License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.23
URN: urn:nbn:de:0030-drops-146047
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14604/
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Caballero, David ; Gomez, Timothy ; Schweller, Robert ; Wylie, Tim

Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete

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Abstract

Staged self-assembly has proven to be a powerful abstract model of self-assembly by modeling laboratory techniques where several nanoscale systems are allowed to assemble separately and then be mixed at a later stage. A fundamental problem in self-assembly is Unique Assembly Verification (UAV), which asks whether a single final assembly is uniquely constructed. This has previously been shown to be Π^{p}₂-hard in staged self-assembly with a constant number of stages, but a more precise complexity classification was left open related to the polynomial hierarchy.
Covert Computation was recently introduced as a way to compute a function while hiding the input to that function for self-assembly systems. These Tile Assembly Computers (TACs), in a growth only negative aTAM system, can compute arbitrary circuits, which proves UAV is coNP-hard in that model. Here, we show that the staged assembly model is capable of covert computation using only 3 stages. We then utilize this construction to show UAV with only 3 stages is Π^{p}₂-hard. We then extend this technique to open problems and prove that general staged UAV is PSPACE-complete. Measuring the complexity of n stage UAV, we show Π^{p}_{n - 1}-hardness. We finish by showing a Π^{p}_{n + 1} algorithm to solve n stage UAV leaving only a constant gap between membership and hardness.

BibTeX - Entry

@InProceedings{caballero_et_al:LIPIcs.ESA.2021.23,
  author =	{Caballero, David and Gomez, Timothy and Schweller, Robert and Wylie, Tim},
  title =	{{Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14604},
  URN =		{urn:nbn:de:0030-drops-146047},
  doi =		{10.4230/LIPIcs.ESA.2021.23},
  annote =	{Keywords: self-assembly, covert computation, staged self-assembly, assembly verification}
}

Keywords: self-assembly, covert computation, staged self-assembly, assembly verification
Collection: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue Date: 2021
Date of publication: 31.08.2021

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