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A Linear Bound for the Size of the Finite Terminal Assembly of a Directed Non-Cooperative Tile Assembly System

Authors Sergiu Ivanov , Damien Regnault

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LIPIcs.ICALP.2026.115.pdf
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ACM Subject Classification
  • Theory of computation → Models of computation
Keywords
  • Models of computation
  • DNA self-assembly
  • aTAM
  • Complexity

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Abstract

Introduced in [Erik Winfree, 1998], the abstract tile assembly model (aTAM) is a model of DNA self-assembly. Most of the studies focus on cooperative aTAM where a form of synchronization between the tiles is possible. Simulating Turing machines is achievable in this context. Few results and constructions are known for the non-cooperative case (a variant of Wang tilings [Hao Wang, 1961] where assemblies do not need to cover the whole plane and some mismatches may occur). For example, assembly of a square of width n is done with 2n-1 tiles types whereas only Θ(log(n)/(log log(n))) are required for the cooperative case [Leonard M. Adleman et al., 2001]. 
Introduced by P.-É. Meunier in [Meunier, 2015], efficient paths are a non-trivial construction for non-cooperative aTAM designed with n different tile types and reaching a distance linearly greater than n. Improved in [Pierre-Étienne Meunier and Damien Regnault, 2019], efficient paths were shown to be able to reach a distance of nlog(n). Assembling them relies heavily on a form of "non-determinism". Indeed, the set of tiles may produce different finite terminal assemblies but they all contain the same efficient path. In this paper, we prove that this non-determinism is strictly necessary for assembling the efficient paths of [Pierre-Étienne Meunier and Damien Regnault, 2019]. More formally, we show that if the terminal assembly of a directed non-cooperative tile assembly system (a model where only one terminal assembly is produced) is finite then its width and length are linear in the number of tiles. This result also implies that the construction of a square of width n using 2n-1 tiles types is asymptotically optimal. Moreover, we hope that the techniques introduced here will lead to a better comprehension of the non-directed case.

Cite As Get BibTex

Sergiu Ivanov and Damien Regnault. A Linear Bound for the Size of the Finite Terminal Assembly of a Directed Non-Cooperative Tile Assembly System. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 115:1-115:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.115

Author Details

Sergiu Ivanov
  • IBISC, Univ Évry, University of Paris-Saclay, Évry, France
Damien Regnault
  • IBISC, Univ Évry, University of Paris-Saclay, Évry, France

Funding

The authors were partially supported by the project CalcADN (ANR-24-RRII-0001), funded by the French National Center for Scientific Research (CNRS).

Acknowledgements

We would like to thank Pierre-Étienne Meunier and Damien Woods for their helpful remarks and comments.

