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The Two-Handed Tile Assembly Model is not Intrinsically Universal

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Abstract

The Two-Handed Tile Assembly Model (2HAM) is a model of algorithmic self-assembly in which large structures, or assemblies of tiles, are grown by the binding of smaller assemblies. In order to bind, two assemblies must have matching glues that can simultaneously touch each other, and stick together with strength that is at least the temperature \(\tau \), where \(\tau \) is some fixed positive integer. We ask whether the 2HAM is intrinsically universal. In other words, we ask: is there a single 2HAM tile set \(U\) which can be used to simulate any instance of the model? Our main result is a negative answer to this question. We show that for all \(\tau ' < \tau \), each temperature-\(\tau '\) 2HAM tile system does not simulate at least one temperature-\(\tau \) 2HAM tile system. This impossibility result proves that the 2HAM is not intrinsically universal and stands in contrast to the fact that the (single-tile addition) abstract Tile Assembly Model is intrinsically universal. On the positive side, we prove that, for every fixed temperature \(\tau \ge 2\), temperature-\(\tau \) 2HAM tile systems are indeed intrinsically universal. In other words, for each \(\tau \) there is a single intrinsically universal 2HAM tile set \(U_{\tau }\) that, when appropriately initialized, is capable of simulating the behavior of any temperature-\(\tau \) 2HAM tile system. As a corollary, we find an infinite set of infinite hierarchies of 2HAM systems with strictly increasing simulation power within each hierarchy. Finally, we show that for each \(\tau \), there is a temperature-\(\tau \) 2HAM system that simultaneously simulates all temperature-\(\tau \) 2HAM systems.

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Notes

  1. Note that this simulation result of Cannon et al. [5] does not imply that the 2HAM is intrinsically universal because (a) it is for 2HAM simulating aTAM, and (b) it is a “for all, there exists\(\ldots \)” statement, whereas intrinsic universality is a “there exists, for all\(\ldots \)” statement.

  2. We do not use this definition in this paper but have included it for the sake of completeness.

  3. with the convention that \(\infty = \infty + 1 = \infty - 1\)

  4. Note that a supertile \(\tilde{\alpha }\) could be non-terminal in the sense that there is a producible supertile \(\tilde{\beta }\) such that \(C^\tau _{\tilde{\alpha },\tilde{\beta }} \ne \emptyset \), yet it may not be possible to produce \(\tilde{\alpha }\) and \(\tilde{\beta }\) simultaneously if some tile types are given finite initial counts, implying that \(\tilde{\alpha }\) cannot be “grown” despite being non-terminal. If the count of each tile type in the initial state is \(\infty \), then all producible supertiles are producible from any state, and the concept of terminal becomes synonymous with “not able to grow”, since it would always be possible to use the abundant supply of tiles to assemble \(\tilde{\beta }\) alongside \(\tilde{\alpha }\) and then attach them.

  5. Note that in the glue-binding pad region there are no “single tile” bumps: this ensures that the simulator tile set \(U_{\tau }\) does not contain strength \(\tau \) glues, which in turn simplifies our construction.

  6. The crawlers, counters, computational primitives (guessing strings, computing simple numerical functions on bit strings, and even simulating Turing machines), and geometric primitives (copying bit sequences around in two-dimensional space) used in this and later constructions are relatively straightforward implementations similar to those used in the aTAM in [15], among others. These primitives are designed to assemble on the edges of existing supertiles (or assemblies in the aTAM), and can be made (and usually already are) “2HAM-safe” (essentially, “polyomino safe” as in [26]), meaning that in the 2HAM they function identically and correctly without danger of unwanted supertiles forming which are unattached to the desired supertiles. The general technique is to limit the number of \(\tau \)-strength glues on any particular tile type which assembles the primitive to \(1\), so that the largest unattached supertile which can form from them is a size \(2\) duple. All other attachments, and even the incorporation of the duples, requires cooperation provided by the surface of the supertile onto which the primitive is intended to form. Since the constructions for these primitives are standard and straightforward, we omit the details here.

