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 n × n squares in near optimal O(log n) 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 τ = 2 systems can be simulated, yielding Turing machine simulation and O(log2 n) assembly of n × n 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.
Cook, Matthew, Yunhui Fu, and Robert Schweller. 2011. “Temperature 1 Self-Assembly: Deterministic Assembly in 3D and Probabilistic Assembly in 2D.” In Proceedings of the 2011 Annual ACM-SIAM Symposium on Discrete Algorithms, 570–89. Proceedings. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973082.45.
Proceedings of the 2011 Annual ACM-SIAM Symposium on Discrete Algorithms