This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM model.In the positive case, the construction relies on a Self-Assembling System in the aTAM which strictly assembles a particular self-similar fractal shape, namely a variant $K^\infty$ of the Sierpinski Carpet. We prove that the aTAM we propose is correct through a novel device, \emph{self-describing circuits} which are generally useful for rigorous yet readable proofs of the behaviour of aTAMs.We then discuss which self-similar fractals can or cannot be strictly self-assembled in the aTAM. It turns out that the ability of iterates of the generator to pass information is crucial: either this \emph{bandwidth} is eventually sufficient in both cardinal directions and $K^\infty$ appears within the fractal pattern after some finite number of iterations, or that bandwidth remains ever insufficient in one direction and any aTAM trying to self-assemble the shape will end up either bounded with an ultimately periodic pattern covering arbitrarily large squares. This is established thanks to a new characterization of the productions of systems whose productions have a uniformly bounded treewidth.

翻译：本文提出一种（多项式时间）算法，用于判定给定离散自相似分形结构能否在aTAM模型中完成组装。在肯定性情形下，该构造依赖于aTAM中的自组装系统，该系统能严格组装特定自相似分形结构——即Sierpinski地毯的变体$K^\infty$。我们通过新型工具\emph{自描述电路}证明了所提aTAM的正确性，该工具通常可用于对aTAM行为进行严谨且可读性强的验证。随后我们讨论了哪些自相似分形能在aTAM中实现严格自组装。结果表明，生成元迭代传递信息的能力至关重要：若该\emph{带宽}最终在基数方向上均充足，则$K^\infty$会在有限次迭代后出现在分形图案中；反之，若带宽在某一方向始终不足，则任何尝试自组装该结构的aTAM最终要么受限，要么产生覆盖任意大正方形的最终周期性图案。这一结论得益于对具有一致有界树宽的系统产出的新表征。