The Two-Handed Tile Assembly Model Is Not Intrinsically Universal

  • Erik D. Demaine
  • Matthew J. Patitz
  • Trent A. Rogers
  • Robert T. Schweller
  • Scott M. Summers
  • Damien Woods
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

Abstract

In this paper, we study the intrinsic universality of the well-studied Two-Handed Tile Assembly Model (2HAM), in which two “supertile” assemblies, each consisting of one or more unit-square tiles, can fuse together (self-assemble) whenever their total attachment strength is at least the global temperature τ. Our main result is that for all τ′ < τ, each temperature-τ′ 2HAM tile system cannot simulate at least one temperature-τ 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal, in stark contrast to the simpler abstract Tile Assembly Model which was shown to be intrinsically universal (The tile assembly model is intrinsically universal, FOCS 2012). On the positive side, we prove that, for every fixed temperature τ ≥ 2, temperature-τ 2HAM tile systems are intrinsically universal: for each τ there is a single universal 2HAM tile set U that, when appropriately initialized, is capable of simulating the behavior of any temperature τ 2HAM tile system. As a corollary of these results we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing power within each hierarchy. Finally, we show how to construct, for each τ, a temperature-τ 2HAM system that simultaneously simulates all temperature-τ 2HAM systems.

Keywords

Cellular Automaton Tile System Tile Type Total Strength Single 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.M., Cheng, Q., Goel, A., Huang, M.-D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pp. 23–32 (2002)Google Scholar
  2. 2.
    Arrighi, P., Schabanel, N., Theyssier, G.: Intrinsic simulations between stochastic cellular automata. arXiv preprint arXiv:1208.2763 (2012)Google Scholar
  3. 3.
    Cannon, S., Demaine, E.D., Demaine, M.L., Eisenstat, S., Patitz, M.J., Schweller, R., Summers, S.M., Winslow, A.: Two hands are better than one (up to constant factors). In: Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science (to appear, 2013)Google Scholar
  4. 4.
    Chacc, E.G., Meunier, P.-E., Rapaport, I., Theyssier, G.: Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci. 412(1-2), 2–21 (2011)CrossRefGoogle Scholar
  5. 5.
    Chen, H.-L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: SODA 2012: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1163–1182. SIAM (2012)Google Scholar
  6. 6.
    Cheng, Q., Aggarwal, G., Goldwasser, M.H., Kao, M.-Y., Schweller, R.T., de Espanés, P.M.: Complexities for generalized models of self-assembly. SIAM Journal on Computing 34, 1493–1515 (2005)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking I: an abstract theory of bulking. Theoretical Computer Science 412(30), 3866–3880 (2011)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking II: Classifications of cellular automata. Theor. Comput. Sci. 412(30), 3881–3905 (2011)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Demaine, E.D., Demaine, M.L., Fekete, S.P., Ishaque, M., Rafalin, E., Schweller, R.T., Souvaine, D.L.: Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Natural Computing 7(3), 347–370 (2008)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pp. 439–446 (October 2012)Google Scholar
  11. 11.
    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
  12. 12.
    Durand, B., Róka, Z.: The game of life: universality revisited. In: Delorme, M., Mazoyer, J. (eds.) Cellular Automata. Kluwer (1999)Google Scholar
  13. 13.
    Fujibayashi, K., Hariadi, R., Park, S.H., Winfree, E., Murata, S.: Toward reliable algorithmic self-assembly of DNA tiles: A fixed-width cellular automaton pattern. Nano Letters 8(7), 1791–1797 (2007)CrossRefGoogle Scholar
  14. 14.
    Lafitte, G., Weiss, M.: Universal tilings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 367–380. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Lafitte, G., Weiss, M.: Simulations between tilings. In: Conference on Computability in Europe (CiE 2008), Local Proceedings, pp. 264–273 (2008)Google Scholar
  16. 16.
    Lafitte, G., Weiss, M.: An almost totally universal tile set. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 271–280. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Luhrs, C.: Polyomino-safe DNA self-assembly via block replacement. In: Goel, A., Simmel, F.C., Sosík, P. (eds.) DNA 14. LNCS, vol. 5347, pp. 112–126. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  18. 18.
    Lund, K., Manzo, A.T., Dabby, N., Micholotti, N., Johnson-Buck, A., Nangreave, J., Taylor, S., Pei, R., Stojanovic, M.N., Walter, N.G., Winfree, E., Yan, H.: Molecular robots guided by prescriptive landscapes. Nature 465, 206–210 (2010)CrossRefGoogle Scholar
  19. 19.
    Ollinger, N.: Intrinsically universal cellular automata. In: The Complexity of Simple Programs, in Electronic Proceedings in Theoretical Computer Science, vol. 1, pp. 199–204 (2008)Google Scholar
  20. 20.
    Ollinger, N., Richard, G.: Four states are enough? Theoretical Computer Science 412(1), 22–32 (2011)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196 (2011)CrossRefGoogle Scholar
  22. 22.
    Qian, L., Winfree, E., Bruck, J.: Neural network computation with DNA strand displacement cascades. Nature 475(7356), 368–372 (2011)CrossRefGoogle Scholar
  23. 23.
    Rothemund, P.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)CrossRefGoogle Scholar
  24. 24.
    Rothemund, P.W., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology 2(12), 2041–2053 (2004)CrossRefGoogle Scholar
  25. 25.
    Seeman, N.C.: Nucleic-acid junctions and lattices. Journal of Theoretical Biology 99, 237–247 (1982)CrossRefGoogle Scholar
  26. 26.
    Wang, H.: Proving theorems by pattern recognition – II. The Bell System Technical Journal XL(1), 1–41 (1961)Google Scholar
  27. 27.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology (June 1998)Google Scholar
  28. 28.
    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
  29. 29.
    Yurke, B., Turberfield, A., Mills Jr., A., Simmel, F., Neumann, J.: A DNA-fuelled molecular machine made of DNA. Nature 406(6796), 605–608 (2000)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Erik D. Demaine
    • 1
  • Matthew J. Patitz
    • 2
  • Trent A. Rogers
    • 3
  • Robert T. Schweller
    • 4
  • Scott M. Summers
    • 5
  • Damien Woods
    • 6
  1. 1.Computer Science and Artificial Intelligence LaboratoryMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Department of Computer Science and Computer EngineeringUniversity of ArkansasUSA
  3. 3.Department of Mathematical SciencesUniversity of ArkansasUSA
  4. 4.Department of Computer ScienceUniversity of Texas-Pan AmericanEdinburgUSA
  5. 5.Department of Computer Science and Software EngineeringUniversity of Wisconsin-PlattevillePlattevilleUSA
  6. 6.Computer ScienceCalifornia Institute of TechnologyUSA

Personalised recommendations