1 David DotyCalifornia Institute of Technology Matthew J. PatitzUniversity of Texas Pan-American Dustin ReishusUniversity of Southern California Robert SchwellerUniversity of Texas Pan-American Scott M. SummersUniversity of Wisconsin-Platteville FOCS 2010 October 25, 2010 Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature

2 Outline Basic Tile Assembly Model Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

3 Tile Assembly Model (Rothemund, Winfree, Adleman) T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 Tile Set: Glue Function: Temperature: x ed cba

4 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

5 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

6 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

7 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

8 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bc Tile Assembly Model (Rothemund, Winfree, Adleman)

9 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

10 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

11 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

12 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 d e x ed cba bca Tile Assembly Model (Rothemund, Winfree, Adleman)

13 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

14 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba x abc d e Tile Assembly Model (Rothemund, Winfree, Adleman)

15 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 abc d e x x ed cba Tile Assembly Model (Rothemund, Winfree, Adleman)

16 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx Tile Assembly Model (Rothemund, Winfree, Adleman)

17 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx x Tile Assembly Model (Rothemund, Winfree, Adleman)

18 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx xx Tile Assembly Model (Rothemund, Winfree, Adleman)

19 Outline Basic Tile Assembly Model Errors! Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

ideal cooperative binding: tile attaches to assembly if and only if it interacts with strength ≥ 2 (such as two matching strength-1 glues)

stable at temperature 1 = temporarily stable at temperature 2 stable at temperature 2 but not producible at temperature 2 a bd a x c a x c a bd a bd 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!

22 Outline Basic Tile Assembly Model Errors! Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

 Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2 abc d e xx abc d e xx xx d e x b

 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 abc d e xx abc d e xx xx d e x b abc d e xx xx

 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 abc d e xx abc d e xx xx d e x b abc d e xx xx abc d e xx xx xxxx abc d e xx xx xxxxxxxx

 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 abc d e xx abc d e xx xx d e x b abc d e xx xx abc d e xx xx xxxx abc d e xx xx xxxxxxxx abc d e xx xx xxxxxxxx

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 Tile set Desired shape Avoid this:

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.

29 T = G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 x ed cba abc d e xx xx Square Building

30 Square Building: Normal Approach n

31 Square Building x Tile Complexity: 2n n

Square Building 0000 log n -Use log n tile types to seed counter:

Square Building 0000 log n -Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting:

Square Building 0000 log n Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting:

Square Building Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting: log n

Square Building Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting:

Square Building n – log n log n x y Tile Complexity: O(log n) (Rothemund, Winfree 2000)

A Fuzzy Fault Tolerant Counter? A counter seems important for efficient assembly of n x n squares Current counter constructions are not fuzzy fault tolerant c c 0 1 nc 1 n 0 01 nn n n 0 1

[Barish, Shulman, Rothemund, Winfree, 2009]

40 Outline Basic Tile Assembly Model Errors! Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

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 bondsStrength-1 bonds

Previous Tile Set Not Fault Tolerant Producible at temperature 1 but stable (and erroneous) at temperature 2

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 glue

Fuzzy Temperature Fault-Tolerant Counter

Square Composed of One Horizontal Counter and Multiple Copies of Vertical Counter

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