Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Fractal Dimension of Assemblies in the Abstract Tile Assembly Model

  • Conference paper
  • First Online:
Unconventional Computation and Natural Computation (UCNC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12984))

Abstract

In this paper, we investigate the power of systems in the abstract Tile Assembly Model to self-assemble shapes having fractal dimensions between 1 and 2. We introduce a concept of sparsity as a tool for investigating such systems and demonstrate its utility by proving how it relates to fractal dimension.

M. J. Patitz—This work was supported in part by National Science Foundation grant CAREER-1553166.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    Here a standard counter gadget refers to commonly used log-width counter gadgets. It is unknown whether or not counter-like gadgets can be implemented in a sparse way.

  2. 2.

    There are universal Turing machines which induce asymptotically smaller runtime blowups, but choosing one with a quadratic blow up makes analysis of the final fractal dimension easier.

References

  1. Barth, K., Furcy, D., Summers, S.M., Totzke, P.: Scaled tree fractals do not strictly self-assemble. In: Ibarra, O.H., Kari, L., Kopecki, S. (eds.) UCNC 2014. LNCS, vol. 8553, pp. 27–39. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08123-6_3

    Chapter  Google Scholar 

  2. Cannon, S., et al.: Two hands are better than one (up to constant factors): self-assembly in the 2HAM vs. aTAM. In: Portier, N., Wilke, T. (eds.) STACS, volume 20 of LIPIcs, pp. 172–184. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)

    Google Scholar 

  3. Chalk, C.T., Fernandez, D.A., Huerta, A., Maldonado, M.A., Schweller, R.T., Sweet, L.: Strict self-assembly of fractals using multiple hands. Algorithmica 76(1), 1–30 (2015). https://doi.org/10.1007/s00453-015-0022-x

    Article  MathSciNet  MATH  Google Scholar 

  4. Demaine, E.D., Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Woods, D.: The two-handed tile assembly model is not intrinsically universal. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013. LNCS, vol. 7965, pp. 400–412. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39206-1_34

    Chapter  Google Scholar 

  5. Doty, D., Gu, X., Lutz, J.H., Mayordomo, E., Moser, P.: Zeta-Dimension. In: Jȩdrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 283–294. Springer, Heidelberg (2005). https://doi.org/10.1007/11549345_25

    Chapter  Google Scholar 

  6. 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. FOCS 2012, pp. 302–310 (2012)

    Google Scholar 

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

    Google Scholar 

  8. Furcy, D., Summers, S.M.: Scaled pier fractals do not strictly self-assemble. Nat. Comput. 16(2), 317–338 (2015). https://doi.org/10.1007/s11047-015-9528-z

    Article  MathSciNet  MATH  Google Scholar 

  9. Hartmanis, J., Stearns, R.E.: On the computational complexity of algorithms. Trans. Am. Math. Soc. 117, 285–306 (1965)

    Article  MathSciNet  Google Scholar 

  10. Hendricks, J., Opseth, J., Patitz, M.J., Summers, S.M.: Hierarchical growth is necessary and (sometimes) sufficient to self-assemble discrete self-similar fractals. In: Doty, D., Dietz, H. (eds.) DNA 2018. LNCS, vol. 11145, pp. 87–104. Springer, Cham (2018). https://doi.org/10.1007/978-3-030-00030-1_6

    Chapter  MATH  Google Scholar 

  11. Hendricks, J., Olsen, M., Patitz, M.J., Rogers, T.A., Thomas, H.: Hierarchical self-assembly of fractals with signal-passing tiles (extended abstract). In: Rondelez, Y., Woods, D. (eds.) DNA 2016. LNCS, vol. 9818, pp. 82–97. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-43994-5_6

    Chapter  Google Scholar 

  12. Hendricks, J., Opseth, J.: Self-assembly of 4-sided fractals in the two-handed tile assembly model. In: Patitz, M.J., Stannett, M. (eds.) UCNC 2017. LNCS, vol. 10240, pp. 113–128. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-58187-3_9

    Chapter  Google Scholar 

  13. Hendricks, J., Patitz, M.J., Rogers, T.A.: Universal simulation of directed systems in the abstract tile assembly model requires undirectedness. In: Proceedings of the 57th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2016), New Brunswick, New Jersey, USA 9–11 October 2016, pp. 800–809 (2016)

    Google Scholar 

  14. Kautz, S.M., Lathrop, J.I.: Self-assembly of the discrete Sierpinski carpet and related fractals. In: Deaton, R., Suyama, A. (eds.) DNA 2009. LNCS, vol. 5877, pp. 78–87. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-10604-0_8

    Chapter  MATH  Google Scholar 

  15. Kautz, S.M., Shutters, B.: Self-assembling rulers for approximating generalized Sierpinski carpets. Algorithmica 67(2), 207–233 (2013)

    Article  MathSciNet  Google Scholar 

  16. 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 

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

    Article  MathSciNet  Google Scholar 

  18. Lutz, J.H., Shutters, B.: Approximate self-assembly of the Sierpinski triangle. Theory Comput. Syst. 51(3), 372–400 (2012)

    Article  MathSciNet  Google Scholar 

  19. 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)

    Article  MathSciNet  Google Scholar 

  20. Patitz, M.J., Summers, S.M.: Self-assembly of discrete self-similar fractals. Nat. Comput. 1, 135–172 (2010)

    Article  MathSciNet  Google Scholar 

  21. Patitz, M.J., Summers, S.M.: Self-assembly of decidable sets. Nat. Comput. 10(2), 853–877 (2011)

    Article  MathSciNet  Google Scholar 

  22. Patitz, M.J., Summers, S.M.: Self-assembly of infinite structures: a survey. Theor. Comput. Sci. 412(1-2), 159–165 (2011). https://doi.org/10.1016/j.tcs.2010.08.015

  23. Rothemund, P.W.K., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), e424 (2004)

    Article  Google Scholar 

  24. 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, Portland, Oregon, United States. ACM (2000)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Winfree, E.: Algorithmic Self-Assembly of DNA. Ph.D. thesis, California Institute of Technology, June 1998

    Google Scholar 

  27. Woods, D., et al.: Diverse and robust molecular algorithms using reprogrammable DNA self-assembly. Nature 567, 366–372 (2019)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the three anonymous reviewers whose comments helped improve the presentation and technical correctness of this paper.

Author information

Authors and Affiliations

  1. University of Arkansas, Fayetteville, AR, 72703, USA

    Daniel Hader & Matthew J. Patitz

  2. University of Wisconsin-Oshkosh, Oshkosh, WI, 54901, USA

    Scott M. Summers

Authors
  1. Daniel Hader
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew J. Patitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Scott M. Summers
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Scott M. Summers .

Editor information

Editors and Affiliations

  1. TU Eindhoven, Eindhoven, The Netherlands

    Irina Kostitsyna

  2. Aalto University, Espoo, Finland

    Pekka Orponen

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Hader, D., Patitz, M.J., Summers, S.M. (2021). Fractal Dimension of Assemblies in the Abstract Tile Assembly Model. In: Kostitsyna, I., Orponen, P. (eds) Unconventional Computation and Natural Computation. UCNC 2021. Lecture Notes in Computer Science(), vol 12984. Springer, Cham. https://doi.org/10.1007/978-3-030-87993-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-87993-8_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-87992-1

  • Online ISBN: 978-3-030-87993-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation