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Verification in staged tile self-assembly

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Abstract

We prove the unique assembly and unique shape verification problems, benchmark measures of self-assembly model power, are \(\textsf {coNP}^{\textsf {NP}}\)-hard and contained in PSPACE (and in \(\mathrm {\Pi }^\textsf {P}_{2s}\) for staged systems with s stages). En route, we prove that unique shape verification problem in the 2HAM is \(\textsf {coNP}^{\textsf {NP}}\)-complete.

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Notes

  1. The original staged model (Demaine et al. 2008) only considered O(1) distinct tile types, and thus for simplicity allowed tiles to be added at any stage (since \(\mathcal {O}(1)\) extra bins could hold the individual tile types to mix at any stage). Because systems here may have super-constant tile complexity, we restrict tiles to only be added at the initial stage.

  2. This is a slight modification of the original staged model (Demaine et al. 2008) in that there is no requirement of a final stage with a single output bin. This may be a slightly more capable model, and so it is considered here. However, all results in this paper apply to both variants of the model.

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Acknowledgements

This research was supported in part by National Science Foundation Grants CCF-1117672 and CCF-1555626 .

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Authors and Affiliations

  1. Department of Computer Science, University of Texas Rio Grande Valley, Edinburg, USA

    Robert Schweller, Andrew Winslow & Tim Wylie

Authors
  1. Robert Schweller
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  2. Andrew Winslow
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  3. Tim Wylie
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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 Unconventional Computation and Natural Computation, LNCS, vol. 10240, pp. 98–112, Springer International Publishing (2017).

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Schweller, R., Winslow, A. & Wylie, T. Verification in staged tile self-assembly. Nat Comput 18, 107–117 (2019). https://doi.org/10.1007/s11047-018-9701-2

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  • DOI: https://doi.org/10.1007/s11047-018-9701-2

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