The tile assembly model is intrinsically universal David Doty, Jack - - PowerPoint PPT Presentation

SLIDE 1

Damien Woods, FOCS 2012

The tile assembly model is intrinsically universal

1

David Doty, Jack H. Lutz, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, Damien Woods Caltech Iowa State

SLIDE 2

Damien Woods, FOCS 2012

Motivation

• Nature computes
• We need a computational theory of self assembly and

molecular interactions

• Perhaps more importantly (for us), we can find

interesting theoretical problems: computation, geometry, asynchronousity, kinetics, thermodynamics

2

Sumazin et al. Cell 147(2). 2011 Soloveichik, Seelig, Winfree. DNA as a Universal Substrate for Chemical Kinetics. PNAS 2010

• Engineers are building nanoscale

molecular (chemical) computers

Rothemund, PWK. Folding DNA to create nanoscale shapes and patterns. Nature 2006 Rothemund, Papadakis, Winfree 2004

SLIDE 3

Damien Woods, FOCS 2012

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Finite set of tile types, unlimited supply of each tile type
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are strong

enough (sum of strengths is at least the temperature)

• Asynchronous

3

Abstract tile assembly model

Winfree, 1998

a b c d

strength 2 strength 1

Rothemund, Papadakis, Winfree 2004

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 4

Damien Woods, FOCS 2012

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 5

Damien Woods, FOCS 2012

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 6

Damien Woods, FOCS 2012

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 b a c

B

a b

A

i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 7

Damien Woods, FOCS 2012

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

b a c

B

a b

A

i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 8

Damien Woods, FOCS 2012

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

b a c

B

a b

A

i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 9

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

b a c

B

a b

A

i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 10

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

b a c

B

a b

A

d e

D

i i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 11

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B

a b

A

d e

D

i i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 12

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B

a b

A

f e E d e

D

i i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 13

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i i a

Seed d

a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 14

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 15

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 16

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 17

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 18

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E b a c

B

a b

A

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 19

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 20

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E b a c

B

a b

A

b a c

B

a b

A

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 21

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 22

Damien Woods, FOCS 2012

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 23

Damien Woods, FOCS 2012

c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 24

Damien Woods, FOCS 2012

d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 25

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 26

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B

a b

A

f e E d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 27

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B F’

f h a b

A

f e E d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 28

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B F’

f h a b

A

f e E

H

h j d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 29

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B F’

f h a b

A

f e E

H

h j d e

D

a

Seed d

d e

D

i i a

Seed d

G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 30

Damien Woods, FOCS 2012

i

E

j d e

D

i c d

C’

a b

A

b a c

B

Abstract tile assembly model

4

• Square units called tiles (non-rotatable)
• Each side has a glue (a,b,c,d,...)
• Each glue has a strength (0,1,2,3,...)
• Growth begins from a seed tile
• Tiles stick to the growing assembly if their glues match, and are

strong enough (sum of strengths is at least the temperature)

• Finite set of tile types, unlimited supply of each tile type

Winfree, 1998 a b

A

c e

C

b a c

B F

f g a b

A

f e E a b

A

b a c

B

a b

A

b a c

B F’

f h a b

A

f e E

H

h j d e

D

a

Seed d

d e

D

i i a

Seed d

Cooperation (adjacent sides) Cooperation (opposite sides) G

g ? e a

Seed d

f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a b c d strength 2 strength 1

• Defn. Tile assembly system: tile set, seed tile, temperature

SLIDE 31

Damien Woods, FOCS 2012

Algorithmic self-assembly with tiles: computation and geometry

5

Rothemund, Papadakis, Winfree 2004 f e E h c Barish, Rothemund, Winfree 2005 Barish, Schulman Rothemund, Winfree 2009 Barish, Schulman Rothemund, Winfree 2009

• Seeman built tiles out of DNA in the laboratory
• Winfree showed that DNA tiles can run algorithms (much like

cellular automata, Wang tiles)

