Tianqi Song Outline Wang tiling Abstract tile assembly model Reversible tile assembly model Kinetic tile assembly model Multiple temperature model qtile assembly model Flexible glue model ID: 810524
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Slide1
Theoretical Tile Assembly Models
Tianqi
Song
Slide2Outline
Wang tiling
Abstract tile assembly model
Reversible tile assembly model
Kinetic tile assembly model
Multiple temperature model
q-tile assembly model
Flexible glue model
Time-dependent glue model
Step-wise assembly model
Staged assembly
model
Slide3Wang Tiling
Proposed by Hao Wang in 1961.
Given a finite set of square tiles with a glue on each side.
Question: whether they can tile the plane with same
abutting glue.
Note that tiles cannot be rotated or reflected and you can use infinite number of copies of each tile.
Slide4Example of Wang Tiling
This tile set contain 13 tiles. They can tile the plane aperiodically as shown in next page.
Designed
by Karel Culik II in 1996. Picture is from
Wikipedia
.
Slide5Example of Wang Tiling
This picture is from Wikipedia.
Slide6Abstract Tile Assembly Model
Proposed by Erik Winfree and Paul W.K. Rothemund in 2000.
A tiling system under aTAM is a quadruple <T, s,
τ
, g>.
T is a set of tiles. A tile is a square with a glue on each side.
E
W
S
N
Slide7Abstract Tile Assembly Model
s is seed tile. The initial configuration has only seed tile.
τ
is temperature: the minimum accumulative strength that can fix a tile in a configuration.
g is glue strength function: G×G->N
+
,
where G is glue set. For any x, y in G, g(x, y)=g(y, x). A configuration is a function: Z×Z->
T
U
{
empty}, where Z
is the set of integers.
Slide8Tile Complexity
Tile complexity or program-size complexity: the minimum number of tile types required to assemble a shape.[
Erik Winfree and Paul W.K. Rothemund
STOC 2000]
A special case: in the linear version proposed by Chandran et al, the tile set is MULTISET.
Slide9Time Complexity
Time complexity or running time: model the assembly process by continuous Markov process.[Adleman et al STOC 2001]
Example of continuous Markov process. Picture is from Wikipedia
Slide10Time Complexity
States
: possible configurations.
Rate between state S1 and S2: if S2 is got by attaching a tile x to S1. The rate is the concentration of tile x. Otherwise, no edge between S1 and S2
.
The time from initial configuration to the terminal configuration is a random variable. Time complexity is the expected value of the random variable
Slide11Reversible Tile Assembly Model
Proposed by Leonard M. Adleman in 1999 to study linear assembly.
Define two functions:
(
1) σ: G×G->[0,1]
(
2) v:
G×G->[0,1] where G is the glue set.
σ
(g1,g2) is the probability of sticking between glue g1 and g2.
v(g1,g2) is the probability of unsticking between glue g1 and g2.
Slide12Kinetic Tile Assembly Model
Proposed by Erik Winfree in 1988.
Four assumptions:
(1) All monomers hold the same constant
concentration.
(2)There are not interactions among
aggregates.
Slide13Kinetic Tile Assembly Model
Four assumptions(continue):
(3) All monomers have the same forward rate
constant.
(4) The reverse rate depends exponentially
on the number of pairs need to be broken.
Slide14Kinetic Tile Assembly Model
Picture is from paper written by Erik Winfree in 1998
Slide15Kinetic Tile Assembly Model
r
f
=
k
f
[monomer]=k
fe-Gmcrr,b=k
r,b
=k
f
e
-bG
se
k
f is the forward rate constant. [monomer] is the concentration of some kind of monomer. Gmc is a measure of entropic cost of fixing a monomer at a particular position. It is decided by the concentration of monomer.
Slide16Kinetic Tile Assembly Model
G
se
is a measure of free energy cost to break a double helix of length s, where
G
se= (4000K/T-11)s.b is the number of s-length double helix that need to be broken.
Slide17Multiple Temperature Model
Proposed by Aggarwal et al. in 2004.
Replace temperature t in the standard model by a sequence of temperatures {τ
i
}
i=1
.
A system that has k temperatures is a k-temperature system.Assembly mechanism: the assembly process of a k-temperature system has k phases as shown in the Figure(next page).
k
Slide18Multiple Temperature Model
Assemble and disassemble under temperature τ
1
for long enough time.
Assemble and disassemble under temperature
τ
k
for long enough time.
Phase 1
Phase k
The terminal product of phase k is the terminal product of this k-temperature system.
Slide19Flexible Glue Model
Proposed by
by Aggarwal et
al.
in 2004.
In standard model: g(x, y)= 0 if x≠y.
In flexible glue model: g(x, y) may not be 0 when x≠y.
Slide20q-Tile Model
Proposed by by Aggarwal et
al.
in 2004.
In standard model: only single tile can attach to the growing supertile containing seed.
In q-tile model: supertiles of size not larger than q can form and attach to the seed supertile if the accumulative glue strength between two supertiles is not less than temperature.
Standard model is exactly 1-tile model.
Slide21Time-dependent Glue Model
Proposed by
Sahu
et al. in 2005.
The glue strength function g is defined differently from standard model:
g: G×G×R -> R. g(x, y, t) is the glue strength
between glue x and glue y when they have
been juxtaposed for t time.
Slide22Time-dependent Glue Model
Picture is from the paper by
Sahu
et al. in 2005
Slide23Time-dependent Glue Model
As shown in the figure, define r: G×G-> R as
time for maximum glue strength
. g(x, y, t) is growing when t< r(x, y). When t≥ r(x, y), g(x, y, t)= g(x, y, r(x, y)).
D
efine u: G×G-> R as
minimum interaction time
between two glues.
Slide24Time-dependent Glue Model
An example from
Sahu’s
paper:
Slide25Step-wise Assembly Model
Proposed by Reif in 1999.
A tiling system under step-wise assembly model is a quadruple S
step
= <{T
i
}
i=1, s, g, {τi}i
=1
>, where k is the number of steps, T
i
is the tile set at step
i
, is
τ
i is the temperature at step i.The assembly process is shown in the Figure(next page).
k
k
Slide26Step-wise Assembly Model
Put in T
1
including s. Assemble under temperature τ
1
for long enough time.
Put in
T
k
in.
Assemble under temperature
τ
k
for long enough time.
step 1 (in tube 1)
step k (in tube k)
terminal product of step 1
terminal product of step k-1
The terminal product of
step
k is the terminal product of this k
-step
system.
Slide27Staged Assembly Model
Proposed by Demaine et al. in 2007.
It is a generalized version of step-wise assembly model.
The assembly process under staged assembly model is shown in the Figure(next page).
Slide28Staged Assembly Model
This picture is from paper by Demaine et al. Vertices of mix graph represent bins for separated assembly reactions. Each bin has its only tile set and temperature. Only terminal product of one bin is delivered to bins in next stage.