# Verification in Staged Tile Self-Assembly

## TL;DR

This paper analyzes the computational complexity of verification problems in staged tile self-assembly, establishing their hardness and containment within certain complexity classes, and proving specific problems are complete for these classes.

## Contribution

It proves the complexity of unique assembly and shape verification problems in staged tile self-assembly models, including their hardness and class containment, and establishes completeness results.

## Key findings

- Unique assembly verification is coNP^NP-hard.
- Shape verification in 2HAM is coNP^NP-complete.
- Verification problems are in PSPACE and in Pi^P_{2s} for staged systems.

## Abstract

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