Complexity of Self-assembled Shapes
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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- 1.Adleman, L., Cheng, Q., Goel, A., Huang, M.-D., Kempe, D., de Espanes, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proc. of STOC (2002)Google Scholar
- 2.Adleman, L.M.: Toward a mathematical theory of self-assembly (extended abstract). Technical report, University of Southern California (1999)Google Scholar
- 3.Adleman, L.M., Cheng, Q., Goel, A., Huang, M.-D.A.: Running time and program size for self-assembled squares. In: ACM Symposium on Theory of Computing, pp. 740–748 (2001)Google Scholar
- 4.Aggarwal, G., Goldwasser, M., Kao, M., Schweller, R.T.: Complexities for generalized models of self-assembly. In: Symposium on Discrete Algorithms (2004)Google Scholar
- 5.Cook, M., Rothemund, P.W.K., Winfree, E.: Self-assembled circuit patterns. In: DNA Based Computers 9, pp. 91–107 (2004)Google Scholar
- 6.Rothemund, P.W.K.: Theory and Experiments in Algorithmic Self-Assembly. PhD thesis, University of Southern California, Los Angeles (2001)Google Scholar
- 7.Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: ACM Symposium on Theory of Computing, pp. 459–468 (2000)Google Scholar
- 13.Wang, H.: Proving theorems by pattern recognition. II. Bell Systems Technical Journal 40, 1–42 (1961)Google Scholar
- 14.Winfree, E.: Simulations of computing by self-assembly, Caltech CS TR, 22 (1998)Google Scholar
- 15.Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, Pasadena (1998)Google Scholar