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Non-cooperatively Assembling Large Structures

Conference paper
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 11648)

Abstract

Algorithmic self-assembly is the study of the local, distributed, asynchronous algorithms ran by molecules to self-organise, in particular during crystal growth. The general cooperative model, also called “temperature 2”, uses synchronisation to simulate Turing machines, build shapes using the smallest possible amount of tile types, and other algorithmic tasks. However, in the non-cooperative (“temperature 1”) model, the growth process is entirely asynchronous, and mostly relies on geometry. Even though the model looks like a generalisation of finite automata to two dimensions, its 3D generalisation is capable of performing arbitrary (Turing) computation [SODA 2011], and of universal simulations [SODA 2014], whereby a single 3D non-cooperative tileset can simulate the dynamics of all possible 3D non-cooperative systems, up to a constant scaling factor.

However, the original 2D non-cooperative model is not capable of universal simulations [STOC 2017], and the question of its computational power is still widely open and it is conjectured to be weaker than “temperature 2” or its 3D counterpart. Here, we show an unexpected result, namely that this model can reliably grow assemblies of diameter \(\varTheta (n \log n)\) with only n tile types, which is the first asymptotically efficient positive construction.

Keywords

Self-assembly aTAM Temperature 1 
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Notes

Acknowledgments

We would like to thank Damien Woods for invaluable discussions, comments, and suggestions.

References

  1. 1.
    Cannon, S., et al.: Two hands are better than one (up to constant factors). In: STACS: Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, pp. 172–184. LIPIcs (2013). arxiv preprint: arXiv:1201.1650
  2. 2.
    Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: SODA: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 570–589 (2011). arxiv preprint: arXiv:0912.0027
  3. 3.
    Demaine, E.D., et al.: One tile to rule them all: simulating any tile assembly system with a single universal tile. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 368–379. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43948-7_31. arxiv preprint: arXiv:1212.4756CrossRefGoogle Scholar
  4. 4.
    Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: FOCS: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pp. 439–446. IEEE, October 2012. arxiv preprint: arXiv:1111.3097
  5. 5.
    Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theor. Comput. Sci. 412(1–2), 145–158 (2011). arxiv preprint: arXiv:0906.3251MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fekete, S.P., Hendricks, J., Patitz, M.J., Rogers, T.A., Schweller, R.T.: Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In: SODA: ACM-SIAM Symposium on Discrete Algorithms, pp. 148–167. SIAM (2015). http://arxiv.org/abs/1408.3351
  7. 7.
    Fekete, S.P., Hendricks, J., Patitz, M.J., Rogers, T.A., Schweller, R.T.: Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In: Indyk, P. (ed.) Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, 4–6 January 2015, pp. 148–167. SIAM (2015).  https://doi.org/10.1137/1.9781611973730.12
  8. 8.
    Gilbert, O., Hendricks, J., Patitz, M.J., Rogers, T.A.: Computing in continuous space with self-assembling polygonal tiles. In: SODA: ACM-SIAM Symposium on Discrete Algorithms, pp. 937–956. SIAM (2016). arxiv preprint: arXiv:1503.00327
  9. 9.
    Gilbert, O., Hendricks, J., Patitz, M.J., Rogers, T.A.: Computing in continuous space with self-assembling polygonal tiles (extended abstract). In: Krauthgamer, R. (ed.) Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, 10–12 January 2016, pp. 937–956. SIAM (2016).  https://doi.org/10.1137/1.9781611974331.ch67
  10. 10.
    Hendricks, J., Patitz, M.J., Rogers, T.A., Summers, S.M.: The power of duples (in self-assembly): it’s not so hip to be square. In: Cai, Z., Zelikovsky, A., Bourgeois, A. (eds.) COCOON 2014. LNCS, vol. 8591, pp. 215–226. Springer, Cham (2014).  https://doi.org/10.1007/978-3-319-08783-2_19. arxiv preprint: arXiv:1402.4515CrossRefGoogle Scholar
  11. 11.
    Maňuch, J., Stacho, L., Stoll, C.: Two lower bounds for self-assemblies at temperature 1. J. Comput. Biol. 17(6), 841–852 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Meunier, P.É., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: SODA: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 752–771 (2014). arxiv preprint: arXiv:1304.1679
  13. 13.
    Meunier, P., Woods, D.: The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation. In: STOC: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 328–341 (2017)Google Scholar
  14. 14.
    Padilla, J.E., Patitz, M.J., Schweller, R.T., Seeman, N.C., Summers, S.M., Zhong, X.: Asynchronous signal passing for tile self-assembly: fuel efficient computation and efficient assembly of shapes. Int. J. Found. Comput. Sci. 25(4), 459–488 (2014). arxiv preprint: arxiv:1202.5012MathSciNetCrossRefGoogle Scholar
  15. 15.
    Patitz, M.J., Schweller, R.T., Summers, S.M.: Exact shapes and turing universality at temperature 1 with a single negative glue. In: Cardelli, L., Shih, W. (eds.) DNA 2011. LNCS, vol. 6937, pp. 175–189. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-23638-9_15. arxiv preprint: arXiv:1105.1215, http://dl.acm.org/citation.cfm?id=2042033.2042050 CrossRefzbMATHGoogle Scholar
  16. 16.
    Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California, December 2001Google Scholar
  17. 17.
    Rothemund, P.W.K.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006).  https://doi.org/10.1038/nature04586CrossRefGoogle Scholar
  18. 18.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 459–468. ACM, Portland (2000). http://doi.acm.org/10.1145/335305.335358
  19. 19.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Thubagere, A.J., et al.: A cargo-sorting DNA robot. Science 357(6356), eaan6558 (2017)CrossRefGoogle Scholar
  21. 21.
    Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998Google Scholar
  22. 22.
    Winfree, E.: Simulations of computing by self-assembly. Technical report, Caltech CS TR:1998.22, California Institute of Technology (1998)Google Scholar
  23. 23.
    Yurke, B., Turberfield, A.J., Mills, A.P., Simmel, F.C., Neumann, J.L.: A DNA-fuelled molecular machine made of DNA. Nature 406(6796), 605–608 (2000)CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Maynooth UniversityMaynoothIreland
  2. 2.IBISC, Univ Évry, Université Paris-SaclayEvryFrance

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