Torus polynomials: an algebraic approach to ACC lower bounds

TL;DR

This paper introduces torus polynomials as an algebraic framework to analyze circuit lower bounds for ACC0, connecting polynomial approximation techniques with new algebraic tools, and explores their limitations for the majority function.

Contribution

It develops the concept of torus polynomials for circuit complexity and demonstrates their application to approximating ACC0 and limitations for majority.

Findings

ACC0 can be approximated by low-degree torus polynomials

Majority cannot be approximated by low-degree symmetric torus polynomials

Framework opens new avenues for circuit lower bound proofs

Abstract

We propose an algebraic approach to proving circuit lower bounds for ACC0 by defining and studying the notion of torus polynomials. We show how currently known polynomial-based approximation results for AC0 and ACC0 can be reformulated in this framework, implying that ACC0 can be approximated by low-degree torus polynomials. Furthermore, as a step towards proving ACC0 lower bounds for the majority function via our approach, we show that MAJORITY cannot be approximated by low-degree symmetric torus polynomials. We also pose several open problems related to our framework.