Annual ACMSIAM Symposium on Discrete Algorithms January 2013 Fuel Efficient Computation in Passive SelfAssembly Robert Schweller University of Texas PanAmerican Michael Sherman University of Texas PanAmerican ID: 709442
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Slide1
1
Proceedings of the 24
th
Annual ACM-SIAM Symposium on Discrete Algorithms
January, 2013
Fuel Efficient Computation in Passive Self-Assembly
Robert
Schweller
University of Texas
Pan-American
Michael Sherman
University of Texas Pan-AmericanSlide2
2
Tile Assembly Model
(Rothemund, Winfree, Adleman)
T =
G(y) =
100%G(g) = 100%
G(r) = 100%G(
b) = 100%G(p) = 5
0%G(w) = 50%
Tile Set:
Glue
Function:
x
e
d
c
b
aSlide3
3
T =
d
e
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide4
4
T =
d
e
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide5
5
T =
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide6
6
T =
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide7
7
T =
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide8
8
T =
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide9
9
T =
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide10
10
T =
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide11
11
T =
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide12
12
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%Slide13
13
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%Slide14
14
T =
a
b
c
d
e
x
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)
G(
y
) =
100%
G(
g
) =
100%
G(
r
) =
100%
G(
b
) =
100%
G(
p
) = 5
0%
G(
w
) =
50%Slide15
15
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%Slide16
16
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%Slide17
17
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%Slide18
18
T =
x
e
d
c
b
a
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
) = 5
0%
G(
w
) =
50%
What is this model capable above?
-efficient assembly of shapes/patterns
-shape and pattern replication
-
computation
Slide19
BEAKER
1
1
0
1
0
1
1
0
_
State: q
3
State: q
2
State: q
2
State: q
3
Goal: Scalable, universal molecular computation
-More than just a (really cool) computer
-Algorithmic manipulation of matter at the
nanoscaleSlide20
Simulation of Cellular Automata
Slide stolen from: Andrew Winslow
[
Rothemund
,
Papadakis
, Winfree
, 2004]Slide21
1
1
0
Turing Machine simulation in the TAM
1
0
1
1
0
_
State: q
0
State: q
3
State: q
2
State: q
7
State: q
7
State: q
2
State: q
3
Slide stolen from: Matt
Patitz
1
0
1
1
0
0
0
1
1
0
-
0
1
1
1
0
-
-
0
1
1
1
0
-
-
-
[
Rothemund
,
Winfree
, 2000]Slide22
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
1
1
0
1
0
1
1
0
_
State: q
3
State: q
2
State: q
2
State: q
3
1
0
1
1
0
0
0
1
1
0
-
0
1
1
1
0
-
-
0
1
1
1
0
-
-
-
Turing Machine simulation in the TAM
[
Rothemund
,
Winfree
, 2000]Slide23
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.Slide24
Negative Glues - Example
200
%
1
00
%
100%
100%
Negative glues previously
considered in:
[
Reif
,
Sahu
, Yin 2005]
[Doty, Kari, Masson 2010]
[
Patitz, Schweller, Summers, 2011]Slide25
Negative Glues - Example
200
%
1
00
%
-50%
100%
-50%
100%
-Negative glues can prevent attachments.
-Can they do anything deeper?Slide26
Negative Glues - Example
200
%
1
00
%
-100%
2
00%
-100%
2
00%
Increase strengthSlide27
Negative Glues - Example
200
%
1
00
%
-100%
2
00%
Key Idea:
-Stable assemblies can combine to form unstable assemblies
-Allows “
diss
-assembly”Slide28
High Level Sketch of Universal Computation
1
0
1
0
0Slide29
High Level Sketch of Universal Computation
1
0
1
0
0Slide30
High Level Sketch of Universal Computation
1
0
1
0Slide31
High Level Sketch of Universal Computation
1
0
1
0Slide32
High Level Sketch of Universal Computation
1
0
1
0
1Slide33
High Level Sketch of Universal Computation
1
0
1
0
1Slide34
Bit Flipping
-30%
1
75%
25%
0
-30%
90
30
70Slide35
Bit Flipping
-30%
1
25%
0
-30%
90
30
70
25
75Slide36
Bit Flipping
1
25%
0
-30%
90
30
70
25
-30%
40%
90%
75Slide37
Bit Flipping
1
25%
0
70
90
30
-30%
25
90
40
75Slide38
Bit Flipping
1
0
90
40
70
25
75Slide39
30%
Bit Flipping
1
15%
70%
90%
75
90
40Slide40
Bit Flipping
1
70%
30%
75
90
40
90
15Slide41
Bit Flipping
1
90
30
70
10%
90%
90%
-60%
75
90
40
15Slide42
Bit Flipping
1
90
30
70
90%
-60%
90
10
90%
75
90
40
15Slide43
Bit Flipping
1
90
30
70
15
75
15
40
10
-60
90
10
90
90
-60
90
40
15
75Slide44
Oscillator
0
1
Expended
fueldSlide45
Oscillator
0
1
1
0
Expended
fueld
Expended
fueldSlide46
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.Slide47
Extension: Double Bit Flipping
1
0
0
1Slide48
Turing Machine Simulation
0
1
0
1
0
Current bit: 0
State:
GREEN
Flip bit to 1, move right, change to state
PURPLE
1
0
Current bit: 0
State:
PURPLE
Flip bit to 1, move left, change to state
ORANGE
1
1
Current bit: 1
State:
ORANGE
Flip bit to 0, move left, change to state
GREEN
0
0
O(1) garbage produced per computation stepSlide49
Tape Extension Gadget
1
1
0
0
0
Also: need an
infinite tapeSlide50
Universal Tile Self-Assembly
O(Tape*Steps) O(Tape)
O(Tape) O(1)
Space Fuel
Old WayNegative Glues
0
1
0
1
0
1
0
1
1
0
0
[
Rothemund
,
Winfree
, 2000]Slide51
Why is Passive, Fuel Efficient Computation Important?
Passive Self-Assembly
Most active models have no current implementation at the nanoscaleInforms when more active components are truly necessary
May lead to connection to active self-assembly: Implement an active model within a passive modelFuel Efficiency
Particle starvation a practical problem in experimentationNecessary for a scalable molecular computerNegative GluesInforms experimentalists that negative glues implementation should be fruitful
Sheds light on natural computation and phenomenaCharged particles, magnetsProtein foldingATP SynthasesSlide52
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 simulationSimulation of active modelsSignal tiles?Fuel Rods?No depletion of monomers