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Sequential non-determinism in tile self-assembly: a general framework and an application to efficient temperature-1 self-assembly of squares

David Furcy, Scott M. Summers

TL;DR

This work introduces sequential non-determinism as a general framework to analyze probabilistic self-assembly in 2DPTAM at temperature 1, showing that the probability of producing a unique correct terminal assembly factors as a finite product over essential points of competition. It then provides a constructive scheme to assemble any $N\times N$ square with probability at least $1-δ$ using only $O(\log N + \log 1/δ)$ tile types, improving prior bounds. The core method combines a gadget-based zigzag counter within a P-shaped domain, a rigorous competition/winner framework, and a pruning/Split-tree technique to reduce global probability analysis to local competition probabilities. The results have implications for scalable, reliable nano-self-assembly and suggest avenues for extending the approach to other probabilistic TAS settings and complexity regimes.

Abstract

In this paper, we work in a 2D version of the probabilistic variant of Winfree's abstract Tile Assembly Model defined by Chandran, Gopalkrishnan and Reif (SICOMP 2012) in which attaching tiles are sampled uniformly with replacement. First, we develop a framework called ``sequential non-determinism'' for analyzing the probabilistic correctness of a non-deterministic, temperature-1 tile assembly system (TAS) in which most (but not all) tile attachments are deterministic and the non-deterministic attachments always occur in a specific order. Our main sequential non-determinism result equates the probabilistic correctness of such a TAS to a finite product of probabilities, each of which 1) corresponds to the probability of the correct type of tile attaching at a point where it is possible for two different types to attach, and 2) ignores all other tile attachments that do not affect the non-deterministic attachment. We then show that sequential non-determinism allows for efficient and geometrically expressive self-assembly. To that end, we constructively prove that for any positive integer $N$ and any real $δ\in (0,1)$, there exists a TAS that self-assembles into an $N \times N$ square with probability at least $1 - δ$ using only $O\left( \log N + \log \frac{1}δ \right)$ types of tiles. Our bound improves upon the previous state-of-the-art bound for this problem by Cook, Fu and Schweller (SODA 2011).

Sequential non-determinism in tile self-assembly: a general framework and an application to efficient temperature-1 self-assembly of squares

TL;DR

This work introduces sequential non-determinism as a general framework to analyze probabilistic self-assembly in 2DPTAM at temperature 1, showing that the probability of producing a unique correct terminal assembly factors as a finite product over essential points of competition. It then provides a constructive scheme to assemble any square with probability at least using only tile types, improving prior bounds. The core method combines a gadget-based zigzag counter within a P-shaped domain, a rigorous competition/winner framework, and a pruning/Split-tree technique to reduce global probability analysis to local competition probabilities. The results have implications for scalable, reliable nano-self-assembly and suggest avenues for extending the approach to other probabilistic TAS settings and complexity regimes.

Abstract

In this paper, we work in a 2D version of the probabilistic variant of Winfree's abstract Tile Assembly Model defined by Chandran, Gopalkrishnan and Reif (SICOMP 2012) in which attaching tiles are sampled uniformly with replacement. First, we develop a framework called ``sequential non-determinism'' for analyzing the probabilistic correctness of a non-deterministic, temperature-1 tile assembly system (TAS) in which most (but not all) tile attachments are deterministic and the non-deterministic attachments always occur in a specific order. Our main sequential non-determinism result equates the probabilistic correctness of such a TAS to a finite product of probabilities, each of which 1) corresponds to the probability of the correct type of tile attaching at a point where it is possible for two different types to attach, and 2) ignores all other tile attachments that do not affect the non-deterministic attachment. We then show that sequential non-determinism allows for efficient and geometrically expressive self-assembly. To that end, we constructively prove that for any positive integer and any real , there exists a TAS that self-assembles into an square with probability at least using only types of tiles. Our bound improves upon the previous state-of-the-art bound for this problem by Cook, Fu and Schweller (SODA 2011).
Paper Structure (30 sections, 44 theorems, 79 equations, 46 figures, 2 tables, 2 algorithms)

