The program-size complexity of self-assembled squares (extended abstract)
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Algorithmic self-assembly of dna
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DISC '08 Proceedings of the 22nd international symposium on Distributed Computing
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UC '09 Proceedings of the 8th International Conference on Unconventional Computation
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Natural Computing: an international journal
Natural Computing: an international journal
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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 驴 ×驴. Our first main theorem says that there is a roughly quadratic function fsuch that a set A驴 驴+is computably enumerable if and only if the set XA= { (f(n), 0) | n驴 A} --- a simple representation of Aas 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驴 驴 ×驴 that do notself-assemble in Winfree's sense.Our first main theorem is established by an explicit translation of an arbitrary Turing machine Mto a modular tile assembly system $\mathcal{T}_M$, together with a proof that $\mathcal{T}_M$ carries out concurrent simulations of Mon all positive integer inputs.