References

  1. Leonard M. Adleman. Molecular computation of solutions to combinatorial problems. Science, 266:1021-4, 1994. Google Scholar
  2. Leonard M. Adleman, Qi Cheng, Ashish Goel, and Ming-Deh Huang. Running time and program size for self-assembled squares. In STOC: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pages 740-748, Hersonissos, Greece, 2001. URL: https://doi.org/10.1145/380752.380881.
  3. Florent Becker, Daniel Hader, and Matthew J. Patitz. Strict self-assembly of discrete self-similar fractals in the abstract tile assembly model. In Yossi Azar and Debmalya Panigrahi, editors, Proceedings of the 2025 Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2025, New Orleans, LA, USA, January 12-15, 2025, pages 2387-2466. SIAM, 2025. URL: https://doi.org/10.1137/1.9781611978322.80.
  4. Bahar Behsaz, Ján Maňuch, and Ladislav Stacho. Turing universality of step-wise and stage assembly at Temperature 1. In DNA18: Proc. of International Meeting on DNA Computing and Molecular Programming, volume 7433 of LNCS, pages 1-11. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-32208-2_1.
  5. Luis Ceze, Jeff Nivala, and Karin Strauss. Molecular digital data storage using DNA. Nature Reviews Genetics, 20(8):456-466, August 2019. Google Scholar
  6. Matthew Cook, Yunhui Fu, and Robert T. Schweller. Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In SODA: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 570-589, 2011. Arxiv preprint: URL: http://arxiv.org/abs/0912.0027.
  7. David Doty, Jack H. Lutz, Matthew J. Patitz, Robert T. Schweller, Scott M. Summers, and Damien Woods. The tile assembly model is intrinsically universal. In FOCS: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science, pages 439-446. IEEE, October 2012. Arxiv preprint: URL: http://arxiv.org/abs/1111.3097.
  8. David Doty, Matthew J. Patitz, and Scott M. Summers. Limitations of self-assembly at temperature 1. Theoretical Computer Science, 412(1-2):145-158, 2011. Arxiv preprint: URL: http://arxiv.org/abs/0903.1857v1.
  9. Phillip Drake, Daniel Hader, and Matthew J. Patitz. Simulation of the Abstract Tile Assembly Model Using Crisscross Slats. In Shinnosuke Seki and Jaimie Marie Stewart, editors, 30th International Conference on DNA Computing and Molecular Programming (DNA 30), volume 314 of Leibniz International Proceedings in Informatics (LIPIcs), pages 3:1-3:25, Dagstuhl, Germany, 2024. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.DNA.30.3.
  10. Sándor P. Fekete, Jacob Hendricks, Matthew J. Patitz, Trent A. Rogers, and Robert T. Schweller. Universal computation with arbitrary polyomino tiles in non-cooperative self-assembly. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 148-167. SIAM, 2015. URL: https://arxiv.org/abs/1408.3351.
  11. David Furcy and Scott M Summers. Optimal self-assembly of finite shapes at temperature 1 in 3D. Algorithmica, 80(6):1909-1963, 2018. URL: https://doi.org/10.1007/S00453-016-0260-6.
  12. David Furcy, Scott M Summers, and Christian Wendlandt. New bounds on the tile complexity of thin rectangles at temperature-1. In DNA25: International Conference on DNA Computing and Molecular Programming, pages 100-119. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-26807-7_6.
  13. Oscar Gilbert, Jacob Hendricks, Matthew J Patitz, and Trent A Rogers. Computing in continuous space with self-assembling polygonal tiles. In SODA: ACM-SIAM Symposium on Discrete Algorithms, pages 937-956. SIAM, 2016. Arxiv preprint. URL: https://arxiv.org/abs/1503.00327.
  14. Sergiu Ivanov and Damien Regnault. A linear bound for the size of the finite terminal assembly of a directed non-cooperative tile assembly system, 2024. URL: https://arxiv.org/abs/2405.18630.
  15. Natasa Jonoska and Daria Karpenko. Active tile self-assembly, part 1: Universality at temperature 1. Int. J. Found. Comput. Sci., 25(2):141-164, 2014. URL: https://doi.org/10.1142/S0129054114500087.
  16. Ján Maňuch, Ladislav Stacho, and Christine Stoll. Two lower bounds for self-assemblies at Temperature 1. Journal of Computational Biology, 17(6):841-852, 2010. URL: https://doi.org/10.1089/CMB.2009.0067.
  17. Pierre-Étienne Meunier. Non-cooperative algorithms in self-assembly. In UCNC: Unconventional Computation and Natural Computation, volume 9252 of LNCS, pages 263-276. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21819-9_20.
