Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D

Published on: Published on::2010/05/13

abstract

[[Has abstract::We investigate the power of the Wang tile self-assembly model at temperature 1, a threshold value that permits attachment between any two tiles that share even a single bond. When restricted to deterministic assembly in the plane, no temperature 1 assembly system has been shown to build a shape with a tile complexity smaller than the diameter of the shape. In contrast, we show that temperature 1 self-assembly in 3 dimensions, even when growth is restricted to at most 1 step into the third dimension, is capable of simulating a large class of temperature 2 systems, in turn permitting the simulation of arbitrary Turing machines and the assembly of <m>n\times n</m> squares in near optimal <m>O(\log n)</m> tile complexity. Further, we consider temperature 1 probabilistic assembly in 2D, and show that with a logarithmic scale up of tile complexity and shape scale, the same general class of temperature <m>\tau=2</m> systems can be simulated with high probability, yielding Turing machine simulation and <m>O(\log^2 n)</m> assembly of <m>n\times n</m> squares with high probability. Our results show a sharp contrast in achievable tile complexity at temperature 1 if either growth into the third dimension or a small probability of error are permitted. Motivated by applications in nanotechnology and molecular computing, and the plausibility of implementing 3 dimensional self-assembly systems, our techniques may provide the needed power of temperature 2 systems, while at the same time avoiding the experimental challenges faced by those systems.]]

authors

{{#arraymap:Matthew Cook, Yunhui Fu, Robert Schweller|,|x|Was written by::x}}

file

Temp1 deterministic3d probablistic2d.pdf