  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| log^2 |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.
