1

Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms

January, 2013

Fuel Efficient Computation in Passive Self-Assembly

Robert Schweller University of Texas Pan-AmericanMichael Sherman University of Texas Pan-American

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Tile Assembly Model(Rothemund, Winfree, Adleman)

T = G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

Tile Set:

Glue Function:

x ed

cba

3

T =

d

e

x ed

cba

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

4

T =

d

e

x ed

cba

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

5

T =

d

e

x ed

cba

b c

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

6

T =

d

e

x ed

cba

b c

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

7

T =

d

e

x ed

cba

b c

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

8

T =

d

e

x ed

cba

b ca

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

9

T =

d

e

x ed

cba

b ca

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

10

T =

d

e

x ed

cba

b ca

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

11

T =

d

e

x ed

cba

b ca

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

12

T =

x ed

cba

a b c

d

e

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

13

T =

x ed

cba

x

a b c

d

e

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

14

T =

a b c

d

e

x

x ed

cba

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

15

T =

x ed

cba

a b c

d

e

x x

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

16

T =

x ed

cba

a b c

d

e

x x

x

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

17

T =

x ed

cba

a b c

d

e

x x

x x

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

18

T =

x ed

cba

a b c

d

e

x x

x x

Tile Assembly Model(Rothemund, Winfree, Adleman)

G(y) = 100%G(g) = 100%G(r) = 100%G(b) = 100%G(p) = 50%G(w) = 50%

What is this model capable above? -efficient assembly of shapes/patterns -shape and pattern replication -computation

BEAKER

1101 0 1 1 0 _

State: q3State: q2State: q2State: q3

Goal: Scalable, universal molecular computation-More than just a (really cool) computer-Algorithmic manipulation of matter at the nanoscale

Simulation of Cellular Automata

Slide stolen from: Andrew Winslow

[Rothemund, Papadakis, Winfree, 2004]

110

Turing Machine simulation in the TAM

1 0 1 1 0 _

State: q0State: q3State: q2State: q7State: q7State: q2State: q3

Slide stolen from: Matt Patitz

1 0 1 1 00 0 1 1 0 -0 1 1 1 0 - -0 1 1 1 0 - - -

[Rothemund, Winfree, 2000]

Limited Scalability Space in-efficient

-Entire history of computation stored in assembly

Fuel Guzzling- Each computation step burns many tiles

Goal: Fuel efficient, space efficient universal computation

1101 0 1 1 0 _

State: q3State: q2State: q2State: q3

1 0 1 1 00 0 1 1 0 -0 1 1 1 0 - -0 1 1 1 0 - - -

Turing Machine simulation in the TAM[Rothemund, Winfree, 2000]

Goal: Fuel efficient, space efficient universal computation

Problem: Assemblies only grow larger

Solution: Negative strength glues

Negative Glues

Our Result: Tile assembly is capable of space efficient, fuel efficient universal computaion with the use of negative and positive strength glues.

Negative Glues - Example

200%

100%100%

100%

Negative glues previouslyconsidered in:[Reif, Sahu, Yin 2005][Doty, Kari, Masson 2010][Patitz, Schweller, Summers, 2011]

Negative Glues - Example

200%

100%

-50%

100%

-50%

100%

-Negative glues can prevent attachments.-Can they do anything deeper?

Negative Glues - Example

200%

100%

-100%

200%

-100%

200%

Increase strength

Negative Glues - Example

200%

100%

-100%

200%

Key Idea: -Stable assemblies can combine to form unstable assemblies-Allows “diss-assembly”

High Level Sketch of Universal Computation

10 1

00

High Level Sketch of Universal Computation

10 1

00

High Level Sketch of Universal Computation

10 1

0

High Level Sketch of Universal Computation

10 1

0

High Level Sketch of Universal Computation

10 1

01

High Level Sketch of Universal Computation

10 1

01

Bit Flipping

-30%

1

75%

25%

0-30%90

30

70

Bit Flipping

-30%

1

25%

0-30%

90

30

70

25

75

Bit Flipping

1

25%

0-30%

90

30

70

25

-30%

40%

90%

75

Bit Flipping

1

25%

0

70

90

30-30%

2590

4075

Bit Flipping

1

090

40 70

25

75

30%

Bit Flipping

1

15%

70%90%75

90

40

Bit Flipping

1

70%30%

75

90

40 90

15

Bit Flipping

190

30

70

10%90% 90%

-60%75

90

40

15

Bit Flipping

190

30

70

90%

-60%

90

10 90%

75

90

40

15

Bit Flipping

190

30

70

15

75

15

40

10

-60

90

10

90

90

-60

90

40

15

75

Oscillator

0

1

Expended fueld

Oscillator

0

1

1

0

Expended fueld

Expended fueld

Graph Walking

0

1

1

0

0 1

Simple Example of Graph Walking:

More General Result:Theorem: For any directed graph G=(V,E), there exists a size O(V+E) tile set that walks graph G in a fuel-efficient manner.

Extension: Double Bit Flipping

1

00 1

Turing Machine Simulation

010 10

Current bit: 0State: GREEN

Flip bit to 1, move right, change to state PURPLE

1 0

Current bit: 0State: PURPLE

Flip bit to 1, move left, change to state ORANGE

1 1

Current bit: 1State: ORANGE

Flip bit to 0, move left, change to state GREEN

00

O(1) garbage produced per computation step

Tape Extension Gadget

1 100 0

Also: need an infinite tape

Universal Tile Self-Assembly

O(Tape*Steps) O(Tape)

O(Tape) O(1)

Space FuelOld Way

Negative Glues

010 101 01 100

[Rothemund, Winfree, 2000]

Why is Passive, Fuel Efficient Computation Important?

• Passive Self-Assembly– Most active models have no current implementation at the nanoscale– Informs when more active components are truly necessary– May lead to connection to active self-assembly: Implement an active

model within a passive model• Fuel Efficiency

– Particle starvation a practical problem in experimentation– Necessary for a scalable molecular computer

• Negative Glues– Informs experimentalists that negative glues implementation should be

fruitful– Sheds light on natural computation and phenomena

• Charged particles, magnets• Protein folding• ATP Synthases

Open Problems• Compact Graph Walking

– Many graphs can likely be fuel efficiently walked by sub linear sized tile systems.

O(log |V|) tiles?

• Negative Glues: Necessary?– Amortized fuel-efficiency?

• Two-tape Turing machine simulation• Simulation of active models

– Signal tiles?• Fuel Rods?

– No depletion of monomers