Natural Computing

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

Self-assembly of decidable sets



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.


  1. Adleman L, Cheng Q, Goel A, Huang M-D (2001) Running time and program size for self-assembled squares. In: STOC ’01: proceedings of the thirty-third annual ACM symposium on theory of computing. ACM, New York, pp 740–748Google Scholar
  2. Adleman LM, Kari J, Kari L, Reishus D, Sosík P (2009) The undecidability of the infinite ribbon problem: implications for computing by self-assembly. SIAM J Comput 38(6):2356–2381MathSciNetMATHCrossRefGoogle Scholar
  3. Barish RD, Schulman R, Rothemund PW, Winfree E (2009) An information-bearing seed for nucleating algorithmic self-assembly. Proc Natl Acad Sci USA 106(15):6054–6059CrossRefGoogle Scholar
  4. Becker F, Rapaport I, Rémila E (2006) Self-assembling classes of shapes with a minimum number of tiles, and in optimal time. In: Foundations of software technology and theoretical computer science (FSTTCS), pp 45–56Google Scholar
  5. Cheng Q, Goel A, de Espanés PM (2004) Optimal self-assembly of counters at temperature two. In: Proceedings of the first conference on foundations of nanoscience: self-assembled architectures and devicesGoogle Scholar
  6. Cheng Q, Aggarwal G, Goldwasser MH, Kao M-Y, Schweller RT, de Espanés PM (2005) Complexities for generalized models of self-assembly. SIAM J Comput 34:1493–1515MathSciNetMATHCrossRefGoogle Scholar
  7. Demaine ED, Demaine ML, Fekete SP, Ishaque M, Rafalin E, Schweller RT, Souvaine DL (2008) Staged self-assembly: nanomanufacture of arbitrary shapes with O(1) glues. Nat Comput 7(3):347–370MathSciNetMATHCrossRefGoogle Scholar
  8. Doty D (2009) Randomized self-assembly for exact shapes. In: Proceedings of the fiftieth IEEE conference on foundations of computer science (FOCS)Google Scholar
  9. Doty D, Patitz MJ (2009) A domain specific language for programming in the tile assembly model. In: Proceedings of the fifteenth international meeting on DNA computing and molecular programming, Fayetteville, Arkansas, USA, June 8–11, 2009, pp 25–34Google Scholar
  10. Doty D, Patitz MJ, Summers SM Limitations of self-assembly at temperature 1. Theor Comput Sci (to appear)Google Scholar
  11. Fu Y, Schweller R (2009) Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. Technical report 0912.0027, Computing Research RepositoryGoogle Scholar
  12. Kao M-Y, Schweller RT (2007) Reducing tile complexity for self-assembly through temperature programming. In: Proceedings of the 17th annual ACM-SIAM symposium on discrete algorithms (SODA 2006), Miami, FL, January 2006, pp 571–580Google Scholar
  13. Kao M-Y, Schweller RT (2008) Randomized self-assembly for approximate shapes. In: International colloqium on automata, languages, and programming (ICALP). Lecture notes in computer science, vol 5125. Springer, pp 370–384Google Scholar
  14. Lathrop JI, Lutz JH, Summers SM (2009) Strict self-assembly of discrete Sierpinski triangles. Theor Comput Sci 410:384–405MathSciNetMATHCrossRefGoogle Scholar
  15. Lathrop JI, Lutz JH, Patitz MJ, Summers SM Computability and complexity in self-assembly. Theory Comput Syst (to appear)Google Scholar
  16. Patitz MJ (2009) Simulation of self-assembly in the abstract tile assembly model with ISU TAS. In: 6th Annual conference on foundations of nanoscience: self-assembled architectures and devices, Snowbird, UT, USA, 20–24 April 2009Google Scholar
  17. Reif JH (1999) Local parallel biomolecular computing. DNA based computers III, vol 48 of DIMACS. American Mathematical Society, pp 217–254Google Scholar
  18. Rothemund PWK (2001) Theory and experiments in algorithmic self-assembly. Ph.D. thesis, University of Southern CaliforniaGoogle Scholar
  19. Rothemund PWK, Winfree E (2000) The program-size complexity of self-assembled squares (extended abstract). In: STOC ’00: Proceedings of the thirty-second annual ACM symposium on theory of computing, New York, NY, USA. ACM, pp 459–468Google Scholar
  20. Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2(12):2041–2053Google Scholar
  21. Soloveichik D, Winfree E (2007) Complexity of self-assembled shapes. SIAM J Comput 36(6):1544–1569MathSciNetMATHCrossRefGoogle Scholar
  22. Wang H (1961) Proving theorems by pattern recognition—II. Bell Syst Tech J XL(1):1–41Google Scholar
  23. Wang H (1963) Dominoes and the AEA case of the decision problem. In: Proceedings of the symposium on mathematical theory of automata, New York, 1962. Polytechnic Press of Polytechnic Inst. of Brooklyn, Brooklyn, pp 23–55Google Scholar
  24. Winfree E (1998) Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of TechnologyGoogle Scholar

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

Personalised recommendations