An Introduction toAlgorithmic Tile Self-Assembly

Simple particles coalescing into complex superstructures.

Self-Assembly

Self-AssemblySimple particles coalescing into

complex superstructures.

Self-AssemblySimple particles coalescing into

complex superstructures.

Self-AssemblySimple particles coalescing into

complex superstructures.

Crystallization

Morphogenesis

Natural Self-Assembly

Synthetic Self-Assembly with DNA

Tile

Glues

Tile

Glues

Strength-1

Strength-2

×∞

×∞

×∞

×∞

×∞

Temperature

Self-assembling tiles: a real thing

Assembling patterned shapes

P. W. K. Rothemund, N. Papadakis, E. Winfree, Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology (12), 2004.

Cellular automata

Tile assembly can simulate CA

Binary counter

Self-assembling tile systems (finite & infinite)

Initiatilization Copy tiles

Increment tiles

Halt tile

Building squares

Build a column and row, then fill L-shape with tiles.

Encode start and end values:

• Start: north glues (0..2t)

• End: # tiles (2t)

Building a rectangle of height h (h in 2t..2t+1)

requires O(t) = O(log(h)) tiles.

Building rectangles

Encode start and end values:

• Start: north glues (0..2t)

• End: # tiles (2t)

Building rectangles

Possible to do better?

• Each tile set encodes the height of the rectangle built.• The tile set needs sufficient information to do so.• Most heights have ≥ 0.5*log2(h) bits of information.

How many bits of information does a tile set with t tile types have? • Let g be number of glue types on the tiles. Then g/4 ≤

t ≤ g4.• So specifying a tile takes ≤ 4*log2(g) ≤ 4*log2(4t) ≤

12*log2(t) bits.• So the entire set has at most t(12*log2(t)) ≤

12t*log2(t) bits.

So most heights need 0.5*log2(h) ≤ 12t*log2(t).By algebra, t = Ω(log(h)/loglog(h)).

Building rectangles and squares

• Possible to build any height rectangle and any size square (at τ= 2) using O(log(h)) tile types.

• More than half of all heights require a tile set of size Ω(log(h)/loglog(h)).

• Possible to build any height rectangle (at τ= 2) using O(log(h)/loglog(h)) tile types.

[Adleman et al. 2001]

[Winfree, Soloveichik 2000]

[Winfree, Soloveichik 2000]

Building rectangles and squares (at τ= 1)

h tile types

h

2n-1 tile types

n

n

Possible to do better? (Open problem)

Computing with self-assembly (via CA)

• Some CA can simulate Turing machines: CA ≥ TM• Can simulate these CA with tiles: SA ≥ CA • So tiles can simulate Turing machines: SA ≥ TM

[Lindgren, Nordahl 1990]

[Winfree 1998]

[http://mathworld.wolfram.com/Rule90.html]

Computing with self-assembly (direct)

Input

Computation

InputMachine

A Turing machine example…[courtesy of Scott Summers]

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘└┘ └┘└┘

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

q01 └┘└┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘C0

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘└┘ └┘└┘

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

q01 └┘└┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘C0

q2

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘└┘ └┘└┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1 1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q2

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘└┘ └┘└┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C2q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘1

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

C3q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘└┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

C4 └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

C5 └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

C6 └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

C7 └┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

q1

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

└┘1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

1

C8 └┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

q0

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

C8

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

q2

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

C8

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

C9 └┘└┘└┘ └┘ └┘ └┘└┘ └┘q2111 1 1

q2

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

q01 └┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

C8

C9

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘q2111 1 1

qhalt

1 └┘ └┘└┘└┘└┘ └┘ └┘└┘ └┘ └┘└┘└┘└┘└┘└┘ └┘

Let M = (Q,Σ,Γ,δ,q0,qhalt), where Q = {q0,q1,q2,qhalt}, Σ = {1}, Γ = {1,└┘}, and δ is defined in the following table…

δ 1 └┘

q0 q2,1,L q1,1,R

q1 q1,1,R q0,1,L

q2 qhalt q1,1,L

111 1

q01 └┘└┘ └┘C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

C8

C9

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘q2111 1 1

Chalt

q01 └┘└┘ └┘C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

C1

C2

C3

C4

C5

C6

C7

C8

C9

q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘

1 └┘1

q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q1└┘1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘111 q01 1

└┘└┘└┘ └┘ └┘ └┘└┘ └┘q2111 1 1

Chalt

“Zig-zag” simulation of a Turing machine

Building any shape optimally

[Soloveichik, Winfree 2007]

• Encode shape via Turing machine. • Do a BFS according to current location.

Arbitrary scale factor!

Temperature-2 systems can require

a tile to use cooperative bonds.

Temperature-1 systems do not have cooperative bonds.

Are τ=1 and τ=2 systems equally powerful?

τ=1 (linear tile types)

τ=2 (logarithmic tile types)

We think building shapes takes linear tile types,

and simulating Turing machines is impossible.

Except in 3D, whereτ=1 can do both of these things…

Abstract Tile Assembly Model (aTAM)

Abstract Tile Assembly Model (aTAM)

A Seedless World?

A Seedless World?

A Seedless World?

A Seedless World?

A Seedless World?

Two-Handed Assembly Model (2HAM)

Two-Handed Assembly Model (2HAM)

How powerful is 2HAM relative to aTAM?

Are there techniques that require a seed?

No.

Every aTAM system can be simulated with a 2HAM system.

Simulation

Simulation

Simulation captures dynamics

Simulation captures production