Self-assembly of shapes at constant scale using repulsive forces

Abstract

The algorithmic self-assembly of shapes has been considered in several models of self-assembly. For the problem of shape construction, we consider an extended version of the two-handed tile assembly model, which contains positive (attractive) and negative (repulsive) interactions. As a result, portions of an assembly can become unstable and detach. In this model, we utilize fuel-efficient computation to perform Turing machine simulations for the construction of the shape. In this paper, we show how an arbitrary shape can be constructed using an asymptotically optimal number of distinct tile types (based on the shape’s Kolmogorov complexity). We achieve this at O(1) scale factor in this straightforward model, whereas all previous results with sublinear scale factors utilize powerful self-assembly models containing features such as staging, tile deletion, chemical reaction networks, and tile activation/deactivation. Furthermore, the computation and construction in our result only creates constant-size garbage assemblies as a byproduct of assembling the shape.

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Notes

  1. 1.

    Note that only matching glues of the same type contribute a non-zero weight, whereas non-equal glues always contribute zero weight to the bond graph. Relaxing this restriction has been considered as well Cheng et al. (2005).

  2. 2.

    The strongest detaching force used in our construction is a \(\tau\) strength detachment, and since the internal bonds of our gadgets are meant to withstand even the strongest repulsive force, it follows that those bonds must be of strength at least \(2\tau\).

  3. 3.

    The formal theorem statement of Schweller and Sherman (2013) cites the product of the states and symbols of the Turing machine as the tile type cost. However, the actual cost is the number of transition rules, which is upper bounded by this product.

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Funding

This research was supported in part by National Science Foundation Grants CCF-1555626 and CCF-1817602.

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Correspondence to Austin Luchsinger.

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Luchsinger, A., Schweller, R. & Wylie, T. Self-assembly of shapes at constant scale using repulsive forces. Nat Comput 18, 93–105 (2019). https://doi.org/10.1007/s11047-018-9707-9

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Keywords

  • Self-assembly
  • Kolmogorov complexity
  • Scaled shapes