Computability and Complexity in Self-assembly

  • James I. Lathrop
  • Jack H. Lutz
  • Matthew J. Patitz
  • Scott M. Summers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)


This paper explores the impact of geometry on computability and complexity in Winfree’s model of nanoscale self-assembly. We work in the two-dimensional tile assembly model, i.e., in the discrete Euclidean plane ℤ ×ℤ. Our first main theorem says that there is a roughly quadratic function f such that a set A ⊆ ℤ +  is computably enumerable if and only if the set X A  = { (f(n), 0) | n ∈ A } – a simple representation of A as a set of points on the x-axis – self-assembles in Winfree’s sense. In contrast, our second main theorem says that there are decidable sets D ⊆ ℤ ×ℤ that do not self-assemble in Winfree’s sense.

Our first main theorem is established by an explicit translation of an arbitrary Turing machine M to a modular tile assembly system \(\mathcal{T}_M\), together with a proof that \(\mathcal{T}_M\) carries out concurrent simulations of M on all positive integer inputs.


computability computational complexity molecular computing self-assembly 


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  1. 1.
    Adleman, L.: Towards a mathematical theory of self-assembly, Tech. report, University of Southern California (2000)Google Scholar
  2. 2.
    Bachrach, J., Beal, J.: Building spatial computers, Tech. report, MIT CSAIL (2007)Google Scholar
  3. 3.
    Beal, J., Sussman, G.: Biologically-inspired robust spatial programming, Tech. report, MIT (2005)Google Scholar
  4. 4.
    Hartmanis, J., Stearns, R.E.: On the computational complexity of algorithms. Transactions of the American Mathematical Society 117, 285–306 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lathrop, J.I., Lutz, J.H., Summers, S.M.: Strict self-assembly of discrete Sierpinski triangles. In: Proceedings of The Third Conference on Computability in Europe, Siena, Italy, June 18-23, 2007 (2007)Google Scholar
  6. 6.
    Reif, J.H.: Molecular assembly and computation: From theory to experimental demonstrations. In: Proceedings of the Twenty-Ninth International Colloquium on Automata, Languages and Programming, pp. 1–21 (2002)Google Scholar
  7. 7.
    Paul, W.K.: Rothemund, Theory and experiments in algorithmic self-assembly, Ph.D. thesis, University of Southern California (December 2001)Google Scholar
  8. 8.
    Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pp. 459–468 (2000)Google Scholar
  9. 9.
    Seeman, N.C.: Nucleic-acid junctions and lattices. Journal of Theoretical Biology 99, 237–247 (1982)CrossRefGoogle Scholar
  10. 10.
    Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. SIAM Journal on Computing 36, 1544–1569 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Wang, H.: Proving theorems by pattern recognition – II. The Bell System Technical Journal XL(1), 1–41 (1961)Google Scholar
  12. 12.
    Wang, H.: Dominoes and the AEA case of the decision problem. In: Proceedings of the Symposium on Mathematical Theory of Automata New York, 1962, Polytechnic Press of Polytechnic Inst. of Brooklyn, pp. 23–55 (1963)Google Scholar
  13. 13.
    Winfree, E.: Algorithmic self-assembly of DNA, Ph.D. thesis, California Institute of Technology (June 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • James I. Lathrop
    • 1
  • Jack H. Lutz
    • 1
  • Matthew J. Patitz
    • 1
  • Scott M. Summers
    • 1
  1. 1.Department of Computer ScienceIowa State UniversityAmesU.S.A.

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