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Directed Non-Cooperative Tile Assembly Is Decidable

Authors Pierre-Étienne Meunier, Damien Regnault

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Author Details

Pierre-Étienne Meunier
  • Albédo Énergie, Le Bourget-du-Lac, France
Damien Regnault
  • IBISC, Université Évry, Université Paris-Saclay, 91025, Evry, France

Funding

  • Meunier, Pierre-Étienne: Supported by H2020-LC-SC3 award number 957823, H2020-LC-SC3 award number 953020, H2020-LC-SC3 award number 101033700, H2020-ERC award number 772766 and SFI grant 18/ERCS/5746 (this manuscript reflects only the authors' view and the European funding institutions are not responsible for any use that may be made of the information it contains).

Acknowledgements

We thank Damien Woods for his support and advice.

Cite AsGet BibTex

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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 205, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.DNA.27.6

Abstract

We provide a complete characterisation of all final states of a model called directed non-cooperative tile self-assembly, also called directed temperature 1 tile assembly, which proves that this model cannot possibly perform Turing computation. This model is a deterministic version of the more general undirected model, whose computational power is still open. Our result uses recent results in the domain, and solves a conjecture formalised in 2011. We believe that this is a major step towards understanding the full model. Temperature 1 tile assembly can be seen as a two-dimensional extension of finite automata, where geometry provides a form of memory and synchronisation, yet the full power of these "geometric blockings" was still largely unknown until recently (note that nontrivial algorithms which are able to build larger structures than the naive constructions have been found).

Subject Classification

ACM Subject Classification
  • Mathematics of computing
  • Theory of computation → Models of computation
Keywords
  • Self-assembly
  • Molecular Computing
  • Models of Computation
  • Computational Geometry

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