Producibility in hierarchical self-assembly

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Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly A is producible. The algorithm can be optimized to use O(|A| log² |A|) time. Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, and Winslow showed that the problem of deciding if an assembly A is the unique producible terminal assembly of a tile system T can be solved in O(|A|² |T| + |A| |T|²) time for the special case of noncooperative "temperature 1" systems. It is shown that this can be improved to O(|A| |T| log |T|) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently -- i.e., if the positions that they share have the same tile type in each assembly -- then their union is also producible.