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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)

Abstract

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.

Keywords

computability computational complexity molecular computing self-assembly 

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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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