Skip to main content
SpringerLink
Log in

Non-cooperative Algorithms in Self-assembly

  • Conference paper
  • First Online:
Unconventional Computation and Natural Computation (UCNC 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9252))

Abstract

Imagine you are left alone in a forest with ogres and wolves, with a paper, a pen and a supply of small stones as your only weapons. How far can you go using a deterministic escape strategy, if you also want to be back in time for dinner (i.e. avoid running periodically)?

The answer to this question has been known for some time (and called the “pumping lemma”) in the simple case where the forest has exactly one self-avoiding trail: after at most \(2^n\) steps (where n is the number of bits writable on your paper) you start running periodically.

However, geometry can sometimes allow for better strategies: in this work, we show the first non-trivial positive algorithmic result (i.e. programs whose output is larger than their size), in a model of self-assembly that has been the center of puzzling open questions for almost 15 years: the planar non-cooperative variant of Winfree’s abstract Tile Assembly Model. Despite significant efforts, very little has been known on this model, until the first fully general results on its computational power, proven recently in SODA 2014.

In this model, tiles can stick to an existing assembly as soon as one of their sides matches the existing assembly. This feature contrasts with the general cooperative model, where it can be required that tiles match on several of their sides in order to bind.

Since the exact computational power of this model is still completely open, we also compare it with classical models from automata theory.

P.-E. Meunier—Dept. of Information and Computer Science, Aalto University, PO Box 15400, FI-00076, Aalto, Finland and Aix Marseille Université, CNRS, LIF UMR 7279, 13288, Marseille, France. Part of this work was carried out while at California Institute of Technology, Pasadena, CA 91125, USA. Supported in part by National Science Foundation Grant CCF-1219274.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Notes

  1. 1.

    We would like to thank an anonymous reviewer for pointing out the equivalence with the known result.

References

  1. Bousquet-Mélou, M.: Families of prudent self-avoiding walks. J. Comb. Theory Ser. A 117(3), 313–344 (2010)

    Article  MATH  Google Scholar 

  2. Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Löding, C., Tison, S., Tommasi, M.: Tree automata techniques and applications (2007). http://www.grappa.univ-lille3.fr/tata. Accessed 12 October 2007

  3. Cook, M., Fu, Y., Schweller, R.T.: Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of SODA 2011, pp. 570–589 (2011). Arxiv preprint: arXiv:0912.0027

  4. Flory, P.J.: Principles of Polymer Chemistry. Cornell University, Ithaca (1953)

    Google Scholar 

  5. Knuth, D.E.: Mathematics and computer science: coping with finiteness. Math. People Prob. Results 2, 210–211 (1984)

    Google Scholar 

  6. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC 2000, Portland, Oregon, United States, pp. 459–468. ACM (2000)

    Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

  9. Winslow, A.: Staged self-assembly and polyomino context-free grammars. In: Soloveichik, D., Yurke, B. (eds.) DNA 2013. LNCS, vol. 8141, pp. 174–188. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

Download references

Acknowledgements

The author thanks Damien Woods for insightful comments and discussions, and one of the anonymous reviewer whose expertise helped improved this paper quite a lot.

Author information

Authors and Affiliations

  1. Department of Information and Computer Science, Aalto University, Espoo, Finland

    Pierre-Étienne Meunier

Authors
  1. Pierre-Étienne Meunier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre-Étienne Meunier .

Editor information

Editors and Affiliations

  1. University of Auckland, Auckland, New Zealand

    Cristian S. Calude

  2. University of Auckland, Auckland, New Zealand

    Michael J. Dinneen

A Printable Version of Fig. 1

A Printable Version of Fig. 1

figure e

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Meunier, PÉ. (2015). Non-cooperative Algorithms in Self-assembly. In: Calude, C., Dinneen, M. (eds) Unconventional Computation and Natural Computation. UCNC 2015. Lecture Notes in Computer Science(), vol 9252. Springer, Cham. https://doi.org/10.1007/978-3-319-21819-9_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21819-9_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21818-2

  • Online ISBN: 978-3-319-21819-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation