Computations by Luminous Robots Giuseppe Prencipe Universit di - - PowerPoint PPT Presentation

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Computations by Luminous Robots Giuseppe Prencipe Universit di Pisa Swarms of robots Many Very Simple Generic Identical Autonomous Silent 21st Oct, 2015 MAC 2015 Swarms of robots A robot - alone - is computationally weak

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Computations by Luminous Robots

Giuseppe Prencipe Università di Pisa

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MAC 2015

Many Very Simple Generic Identical Autonomous Silent Swarms of robots

21st Oct, 2015

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MAC 2015

  • Cooperation of robots is essential

to perform complex tasks

  • A robot - alone - is computationally weak

Swarms of robots

21st Oct, 2015

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MAC 2015

viewed as points

Mobile Robots

MAC2014

x y Unit

Algorithm: 1. 2. 3. ….

Autonomous Homogeneous Identical Silent

21st Oct, 2015

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MAC 2015

LOOK COMPUTE MOVE SLEEP

Robot's behaviour : Life Cycle

LOOK

21st Oct, 2015

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MAC 2015

Powerful? Restricted?

21st Oct, 2015

  • Precision
  • Dimensionless
  • Communication
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MAC 2015

Luminous Robots

The robots are still oblivious But they are enhanced with VISIBLE LIGHTS that can change color

21st Oct, 2015

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MAC 2015

Robots’ swarm with Lights

21st Oct, 2015

S-bots: Light ring for communication. Each of the 8 sectors of the light ring can take RGB colours and can blink at different frequencies (the ring is observed with the omnidirectional camera)

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MAC 2015 21st Oct, 2015

S-bots: Light ring for communication. Each of the 8 sectors of the light ring can take RGB colours and can blink at different frequencies (the ring is observed with the omnidirectional camera).

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MAC 2015

Robots’ swarm with Lights

21st Oct, 2015

Robot Swarm, currently at the MoMath’s exhibit in NY: Robots who react to your presence and communicate with each other, chasing after you or zooming away as you move across the floor.

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MAC 2015 21st Oct, 2015

Display Swarm (Disney research): new kind of display composed of a mobile robot swarm (Pixelbots). Each robot acts as an individual pixel and has controllable color. The swarm is used to make representational images and animated movies.

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MAC 2015

Computational model

21st Oct, 2015

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MAC 2015

LOOK COMPUTE MOVE SLEEP

Uses its sensors to observe the world. result = colored SNAPSHOT

  • f the world

Robot's behaviour : Life Cycle

21st Oct, 2015

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MAC 2015

LOOK COMPUTE MOVE SLEEP

input = position of the other robots with their color result = destination point

Robot's behaviour : Life Cycle

21st Oct, 2015

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MAC 2015

LOOK COMPUTE MOVE SLEEP

Possibly change color of light

Robot's behaviour : Life Cycle

21st Oct, 2015

input = position of the other robots with their color result = destination point

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MAC 2015

LOOK COMPUTE MOVE SLEEP

The robot moves towards the computed destination (it might not reach it)

Robot's behaviour : Life Cycle

21st Oct, 2015

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MAC 2015

LOOK COMPUTE MOVE SLEEP

The robot may be idle (e.g., to recharge battery)

Robot's behaviour : Life Cycle

21st Oct, 2015

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MAC 2015

There is a global clock tick reaching all robots simultaneously At each clock tick every robot is either active or inactive, and

  • nly active robots perform their cycle atomically

Time – Synchronous/Semi-Synchronous

In Fully Synchronous: all robots are active at each step In Semi-Syncrhonous: a subset of robots is active in each step

21st Oct, 2015

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MAC 2015

There is no global clock Robots are active and perform their operations independently and asynchronously Compute and Move take a bounded but unpredictable amount

  • f time

Time – Asynchronous (ASYNC)

21st Oct, 2015

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MAC 2015

Luminosity and Synchrony

21st Oct, 2015

[Das, Flocchini, Prencipe, Santoro, Yamashita, TCS 2015]

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MAC 2015

For non-luminous robots

FSYNCH > SSYNCH > ASYNCH

21st Oct, 2015

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MAC 2015

Gathering of 2 robots Move-and-switch (non oblivious)

For non-luminous robots

21st Oct, 2015

FSYNCH > SSYNCH > ASYNCH

  • Move along x axis
  • Switch to moving along y only after
  • 1. I observed all others in at least 3 different positions, and
  • 2. I have been observed in at least 3 different positions
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MAC 2015

  • 1. Any problem solvable in SSYNC without

lights is also solvable in ASYNC with lights

ASYNC + lights >= SSYNC

Impact of Lights

Let A be an algorithm that solves problem P in SSYNC There exists an algorithm B in ASYNC + lights (5 colors) in which every execution is equivalent to a SSYNC execution of A

21st Oct, 2015

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MAC 2015

To simulate a SSYNC execution: I COMPUTE and MOVE according to algorithm A

  • nly if the colors I see indicate that

my movement is consistent with a SSYNC execution

There exists an algorithm B in ASYNC+lights (5 colors) in which every execution is equivalent to a SSYNC execution of A

LOOK

21st Oct, 2015

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MAC 2015

If it is a safe-snapshot

21st Oct, 2015

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MAC 2015 21st Oct, 2015

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MAC 2015 21st Oct, 2015

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MAC 2015

If it is not a safe snapshot (i.e. someone is moving on the basis

  • f a different snapshot)

21st Oct, 2015

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MAC 2015

Wait next turn

21st Oct, 2015

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MAC 2015 21st Oct, 2015

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MAC 2015 21st Oct, 2015

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MAC 2015

T M S F W T, S M, W, S S, F F, T

21st Oct, 2015

States:

