Deterministic 2-Dimensional Temperature-1 Tile Assembly Systems Cannot Compute
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We consider non cooperative binding in so called `temperature 1', in deterministic (here called confluent) tile self-assembly systems (1-TAS) and prove the standing conjecture that such systems do not have universal computational power. We call a TAS whose maximal assemblies contain at least one ultimately periodic assembly path para-periodic. We observe that a confluent 1-TAS has at most one maximal producible assembly, α_max, that can be considered a union of path assemblies, and we show that such a system is always para-periodic. This result is obtained through a superposition and a combination of two paths that produce a new path with desired properties, a technique that we call co-grow of two paths. Moreover we provide a characterization of an α_max of a confluent 1-TAS as one of two possible cases, so called, a grid or a disjoint union of combs. To a given α_max we can associate a finite labeled graph, called quipu, such that the union of all labels of paths in the quipu equals α_max, therefore giving a finite description for α_max. This finite description implies that α_max is a union of semi-affine subsets of Z^2 and since such a finite description can be algorithmicly generated from any 1-TAS, 1-TAS cannot have universal computational power.
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