References

  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)

  2. Arrighi, P., Schabanel, N., Theyssier, G.: Intrinsic simulations between stochastic cellular automata. In: Automata & JAC: Proceedings of the 18th International Workshop on Cellular Automata and Discrete Complex Systems and the 3rd International Symposium Journées Automates Cellulaires, EPTCS, vol. 90, pp. 208–224 (2012). Arxiv preprint: arXiv:1208.2763

  3. Barish, R.D., Rothemund, P.W., Winfree, E.: Two computational primitives for algorithmic self-assembly: copying and counting. Nano Lett. 5(12), 2586–2592 (2005)

    Article  Google Scholar 

  4. Barish, R.D., Schulman, R., Rothemund, P.W., Winfree, E.: An information-bearing seed for nucleating algorithmic self-assembly. Proc. Natl. Acad. Sci 106(15), 6054–6059 (2009)

    Article  Google Scholar 

  5. 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: STACS: Proceedings of the Thirtieth International Symposium on Theoretical Aspects of Computer Science, pp. 172–184 (2013). Arxiv preprint: arXiv:1201.1650

  6. Goles, E., Meunier, P.E., Rapaport, I., Theyssier, G.: Communication complexity and intrinsic universality in cellular automata. Theor. Comput. Sci. 412(1—-2), 2–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen, H.L., Doty, D.: Parallelism and time in hierarchical self-assembly. In: Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1163–1182. Society for Industrial and Applied Mathematics (2012)

  8. 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 J. Comput. 34, 1493–1515 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cook, M., Fu, Y., Schweller, R.T.: 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, pp. 570–589. Society for Industrial and Applied Mathematics (2011)

  10. Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking I: an abstract theory of bulking. Theor. Comput. Sci. 412(30), 3866–3880 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Delorme, M., Mazoyer, J., Ollinger, N., Theyssier, G.: Bulking II: classifications of cellular automata. Theor. Comput. Sci. 412(30), 3881–3905 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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. Nat. Comput. 7(3), 347–370 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Demaine, E.D., Demaine, M.L., Fekete, S.P., Patitz, M.J., Schweller, R.T., Winslow, A., Woods, D.: One tile to rule them all: simulating any Turing machine, tile assembly system, or tiling system with a single puzzle piece. In: ICALP: Proceedings of the 41st International Colloquium on Automata, Languages, and Programming, LNCS, vol. 8572, pp. 368–379. Springer (2014). Arxiv preprint: arXiv:1212.4756

  14. Demaine, E.D., Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Woods, D.: The two-handed assembly model is not intrinsically universal. Tech. rep., Computing Research Repository (2013). arXiv:1306.6710 [cs.CG]

  15. 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 (2012). Arxiv preprint: arXiv:1111.3097

  16. Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: STACS: Proceedings of the 27th International Symposium on Theoretical Aspects of Computer Science, pp. 275–286 (2009). Arxiv preprint: arXiv:1001.0208

  17. 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)

  18. Durand, B., Róka, Z.: The game of life: universality revisited. In: Delorme, M., Mazoyer, J. (eds.) Cellular Automata. Kluwer, Dordrecht (1999)

    Google Scholar 

  19. Evans, C.G.: Crystals that count! Physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, California Institute of Technology (2014)

  20. 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 Lett. 8(7), 1791–1797 (2007)

    Article  Google Scholar 

  21. Hendricks, J., Patitz, M.J., Rogers, T.A.: Doubles and negatives are positive (in self-assembly). In: UCNC: Proceeding of Unconventional Computation and Natural Computation, LNCS, vol. 8553, pp. 190–202. Springer (2014)

  22. Lafitte, G., Weiss, M.: Universal tilings. In: Thomas, W., Weil, P. (eds.) STACS 2007, 24th Annual Symposium on Theoretical Aspects of Computer Science, Aachen, Germany, February 22–24, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4393, pp. 367–380. Springer (2007). http://dx.doi.org/10.1007/978-3-540-70918-3_32

