Computability and Complexity in Self-assembly

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...

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Published inTheory of computing systems Vol. 48; no. 3; pp. 617 - 647
Main Authors Lathrop, James I., Lutz, Jack H., Patitz, Matthew J., Summers, Scott M.
Format Journal Article
LanguageEnglish
Published New York Springer-Verlag 01.04.2011
Springer Nature B.V
Subjects
Assembly
Complexity
Complexity theory
Computer Science
Computer simulation
Euclidean space
Geometry
Modular
Nanomaterials
Nanostructure
Nanotechnology
Self assembly
Studies
Theorems
Theory of Computation
Self-assembly
Molecular computing
Computational complexity
Computability
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Summary: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 , together with a proof that carries out concurrent simulations of M on all positive integer inputs.
Bibliography:ObjectType-Article-2
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-010-9252-0