• A variety of algorithmic tile assembly systems have been built

from DNA

SLIDE 32

Damien Woods, FOCS 2012

Intrinsic universality

6

http://otcametapixel.blogspot.com/ http://www.youtube.com/watch?v=xP5-iIeKXE8

SLIDE 33

Damien Woods, FOCS 2012

Intrinsic universality

6

http://otcametapixel.blogspot.com/ http://www.youtube.com/watch?v=xP5-iIeKXE8

SLIDE 34

Damien Woods, FOCS 2012

• Conway’s Game of Life is an intrinsically universal cellular

automaton

Durand, Roka, The game of life: universality revisited. 1999

Intrinsic universality

6

http://otcametapixel.blogspot.com/ http://www.youtube.com/watch?v=xP5-iIeKXE8

SLIDE 35

Damien Woods, FOCS 2012

Intrinsic universality

• An intrinsically universal CA is one that “simulates” any other:
• Appropriate initial conditions
• Encodes many properties of the simulated CA
• Theory of intrinsic universality in CA
• Applications: non-intrinsic universality results to prove

weakness of a model

7

SLIDE 36

Damien Woods, FOCS 2012

Is the abstract tile assembly model intrinsically universal?

8

SLIDE 37

Damien Woods, FOCS 2012

Is the abstract tile assembly model intrinsically universal?

Yes!

8

SLIDE 38

Damien Woods, FOCS 2012

Is the abstract tile assembly model intrinsically universal?

Yes!

8

There is a single universal tile set that simulates any tile assembly system

SLIDE 39

Damien Woods, FOCS 2012

Simulation

• We want a definition of simulation that captures the notion of
• ne TAS simulating the “behavior” of another
• T is a (any) simulated tile assembly system,
• T = (tileset T, seed assembly σ, temperature τ)
• Tile assembly system UT = (U, σT, 2) simulates T if:
• Tiles from T are simulated by m x m supertiles in UT, i.e., a

representation function R maps supertiles in assemblies of UT, to tiles in T. Oh, and R is “clean”.

• The producible assemblies in UT represent exactly the

producible assemblies in T.

• Ignoring m x m scaling, dynamics are equivalent in UT & T.

9

SLIDE 40

Damien Woods, FOCS 2012

Simulation with tiles

10

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 41

Damien Woods, FOCS 2012

Simulation with tiles

10

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 42

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 43

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 44

Damien Woods, FOCS 2012

Simulation with tiles

10

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 45

Damien Woods, FOCS 2012

Simulation with tiles

10

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 46

Damien Woods, FOCS 2012

Simulation with tiles

10

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 47

Damien Woods, FOCS 2012

Simulation with tiles

10

c e

C

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 48

Damien Woods, FOCS 2012

Simulation with tiles

10

c e

C

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 49

Damien Woods, FOCS 2012

Simulation with tiles

10

c e

C

b a c

B

a b

A

a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 50

Damien Woods, FOCS 2012

Simulation with tiles

10

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 51

Damien Woods, FOCS 2012

Simulation with tiles

10

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 52

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 53

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 54

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 55

Damien Woods, FOCS 2012

Simulation with tiles

10

b a c

B

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 56

Damien Woods, FOCS 2012

Simulation with tiles

10

b a c

B

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 57

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 58

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B

a b

A

d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 59

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B

a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 60

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B

a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 61

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 62

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 63

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 64

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 65

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

etc.

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

SLIDE 66

Damien Woods, FOCS 2012

Simulation with tiles

10

a b

A

b a c

B

a b

A

c e

C

b a c

B F

f g a b

A

f e E d e

D

i a

Seed d

Preassembled seed structure

• Tiles are simulated by supertiles
• For each assembly sequence in the

simulated tile system, there is an assembly sequence in the simulator, and vice-versa

etc.

Temperature = 2

Simulator tile set

a b

A

Simulated tile Simulator tiles

Ignoring m x m scaling, dynamics are equivalent in the simulated system and simulator

SLIDE 67

Damien Woods, FOCS 2012

seed supertile

glue frame tile lookup table blank probe region tile lookup table frame probe table glue glue frame tile lookup table blank probe region tile lookup table frame probe table glue glue frame tile lookup table blank probe region tile lookup table frame probe table glue glue frame tile lookup table blank probe region tile lookup table frame probe table glue

11

Preassembled seed structure

a

Seed d

SLIDE 68

Damien Woods, FOCS 2012

Superside

glue frame tile lookup table blank probe region tile lookup table frame probe table glue

4 O(log |T|) O(|T|4 log |T|) O(|T|2) O(|T|2) O(|T|2) O(|T|4 log |T|) 4 O(log |T|)

|T| is number of tiles in the simulated tileset T.

