On Lifting Lower Bounds for Noncommutative Circuits using Automata

TL;DR

This paper provides a simpler automata-theoretic proof for lifting lower bounds from noncommutative circuits to larger polynomial families, building on prior results that connect circuit complexity with automata theory.

Contribution

It introduces a more straightforward, automata-based proof of existing lower bound lifting results for noncommutative circuits, enhancing conceptual understanding.

Findings

Automata-theoretic proof simplifies previous arguments.

Lifting technique extends lower bounds to larger polynomial families.

Connects automata theory with noncommutative circuit complexity.

Abstract

We revisit the main result of Carmosino et al \cite{CILM18} which shows that an $\Omega(n^{\omega/2+\epsilon})$ size noncommutative arithmetic circuit size lower bound (where $\omega$ is the matrix multiplication exponent) for a constant-degree $n$-variate polynomial family $(g_n)_n$, where each $g_n$ is a noncommutative polynomial, can be ``lifted'' to an exponential size circuit size lower bound for another polynomial family $(f_n)$ obtained from $(g_n)$ by a lifting process. In this paper, we present a simpler and more conceptual automata-theoretic proof of their result.