Cameron T Chalk Bin Fu Alejandro Huerta Mario A Maldonado Eric Martinez Robert T Schweller Tim Wylie Funding by NSF Grant CCF1117672 NSF Early Career Award 0845376 Introduction ID: 776448
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Slide1
Flipping Tiles: Concentration Independent Coin Flips in Tile Self-Assembly
Cameron T. Chalk, Bin Fu, Alejandro Huerta, Mario A. Maldonado, Eric Martinez, Robert T. Schweller, Tim WylieFunding by NSF Grant CCF-1117672NSF Early Career Award 0845376
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Slide2Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide3?
?
Slide4Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide5Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
Slide6Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide7Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide8Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide9Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide10Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide11Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide12Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide13Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide14Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide15Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide16Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide17Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
Slide18Tile Assembly Model(Rothemund, Winfree, Adleman)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
y
) = 2
G(
r
) = 2
G(
b
) = 1
G(
p
) = 1
S
Temperature: 2
Seed:
S
TERMINAL
Slide19Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
p
) = 2
G(b) = 2
S
Temperature: 2
Seed:
.1
.2
.2
.2
.3
Slide20Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
p
) = 2
G(b) = 2
S
Temperature: 2
Seed:
S
.1
.2
.2
.2
.3
Slide21Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
p
) = 2
G(b) = 2
S
Temperature: 2
Seed:
S
S
S
.1
.2
.2
.2
.3
Slide22Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
p
) = 2
G(b) = 2
S
Temperature: 2
Seed:
.1
.2
.2
.2
.3
S
S
.2
.2 + .3
S
=
.4
Slide23Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
Tileset
:
S
Glue:
G(
g
) = 2
G(
o
) = 2
G(
p
) = 2
G(b) = 2
S
Temperature: 2
Seed:
.1
.2
.2
.2
.3
S
S
.2
.2 + .3
S
=
.4
.3
.2 + .3
=
.6
Slide24Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
Slide25Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
Slide26Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
S
S
1
.5
.5
1
Slide27Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
S
S
1
.5
.5
1
(.4)(.5)(1)
Slide28Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
S
S
1
.5
.5
1
(.4)(.5)(1)
+
(.4)(.5)(.5)
Slide29Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
S
S
1
.5
.5
1
(.4)(.5)(1)
+ (.4)(.5)(.5)
+
(.6)(.5)(.5)
.45
Slide30Probabilistic Tile Assembly Model(Becker, Remila, Rapaport)
S
.1
.2
.2
.2
.3
.4
.6
S
S
S
S
S
S
.5
.5
.5
.5
S
S
1
.5
.5
1
+ (.6)(.5)(.5)
+ (.6)(.5)(1)
.55
(.4)(.5)(1)
+ (.4)(.5)(.5)
+ (.6)(.5)(.5)
.45
(.4)(.5)(.5)
Slide31Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide32Concentration Independent Coin Flipping
(TAS, C)
Slide33Concentration Independent Coin Flipping
(TAS, C) { , , , , }
Slide34Concentration Independent Coin Flipping
(TAS, C) { , , , , }
Slide35Concentration Independent Coin Flipping
(TAS, C) { , , , , }
P( ) + P( )
+
P(
) = .5
Slide36Concentration Independent Coin Flipping
(TAS, C) { , , , , }
P( ) + P( )
+
P(
) = .5
P( ) + P(
) = .5
Slide37Concentration Independent Coin Flipping
(TAS, C) { , , , , }
P( ) + P( )
+
P(
) = .5
P( ) + P(
) = .5
For ALL C
Slide38.5
.5
Slide39.5
.5
.5
.5
Slide40.5
.5
.5
.5
P(
) = .5
P(
) = .5
Slide41.7
.3
.3
.7
Slide42.7
.3
.3
.7
P(
) = .3
P(
) = .7
Slide43Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide44x
y
Slide45x
y
Slide46x
y
Slide47x
y
Slide48x
y
Slide49x
y
x
x
+y
y
y+y
1
P( ) =
x
x
+y
y
y+y
Slide50x
y
x
x
+y
y
y+y
1
P( ) =
xy
2y(
x+y
)
Slide51x
y
x
x
+y
P( ) =
xy
y
y+y
y
x
+y
+
x
x
+y
y
y+y
y
x
+y
2y(
x+y
)
Slide52x
y
x
x
+y
P( ) =
xy
y
y+y
y
x
+y
+
xy
2
2y(
x+y
)
2
2y(
x+y
)
Slide53x
y
P( ) =
xy
y
x
+y
+
xy
2
2y(
x+y
)
2
x
x+x
y
x
+y
2y(
x+y
)
Slide54x
y
P( ) =
xy
y
x
+y
+
xy
2
2y(
x+y
)
2
x
x+x
y
x
+y
y
x+y
+
x
x+x
y
x
+y
2y(
x+y
)
Slide55x
y
P( ) =
xy
y
x
+y
+
xy
2
2y(
x+y
)
2
x
x+x
y
x
+y
xy
2
+
2x(
x+y
)
2
2y(
x+y
)
Slide56P( ) =
xy
+
2y(
x+y
)
2
xy
2
+
2x(
x+y
)
2
2y(
x+y
)
x
2
+ 2xy + y
2
2(
x+y
)
2
=
=
(
x+y
)
2
2(
x+y
)
2
=
1
2
xy
2
Slide57Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide58S
1
Slide59S
1
1
2
Slide60S
1
1
2
2
3
Slide61S
1
1
2
2
3
3
4
Slide62S
1
1
2
2
3
3
4
4
5
Slide63S
1
1
2
2
3
3
4
4
5
5
6
Slide64S
1
1
2
2
3
3
4
4
5
5
6
6
7
Slide65S
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
Slide66S
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
Slide67S
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
Slide68S
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
Slide69S
