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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
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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%
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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
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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
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Simulation of Cellular Automata
Slide stolen from: Andrew Winslow
[Rothemund, Papadakis, Winfree, 2004]
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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]
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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]
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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.
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Negative Glues - Example
200%
100%100%
100%
Negative glues previouslyconsidered in:[Reif, Sahu, Yin 2005][Doty, Kari, Masson 2010][Patitz, Schweller, Summers, 2011]
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Negative Glues - Example
200%
100%
-50%
100%
-50%
100%
-Negative glues can prevent attachments.-Can they do anything deeper?
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Negative Glues - Example
200%
100%
-100%
200%
-100%
200%
Increase strength
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Negative Glues - Example
200%
100%
-100%
200%
Key Idea: -Stable assemblies can combine to form unstable assemblies-Allows “diss-assembly”
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High Level Sketch of Universal Computation
10 1
00
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High Level Sketch of Universal Computation
10 1
00
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High Level Sketch of Universal Computation
10 1
0
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High Level Sketch of Universal Computation
10 1
0
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High Level Sketch of Universal Computation
10 1
01
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High Level Sketch of Universal Computation
10 1
01
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Bit Flipping
-30%
1
75%
25%
0-30%90
30
70
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Bit Flipping
-30%
1
25%
0-30%
90
30
70
25
75
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Bit Flipping
1
25%
0-30%
90
30
70
25
-30%
40%
90%
75
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Bit Flipping
1
25%
0
70
90
30-30%
2590
4075
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Bit Flipping
1
090
40 70
25
75
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30%
Bit Flipping
1
15%
70%90%75
90
40
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Bit Flipping
1
70%30%
75
90
40 90
15
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Bit Flipping
190
30
70
10%90% 90%
-60%75
90
40
15
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Bit Flipping
190
30
70
90%
-60%
90
10 90%
75
90
40
15
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Bit Flipping
190
30
70
15
75
15
40
10
-60
90
10
90
90
-60
90
40
15
75
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Oscillator
0
1
Expended fueld
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Oscillator
0
1
1
0
Expended fueld
Expended fueld
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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.
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Extension: Double Bit Flipping
1
00 1
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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
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Tape Extension Gadget
1 100 0
Also: need an infinite tape
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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]
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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
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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