Document Type
Article
Publication Date
2021
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.
Recommended Citation
Caballero, David, Timothy Gomez, Robert Schweller, and Tim Wylie. 2021. “Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete.” In 29th Annual European Symposium on Algorithms (ESA 2021), edited by Petra Mutzel, Rasmus Pagh, and Grzegorz Herman, 204:23:1-23:18. Leibniz International Proceedings in Informatics (LIPIcs). Dagstuhl, Germany: Schloss Dagstuhl – Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.ESA.2021.23.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Publication Title
29th Annual European Symposium on Algorithms (ESA 2021)
DOI
10.4230/LIPIcs.ESA.2021.23
Comments
© David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie