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An Introduction toAlgorithmic Tile Self-Assembly
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Simple particles coalescing into complex superstructures.
Self-Assembly
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Self-AssemblySimple particles coalescing into
complex superstructures.
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Self-AssemblySimple particles coalescing into
complex superstructures.
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Self-AssemblySimple particles coalescing into
complex superstructures.
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Crystallization
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Morphogenesis
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Natural Self-Assembly
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Synthetic Self-Assembly with DNA
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Tile
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Glues
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Tile
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Glues
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Strength-1
Strength-2
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×∞
×∞
×∞
×∞
×∞
Temperature
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Self-assembling tiles: a real thing
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Assembling patterned shapes
P. W. K. Rothemund, N. Papadakis, E. Winfree, Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology (12), 2004.
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Cellular automata
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Tile assembly can simulate CA
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Binary counter
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Self-assembling tile systems (finite & infinite)
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Initiatilization Copy tiles
Increment tiles
Halt tile
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Building squares
Build a column and row, then fill L-shape with tiles.
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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
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Encode start and end values:
• Start: north glues (0..2t)
• End: # tiles (2t)
Building rectangles
Possible to do better?
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• 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)).
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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]
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Building rectangles and squares (at τ= 1)
h tile types
h
2n-1 tile types
n
n
Possible to do better? (Open problem)
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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]
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Computing with self-assembly (direct)
Input
Computation
InputMachine
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A Turing machine example…[courtesy of Scott Summers]
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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
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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
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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
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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└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘
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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
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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
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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
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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
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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
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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
![Page 61: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/61.jpg)
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
![Page 62: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/62.jpg)
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
![Page 63: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/63.jpg)
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
![Page 64: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/64.jpg)
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
![Page 65: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/65.jpg)
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
![Page 66: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/66.jpg)
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
![Page 67: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/67.jpg)
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
![Page 68: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/68.jpg)
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
![Page 69: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/69.jpg)
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
![Page 70: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/70.jpg)
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
![Page 71: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/71.jpg)
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
![Page 72: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/72.jpg)
q01 └┘└┘ └┘C0 └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘
C1
C2
C3
C4
C5
C6
C7
C8
C9
q1└┘ └┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘
1q2└┘ └┘ └┘└┘└┘└┘└┘ └┘ └┘ └┘└┘ └┘
1 └┘1
q0└┘ └┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11
└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11
└┘└┘└┘ └┘ └┘ └┘└┘ └┘1 └┘11 q11
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Chalt
“Zig-zag” simulation of a Turing machine
![Page 73: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/73.jpg)
Building any shape optimally
[Soloveichik, Winfree 2007]
• Encode shape via Turing machine. • Do a BFS according to current location.
Arbitrary scale factor!
![Page 74: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/74.jpg)
Temperature-2 systems can require
a tile to use cooperative bonds.
![Page 75: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/75.jpg)
Temperature-1 systems do not have cooperative bonds.
![Page 76: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/76.jpg)
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…
![Page 77: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/77.jpg)
Abstract Tile Assembly Model (aTAM)
![Page 78: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/78.jpg)
Abstract Tile Assembly Model (aTAM)
![Page 79: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/79.jpg)
![Page 80: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/80.jpg)
A Seedless World?
![Page 81: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/81.jpg)
A Seedless World?
![Page 82: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/82.jpg)
A Seedless World?
![Page 83: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/83.jpg)
A Seedless World?
![Page 84: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/84.jpg)
A Seedless World?
![Page 85: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/85.jpg)
Two-Handed Assembly Model (2HAM)
![Page 86: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/86.jpg)
Two-Handed Assembly Model (2HAM)
![Page 87: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/87.jpg)
How powerful is 2HAM relative to aTAM?
Are there techniques that require a seed?
No.
![Page 88: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/88.jpg)
Every aTAM system can be simulated with a 2HAM system.
![Page 89: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/89.jpg)
Simulation
![Page 90: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/90.jpg)
Simulation
![Page 91: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/91.jpg)
Simulation captures dynamics
![Page 92: An Introduction to Algorithmic Tile Self-Assembly](https://reader036.vdocuments.mx/reader036/viewer/2022062301/5697c00b1a28abf838cc8514/html5/thumbnails/92.jpg)
Simulation captures production