Limitations of Self-assembly at Temperature One

  • David Doty
  • Matthew J. Patitz
  • Scott M. Summers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5877)

Abstract

We prove that if a set X ⊆ ℤ2 weakly self-assembles at temperature 1 in a deterministic (Winfree) tile assembly system satisfying a natural condition known as pumpability, then X is a finite union of semi-doubly periodic sets. This shows that only the most simple of infinite shapes and patterns can be constructed using pumpable temperature 1 tile assembly systems, and gives evidence for the thesis that temperature 2 or higher is required to carry out general-purpose computation in a tile assembly system. Finally, we show that general-purpose computation is possible at temperature 1 if negative glue strengths are allowed in the tile assembly model.

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References

  1. 1.
    Adleman, L.M., Kari, J., Kari, L., Reishus, D.: 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 (2002)Google Scholar
  2. 2.
    Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theoretical Computer Science 410, 384–405 (2009)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Patitz, M.J., Summers, S.M.: Self-assembly of decidable sets. In: Calude, C.S., Costa, J.F., Freund, R., Oswald, M., Rozenberg, G. (eds.) UC 2008. LNCS, vol. 5204, pp. 206–219. Springer, Heidelberg (2008)Google Scholar
  4. 4.
    Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California (December 2001)Google Scholar
  5. 5.
    Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of dna sierpinski triangles. PLoS Biology 2(12) (2004)Google Scholar
  6. 6.
    Seeman, N.C.: Nucleic-acid junctions and lattices. Journal of Theoretical Biology 99, 237–247 (1982)CrossRefGoogle Scholar
  7. 7.
    Wang, H.: Proving theorems by pattern recognition – II. The Bell System Technical Journal XL(1), 1–41 (1961)Google Scholar
  8. 8.
    Wang, H.: Dominoes and the AEA case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata, New York, pp. 23–55. Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn (1962/1963)Google Scholar
  9. 9.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (June 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • David Doty
    • 1
  • Matthew J. Patitz
    • 1
  • Scott M. Summers
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesUSA

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