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Intrinsic Universality in Tile Self-Assembly Requires Cooperation

Intrinsic Universality in Tile Self-Assembly Requires Cooperation. Pierre-Etienne Meunier Matthew J. Patitz Scott M. Summers Guillaume Theyssier Damien Woods Andrew Winslow. Tile. Glues. Strength-2. Strength-1. ×∞. ×∞. ×∞. ×∞. ×∞. Temperature.

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Intrinsic Universality in Tile Self-Assembly Requires Cooperation

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  1. Intrinsic Universality in Tile Self-Assembly Requires Cooperation Pierre-Etienne Meunier Matthew J. Patitz Scott M. Summers Guillaume Theyssier Damien Woods Andrew Winslow

  2. Tile

  3. Glues

  4. Strength-2 Strength-1

  5. ×∞ ×∞ ×∞ ×∞ ×∞ Temperature

  6. Self-assembling tiles: a real thing

  7. 0 0 0

  8. 0 0 1

  9. 0 1 0

  10. 1 1 1

  11. Temperature-2 systems can require a tile to use cooperative bonds.

  12. Temperature-1 systems do not have cooperative bonds.

  13. Are temp-1 and temp-2 systems equally powerful? Conjectured not.

  14. What does “power” mean? Power = ability to simulate.

  15. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  16. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  17. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  18. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  19. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  20. Simulation Definition 1. Pick scale factor k. 2. Pick special marker tiles to encode tile type of each macrotile. 3. Tiles only in macrotiles encoding tiles and neighbors. 4. Macrotilesare placed in same order as simulated tiles.

  21. Encoded as Assembles Mixed with Given assembling at

  22. Encoded as Assembles Mixed with Given assembling at

  23. This system is instrinsically universal. Encoded as Assembles Mixed with Given assembling at

  24. Intrinsic universality: simulating a whole class of things

  25. There is a temp-2 system that is intrinsically universal for all tile systems!

  26. There is a temp-2 system that is intrinsically universal for all tile systems! Dave Doty Jack Lutz Robbie Schweller Scott Summers Matt Patitz Damien Woods

  27. Is there a temp-1 system that is intrinsically universal for all tile systems? No. Is there a temp-1 system that is intrinsically universal for temp-1 systems? Yes (but 3D).

  28. Keystone

  29. Flag

  30. 18-tile temp-2 system a c assembles b For all a, b, c ≥ 0

  31. We show no temp-1 system simulates temp-2 system

  32. Breaking any simulation

  33. Breaking any simulation

  34. Breaking any simulation

  35. Breaking any simulation

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