Encoded glue of this superside (e.g. “a”) Encoding of the entire simulated tile assembly system

12

a Seed d f e E a b

A

b a c

B

c e

C

d e

D

i

F’

f h i

E

j c d

C’ H

h j

F

f g

Temperature = 2

G

g ? e

a

1

SLIDE 69

Damien Woods, FOCS 2012

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

SLIDE 70

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

SLIDE 71

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

SLIDE 72

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

SLIDE 73

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

“Genome” is copied “Genome” is read crawler encodes glue of south superside

SLIDE 74

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

“Genome” is copied “Genome” is read crawler encodes glue of south superside

SLIDE 75

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1

One-sided binding with a single strength-τ south superside

13

a

1

“Genome” is copied “Genome” is read crawler encodes glue of south superside

SLIDE 76

Damien Woods, FOCS 2012

a

2

glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue glue tile lookup table blank probe region tile lookup table probe table glue

Supertile 1 Nondeterminism Rotations

One-sided binding with a single strength-τ south superside

13

a

1

“Genome” is copied “Genome” is read crawler encodes glue of south superside

SLIDE 77

Damien Woods, FOCS 2012

                                 

 

      

  

 

             

                          



  

 

   

crawler encodes “input” glues crawler encodes “output” tile type

Crawler doing a tile lookup

14

SLIDE 78

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a

SLIDE 79

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 80

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 81

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 82

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 83

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 84

Damien Woods, FOCS 2012

glue glue glue glue glue glue probe region probe region blank blank

tile lookup table tile lookup table tile lookup table tile lookup table probe table probe table

blank

tile lookup table tile lookup table probe table

glue glue probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with adjacent cooperating supersides

15

3

b c

1

a e a

4

c

SLIDE 85

Damien Woods, FOCS 2012

S

S

N

N

E W W

? ? ? ?

WE

W

S

S

S

N

N

N ?

? ? ?

Better luck next time! Uh oh!

A key problem

16

No tile fits A tile fits, but the path to the right is blocked!

SLIDE 86

Damien Woods, FOCS 2012

2

blank

probe table tile lookup table tile lookup table tile lookup table tile lookup table probe table

blank glue glue glue glue glue glue glue glue probe region probe region

collect north & south glues, & random number

blank

tile lookup table tile lookup table probe table

probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with opposite cooperating supersides

17

1

a b

SLIDE 87

Damien Woods, FOCS 2012

b a

3 2

blank

probe table tile lookup table tile lookup table tile lookup table tile lookup table probe table

blank glue glue glue glue glue glue glue glue probe region probe region

collect north & south glues, & random number

blank

tile lookup table tile lookup table probe table

probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with opposite cooperating supersides

17

1

a b

SLIDE 88

Damien Woods, FOCS 2012

b a

3 2

blank

probe table tile lookup table tile lookup table tile lookup table tile lookup table probe table

blank glue glue glue glue glue glue glue glue probe region probe region

collect north & south glues, & random number

blank

tile lookup table tile lookup table probe table

probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with opposite cooperating supersides

17

1

a b

SLIDE 89

Damien Woods, FOCS 2012

b a

3 2

blank

probe table tile lookup table tile lookup table tile lookup table tile lookup table probe table

blank glue glue glue glue glue glue glue glue probe region probe region

collect north & south glues, & random number

blank

tile lookup table tile lookup table probe table

probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with opposite cooperating supersides

17

1

a b

SLIDE 90

Damien Woods, FOCS 2012

b a

3 2

blank

probe table tile lookup table tile lookup table tile lookup table tile lookup table probe table

blank glue glue glue glue glue glue glue glue probe region probe region

collect north & south glues, & random number

blank

tile lookup table tile lookup table probe table

probe region blank

tile lookup table tile lookup table probe table

probe region

Two-sided binding with opposite cooperating supersides

17

1

a b

SLIDE 91

Damien Woods, FOCS 2012

2

b

probe table tile lookup table tile lookup table

blank g l u e g l u e glue glue blank

tile lookup table tile lookup table probe table

glue glue probe region probe region blank

tile lookup table tile lookup table probe table

glue glue probe region

continues here because probes will not meet

b l a n k

tile lookup table tile lookup table probe table

p r

• b

e r e g i

• n

3-sided “uh-oh” example: probes miss each other

18

1

a

x

S

S

N

N

E W W

? ? ? ?

