Strict Self-Assembly of Discrete Self-Similar Fractal Shapes
<|reference_start|>Strict Self-Assembly of Discrete Self-Similar Fractal Shapes: 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.<|reference_end|>
@article{becker2024strict,
    title={Strict Self-Assembly of Discrete Self-Similar Fractal Shapes},
    author={Florent Becker (LIFO)},
    journal={arXiv preprint arXiv:2405.08548},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.08548},
    primaryClass={cs.DM}
}