Complexity of Self-assembled Shapes

(Extended Abstract)
  • David Soloveichik
  • Erik Winfree
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3384)

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 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. As part of the proof of the main result, we sketch a general method for converting a program outputting a shape as a list of locations into a set of tile types that self-assembles into a scaled up version of that shape. Our result implies, somewhat counter-intuitively, that self-assembly of a scaled up version of a shape often requires fewer tile types, and suggests that the independence of scale in self-assembly theory plays the same crucial role as the independence of running time in the theory of computability.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • David Soloveichik
    • 1
  • Erik Winfree
    • 1
  1. 1.California Institute of TechnologyPasadenaUSA

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