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11
Slide70S
Slide71S
Slide72S
S
S
Slide73S
S
S
S
S
S
Slide74S
S
S
S
S
S
S
S
Slide75Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide76Simulation
(TAS, C) { , , , , }
Slide77Simulation
(TAS, C) { , , , , }
(
TAS’, for all C
) { , , , , }
with m x n scale factor
Slide78Simulation
(TAS, C) { , , , , }
(
TAS’, for all C
) { , , , , }
with m x n scale factor
P(TAS, C) = P(TAS’, for all C)
Slide79Simulation
(TAS, C) { , , , , }
(
TAS’, for all C
) { , , , , }
with m x n scale factor
P(TAS, C) = P(TAS’, for all C)
TAS’ robustly simulates TAS for C at scale factor
m,n
Slide80Simulation
Unidirectional two-choice linear assembly systems:
Grows in one direction from seed
Non-determinism between at most 2 tiles
Slide81Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
Slide82Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
a
b
S
S
Slide83Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
a
b
S
S
a
S
Slide84Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
a
b
S
S
a
S
b
S
c
b
S
d
Slide85Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
a
b
S
S
a
S
b
S
c
b
S
d
b
S
d
Slide86Simulation
Unidirectional two-choice linear assembly systems:Grows in one direction from seedNon-determinism between at most 2 tiles
S
a
b
S
S
a
S
b
S
c
b
S
d
b
S
d
For any unidirectional two-choice linear assembly system X, there exists a tile assembly system X’ which robustly simulates X for the uniform concentration distribution at scale factor 5,4.
Slide87Simulation
S
S
S
Slide88Simulation
S
S
S
S
Slide89Simulation
S
S
S
S
S
S
Slide90Simulation
S
S
S
S
S
S
Slide91Simulation
S
S
S
S
S
S
Slide92Simulation
S
S
S
S
S
S
Slide93Simulation
S
S
S
S
S
S
Slide94Simulation
S
S
S
S
S
S
Slide95Simulation
S
S
S
S
S
S
Slide96Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide97Simulation Application
There exists a TAS which assembles an expected length
N
linear assembly using
Θ
(
log
N
) tile types.
(
Chandran
,
Gopalkrishnan
,
Reif
)
Slide98Simulation Application
There exists a TAS which assembles an expected length
N
linear assembly using
Θ
(
log
N
) tile types.
(
Chandran
,
Gopalkrishnan
,
Reif
)
The construction is a unidirectional two-choice linear assembly system
Applies to uniform concentration distribution
Slide99Simulation Application
There exists a TAS which assembles an expected length
N
linear assembly using
Θ
(
log
N
) tile types.
(
Chandran
,
Gopalkrishnan
,
Reif
)
The construction is a unidirectional two-choice linear assembly system
Applies to uniform concentration distribution
Corollary to simulation technique:
There exists a TAS which assembles a width-4 expected length
N
assembly for all concentration distributions using O(
log
N
) tile types.
Slide100Simulation Application
There exists a TAS which assembles an expected length
N
linear assembly using
Θ
(
log
N
) tile types.
(
Chandran
,
Gopalkrishnan
,
Reif
)
The construction is a unidirectional two-choice linear assembly system
Applies to uniform concentration distribution
Corollary to simulation technique:
There exists a TAS which assembles a width-4 expected length
N
assembly for all concentration distributions using O(
log
N
) tile types.
Further, there is no PTAM tile system which generates width-1 expected length
N
assemblies for all concentration distributions with less than
N
tile types.
Slide101Introduction
Models
Concentration Independent Coin Flip
Big Seed, Temperature 1
Single Seed, Temperature 2
Simulation
Simulation Application
Unstable Concentrations
Summary
Slide102Unstable Concentrations
(TAS, C) { , , , , }
P( ) + P( )
+
P(
) = .5
P( ) + P(
) = .5
Slide103Unstable Concentrations
(TAS, C) { , , , , }
P( ) + P( )
+
P(
) = .5
P( ) + P(
) = .5
At each assembly stage, C changes
Slide104Unstable Concentrations
Impossible in the
aTAM
(in bounded space
)
Slide105Unstable ConcentrationsImpossible in the aTAM (in bounded space)Possible (and easy) in some extended models:
aTAM
with neg.
Interactions, big seed
Slide106Unstable ConcentrationsImpossible in the aTAM (in bounded space)Possible (and easy) in some extended models:
aTAM
with neg. Interactions, big seed
Hexagonal TAM with neg.
Interactions
Slide107Unstable ConcentrationsImpossible in the aTAM (in bounded space)Possible (and easy) in some extended models:
aTAM
with neg. Interactions, big seed
Hexagonal TAM with neg.Interactions
Polyomino
TAM
Slide108Unstable ConcentrationsImpossible in the aTAM (in bounded space)Possible (and easy) in some extended models:
aTAM
with neg. Interactions, big seed
Hexagonal TAM with neg.Interactions
Polyomino TAM
Geometric TAM,
big seed
Slide109Summary
Unstable Concentrations
Impossible in the aTAM (in bounded space)Extended models:Future Work:Single seed temperature 1Other simulations (other TASs/Boolean circuits)Uniform random number generationRandomized algorithms
Simulation
Concentration Independent Coin FlipsLarge seed, temperature 1Single seed, temperature 2
S
S
S
S
S
S
S