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David Doty California Institute of Technology

Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature. FOCS 2010 October 25, 2010. David Doty California Institute of Technology Matthew J. Patitz University of Texas Pan-American Dustin Reishus University of Southern California

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David Doty California Institute of Technology

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  1. Strong Fault-Tolerance for Self-Assembly with Fuzzy Temperature FOCS 2010 October 25, 2010 • David Doty California Institute of Technology • Matthew J. Patitz University of Texas Pan-American • Dustin Reishus University of Southern California • Robert Schweller University of Texas Pan-American • Scott M. Summers University of Wisconsin-Platteville

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

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

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

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

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

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

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

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

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

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

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

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

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

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

  16. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  17. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  18. Tile Assembly Model (Rothemund, Winfree, Adleman) a b c x d e x x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

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

  20. stable at temperature 2 stable at temperature 2 c a c b d a c a c a a a c b d b b d b d b d unstable at temperature 2 a c a c b x a c x a c a a a x b d b d b d 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 c c x d a a c a c c a a x x d b d b d b 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

  23. ·Dependably producible (DP): the set of supertiles that can be assembled at temperature = 2 x x e e x x x x d d a a b b c c e x d 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 x x x x e e e x x x x x x d d d a a a b b b c c c e x d b

  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 x x x x x x x x e e e e e x x x x x x x x x x d d d d d a a a a a b b b b b c c c c c e x x x x x x x x x x x x x d b

  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 x x x x x x x x x x e e e e e e x x x x x x x x x x x x d d d d d d a a a a a a b b b b b b c c c c c c e x x x x x x x x x x x x x x x x x x x x x d b

  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 Desired shape Tile set Avoid this: 2 2 2 1 1 1 1 1 1 0 0 0 2 2 2

  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. Square Building a b c x d e x x e x x d a b c G(y) = 2 G(g) = 2 G(r) = 2 G(b) = 2 G(p) = 1 G(w) = 1 t = 2 T =

  30. Square Building: Normal Approach n

  31. Square Building Tile Complexity: 2n n x

  32. Square Building -Use log n tile types to seed counter: 0 0 0 0 log n

  33. Square Building -Use log n tile types capable of Binary counting: -Use 8 additional tile types capable of binary counting: 0 0 0 0 log n

  34. Square Building -Use log n tile types capable of Binary counting: -Use 8 additional tile types capable of binary counting: 1 1 0 0 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 log n

  35. Square Building -Use log n tile types capable of Binary counting: -Use 8 additional tile types capable of binary counting: 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 log n

  36. Square Building -Use log n tile types capable of Binary counting: -Use 8 additional tile types capable of binary counting: 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  37. Square Building 1 1 1 1 Tile Complexity: O(log n) 1 1 1 0 1 1 0 1 (Rothemund, Winfree 2000) 1 1 0 0 1 0 1 1 1 0 1 0 n – log n 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 x 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 log n 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 y 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

  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 1 0 1 0 0 1 0 1 n n n c n c n n 0 0 1 1 0 c 0 0 0 0 0 0 1 0 0 0 0

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

  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 bonds Strength-1 bonds

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

  43. Add more synchronization Strength-1 glue Strength-2 glue • 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.

  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

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