1. 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. 2 Outline Basic Tile Assembly Model Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 19 Outline Basic Tile Assembly Model Errors! Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

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

21. 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. 22 Outline Basic Tile Assembly Model Errors! Fuzzy Fault Tolerance Efficient, Fault Tolerant Results

23.  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

24.  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

25.  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

26.  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

27. 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 01 2 1 2 0 1 2 1 2 01 2 1 2 Tile set Desired shape Avoid this:

28. 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. 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. 30 Square Building: Normal Approach n

31. 31 Square Building x Tile Complexity: 2n n

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

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

34. Square Building 0000 log n 000 00 000 1010 1100 1110 000 01 10 1 1 11 1 0001 -Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting:

35. Square Building 0000 000 000 00 000 1010 1100 1110 0001 0011 0101 0111 1001 1011 1111 1101 1 1 11 1 -Use 8 additional tile types capable of binary counting: -Use log n tile types capable of Binary counting: log n

36. Square Building 0000 000 000 00 000 1010 1100 1110 0001 0011 0101 0111 1001 1011 1111 1101 1 1 11 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:

37. Square Building 0000 000 000 00 000 1010 1100 1110 0001 0011 0101 0111 1001 1011 1111 1101 1 1 11 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)

38. 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 0001 0 0000 10 c 000 10 c 0 1 nc 1 n 0 01 nn n n 0 1

39. [Barish, Shulman, Rothemund, Winfree, 2009]

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

41. 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

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

43. 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

44. Fuzzy Temperature Fault-Tolerant Counter

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

46. 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