Turing Universality of Step-Wise and Stage Assembly at Temperature 1
In this paper, we investigate the computational power of two variants of Winfree’s abstract Tile Assembly Model  at temperature 1: the Stage Tile Assembly Model and the Step-wise Tile Assembly Model. In the Stage Tile Assembly Model, the intermediate assemblies are assembled in several “bins” and they can be mixed in prescribed order and attach together to form more complex structures. The Step-wise Tile Assembly Model is a simplified model of stage assembly in which only one bin is used and assembly happens by attaching tiles one by one to the growing structure.
An interesting and still open question is whether the abstract Tile Assembly Model at temperature 1 is Turing Universal, i.e., it can simulate a Turing machine. It is known that various slight modifications of the model are indeed Turing Universal. Namely, deterministic self-assembly in 3D and probabilistic self-assembly in 2D at temperature 1  and self-assembly model at temperature 1 with a single negative glue  are known to be able to simulate a Turing machine. In this paper we show that the Step-wise Tile Assembly Model and the Stage Tile Assembly Model are also Turing Universal at temperature 1.
KeywordsTuring Machine Assembly Model South Side Tile Type Tile Assembly Model
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