Complexity of Self‐Assembled Shapes

Abstract

Abstract.The connection between self-assembly and computation suggests that a shape can be considered the output of a self-assembly "program," a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self-assembly program that builds a shape and the shape's descriptional (Kolmogorov) complexity should be related. We show that when using a 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 a shape, at some scale, can be bounded both above and below in terms of the shape's Kolmogorov complexity. As part of the proof, we develop a universal constructor for this model of self-assembly that can execute an arbitrary Turing machine program specifying how to grow a shape. Our result implies, somewhat counterintuitively, that self-assembly of a scaled-up version of a shape often requires fewer tile types. Furthermore, the independence of scale in self-assembly theory appears to play the same crucial role as the independence of running time in the theory of computability. This leads to an elegant formulation of languages of shapes generated by self-assembly. Considering functions from bit strings to shapes, we show that the running-time complexity, with respect to Turing machines, is polynomially equivalent to the scale complexity of the same function implemented via self-assembly by a finite set of tile types. Our results also hold for shapes defined by Wang tiling-where there is no sense of a self-assembly process-except that here time complexity must be measured with respect to nondeterministic Turing machines.

Reducing Facet Nucleation during Algorithmic Self-Assembly

“…In theory, this simple mechanism is sufficient to produce arbitrarily complex shapes and patterns. [9][10][11][12] In practice, however, self-assembly is stochastic, and unfavorable attachments of tiles with one or more incorrect or absent inputs also occur. 9,19 While tiles that attach unfavorably usually fall off quickly, occasionally such a tile is locked in by the subsequent favorable attachment of an adjacent tile, an event we will call an insufficient attachment.…”

Computing by polymerase chain reaction

“…Since then, a great deal of research has been carried on and many technical and theoretical achievements have been reached in DNA computing [12,[3][4][5]. Recently, new research perspectives have emerged that widen the possibilities of this field, among them: DNA self-assembly [21,22,15], DNA automata [2], and tools for DNA and RNA manipulation inspired by algorithmic analyses [17,11].…”

Strict self-assembly of discrete Sierpinski triangles

“…Our tile set that self-assembles T consists of 51 tile types, and we prove that our construction is correct using the method of local determinism due to Soloveichik and Winfree [15].…”

“…Winfree [15]. Local determinism is a method for proving the correctness of tile assembly systems in which "irregular" shapes self-assemble.…”

“…The TAM was first introduced by Winfree [18] and further developed by Rothemund and Winfree [13,12]. Similar developments of the TAM can be found in [1,11,15,3]. In the TAM, molecules are represented by tile types (i.e., squares that cannot be rotated) that each have a "glue strength" and a "color" on all four sides.…”