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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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Nearly Constant Tile Complexity for any Shape in Two-Handed Tile Assembly
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.
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- Award ID(s):
- 1817602
- PAR ID:
- 10099123
- Date Published:
- Journal Name:
- Algorithmica
- ISSN:
- 1432-0541
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Many models of self-assembly have been shown to be capable of performing computation. Tile Automata was recently introduced combining features of both Celluar Automata and the 2-Handed Model of self-assembly both capable of universal computation. In this work we study the complexity of Tile Automata utilizing features inherited from the two models mentioned above. We first present a construction for simulating Turing Machines that performs both covert and fuel efficient computation. We then explore the capabilities of limited Tile Automata systems such as 1-Dimensional systems (all assemblies are of height $$1$$) and freezing Systems (tiles may not repeat states). Using these results we provide a connection between the problem of finding the largest uniquely producible assembly using $$n$$ states and the busy beaver problem for non-freezing systems and provide a freezing system capable of uniquely assembling an assembly whose length is exponential in the number of states of the system. We finish by exploring the complexity of the Unique Assembly Verification problem in Tile Automata with different limitations such as freezing and systems without the power of detachment.more » « less
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Self-assembly is the process by which a system of particles randomly agitate and combine, through local interactions, to form larger complex structures. In this work, we fuse a particular well-studied generalization of tile assembly (the 2-Handed or Hierarchical Tile Assembly Model) with concepts from cellular automata such as states and state transitions characterized by neighboring states. This allows for a simplification of the concepts from active self-assembly, and gives us machinery to relate the disparate existing models. We show that this model, coined Tile Automata, is invariant with respect to freezing and non-freezing transition rules via a simulation theorem showing that any non-freezing tile automata system can be simulated by a freezing one. Freezing tile automata systems restrict state transitions such that each tile may visit a state only once, i.e., a tile may undergo only a finite number of transitions. We conjecture that this result can be used to show that the Signal-passing Tile Assembly Model is also invariant to this constraint via a series of simulation results between that model and the Tile Automata model. Further, we conjecture that this model can be used to consolidate the several oft-studied models of self-assembly wherein assemblies may break apart, such as the Signal-passing Tile Assembly Model, the negative-glue 2-Handed Tile Assembly Model, and the Size-Dependent Tile Assembly Model. Lastly, the Tile Automata model may prove useful in combining results in cellular automata with self-assembly.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/https://doi.org/10.1007/s00453-019-00573-w
