Computer Science > Discrete Mathematics
[Submitted on 14 May 2024 (v1), last revised 21 May 2024 (this version, v2)]
Title:Strict Self-Assembly of Discrete Self-Similar Fractal Shapes
View PDFAbstract:This paper gives a (polynomial time) algorithm to decide whether a given Discrete Self-Similar Fractal Shape can be assembled in the aTAM this http URL 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.
Submission history
From: Florent Becker [view email] [via CCSD proxy][v1] Tue, 14 May 2024 12:37:19 UTC (92 KB)
[v2] Tue, 21 May 2024 08:01:57 UTC (1,638 KB)
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