T(rying) M(oving) S(topped) F(inished) W(aiting)

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MAC 2015

  • 2. There are problems unsolvable in SSYNC

without lights, but solvable in ASYNC with lights

ASYNC + light > SSYNC

Impact of Lights

Gathering of two robots

21st Oct, 2015

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MAC 2015

FSYNCH SSYNCH ASYNCH yes impossible impossible

Gathering of 2 robots without lights

Suzuki, Yamashita, SIAM J. Comp 1999

Exactly in the same point 21st Oct, 2015

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MAC 2015

FSYNCH SSYNCH ASYNCH yes yes yes

MAC 2014

Gathering of 2 robots with lights

[Das, Flocchini, Prencipe, Santoro, Yamashita, TCS 2015] (4 colors) [Viglietta, ALGOSENSORS 2013] (2 colors, optimal)

ASYNCH + lights > SSYNC

21st Oct, 2015

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MAC 2015

Main Idea

ASYNCH If we are both white, I move towards you, becoming red If this combinationoccurs, we get CLOSER to the solution

21st Oct, 2015

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MAC 2015

Main Idea

ASYNCH If we are both white, I move towards you, becoming red If this combinationoccurs, we get CLOSER to the solution

21st Oct, 2015

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MAC 2015

Main Idea

ASYNCH If we are both white, I move towards you, becoming red If this combinationoccurs, we get CLOSER to the solution

21st Oct, 2015

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MAC 2015

if I am white and I see you blue I move towards you If this combinationoccurs, we get CLOSER to the solution if I am blue and I see you white, I don’t move.

Main Idea

21st Oct, 2015

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MAC 2015

If this combinationoccurs, we get CLOSER to the solution if I am blue and I see you white, I don’t move.

Main Idea

if I am white and I see you blue I move towards you

21st Oct, 2015

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MAC 2015

The algorithm guarantees that those combinations occur periodically until the two robots eventually gather Main Idea

21st Oct, 2015

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MAC 2015

Starting Moving Waiting for you Not Waiting anymore, will become White again

21st Oct, 2015

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MAC 2015

If other = WHITE Become RED and Move to half BLUE /GREEN GREEN/WHITE If other= BLUE Become RED and Move to other

COMPUTE If other = WHITE or RED

BLUE waits

21st Oct, 2015

If gathered: STOP

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MAC 2015

A B

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

If they both see each other WHITE and they both move all the way to half, gathering is achieved

go-to-half go-to-half

21st Oct, 2015

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MAC 2015

A B

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

go-to-half

21st Oct, 2015

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MAC 2015

A B

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

go-to-half B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

Another scenario

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

See blue, go-to-other

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

go-to-half go-to-half B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

Yet another scenario

(not-rigid)

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

go-to-half

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MAC 2015

A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

go-to-half B

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

21st Oct, 2015

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MAC 2015

B A

OFF Go-to half BLUE/GREEN GREEN OFF BLUE Go-to

  • ther

Restarting from a closer distance

21st Oct, 2015

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MAC 2015

ASYNC + lights = SSYNC + lights

  • 3. SSYNC with lights and ASYNC with lights

have the same computational power Impact of Lights

21st Oct, 2015

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MAC 2015

ASYNC + lights = SSYNC + lights

  • 3. SSYNC with lights and ASYNC with lights

have the same computational power Impact of Lights

21st Oct, 2015

SSYNC + lights >= ASYNC + lights ASYNC + lights >= SSYNC + lights

Obvious

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MAC 2015

ASYNC + lights = SSYNC + lights

  • 3. SSYNC with lights and ASYNC with lights

have the same computational power Impact of Lights

21st Oct, 2015

ASYNC + lights >= SSYNC + lights

Based on the simulation of SSYNC with ASYNC with 5 lights

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MAC 2015

  • 4. FSYNC is not more powerful than

ASYNC with lights Impact of Lights

21st Oct, 2015

Oscillating Points problem (OSP): two robots are required to alternately come closer and move further from each other

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MAC 2015

  • 4. FSYNC is not more powerful than

ASYNC with lights Impact of Lights

21st Oct, 2015

FSYNC: the robots cannot distinguish whether they are getting closer or moving further away Oscillating Points problem (OSP): two robots are required to alternately come closer and move further from each other

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MAC 2015

  • 4. FSYNC is not more powerful than

ASYNC with lights Impact of Lights

21st Oct, 2015

FSYNC: the robots cannot distinguish whether they are getting closer or moving further away Oscillating Points problem (OSP): two robots are required to alternately come closer and move further from each other ASYNC: it can be done with 4 lights

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MAC 2015

Impact of Lights

21st Oct, 2015

ASYNC: it can be done with 4 lights

If other = WHITE/RED Become RED and Move towards If other = RED/BLUE Become BLUE If other = GREEN/WHITE Become WHITE If other = BLUE/GREEN Become GREEN and Move away

  • 4. FSYNC is not more powerful than

ASYNC with lights

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ASYNC+lights = SSYNC+lights

MAC 2015

FSYNC SSYNC ASYNC

?

OSP

Impact of Lights

Gathering-2

21st Oct, 2015

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MAC 2015

Availabilityof a single snapshot renders ASYNC with lights more powerful than FSYNC without lights. Are there weaker conditions ?

Open Problems

Are there problems solvable in FSYNC without lights, but not solvable in ASYNC with lights (i.e., FSYNC and ASYNC with lights

  • rthogonal) ?

21st Oct, 2015

Yes, in discrete

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