# A Note on the Power of Non-Deterministic Circuits with Gate Restrictions

## TL;DR

This paper explores how non-deterministic circuits with limited gate sets compare to deterministic ones, revealing a dichotomy where their relative power depends on the gate base, with implications for computational complexity.

## Contribution

It provides a clear characterization of when non-deterministic circuits outperform deterministic circuits under gate restrictions, based on the strength of the base gates.

## Key findings

- Non-deterministic circuits are equivalent to deterministic ones over weak gate bases.
- Over stronger bases, non-deterministic circuits are super-polynomially more efficient.
- A precise boundary between the two regimes is established.

## Abstract

We investigate the power of non-deterministic circuits over restricted sets of base gates. We note that the power of non-deterministic circuits exhibit a dichotomy, in the following sense: For weak enough bases, non-deterministic circuits are no more powerful than deterministic circuits, and for the remaining bases, non-deterministic circuits are super polynomial more efficient than deterministic circuits (under the assumption that $P/poly \neq NP/poly$). Moreover, we give a precise characterization of the borderline between the two situations.