  18. Pierre-Étienne Meunier, Matthew J. Patitz, Scott M. Summers, Guillaume Theyssier, Andrew Winslow, and Damien Woods. Intrinsic universality in tile self-assembly requires cooperation. In SODA: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pages 752-771, 2014. Arxiv preprint. URL: https://arxiv.org/abs/1304.1679.
  19. Pierre-Étienne Meunier and Damien Regnault. Non-cooperatively assembling large structures. In DNA Computing and Molecular Programming - 25th International Conference, DNA 25, Seattle, WA, USA, August 5-9, 2019, Proceedings, 2019. URL: https://doi.org/10.1007/978-3-030-26807-7_7.
  20. Pierre-Étienne Meunier and Damien Regnault. Directed Non-Cooperative Tile Assembly Is Decidable. In 27th International Conference on DNA Computing and Molecular Programming (DNA 27), volume 205 of Leibniz International Proceedings in Informatics (LIPIcs), pages 6:1-6:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.DNA.27.6.
  21. Pierre-Étienne Meunier, Damien Regnault, and Damien Woods. The program-size complexity of self-assembled paths. In Proccedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, Chicago, IL, USA, June 22-26, 2020, pages 727-737. ACM, 2020. Arxiv preprint: https://arxiv.org/abs/2002.04012 [cs.CC]. URL: https://doi.org/10.1145/3357713.3384263.
  22. Pierre-Étienne Meunier and Damien Woods. The non-cooperative tile assembly model is not intrinsically universal or capable of bounded Turing machine simulation. In STOC: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 328-341, Montreal, Canada, 2017. ACM. Arxiv preprint with full proofs: https://arxiv.org/abs/1702.00353v2 [cs.CC].
  23. Dionis Minev, Christopher M. Wintersinger, Anastasia Ershova, and William M. Shih. Robust nucleation control via crisscross polymerization of dna slats. bioRxiv, 2019. URL: https://doi.org/10.1101/2019.12.11.873349.
  24. Matthew J. Patitz, Robert T. Schweller, and Scott M. Summers. Exact shapes and Turing universality at Temperature 1 with a single negative glue. In DNA 17: Proceedings of the Seventeenth International Conference on DNA Computing and Molecular Programming, LNCS, pages 175-189. Springer, September 2011. Arxiv preprint. URL: https://arxiv.org/abs/1105.1215.
  25. Lulu Qian, Erik Winfree, and Jehoshua Bruck. Neural network computation with DNA strand displacement cascades. Nature, 475(7356):368-372, 2011. URL: https://doi.org/10.1038/NATURE10262.
  26. Paul W. K. Rothemund. Folding DNA to create nanoscale shapes and patterns. Nature, 440(7082):297-302, March 2006. URL: https://doi.org/10.1038/nature04586.
  27. Paul W. K. Rothemund and Erik Winfree. The program-size complexity of self-assembled squares (extended abstract). In STOC: Proceedings of the thirty-second annual ACM Symposium on Theory of Computing, pages 459-468, Portland, Oregon, 2000. ACM. URL: https://doi.org/10.1145/335305.335358.
  28. Paul W.K. Rothemund, Nick Papadakis, and Erik Winfree. Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biology, 2(12):2041-2053, 2004. Google Scholar
  29. David Soloveichik and Erik Winfree. Complexity of self-assembled shapes. SIAM Journal on Computing, 36(6):1544-1569, 2007. URL: https://doi.org/10.1137/S0097539704446712.
  30. Hao Wang. Proving theorems by pattern recognition - II. The Bell System Technical Journal, XL(1):1-41, 1961. Google Scholar
  31. Elena M. Willner, Fenna Kolbe, Frank Momburg, Ulrike Protzer, and Hendrik Dietz. Hepatitis b virus neutralization with dna origami nanoshells. ACS Applied Materials and Interfaces, 16:25836-25842, 2024. Google Scholar
  32. Erik Winfree. Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology, June 1998. Google Scholar
  33. Christopher M. Wintersinger, Dionis Minev, Anastasia Ershova, Hiroshi M. Sasaki, Gokul Gowri, Jonathan F. Berengut, F. Eduardo Corea-Dilbert, Peng Yin, and William M. Shih. Multi-micron crisscross structures grown from dna-origami slats. Nature Nanotechnology, 18:281-289, 2023. Google Scholar
  34. Damien Woods, David Doty, Cameron Myhrvold, Joy Hui, Felix Zhou, Peng Yin, and Erik Winfree. Diverse and robust molecular algorithms using reprogrammable dna self-assembly. Nature, 567:366-372, March 2019. URL: https://doi.org/10.1038/s41586-019-1014-9.
  35. Bernard Yurke, Andrew J Turberfield, Allen P Mills, Friedrich C Simmel, and Jennifer L Neumann. A DNA-fuelled molecular machine made of DNA. Nature, 406(6796):605-608, 2000. Google Scholar
  36. Pierre Étienne Meunier and Damien Regnault. A pumping lemma for non-cooperative self-assembly, 2013. URL: https://arxiv.org/abs/1312.6668.
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