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In the abstract Tile Assembly Model, self-assembling systems consisting of tiles of different colors can form structures on which colored patterns are “painted.” We explore the complexity, in terms of the numbers of unique tile types required, of assembling various patterns. We first demonstrate how to efficiently self-assemble a set of simple patterns, then show tight bounds on the tile type complexity of self-assembling multi-colored patterns on the surfaces of square assemblies. Finally, we demonstrate an exponential gap in tile type complexity of self-assembling an infinite series of patterns between systems restricted to one plane versus those allowed two planes. This paper is an expansion over a version in the proceedings of the 21st International Conference on Unconventional Computation and Natural Computation (UCNC 2024), with several improved results and full details of all proofs.
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Self-Assembly of Any Shape with Constant Tile Types using High Temperature
Inspired by nature and motivated by a lack of top-down tools for precise nanoscale manufacture, self-assembly is a bottom-up process where simple, unorganized components autonomously combine to form larger more complex structures. Such systems hide rich algorithmic properties - notably, Turing universality - and a self-assembly system can be seen as both the object to be manufactured as well as the machine controlling the manufacturing process. Thus, a benchmark problem in self-assembly is the unique assembly of shapes: to design a set of simple agents which, based on aggregation rules and random movement, self-assemble into a particular shape and nothing else. We use a popular model of self-assembly, the 2-handed or hierarchical tile assembly model, and allow the existence of repulsive forces, which is a well-studied variant. The technique utilizes a finely-tuned temperature (the minimum required affinity required for aggregation of separate complexes). We show that calibrating the temperature and the strength of the aggregation between the tiles, one can encode the shape to be assembled without increasing the number of distinct tile types. Precisely, we show one tile set for which the following holds: for any finite connected shape S, there exists a setting of binding strengths between tiles and a temperature under which the system uniquely assembles S at some scale factor. Our tile system only uses one repulsive glue type and the system is growth-only (it produces no unstable assemblies). The best previous unique shape assembly results in tile assembly models use O(K(S)/(log K(S))) distinct tile types, where K(S) is the Kolmogorov (descriptional) complexity of the shape S.
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- Award ID(s):
- 1817602
- PAR ID:
- 10099116
- Date Published:
- Journal Name:
- Leibniz international proceedings in informatics
- Volume:
- 112
- ISSN:
- 1868-8969
- Page Range / eLocation ID:
- 14:1--14:14
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Tile self-assembly is a well-studied theoretical model of geometric computation based on nanoscale DNA-based molecular systems. Here, we study the two-handed tile self-assembly model or 2HAM at general temperatures, in contrast with prior study limited to small constant temperatures, leading to surprising results. We obtain constructions at larger (i.e., hotter) temperatures that disprove prior conjectures and break well-known bounds for low-temperature systems via new methods of temperature-encoded information. In particular, for all n∈N , we assemble n×n squares using O(2log∗n) tile types, thus breaking the well-known information theoretic lower bound of Rothemund and Winfree. Using this construction, we then show how to use the temperature to encode general shapes and construct them at scale with O(2log∗K) tiles, where K denotes the Kolmogorov complexity of the target shape. Following, we refute a long-held conjecture by showing how to use temperature to construct n×O(1) rectangles using only O(logn/loglogn) tile types. We also give two small systems to generate nanorulers of varying length based solely on varying the system temperature. These results constitute the first real demonstration of the power of high temperature systems for tile assembly in the 2HAM. This leads to several directions for future explorations which we discuss in the conclusion.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript
- Conference Paper:
- https://doi.org/10.4230/LIPIcs.ESA.2018.14
