# Self-Assembly of Infinite Structures

@inproceedings{Patitz2009SelfAssemblyOI, title={Self-Assembly of Infinite Structures}, author={Matthew J. Patitz and Scott M. Summers}, booktitle={CSP}, year={2009} }

We review some recent results related to the self-assembly of infinite structures in the Tile Assembly Model. These results include impossibility results, as well as novel tile assembly systems in which shapes and patterns that represent various notions of computation self-assemble. Several open questions are also presented and motivated.

## References

SHOWING 1-10 OF 33 REFERENCES

### Self-assembly of decidable sets

- MathematicsNatural Computing
- 2010

The wedge construction is extended to prove the following result: if a set of natural numbers is decidable, then it and its complement’s canonical two-dimensional representation self-assemble, which leads to a novel characterization of decidable sets of naturalNumbers in terms of self-assembly.

### The program-size complexity of self-assembled squares (extended abstract)

- Computer ScienceSTOC '00
- 2000

A formal model of pseudocrystalline self-assembly, called the Tile Assembly Model, in which a tile may be added to the growing object when the total interaction strength with its neighbors exceeds a parameter Τ is studied, which finds a dramatic decrease in complexity.

### Complexity of Self-Assembled Shapes

- Materials ScienceSIAM J. Comput.
- 2004

It is shown that under the notion of a shape that is independent of scale this is indeed so: in the Tile Assembly Model, the minimal number of distinct tile types necessary to self-assemble an arbitrarily scaled shape can be bounded both above and below in terms of the shape's Kolmogorov complexity.

### Randomized Self-assembly for Approximate Shapes

- Materials ScienceICALP
- 2008

T tile self-assembly systems which assemble arbitrarily close approximations to target squares with arbitrarily high probability are designed, in contrast to previous work which has only considered deterministic assemblies of a single shape.

### Combinatorial optimization problems in self-assembly

- Computer ScienceSTOC '02
- 2002

Two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self- assembly proposed by Rothemund and Winfree are studied, and it is proved that the first problem is NP-complete in general, and polynomial time solvable on trees and squares.

### Self-assemblying Classes of Shapes with a Minimum Number of Tiles, and in Optimal Time

- Mathematics, Computer ScienceFSTTCS
- 2006

This paper constructs fixed finite tile systems that assemble into particular classes of shapes that are optimal for rectangles and squares and introduces the notion of parallel time, which is a good approximation of the classical asynchronous time.

### On the decidability of self-assembly of infinite ribbons

- MathematicsThe 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings.
- 2002

It is proved that it is undecidable whether an arbitrary finite set of tiles with glues can be used to assemble an infinite ribbon, a non-self-crossing sequence of tiles on the plane, in which successive tiles are adjacent along an edge, and abutting edges of consecutive tiles have matching glues.

### Simulations of Computing by Self-Assembly

- Materials Science
- 1998

Winfree (1996) proposed a Turing-universal model of DNA self-assembly. In this abstract model, DNA double-crossover molecules self-assemble to form an algorithmically-patterned two-dimensional…

### Reducing tile complexity for self-assembly through temperature programming

- Computer ScienceSODA '06
- 2006

This work suggests that temperature change can constitute a natural, dynamic method for providing input to self-assembly systems that is potentially superior to the current technique of designing large tile sets with specific inputs hardwired into the tileset.

### Computability and Complexity in Self-assembly

- MathematicsTheory of Computing Systems
- 2010

The impact of geometry on computability and complexity in Winfree’s model of nanoscale self-assembly in the two-dimensional tile assembly model in the discrete Euclidean plane is explored.