  23. Lafitte, G., Weiss, M.: Simulations between tilings. In: Conference on Computability in Europe, Local Proceedings, pp. 264–273 (2008)

  24. Lafitte, G., Weiss, M.: An almost totally universal tile set. In: Chen, J., Cooper, S.B. (eds.) Theory and Applications of Models of Computation, 6th Annual Conference, TAMC 2009, Changsha, China, May 18–22, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5532, pp. 271–280. Springer (2009). http://dx.doi.org/10.1007/978-3-642-02017-9

  25. 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  MATH  Google Scholar 

  26. Luhrs, C.: Polyomino-safe DNA self-assembly via block replacement. In: Goel, A., Simmel, F.C., Sosík, P. (eds.) DNA14, Lecture Notes in Computer Science, vol. 5347, pp. 112–126. Springer (2008). doi:10.1007/978-3-642-03076-5

  27. Meunier, P.E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: SODA: ACM-SIAM Symposium on Discrete Algorithms, pp. 752–771, Portland, OR, USA, January 5–7, 2014. Society for Industrial and Applied Mathematics (2014). Arxiv preprint: arXiv:1304.1679

  28. 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)

  29. Ollinger, N., Richard, G.: Four states are enough!. Theor. Comput. Sci. 412(1), 22–32 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rothemund, P.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)

    Article  Google Scholar 

  31. Rothemund, P.W., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), 2041–2053 (2004)

    Article  Google Scholar 

  32. Rothemund, P.W.K.: Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern California (2001)

  33. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  34. Wang, H.: Proving theorems by pattern recognition—II. Bell Syst. Tech. J. XL(1), 1–41 (1961)

    Article  Google Scholar 

  35. Winfree, E.: Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of Technology (1998)

  36. Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–44 (1998)

    Article  Google Scholar 

  37. Woods, D.: Intrinsic universality and the computational power of self-assembly. In: MCU: Proceedings of Machines, Computations and Universality, Electronic Proceedings in Theoretical Computer Science, vol. 128, pp. 16–22. Open Publishing Association, Univ. of Zürich, Switzerland, Sept. 9–12 (2013). doi:10.4204/EPTCS.128.5

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Acknowledgments

This work was initiated at the 27th Bellairs Winter Workshop on Computational Geometry held on February 11–17, 2012 in Holetown, Barbados. We thank the other participants of that workshop for a fruitful and collaborative environment. We would also like to thank an anonymous reviewer for very thorough and insightful comments, helping us to improve this version of the paper.

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Authors and Affiliations

  1. Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, 32 Vassar St., Cambridge, MA, 02139, USA

    Erik D. Demaine

  2. Department of Computer Science and Computer Engineering, University of Arkansas, 504 J. B. Hunt Building, 1 University of Arkansas, Fayetteville, AR, 72701, USA

    Matthew J. Patitz & Trent A. Rogers

  3. Department of Computer Science, University of Texas–Pan American, 1201 W University Drive, Edinburg, TX, 78539, USA

    Robert T. Schweller

  4. Department of Computer Science, University of Wisconsin–Oshkosh, 800 Algoma Blvd., Oshkosh, WI, 54901, USA

    Scott M. Summers

  5. Computer Science, California Institute of Technology, 1200 E California Blvd., MC 305-16, Pasadena, CA, 91125, USA

    Damien Woods

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  1. Erik D. Demaine
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Corresponding author

Correspondence to Trent A. Rogers.

Additional information

Matthew J. Patitz’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1422152.

Trent A. Rogers’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grants CCF-1117672 and CCF-1422152.

Robert T. Schweller’s research was supported in part by National Science Foundation Grant CCF-1117672.

Damien Woods’s research was supported by National Science Foundation Grants CCF-1219274, 0832824 (The Molecular Programming Project), CCF-1219274, and CCF-1162589.

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Demaine, E.D., Patitz, M.J., Rogers, T.A. et al. The Two-Handed Tile Assembly Model is not Intrinsically Universal. Algorithmica 74, 812–850 (2016). https://doi.org/10.1007/s00453-015-9976-y

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