-Tolerant Simulation of Population Protocols

slide-29
SLIDE 29

Simulating TW protocols with weaker ones a b c

4

w

b

5 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /Occasionally, changes in the simulated state may occur./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 30

Simulating TW protocols with weaker ones a b c

4

w

d

6 ′

w

c ) = a, b (

s

f d ) = a, b (

r

f /These have to mimic transitions in the simulated TW protocol./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 31

Simulating TW protocols with weaker ones c d c

4

w

d

6 ′

w

/Globally, we want to pair up simulated states transitions.../

Fault-Tolerant Simulation of Population Protocols

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SLIDE 32

Simulating TW protocols with weaker ones c d c

4

w

d

6 ′

w

/...in a way that is compatible with the simulated TW protocol./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 33

Results: infinite memory

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/Suppose the simulating agents have infinite memory: what models can simulate all TW population protocols?/

Fault-Tolerant Simulation of Population Protocols

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SLIDE 34

Results: infinite memory

I2 TW T3 IT I4 I3 T2 I1 T1 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

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SLIDE 35

Results: infinite memory

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/In T3, it is impossible to simulate a two-way protocol for the pairing problem./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 36

Results: infinite memory

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/As a consequence, simulation is impossible also in the weaker interaction models./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 37

Results: unique IDs

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/Suppose the simulating agents have unique IDs as part of their initial state./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 38

Results: unique IDs

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/In IO, we can implement a locking mechanism, along with a rollback process to avoid deadlocks./

Fault-Tolerant Simulation of Population Protocols

slide-39
SLIDE 39

Results: unique IDs

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/As a consequence, simulation is possible also in the stronger interaction models./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 40

Results: knowledge of n

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/Suppose the simulating agents know the size of the system, n, and have O(log n) bits of internal memory./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 41

Results: knowledge of n

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/In IO, we can implement a naming algorithm that eventually gives each agent a unique ID./

Fault-Tolerant Simulation of Population Protocols

slide-42
SLIDE 42

Results: knowledge of n

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/When an agent has ID n, the system starts executing the previous unique-ID simulation protocol./

Fault-Tolerant Simulation of Population Protocols

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SLIDE 43

Results: knowledge on omissions

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/Suppose that the simulating agents are given an upper bound b

  • n the number of faulty interactions in the system./

Fault-Tolerant Simulation of Population Protocols

slide-44
SLIDE 44

Results: knowledge on omissions

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/In I3 and I4, we can extend the token-passing technique by splitting each token into b + 1 parts./

Fault-Tolerant Simulation of Population Protocols

slide-45
SLIDE 45

Results: knowledge on omissions

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/In T1, I1, and I2, it is impossible to simulate the pairing protocol, even for b = 1./

Fault-Tolerant Simulation of Population Protocols

slide-46
SLIDE 46

Results: knowledge on omissions

I2 TW T3 IT I4 I3 T2 I1 T1 IO

/Open problem: is it possible to simulate all TW protocols in T2, given an upper bound on the number of faulty interactions?/

Fault-Tolerant Simulation of Population Protocols