WE

W

S

uh oh!

SLIDE 92

Damien Woods, FOCS 2012

2

b

probe table tile lookup table tile lookup table

blank g l u e g l u e glue glue blank

tile lookup table tile lookup table probe table

glue glue probe region probe region blank

tile lookup table tile lookup table probe table

glue glue probe region

continues here because probes will not meet

b l a n k

tile lookup table tile lookup table probe table

p r

• b

e r e g i

• n

3-sided “uh-oh” example: probes miss each other

18

3

b c

1

a

x

S

S

N

N

E W W

? ? ? ?

WE

W

S

uh oh!

SLIDE 93

Damien Woods, FOCS 2012

2

b e a

4

c

probe table tile lookup table tile lookup table

blank g l u e g l u e glue glue blank

tile lookup table tile lookup table probe table

glue glue probe region probe region blank

tile lookup table tile lookup table probe table

glue glue probe region

continues here because probes will not meet

b l a n k

tile lookup table tile lookup table probe table

p r

• b

e r e g i

• n

3-sided “uh-oh” example: probes miss each other

18

3

b c

1

a

x

S

S

N

N

E W W

? ? ? ?

WE

W

S

uh oh!

SLIDE 94

Damien Woods, FOCS 2012

2

b e a

4

c

probe table tile lookup table tile lookup table

blank g l u e g l u e glue glue blank

tile lookup table tile lookup table probe table

glue glue probe region probe region blank

tile lookup table tile lookup table probe table

glue glue probe region

continues here because probes will not meet

b l a n k

tile lookup table tile lookup table probe table

p r

• b

e r e g i

• n

3-sided “uh-oh” example: probes miss each other

18

3

b c

1

a

x

S

S

N

N

E W W

? ? ? ?

WE

W

S

uh oh!

SLIDE 95

Damien Woods, FOCS 2012

2

b e a

4

c

probe table tile lookup table tile lookup table

blank g l u e g l u e glue glue blank

tile lookup table tile lookup table probe table

glue glue probe region probe region blank

tile lookup table tile lookup table probe table

glue glue probe region

continues here because probes will not meet

b l a n k

tile lookup table tile lookup table probe table

p r

• b

e r e g i

• n

3-sided “uh-oh” example: probes miss each other

18

3

b c

1

a

x

S

S

N

N

E W W

? ? ? ?

WE

W

S

uh oh!

SLIDE 96

Damien Woods, FOCS 2012

• Variety of cases for different
• rders of superside arrival
• Superside win/lose

configurations and crawler initiation locations (green)

• Proof analogy: Distributed

game, computation & geometry

                                                                                                         

19

SLIDE 97

Damien Woods, FOCS 2012

Future work & conclusions

• Can we use intrinsic universality as a tool in tile assembly to prove

negative results on the power of tile assembly systems?

• Conjecture: The temperature 1 tile assembly model is not

intrinsically universal for all tile assembly systems

• As has been done in the theory of cellular automata, can we use

intrinsic universality as a tool to classify tile assembly systems into hierarchies based on their ability to simulate each other?

• Application to other tile assembly models: roughly speaking, in any

tile assembly model that simulates the aTAM (in a ‘for all, exists’ sense), our intrinsically universal tile set immediately implies that there is one tile set in that model that simulates all aTAM tile assembly systems

20

SLIDE 98

Damien Woods, FOCS 2012

Intrinsic universality: Who dunnit?

21

SLIDE 99

Damien Woods, FOCS 2012

Where?

22

Who paid? NSF: Molecular programming project grant 0832824, CCF-1219274 and CCF-1162589; Science Foundation Ireland, Junta de Andalucia Spain, Canada. Special thanks: Matthew Cook, David Soloveichik, Erik Winfree, Paul Rothemund, Shuki Bruck. Also the members of Erik Winfree’s, Shuki Bruck’s, and Paul Rothemund’s groups at Caltech

SLIDE 100

Damien Woods, FOCS 2012

f i n