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Strict Self-Assembly of Discrete Self-Similar Fractal Shapes

Florent Becker

TL;DR

This work shows that strict self-assembly of discrete self-similar fractal shapes in the aTAM is decidable in polynomial time, and it provides a constructive positive instance by strictly self-assembling the Sierpinski Cacarpet variant $K^\infty$ using a self-describing, multi-layer circuit construction. The core techniques combine a Tree Pump Lemma to bound computation, and a framework of Self-Describing Embedded Circuits with Locally Deterministic Patterns to certify correct, readable proofs of assembly behavior. The results identify a sharp dichotomy for generators: those whose iterates sustain bandwidth in both directions permit strict assembly, while otherwise assemblies become bounded or ultimately periodic; a polynomial-time procedure decides which regime applies. The combination of algebraic substitutions, circuit-level self-description, and a rigorous geometric-bandwidth analysis advances understanding of the limits and possibilities of strict self-assembly in aTAM, with potential implications for HDL-like verification of tile-based constructions.

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.

Strict Self-Assembly of Discrete Self-Similar Fractal Shapes

TL;DR

This work shows that strict self-assembly of discrete self-similar fractal shapes in the aTAM is decidable in polynomial time, and it provides a constructive positive instance by strictly self-assembling the Sierpinski Cacarpet variant using a self-describing, multi-layer circuit construction. The core techniques combine a Tree Pump Lemma to bound computation, and a framework of Self-Describing Embedded Circuits with Locally Deterministic Patterns to certify correct, readable proofs of assembly behavior. The results identify a sharp dichotomy for generators: those whose iterates sustain bandwidth in both directions permit strict assembly, while otherwise assemblies become bounded or ultimately periodic; a polynomial-time procedure decides which regime applies. The combination of algebraic substitutions, circuit-level self-description, and a rigorous geometric-bandwidth analysis advances understanding of the limits and possibilities of strict self-assembly in aTAM, with potential implications for HDL-like verification of tile-based constructions.

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 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 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.
Paper Structure (24 sections, 40 theorems, 32 equations, 18 figures)

This paper contains 24 sections, 40 theorems, 32 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Definitions and statement of the main results
  3. Notations
  4. Self-assembly and the Abstract Tile-Assembly Model
  5. Definitions
  6. The Tree Pump Lemma
  7. Strict versus Weak Assembly of shapes
  8. Discrete Self-Similar Fractal Shapes
  9. The Sierpinski Cacarpet can be Strictly Self-Assembled
  10. Abstract models: Self-Describing Embedded Circuits and Locally Deterministic Patterns
  11. Embedded circuits
  12. Self-description
  13. Locally deterministic patterns
  14. A Self-Describing Embedded Circuit for the Sierpinski Cacarpet
  15. The wiring layer
  16. ...and 9 more sections

Key Result

Lemma 7

For any aTAM system $\mathcal{S}$ with $\mathcal{S}{\cdot}\operatorname{seed}$ finite and connected, define the following sets of assemblies: Then, there is a function $F: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ such that for aTAM system $\mathcal{S}$ with $n$ tiles and a 1-tile seed, integer $m$ and unit vector $\vec{d}$ of $\mathbb{R}^2$,

Figures (18)

  • Figure 1: The three sets defined by \ref{['lem:tree_pump']}: $C_{m}[\mathcal{S}]$ is the assemblies which encircle an $m \times m$ square, $B_{F(n, m), \vec{d}}[\mathcal{S}]$ is the set of assemblies which do not reach further than $F(n,m)$ in direction $\vec{d}$, and $P_{\vec{d}}[\mathcal{S}]$ is the assemblies which contain a periodic path with period $\vec{p}$ such that $\vec{p}\cdot\vec{d} > 0$.
  • Figure 2: A colored substitution on a 3-colored alphabet $\Sigma = \{ , , \}$ is defined from a shape $G = $ and a coloring of $G$ for each color of $\Sigma$. It can then be applied to any colored shape (right).
  • Figure 3: The substitution $\kappa$ of the Sierpinski's Cacarpet.
  • Figure 4: The graphical representation of three wirings: $w_1 = \{ {} \operatorname{Inputs} = (W), {} \operatorname{Outputs} = \{ N, S, E \}, {} \operatorname{output-num} = \{ N \to 1, E \to 0, S \to 1 \}, {} \operatorname{outwires} = \{ Se, W, wN \} \}$, $w_2 = \{ {} \operatorname{Inputs} = (W, S), {} \operatorname{Outputs} = \{N\}, {} \operatorname{output-num} = \{ N \to 0 \}, {} \operatorname{outwires} = \{ Sn \} \}$ and $w_3 = \{ {} \operatorname{Inputs} = (W, E), {} \operatorname{Outputs} = \{ S \}, {} \operatorname{output-num} = \{ S \to 0 \}, {} \operatorname{outwires} = \{ wN \} \}$. Neighboring input wires bend to come together into the wiring. Double arrows indicate a pair of opposite input arrows. The output wire in direction $d$ bends to mirror $w{\cdot}\operatorname{outwires} (d)$.
  • Figure 5: A multiplier circuit $M_2^1$ and the definition of the functions of its gates. The alphabet is the set of digits $\mathbb{Z} / 10 \mathbb{Z}$. The values in the grey squares are an example of evaluation: $32 \times 7 = 224$
  • ...and 13 more figures

Theorems & Definitions (82)

  • Definition 1: Wang Tile
  • Definition 2: Assembly
  • Definition 3: aTAM
  • Definition 4: Binding
  • Definition 5: Stable assembly
  • Definition 6: Attachment, Assembly Sequence
  • Definition 7: Production, Terminal Production
  • Lemma 7: Tree Pump
  • Definition 8: Strict Assembly
  • Definition 9: Weak Assembly
  • ...and 72 more