1. Flipping Tiles: Concentration Independent Coin Flips in Tile Self- Assembly Cameron T. Chalk, Bin Fu, Alejandro Huerta, Mario A. Maldonado, Eric Martinez, Robert T. Schweller, Tim Wylie Funding by NSF Grant CCF-1117672 NSF Early Career Award 0845376 ? ?

2. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

3. ? ?

4. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

5. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed:

6. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

7. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

8. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

9. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

10. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

11. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

12. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

13. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

14. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

15. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

16. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

17. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S

18. Tile Assembly Model (Rothemund, Winfree, Adleman) Tileset: S Glue: G(g) = 2 G(o) = 2 G(y) = 2 G(r) = 2 G(b) = 1 G(p) = 1 S Temperature: 2 Seed: S TERMINAL

19. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) Tileset: S Glue: G(g) = 2 G(o) = 2 G(p) = 2 G(b) = 2 S Temperature: 2 Seed:.1.2.3

20. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) Tileset: S Glue: G(g) = 2 G(o) = 2 G(p) = 2 G(b) = 2 S Temperature: 2 Seed: S.1.2.3

21. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) Tileset: S Glue: G(g) = 2 G(o) = 2 G(p) = 2 G(b) = 2 S Temperature: 2 Seed: S S S.1.2.3

22. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) Tileset: S Glue: G(g) = 2 G(o) = 2 G(p) = 2 G(b) = 2 S Temperature: 2 Seed:.1.2.3 S S.2.2 +.3 S =.4

23. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) Tileset: S Glue: G(g) = 2 G(o) = 2 G(p) = 2 G(b) = 2 S Temperature: 2 Seed:.1.2.3 S S.2.2 +.3 S =.4.3.2 +.3 =.6

24. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S

25. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5

26. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5 S S 1 1

27. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5 S S 1 1 (.4)(.5)(1)

28. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5 S S 1 1 (.4)(.5)(1) + (.4)(.5)(.5)

29. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5 S S 1 1 (.4)(.5)(1) + (.4)(.5)(.5) + (.6)(.5)(.5).45

30. Probabilistic Tile Assembly Model (Becker, Remila, Rapaport) S.1.2.3.4.6 S S S S S S.5 S S 1 1 + (.6)(.5)(.5) + (.6)(.5)(1).55 (.4)(.5)(1) + (.4)(.5)(.5) + (.6)(.5)(.5).45 (.4)(.5)(.5)

31. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

32. Concentration Independent Coin Flipping (TAS, C)

33. Concentration Independent Coin Flipping (TAS, C) {,,,, }

34. Concentration Independent Coin Flipping (TAS, C) {,,,, }

35. Concentration Independent Coin Flipping (TAS, C) {,,,, } P( ) + P( ) + P( ) =.5

36. Concentration Independent Coin Flipping (TAS, C) {,,,, } P( ) + P( ) + P( ) =.5P( ) + P( ) =.5

37. Concentration Independent Coin Flipping (TAS, C) {,,,, } P( ) + P( ) + P( ) =.5P( ) + P( ) =.5 For ALL C

38. .5

40. P( ) =.5

41. .7.3.7

42. .3.7 P( ) =.3 P( ) =.7

43. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

44. x y

45. x y

46. x y

47. x y

48. x y

49. x y x x+y y y+y 1 P( ) = x x+y y y+y

50. x y x x+y y y+y 1 P( ) = xy 2y(x+y)

51. x y x x+y P( ) = xy y y+y y x+y + x y y+y y x+y2y(x+y)

52. x y x x+y P( ) = xy y y+y y x+y + xy 2 2y(x+y) 2 2y(x+y)

53. x y P( ) = xy y x+y + xy 2 2y(x+y) 2 x x+x y x+y 2y(x+y)

54. x y P( ) = xy y x+y + xy 2 2y(x+y) 2 x x+x y x+y y + x x+x y x+y 2y(x+y)

55. x y P( ) = xy y x+y + xy 2 2y(x+y) 2 x x+x y x+y xy 2 + 2x(x+y) 2 2y(x+y)

