# Characterizing PSPACE with shallow non-confluent P systems

## TL;DR

This paper proves that shallow non-confluent P systems with active membranes exactly characterize the complexity class PSPACE, clarifying the computational power of non-confluence in these systems.

## Contribution

It establishes that non-confluent shallow P systems precisely characterize PSPACE, linking non-confluence and membrane nesting depth in computational power.

## Key findings

- Non-confluent shallow P systems characterize PSPACE.
- PSPACE is both an upper and lower bound for these systems.
- Non-confluence provides computational power equivalent to deep confluent systems with membrane division.

## Abstract

In P systems with active membranes, the question of understanding the power of non-confluence within a polynomial time bound is still an open problem. It is known that, for shallow P systems, that is, with only one level of nesting, non-confluence allows them to solve conjecturally harder problems than confluent P systems, thus reaching PSPACE. Here we show that PSPACE is not only a bound, but actually an exact characterization. Therefore, the power endowed by non-confluence to shallow P systems is equal to the power gained by confluent P systems when non-elementary membrane division and polynomial depth are allowed, thus suggesting a connection between the roles of non-confluence and nesting depth.