Optimal Lower Bounds for Symmetric Modular Circuits

TL;DR

This paper establishes subexponential lower bounds for symmetric modular circuits computing the Boolean AND function, showing optimal size is achieved at depth 2, advancing understanding in circuit complexity.

Contribution

It provides the first nontrivial lower bounds for symmetric modular circuits computing AND, matching known upper bounds and analyzing depth and symmetry constraints.

Findings

Subexponential lower bounds for symmetric MOD$_m$-circuits computing AND.

Optimal symmetric circuit size achieved at depth 2.

Tight size bounds for circuits with nested block symmetry.

Abstract

A notorious open question in circuit complexity is whether Boolean operations of arbitrary arity can efficiently be expressed using modular counting gates only. H{\aa}stad's celebrated switching lemma yields exponential lower bounds for the dual problem - realising modular arithmetic with Boolean gates - but, a similar lower bound for modular circuits computing the Boolean AND function has remained elusive for almost 30 years. We solve this problem for the restricted model of symmetric circuits: We consider MOD$_m$-circuits of arbitrary depth, and for an arbitrary modulus $m \in \mathbb{N}$, and obtain subexponential lower bounds for computing the $n$-ary Boolean AND function, under the assumption that the circuits are syntactically symmetric under all permutations of their $n$ input gates. This lower bound is matched precisely by a construction due to (Idziak, Kawa{\l}ek, Krzaczkowski, LICS'22), leading to the surprising conclusion that the optimal symmetric circuit size is already achieved with depth $2$. Motivated by another construction from (LICS'22), which achieves smaller size at the cost of greater depth, we also prove tight size lower bounds for circuits with a more liberal notion of symmetry characterised by a nested block structure on the input variables.