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.
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The research of J.H. Lutz was supported in part by National Science Foundation Grants 0344187, 0652569 and 0728806 and by Spanish Government MEC Project TIN 2005-08832-C03-02.
The research of S.M. Summers was supported in part by NSF-IGERT Training Project in Computational Molecular Biology Grant number DGE-0504304.
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Lathrop, J.I., Lutz, J.H., Patitz, M.J. et al. Computability and Complexity in Self-assembly. Theory Comput Syst 48, 617–647 (2011). https://doi.org/10.1007/s00224-010-9252-0
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DOI: https://doi.org/10.1007/s00224-010-9252-0