56. P( ) = xy + 2y(x+y) 2 xy 2 + 2x(x+y) 2 2y(x+y) x 2 + 2xy + y 2 2(x+y) 2 = = (x+y) 2 2(x+y) 2 = 1 2 xy 2

57. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

58. S 1

59. S 1 1 2

60. S 1 1 2 2 3

61. S 1 1 2 2 3 34

62. S 1 1 2 2 3 34 4 5

63. S 1 1 2 2 3 34 4 5 5 6

64. S 1 1 2 2 3 34 4 5 5 6 6 7

65. S 1 1 2 2 3 34 4 5 5 6 6 7 7 8

66. S 1 1 2 2 3 34 4 5 5 6 6 7 7 8 8 9

67. S 1 1 2 2 3 34 4 5 5 6 6 7 7 8 8 9 9 10

68. S 1 1 2 2 3 34 4 5 5 6 6 7 7 8 8 9 9 11

69. S 1 1 2 2 3 34 4 5 5 6 6 7 7 8 8 9 9 10 11

70. S

71. S

72. SSS

73. SSSSSS

74. SSSSSSSS

75. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

76. Simulation (TAS, C) {,,,, }

77. Simulation (TAS, C) {,,,, } (TAS’, for all C) {,,,, } with m x n scale factor

78. Simulation (TAS, C) {,,,, } (TAS’, for all C) {,,,, } with m x n scale factor P(TAS, C) = P(TAS’, for all C)

79. Simulation (TAS, C) {,,,, } (TAS’, for all C) {,,,, } with m x n scale factor P(TAS, C) = P(TAS’, for all C) TAS’ robustly simulates TAS for C at scale factor m,n

80. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles

81. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S

82. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S a b S S

83. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S a b S S aS

84. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S a b S S aS b Sc b Sd

85. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S a b S S aS b Sc b Sd b Sd

86. Simulation Unidirectional two-choice linear assembly systems: Grows in one direction from seed Non-determinism between at most 2 tiles S a b S S aS b Sc b Sd b Sd For any unidirectional two-choice linear assembly system X, there exists a tile assembly system X’ which robustly simulates X for the uniform concentration distribution at scale factor 5,4.

87. Simulation S S S

88. S S S S

89. S S S SSS

90. S S S SSS

91. S S S SSS

92. S S S SSS

93. S S S SSS

94. S S S SSS

95. S S S SSS

96. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

97. Simulation Application There exists a TAS which assembles an expected length N linear assembly using Θ(logN) tile types. (Chandran, Gopalkrishnan, Reif)

98. Simulation Application There exists a TAS which assembles an expected length N linear assembly using Θ(logN) tile types. (Chandran, Gopalkrishnan, Reif) The construction is a unidirectional two-choice linear assembly system Applies to uniform concentration distribution

99. Simulation Application There exists a TAS which assembles an expected length N linear assembly using Θ(logN) tile types. (Chandran, Gopalkrishnan, Reif) The construction is a unidirectional two-choice linear assembly system Applies to uniform concentration distribution Corollary to simulation technique: There exists a TAS which assembles a width-4 expected length N assembly for all concentration distributions using O(logN) tile types.

100. Simulation Application There exists a TAS which assembles an expected length N linear assembly using Θ(logN) tile types. (Chandran, Gopalkrishnan, Reif) The construction is a unidirectional two-choice linear assembly system Applies to uniform concentration distribution Corollary to simulation technique: There exists a TAS which assembles a width-4 expected length N assembly for all concentration distributions using O(logN) tile types. Further, there is no PTAM tile system which generates width-1 expected length N assemblies for all concentration distributions with less than N tile types.

101. Introduction Models Concentration Independent Coin Flip Big Seed, Temperature 1 Single Seed, Temperature 2 Simulation Simulation Application Unstable Concentrations Summary

102. Unstable Concentrations (TAS, C) {,,,, } P( ) + P( ) + P( ) =.5P( ) + P( ) =.5

103. Unstable Concentrations (TAS, C) {,,,, } P( ) + P( ) + P( ) =.5P( ) + P( ) =.5 At each assembly stage, C changes

104. Unstable Concentrations Impossible in the aTAM (in bounded space)

105. Unstable Concentrations Impossible in the aTAM (in bounded space) Possible (and easy) in some extended models: aTAM with neg. Interactions, big seed

106. Unstable Concentrations Impossible in the aTAM (in bounded space) Possible (and easy) in some extended models: aTAM with neg. Interactions, big seed Hexagonal TAM with neg. Interactions

107. Unstable Concentrations Impossible in the aTAM (in bounded space) Possible (and easy) in some extended models: aTAM with neg. Interactions, big seed Hexagonal TAM with neg. Interactions Polyomino TAM

108. Unstable Concentrations Impossible in the aTAM (in bounded space) Possible (and easy) in some extended models: aTAM with neg. Interactions, big seed Hexagonal TAM with neg. Interactions Polyomino TAM Geometric TAM, big seed

109. Summary Unstable Concentrations Impossible in the aTAM (in bounded space) Extended models: Future Work: -Single seed temperature 1 -Other simulations (other TASs/Boolean circuits) -Uniform random number generation -Randomized algorithms Simulation Concentration Independent Coin Flips Large seed, temperature 1 Single seed, temperature 2 S S S S S S S