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Geometric Tiles and Powers and Limitations of Geometric Hindrance in Self-assembly

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Book cover Unconventional Computation and Natural Computation (UCNC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11493))

Abstract

Tile-based self-assembly systems are capable of universal computation and algorithmically-directed growth. Systems capable of such behavior typically make use of “glue cooperation” in which the glues on at least 2 sides of a tile must match and bind to those exposed on the perimeter of an assembly for that tile to attach. However, several models have been developed which utilize “weak cooperation”, where only a single glue needs to bind but other preventative forces (such as geometric, or steric, hindrance) provide additional selection for which tiles may attach, and where this allows for algorithmic behavior. In this paper we first work in a model where tiles are allowed to have geometric bumps and dents on their edges. We show how such tiles can simulate systems of square tiles with complex glue functions (using asymptotically optimal sizes of bumps and dents). We also show that with only weak cooperation via geometric hindrance, no system in any model can simulate even a class of tightly constrained, deterministic cooperative systems, further defining the boundary of what is possible using this tool.

This author’s research was supported in part by National Science Foundation Grants CCF-1422152 and CAREER-1553166.

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Change history

  • 27 May 2019

    The original version of this paper contained a mistake. Theorem 3 in “Section 4,” which claimed that for every temperature-1 system in the Dupled abstract Tile Assembly Model (DaTAM) there exists a temperature-1 system in the Geometric Tile Assembly Model (GTAM) which simulates it at scale factor 1 and using only 2 glues, was incorrect. That theorem and the section that contained it have been removed. Please note that that result was independent of all other results, and the fact that it was incorrect does not impact them or any of the other remaining portions of the paper.

Notes

  1. 1.

    Note that rectilinear systems can also be simulated, and they have similar properties except that they may have multiple frontier locations available.

  2. 2.

    Note that with negative glues, more complex dynamics, which include the breaking apart of assemblies, are possible [13].

References

  1. Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: SODA 2011: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM (2011)

    Google Scholar 

  2. 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

    Chapter  Google Scholar 

  3. Doty, D., Kari, L., Masson, B.: Negative interactions in irreversible self-assembly. Algorithmica 66(1), 153–172 (2013)

    Article  MathSciNet  Google Scholar 

  4. Doty, D., Patitz, M.J., Summers, S.M.: Limitations of self-assembly at temperature 1. Theor. Comput. Sci. 412, 145–158 (2011)

    Article  MathSciNet  Google Scholar 

  5. 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: 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 (2015). https://doi.org/10.1137/1.9781611973730.12

  6. Fu, B., Patitz, M.J., Schweller, R.T., Sheline, R.: Self-assembly with geometric tiles. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012. LNCS, vol. 7391, pp. 714–725. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31594-7_60

    Chapter  Google Scholar 

  7. Gilber, O., Hendricks, J., Patitz, M.J., Rogers, T.A.: Computing in continuous space with self-assembling polygonal tiles. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), Arlington, VA, USA, 10–12 January 2016, pp. 937–956 (2016)

    Google Scholar 

  8. Hader, D., Patitz, M.J.: Geometric tiles and powers and limitations of geometric hindrance in self-assembly. Technical report 1903.05774, Computing Research Repository (2019). http://arxiv.org/abs/1903.05774

    Chapter  Google Scholar 

  9. Hendricks, J., Patitz, M.J., Rogers, T.A.: Doubles and negatives are positive (in self-assembly). Nat. Comput. 15(1), 69–85 (2016). https://doi.org/10.1007/s11047-015-9513-6

    Article  MathSciNet  MATH  Google Scholar 

  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. Theor. Comput. Sci. (2015). https://doi.org/10.1016/j.tcs.2015.12.008. http://www.sciencedirect.com/science/article/pii/S030439751501169X

    Article  MathSciNet  Google Scholar 

  11. Lathrop, J.I., Lutz, J.H., Patitz, M.J., Summers, S.M.: Computability and complexity in self-assembly. Theory Comput. Syst. 48(3), 617–647 (2011)

    Article  MathSciNet  Google Scholar 

  12. Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. Theor. Comput. Sci. 410, 384–405 (2009)

    Article  MathSciNet  Google Scholar 

  13. Luchsinger, A., Schweller, R., Wylie, T.: Self-assembly of shapes at constantscale using repulsive forces. Nat. Comput. (2018). https://doi.org/10.1007/s11047-018-9707-9

    Article  MathSciNet  Google Scholar 

  14. Luchsinger, A., Schweller, R., Wylie, T.: Self-assembly of shapes at constant scale using repulsive forces. In: Patitz, M.J., Stannett, M. (eds.) UCNC 2017. LNCS, vol. 10240, pp. 82–97. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58187-3_7

    Chapter  Google Scholar 

  15. Meunier, P.E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA 2014), Portland, OR, USA, 5–7 January 2014, pp. 752–771 (2014)

    Google Scholar 

  16. Meunier, P., Woods, D.: The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, 19–23 June 2017, pp. 328–341 (2017). https://doi.org/10.1145/3055399.3055446

  17. 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

    Chapter  MATH  Google Scholar 

  18. 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, pp. 459–468. ACM, Portland (2000). https://doi.org/10.1145/335305.335358

  19. Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM J. Comput. 36(6), 1544–1569 (2007)

    Article  MathSciNet  Google Scholar 

  20. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Daniel Hader & Matthew J. Patitz

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  1. Daniel Hader
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  2. Matthew J. Patitz
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Correspondence to Matthew J. Patitz .

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Editors and Affiliations

  1. University of Saskatchewan, Saskatoon, SK, Canada

    Ian McQuillan

  2. University of Electro-Communications, Tokyo, Japan

    Shinnosuke Seki

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Hader, D., Patitz, M.J. (2019). Geometric Tiles and Powers and Limitations of Geometric Hindrance in Self-assembly. In: McQuillan, I., Seki, S. (eds) Unconventional Computation and Natural Computation. UCNC 2019. Lecture Notes in Computer Science(), vol 11493. Springer, Cham. https://doi.org/10.1007/978-3-030-19311-9_16

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  • DOI: https://doi.org/10.1007/978-3-030-19311-9_16

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  • Online ISBN: 978-3-030-19311-9

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