# Solving QSAT in sublinear depth

## TL;DR

This paper demonstrates that the QSAT problem, a PSPACE-complete problem, can be solved using P systems with active membranes at a nesting depth significantly less than linear, specifically of order n/log n.

## Contribution

The authors introduce a novel construction showing QSAT can be solved with sublinear depth in P systems, improving upon previous linear-depth requirements.

## Key findings

- QSAT solvable with sublinear depth of order n/log n
- P systems with active membranes are effective for complex problem solving
- Depth reduction does not compromise the computational power for QSAT

## Abstract

Among $\mathbf{PSPACE}$-complete problems, QSAT, or quantified SAT, is one of the most used to show that the class of problems solvable in polynomial time by families of a given variant of P systems includes the whole $\mathbf{PSPACE}$. However, most solutions require a membrane nesting depth that is linear with respect to the number of variables of the QSAT instance under consideration. While a system of a certain depth is needed, since depth 1 systems only allows to solve problems in $\mathbf{P^{\#P}}$, it was until now unclear if a linear depth was, in fact, necessary. Here we use P systems with active membranes with charges, and we provide a construction that proves that QSAT can be solved with a sublinear nesting depth of order $\frac{n}{\log n}$, where $n$ is the number of variables in the quantified formula given as input.