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Complexity of Verification in Self-Assembly with Prebuilt Assemblies

Authors David Caballero, Timothy Gomez, Robert Schweller, Tim Wylie

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Author Details

David Caballero
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Timothy Gomez
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Robert Schweller
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA
Tim Wylie
  • Department of Computer Science, University of Texas Rio Grande Valley, TX, USA

Funding

This research was supported in part by National Science Foundation Grant CCF-1817602.

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David Caballero, Timothy Gomez, Robert Schweller, and Tim Wylie. Complexity of Verification in Self-Assembly with Prebuilt Assemblies. In 1st Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 221, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.SAND.2022.8

Abstract

We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger pre-built assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP^{NP}-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is in coNP for 𝒪(1) bounded temperature and coNP-complete when temperature is part of the input. We further provide preliminary results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, and provide polynomial time solutions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Applied computing → Computational biology
Keywords
  • 2-handed assembly
  • verification
  • prebuilt

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