Natural Computing

, Volume 10, Issue 2, pp 853–877 | Cite as

Self-assembly of decidable sets

  • Matthew J. Patitz
  • Scott M. Summers


The theme of this paper is computation in Winfree’s Abstract Tile Assembly Model (TAM). We first review a simple, well-known tile assembly system (the “wedge construction”) that is capable of universal computation. We then extend the wedge construction to prove the following result: if a set of natural numbers is decidable, then it and its complement’s canonical two-dimensional representation self-assemble. This leads to a novel characterization of decidable sets of natural numbers in terms of self-assembly. Finally, we show that our characterization is robust with respect to various (restrictive) geometrical constraints.


Computability Tile assembly Self-assembly DNA computing Decidability Universality Space complexity 



Both authors wish to thank David Doty, Jack Lutz and Damien Woods for useful discussions. This research was supported in part by National Science Foundation Grants 0652569 and 0728806. A preliminary version of this research was presented at the Sixth International Conference on Unconventional Computation, August 25–28 2008, Vienna, Austria. Scott M. Summers’s research was supported in part by NSF-IGERT Training Project in Computational Molecular Biology Grant number DGE-0504304.


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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Texas–Pan AmericanEdinburgUSA
  2. 2.Department of Computer Science and Software EngineeringUniversity of Wisconsin–PlattevillePlattevilleUSA

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