Strict Self-Assembly of Discrete Self-Similar Fractal Shapes

TL;DR

This paper presents a polynomial-time algorithm to determine if a discrete self-similar fractal shape can be strictly self-assembled in the aTAM model, introducing new methods and characterizations for self-assembly capabilities.

Contribution

It provides the first polynomial-time decision algorithm for strict self-assembly of self-similar fractals in the aTAM and introduces self-describing circuits for rigorous proofs.

Findings

The aTAM can strictly assemble a variant of the Sierpinski Carpet.

The ability to pass information in the fractal generator determines self-assembly feasibility.

Bounded treewidth productions limit the complexity of self-assembled patterns.

Abstract

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.