Advertisement

Exact Shapes and Turing Universality at Temperature 1 with a Single Negative Glue

  • Matthew J. Patitz
  • Robert T. Schweller
  • Scott M. Summers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6937)

Abstract

Is Winfree’s abstract Tile Assembly Model (aTAM) “powerful?” Well, if certain tiles are required to “cooperate” in order to be able to bind to a growing tile assembly (a.k.a., temperature 2 self-assembly), then Turing universal computation and the efficient self-assembly of N ×N squares is achievable in the aTAM (Rotemund and Winfree, STOC 2000). So yes, in a computational sense, the aTAM is quite powerful! However, if one completely removes this cooperativity condition (a.k.a., temperature 1 self-assembly), then the computational “power” of the aTAM (i.e., its ability to support Turing universal computation and the efficient self-assembly of N ×N squares) becomes unknown. On the plus side, the aTAM, at temperature 1, is not only Turing universal but also supports the efficient self-assembly N ×N squares if self-assembly is allowed to utilize three spatial dimensions (Fu, Schweller and Cook, SODA 2011). In this paper, we investigate the theoretical “power” of a seemingly simple, restrictive variant of Winfree’s aTAM in which (1) the absolute value of every glue strength is 1, (2) there is a single negative strength glue type and (3) unequal glues cannot interact (i.e., glue functions must be “diagonal”). We call this abstract model of self-assembly the restricted glue Tile Assembly Model (rgTAM). We achieve two positive results. First, we show that the tile complexity of uniquely producing an N ×N square in the rgTAM is O(logN). In our second result, we prove that the rgTAM is Turing universal.

Keywords

Binary String Tile Type Binary Counter Tile Assembly Model Green Tile 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adleman, L., Cheng, Q., Goel, A., Huang, M.-D.: Running time and program size for self-assembled squares. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 740–748. ACM, New York (2001)Google Scholar
  2. 2.
    Adleman, L.M., Kari, J., Kari, L., Reishus, D., Sosík, P.: The undecidability of the infinite ribbon problem: Implications for computing by self-assembly. SIAM Journal on Computing 38(6), 2356–2381 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barish, R.D., Schulman, R., Rothemund, P.W., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proceedings of the National Academy of Sciences 106(15), 6054–6059 (2009)CrossRefGoogle Scholar
  4. 4.
    Chen, H.-L., Schulman, R., Goel, A., Winfree, E.: Reducing facet nucleation during algorithmic self-assembly. Nano Letters 7(9), 2913–2919 (2007)CrossRefGoogle Scholar
  5. 5.
    Cook, M., Fu, Y., Schweller, R.: Temperature 1 self-assembly: Deterministic assembly in 3d and probabilistic assembly in 2d. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (2011)Google Scholar
  6. 6.
    Doty, D.: Randomized self-assembly for exact shapes. SIAM Journal on Computing 39(8), 3521–3552 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. In: Sakakibara, Y., Mi, Y. (eds.) DNA 16 2010. LNCS, vol. 6518, pp. 37–48. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. 8.
    Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, pp. 275–286 (2009)Google Scholar
  9. 9.
    Doty, D., Patitz, M.J., Reishus, D., Schweller, R.T., Summers, S.M.: Strong fault-tolerance for self-assembly with fuzzy temperature. In: Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010), pp. 417–426 (2010)Google Scholar
  10. 10.
    Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theoretical Computer Science 412, 145–158 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kao, M.-Y., Schweller, R.T.: 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, January 2006, pp. 571-580 (2007)Google Scholar
  12. 12.
    Kao, M.-Y., Schweller, R.T.: Randomized self-assembly for approximate shapes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 370–384. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  13. 13.
    Kinsella, J.M., Ivanisevic, A.: Enzymatic clipping of dna wires coated with magnetic nanoparticles. Journal of the American Chemical Society 127(10), 3276–3277 (2005)CrossRefGoogle Scholar
  14. 14.
    Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theoretical Computer Science 410, 384–405 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Luhrs, C.: Polyomino-safe dna self-assembly via block replacement. DNA, 112–126 (2008)Google Scholar
  16. 16.
    Majumder, U., Reif, J.: A framework for designing novel magnetic tiles capable of complex self-assemblies. In: Calude, C.S., Costa, J.F., Freund, R., Oswald, M., Rozenberg, G. (eds.) UC 2008. LNCS, vol. 5204, pp. 129–145. Springer, Heidelberg (2008)Google Scholar
  17. 17.
    Mao, C., Sun, W., Seeman, N.C.: Designed two-dimensional DNA holliday junction arrays visualized by atomic force microscopy. Journal of the American Chemical Society 121(23), 5437–5443 (1999)CrossRefGoogle Scholar
  18. 18.
    Reif, J., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. In: Carbone, A., Pierce, N.A. (eds.) DNA 2005. LNCS, vol. 3892, pp. 257–274. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Rickwood, D., Lund, V.: Attachment of dna and oligonucleotides to magnetic particles: methods and applications. Fresenius’ Journal of Analytical Chemistry 330, 330–330 (1988), doi:10.1007/BF00469247CrossRefGoogle Scholar
  20. 20.
    Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly, Ph.D. thesis, University of Southern California (December 2001)Google Scholar
  21. 21.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC 2000: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, Portland, Oregon, United States, pp. 459–468. ACM, New York (2000)CrossRefGoogle Scholar
  22. 22.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 2041–2053 (2004)CrossRefGoogle Scholar
  23. 23.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM Journal on Computing 36(6), 1544–1569 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Winfree, E.: Algorithmic self-assembly of DNA, Ph.D. thesis, California Institute of Technology (June 1998)Google Scholar
  25. 25.
    Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–544 (1998)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Matthew J. Patitz
    • 1
  • Robert T. Schweller
    • 1
  • Scott M. Summers
    • 2
  1. 1.Department of Computer ScienceUniversity of Texas–Pan AmericanEdinburgUSA
  2. 2.Department of Computer Science and Software EngineeringUniversity of WisconsinPlattevilleUSA

Personalised recommendations