Matthew J Patitz University of Texas PanAmerican Dustin Reishus University of Southern California Robert Schweller University of Texas PanAmerican Scott M Summers University of WisconsinPlatteville ID: 373635
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1
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 2010
October 25, 2010
Strong Fault-Tolerance for Self-Assembly with Fuzzy TemperatureSlide2
2
OutlineBasic Tile Assembly ModelFuzzy Fault ToleranceEfficient, Fault Tolerant ResultsSlide3
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
e
d
c
b
aSlide4
4
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide5
5
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide6
6
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide7
7
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide8
8
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide9
9
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide10
10
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide11
11
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide12
12
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
d
e
x
e
d
c
b
a
b
c
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide13
13
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
a
b
c
d
e
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide14
14
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
x
a
b
c
d
e
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide15
15
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
a
b
c
d
e
x
x
e
d
c
b
a
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide16
16
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
a
b
c
d
e
x
x
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide17
17
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
a
b
c
d
e
x
x
x
Tile Assembly Model
(Rothemund, Winfree, Adleman)Slide18
18
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
a
b
c
d
e
x
x
x
x
Tile
Assembly Model
(
Rothemund
,
Winfree
,
Adleman
)Slide19
19
OutlineBasic Tile Assembly ModelErrors!Fuzzy Fault ToleranceEfficient, Fault Tolerant ResultsSlide20
a
b
c
stable at
temperature 2
stable at
temperature 2
unstable at
temperature 2
a
b
d
a
b
d
a
b
c
c
d
a
b
d
a
b
c
d
c
a
b
c
a
x
c
a
b
d
a
b
d
a
x
c
a
x
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)Slide21
stable at
temperature 1 =
temporarily
stable at temperature 2
stable at temperature 2 but not producible at temperature 2
a
b
d
a
x
c
a
x
c
a
b
d
a
b
d
a
x
c
c
d
c
d
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!Slide22
22
OutlineBasic Tile Assembly ModelErrors!Fuzzy Fault ToleranceEfficient, Fault Tolerant ResultsSlide23
·
Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2
a
b
c
d
e
x
x
a
b
c
d
e
x
x
x
x
d
e
x
bSlide24
·
Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2
·
Dependably terminal
(DT):
the subset of DP
supertiles
that are terminal at temperature = 2
a
b
c
d
e
x
x
a
b
c
d
e
x
x
x
x
d
e
x
b
a
b
c
d
e
x
x
x
xSlide25
·
Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2
·
Dependably terminal
(DT):
the subset of DP
supertiles
that are terminal at temperature = 2
·
Plausibly producible
(PP):
the set of
supertiles
that can be assembled at temperature = 1
a
b
c
d
e
x
x
a
b
c
d
e
x
x
x
x
d
e
x
b
a
b
c
d
e
x
x
x
x
a
b
c
d
e
x
x
x
x
x
x
x
x
a
b
c
d
e
x
x
x
x
x
x
x
x
x
x
x
xSlide26
·
Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2
·
Dependably terminal
(DT):
the subset of DP
supertiles
that are terminal at temperature = 2
·
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
c
d
e
x
x
x
x
d
e
x
b
a
b
c
d
e
x
x
x
x
a
b
c
d
e
x
x
x
x
x
x
x
x
a
b
c
d
e
x
x
x
x
x
x
x
x
x
x
x
x
a
b
c
d
e
x
x
x
x
x
x
x
x
x
x
x
xSlide27
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:Slide28
28
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.Slide29
29
T = G(y) = 2G(g) = 2G(r) = 2G(b) = 2G(p) = 1G(w) = 1 t = 2
x
e
d
c
b
a
a
b
c
d
e
x
x
x
x
Square BuildingSlide30
30
Square Building: Normal Approach
nSlide31
31
Square Building
x
Tile Complexity:
2n
nSlide32
Square Building
0
0
0
0
log n
-Use
log n
tile types to seedcounter:Slide33
Square Building
0
0
0
0
log n
-Use
8
additional tile types capable of binary counting:
-Use log n tile types capable ofBinary counting:Slide34
Square Building
0
0
0
0
log n
0
0
0
0
0
0
0
0
1
0
1
0
1
1
0
0
1
1
1
0
0
0
0
0
1
1
0
1
1
1
1
1
0
0
0
1
-Use
8
additional tile types
capable of binary counting:
-Use
log n
tile types capable of
Binary counting:Slide35
Square Building
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
-Use
8
additional tile types
capable of binary counting:
-Use
log n
tile types capable of
Binary counting:
log nSlide36
Square Building
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
-Use
8
additional tile types
capable of binary counting:
-Use
log n
tile types capable of
Binary counting:Slide37
Square Building
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
1
0
1
1
0
0
1
1
1
0
0
0
0
1
0
0
1
1
0
1
0
1
0
1
1
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
0
0
0
0
1
0
0
0
0
1
0
0
1
1
0
0
0
0
1
0
1
0
1
0
0
1
1
0
1
1
1
0
0
0
0
1
1
0
0
1
0
1
0
1
1
1
0
1
0
0
1
1
1
0
1
1
0
1
1
1
1
1
1
1
n – log n
log n
x
y
Tile Complexity:
O(log n)
(Rothemund, Winfree 2000)Slide38
A Fuzzy Fault Tolerant Counter?
A counter seems important for efficient assembly of n x n squares Current counter constructions are not fuzzy fault tolerant00
0
1
0
0
0
0
0
1
0
c
0
0
0
1
0
c
0
1
n
c
1
n
0
0
1
n
n
n
n
0
1Slide39
[
Barish, Shulman, Rothemund, Winfree, 2009]Slide40
40
OutlineBasic Tile Assembly ModelErrors!Fuzzy Fault ToleranceEfficient, Fault Tolerant ResultsSlide41
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 bondsSlide42
Previous Tile Set Not Fault Tolerant
Producible at temperature 1 but stable (and erroneous) at temperature 2Slide43
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 glueSlide44
Fuzzy Temperature Fault-Tolerant CounterSlide45
Square Composed of One Horizontal Counter and Multiple Copies of Vertical CounterSlide46
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