Self-assembly of discrete self-similar fractals (original) (raw)
Abstract
In this paper, we search for theoretical limitations of the Tile Assembly Model (TAM), along with techniques to work around such limitations. Specifically, we investigate the self-assembly of fractal shapes in the TAM. We prove that no self-similar fractal weakly self-assembles at temperature 1 in a locally deterministic tile assembly system, and that certain kinds of discrete self-similar fractals do not strictly self-assemble at any temperature. Additionally, we extend the fiber construction of Lathrop et al. (2009) to show that any discrete self-similar fractal belonging to a particular class of “nice” discrete self-similar fractals has a fibered version that strictly self-assembles in the TAM.
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References
- Adleman LM, Cheng Q, Goel A, Huang M-D, Kempe D, de Espanés PM, Rothemund PWK (2002) Combinatorial optimization problems in self-assembly. In: Proceedings of the thirty-fourth annual ACM symposium on theory of computing, pp 23–32
- Adleman LM, Kari J, Kari L, Reishus D (2002) On the decidability of self-assembly of infinite ribbons. In: Proceedings of the 43rd annual IEEE symposium on foundations of computer science, pp 530–537
- Aggarwal G, Goldwasser MH, Kau M-Y, Schweller RT (2004) Complexities for generalized models of self-assembly. In: Proceedings of ACM-SIAM symposium on discrete algorithms
- Doty D, Gu X, Lutz JH, Mayordomo E, Moser P (2005) Zeta-dimension. In: Proceedings of the thirtieth international symposium on mathematical foundations of computer science, Springer-Verlag, NY, pp 283–294
- Graham RL, Knuth DE, Patashnik O (1994) Concrete mathematics. Addison-Wesley, Reading
- Kao M-Y, Schweller RT (2007) Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th annual ACM-SIAM symposium on discrete algorithms (SODA 2006), Miami, Florida, Jan 2006, pp 571–580
- Kao M-Y, Schweller RT (2008) Randomized self-assembly for approximate shapes, International Colloqium on Automata, Languages, and Programming (ICALP) In: Aceto L, Damgård I, Goldberg LA, Halldórsson MM, Ingólfsdóttir A, Walukiewicz I (eds) Lecture notes in computer science, vol 5125. Springer, Berlin, pp 370–384
- Kautz SM, Lathrop JI (2009) Self-assembly of the Sierpinski carpet and related fractals. In: Proceedings of the fifteenth international meeting on DNA computing and molecular programming, Fayetteville, Arkansas, USA, 8–11 June 2009 (to appear)
- Lathrop JI, Lutz JH, Patitz Matthew J, Summers SM (2008) Computability and complexity in self-assembly. In: Proceedings of the fourth conference on computability in Europe Athens, Greece, 15–20 June 2008
- Lathrop JI, Lutz JH, Summers SM (2009) Strict self-assembly of discrete Sierpinski triangles. Theor Comp Sci 410:384–405
Article MATH MathSciNet Google Scholar - Majumder U, LaBean TH, and Reif JH (2007) Activatable tiles for compact error-resilient directional assembly. In: Thirteenth international meeting on DNA computing (DNA 13), Memphis, Tennessee, 4–8 June 2007
- Rothemund PWK (2001) Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California, December
- Rothemund PWK, Winfree E (2000) The program-size complexity of self-assembled squares (extended abstract), STOC ’00. In: Proceedings of the thirty-second annual ACM symposium on theory of computing, New York, USA, ACM, pp 459–468
- Seeman NC (1982) Nucleic-acid junctions and lattices. J Theor Biol 99:237–247
Article Google Scholar - Soloveichik D, Winfree E (2007) Complexity of self-assembled shapes. SIAM J Comp 36(6):1544–1569
Article MATH MathSciNet Google Scholar - Wang H (1961) Proving theorems by pattern recognition—II. Bell Syst Tech J XL(1):1–41
Google Scholar - Wang H (1963) Dominoes and the AEA case of the decision problem. In: Proceedings of the symposium on mathematical theory of automata, New York, 1962, Polytechnic Press of Polytechnic Institute of Brooklyn, Brooklyn, NY, pp 23–55
- Winfree E (1998) Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June
Acknowledgments
We thank David Doty, Jim Lathrop, Jack Lutz, and Aaron Sterling for useful discussions. We would especially like to thank an anonymous reviewer whose detailed comments have substantially improved the final version of this paper.
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Authors and Affiliations
- Department of Computer Science, Iowa State University, Ames, IA, 50011, USA
Matthew J. Patitz & Scott M. Summers
Authors
- Matthew J. Patitz
- Scott M. Summers
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Correspondence toMatthew J. Patitz.
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Preliminary version appeared in the Proceedings of The Fourteenth International Meeting on DNA Computing (DNA 14), Prague, Czech Republic, June 2–6, 2008. This author’s research was supported in part by NSF-IGERT Training Project in Computational Molecular Biology Grant number DGE-0504304.
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Patitz, M.J., Summers, S.M. Self-assembly of discrete self-similar fractals.Nat Comput 9, 135–172 (2010). https://doi.org/10.1007/s11047-009-9147-7
- Received: 01 December 2008
- Accepted: 19 July 2009
- Published: 19 August 2009
- Issue date: March 2010
- DOI: https://doi.org/10.1007/s11047-009-9147-7