David DotyCalifornia Institute of Technology

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David Doty California Institute of TechnologyMatthew J. Patitz University of Texas Pan-AmericanDustin Reishus University of Southern CaliforniaRobert Schweller University of Texas Pan-AmericanScott M. Summers University of Wisconsin-Platteville

FOCS 2010October 25, 2010

Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

2

Outline

• Basic Tile Assembly Model• Fuzzy Fault Tolerance• Efficient, Fault Tolerant Results

3

Tile Assembly Model(Rothemund, Winfree, Adleman)

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

Tile Set:

Glue Function:

Temperature:

x ed

cba

4

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

d

e

x ed

cba


5

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba


6

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b c


7

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b c


8

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b c


9

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b ca


10

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b ca


11

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b ca


12

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2d

e

x ed

cba

b ca


13

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e


14

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

x

a b c

d

e


15

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

a b c

d

e

x

x ed

cba


16

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e

x x


17

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e

x x

x


18

T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e

x x

x x


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Outline

• Basic Tile Assembly Model• Errors!• Fuzzy Fault Tolerance• Efficient, Fault Tolerant Results

ab

c stable attemperature 2

stable attemperature 2

unstable attemperature 2

a

b d

a

b d

ab

c

cd

a

b d

ab

cd

c

ab

c

ax

ca

b d

a

b d

ax

c

ax

c

a

b d

ideal cooperative binding: tile attaches to assembly if and only if it interacts with strength ≥ 2 (such as two matching strength-1 glues)

stable attemperature 1 = temporarily stable at temperature 2

stable at temperature 2 but not producible at temperature 2

a

b d

ax

c

ax

ca

b d

a

b d

ax

c

cd

cd

more realistic kinetic model: tile attaches to assembly but detaches "quickly" if attached with only strength 1 (and detaches "slowly" if attached with strength 2)

• insufficient attachment...

becomes stabilized by subsequent attachment: permanent error!

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Outline


· Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2

a b c

d

e

x x

a b cdex xx x

dex

b


· Dependably terminal (DT): the subset of DP supertiles that are terminal at temperature = 2

a b c

d

e

x x

a b cdex xx x

dex

b

a b cdex xx x



· Plausibly producible (PP): the set of supertiles that can be assembled at temperature = 1

a b c

d

e

x x

a b cdex xx x

dex

b

a b cdex xx x

a b cdex xx x x x x

x

a b cdex xx x

x x

xx

xx xx



· Plausibly producible (PP): the set of supertiles that can be assembled at temperature = 1

· Plausibly stable (PS): the set of supertiles in PP that are stable at temperature = 2

a b c

d

e

x x

a b cdex xx x

dex

b

a b cdex xx x

a b cdex xx x x x x

x

a b cdex xx x

x x

xx

xx xx

a b cdex xx x

x x

xx

xx xx

The Fuzzy Temperature Fault-Tolerance Design Problem

Given a target shape X, design a tile set such that:•Every PS supertile can grow into a DT supertile•Every DT supertile has the shape X

0 1 2

1

2

0 1 2

1

2

0 1 2

1

2

Tile set Desired shape Avoid this:

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Goal:

• Design an efficient tile system for the assembly of a n x n square that is fuzzy fault tolerant.

• Result: O(log n) tile complexity construction for n x n squares that is fuzzy fault tolerant.

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T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1

t = 2

x ed

cba

a b c

d

e

x x

x x

Square Building

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Square Building: Normal Approach

n

31

Square Building

x

Tile Complexity:2n

n

Square Building

0 0 00

log n

-Use log n tile types to seedcounter:

Square Building

0 0 00

log n

-Use 8 additional tile types capable of binary counting:

-Use log n tile types capable ofBinary counting:

Square Building

0 0 00

log n

000000 00

1 0 101 1 001 1 100 0 00 11 01

111

1

000 1



Square Building

0 0 00000000

000 00

1 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

1111

1



log n

Square Building

0 0 00000000

000 00

1 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

1111

1

0000

1000

0100

1100

0010

1010

0110

1110

0001

1001

0101

1101

0011

1011

0111

1111



Square Building

0 0 00000000

000 00

1 0 101 1 001 1 100 0 010 0 110 1 010 1 111 0 011 0 11

1 1 111 1 01

1111

1

0000

1000

0100

1100

0010

1010

0110

1110

0001

1001

0101

1101

0011

1011

0111

1111

n – log n

log n

x

y

Tile Complexity:O(log n)

(Rothemund, Winfree 2000)

A Fuzzy Fault Tolerant Counter?

• A counter seems important for efficient assembly of n x n squares

• Current counter constructions are not fuzzy fault tolerant

0 0 0 1

0

0 0 0 0

1 0

c

000

10 c

0

1

n c

1

n

0

0 1

nnnn

0 1

[Barish, Shulman, Rothemund, Winfree, 2009]

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Outline


Strength-2 growth is error-free

Idea: use nondeterministic strength-2 growth to guess numbers in counter and use geometric blocking (“steric hindrance”) to ensure they only come together in proper places.

Strength-1 bonds used to enforce bumps are present when binding occurs

Strength-2 bonds Strength-1 bonds

Previous Tile Set Not Fault Tolerant

Producible at temperature 1 but stable (and erroneous) at temperature 2

Add more synchronization

Each counter column is composed of 2 sub-columns, each which contributes a single strength bond at the top.

Each must be fully complete for them to bind.

Strength-1 glue

Strength-2 glue

Fuzzy Temperature Fault-Tolerant Counter

Square Composed of One Horizontal Counter and Multiple Copies of Vertical Counter

Open Problems

• Make construction robust to non-rigidity of DNA tiles to enhance effectiveness of “programmed steric hindrance”

• Experimentally determine the largest size of supertiles that reliably attach

• Universal Computation and Fuzzy-Fault Tolerance?

• Assembly Time

Documents

David DotyCalifornia Institute of Technology