This paper contains 30 sections, 44 theorems, 79 equations, 46 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Formal description of the abstract Tile Assembly Model
  4. The probabilistic aTAM or 2DPTAM
  5. Defining sequential non-determinism
  6. Proof of Theorem 1
  7. Efficient, sequentially non-deterministic self-assembly of an $N \times N$ square with high probability
  8. Overview of the construction
  9. Statement and proof sketch of the main lemma
  10. First two steps in the proof of the main lemma
  11. Description of producible assemblies that are not $w$-correct
  12. Last four steps in the proof of the main lemma
  13. $\bullet$ Condition \ref{['def:snd-1']} of Definition \ref{['def:seq-non-deterministic']}.
  14. (a) Points covered by gadgets that self-assemble before a tile is placed at the first POC.
  15. (b) All POCs associated with $b_1$ in increment steps.
  16. ...and 15 more sections

Key Result

Lemma 1

If $\vec{y} \in \mathbb{Z}^2$ is any POC in $\mathcal{T}$ with corresponding starting point $\vec{x} \in \mathbb{Z}^2$ and corresponding competing paths $\pi$ and $\pi'$, then $\pi$ and $\pi'$ are the only two paths competing for $\vec{y}$, from $\vec{x}$ in $\mathcal{T}$.

Figures (46)

  • Figure 1: Example of a TAS $\mathcal{T}$ with the tile set shown in (a), in which the temperature is 1 and s is the seed tile. Sub-figures (b) and (c) show assembly sequences of $\mathcal{T}$ that result in two different terminal assemblies.
  • Figure 2: $\mathcal{M}_{\mathcal{T}}$ corresponding to the TAS in Figure \ref{['fig:intro-example1']}. Each path from the root to a leaf node corresponds to an assembly sequence of $\mathcal{T}$. For the sake of simplification, the label of each node is merely the type of the last tile to attach in the assembly sequence to which that node corresponds, rather than the full assembly sequence, which can easily be reconstructed by following (and then reversing) the path back up to the root. Each leaf node corresponds to an assembly sequence that results in a terminal assembly. The number under each leaf node is the probability of the corresponding sequence being produced by $\mathcal{T}$.
  • Figure 3: Example of another TAS $\mathcal{T}$ with the tile set shown in (a), in which the temperature is 1 and s is the seed tile. Sub-figures (b) and (c) show assembly sequences of $\mathcal{T}$ that result in two different terminal assemblies.
  • Figure 4: Markov chain $\mathcal{M}_{\mathcal{T}}$, where $\mathcal{T}$ is the TAS from Figure \ref{['fig:intro-example2']}. Note that this tree is not quite complete because it only contains the assembly sequences that result in the terminal assembly that contains the 4 tile. All of the sub-trees whose root is labeled with a $3$ and does not appear on the lowest level of the tree have been pruned below their root node (e.g., the leftmost node in the figure).
  • Figure 5: The TAS $\mathcal{T}=(T,\sigma,1)$ we are using as a running example throughout this and the next section, with $|\mathcal{A}_{\Box}[\mathcal{T}]|$ = 4 (a) Depiction of the tile set $T=\{s,1,2,3,4,5,6,$$7,8,9,A,B\}$, in which $s$ is the seed tile and the other tiles are labeled with either an uppercase letter or a positive integer; all glue strengths are 1; the different glue labels are depicted with shades of gray or tiling patterns (b) Depiction (in the middle) of the points in $\mathbb{Z}^2$ at which tiles in $T$ may attach; $\vec{s}$ is the point where the seed tile $s$ is always placed in this example, i.e., $\sigma=\{(\vec{s},s)\}$; the tiles in $T$ are shown either above or below the points, with an arrow indicating the unique point at which the tile may attach; $\vec{y}_1$ and $\vec{y}_2$ are the only points where more than one tile may attach (c) $\alpha_{3,7}$ is the assembly in $\mathcal{A}_{\Box}[\mathcal{T}]$ containing the two tiles $(\vec{y}_1,3)$ and $(\vec{y}_2,7)$ (d) $\alpha_{3,9}$ is the assembly in $\mathcal{A}_{\Box}[\mathcal{T}]$ containing the two tiles $(\vec{y}_1,3)$ and $(\vec{y}_2,9)$ (e) $\alpha_{4,7}$ is the assembly in $\mathcal{A}_{\Box}[\mathcal{T}]$ containing the two tiles $(\vec{y}_1,4)$ and $(\vec{y}_2,7)$ (f) $\alpha_{4,9}$, is the assembly in $\mathcal{A}_{\Box}[\mathcal{T}]$ containing the two tiles $(\vec{y}_1,4)$ and $(\vec{y}_2,9)$
  • ...and 41 more figures

Theorems & Definitions (106)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 2
  • proof
  • ...and 96 more