Supercritical Tradeoffs for Monotone Circuits

TL;DR

This paper demonstrates a fundamental size-depth tradeoff in monotone circuits, showing that certain functions require exponential size at polynomial depth, revealing limits of monotone circuit efficiency.

Contribution

It introduces the first size-depth tradeoff result for monotone circuits in the supercritical regime, linking circuit complexity with proof complexity via new bracket formulas.

Findings

Existence of a monotone function with quasipolynomial size circuits but exponential size at polynomial depth

Introduction of bracket formulas with quasipolynomial resolution refutations

Establishment of size-depth tradeoffs in monotone circuit complexity

Abstract

We exhibit a monotone function computable by a monotone circuit of quasipolynomial size such that any monotone circuit of polynomial depth requires exponential size. This is the first size-depth tradeoff result for monotone circuits in the so-called supercritical regime. Our proof is based on an analogous result in proof complexity: We introduce a new family of unsatisfiable 3-CNF formulas (called bracket formulas) that admit resolution refutations of quasipolynomial size while any refutation of polynomial depth requires exponential size.