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Resiliency to multiple nucleation in temperature-1 self-assembly

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Abstract

We consider problems in variations of the two-handed abstract Tile Assembly Model (2HAM), a generalization of Erik Winfree’s abstract Tile Assembly Model (aTAM). In the latter, tiles attach one-at-a-time to a seed-containing assembly. In the former, tiles aggregate into supertiles that then further combine to form larger supertiles; hence, constructions must be robust to the choice of seed (nucleation) tiles. We obtain three distinct temperature-1 results in two 2HAM variants whose aTAM siblings are well-studied. In the first variant, called the restricted glue 2HAM (rg2HAM), glue strengths are restricted to \(-\,1\), 0, or 1. We prove this model is Turing universal, overcoming undesired growth by breaking apart undesired computation assembly via repulsive forces. In the second 2HAM variant, the 3D 2HAM (3D2HAM), tiles are (three-dimensional) cubes. We prove that assembling a (roughly two-layer) \(n \times n\) square in this model at temperature 1 is possible with \(O(\log ^2{n})\) tile types. The construction uses “cyclic, colliding” binary counters, and assembles the shape non-deterministically. Finally, we prove that there exist 3D2HAM systems that only assemble infinite aperiodic shapes.

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Notes

  1. Such a distinction is only needed in two-handed models, where the seed cannot be used as a “reference point”.

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Acknowledgements

Matthew J. Patitz: This author’s research was supported in part by National Science Foundation Grants CCF-1117672, CCF-1422152, and CAREER-1553166. Robert Schweller: This author’s research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1555626. Trent A. Rogers: This author’s research was supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE-1450079, and National Science Foundation Grant CCF-1422152

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Authors and Affiliations

  1. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Matthew J. Patitz & Trent A. Rogers

  2. Department of Computer Science, University of Texas–Rio Grande Valley, Edinburg, TX, USA

    Robert Schweller & Andrew Winslow

  3. Computer Science Department, University of Wisconsin–Oshkosh, Oshkosh, WI, USA

    Scott M. Summers

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  1. Matthew J. Patitz
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  2. Robert Schweller
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  3. Trent A. Rogers
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  4. Scott M. Summers
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  5. Andrew Winslow
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Corresponding author

Correspondence to Andrew Winslow.

Additional information

An extended abstract of this work that omitted several proofs and details was previously published in DNA Computing and Molecular Programming, LNCS, vol. 9818, pp. 98–113, Springer International Publishing (2016).

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Patitz, M.J., Schweller, R., Rogers, T.A. et al. Resiliency to multiple nucleation in temperature-1 self-assembly. Nat Comput 17, 31–46 (2018). https://doi.org/10.1007/s11047-017-9662-x

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  • DOI: https://doi.org/10.1007/s11047-017-9662-x

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