On Lower Bounds for Constant Width Arithmetic Circuits

TL;DR

This paper investigates the complexity of constant-width arithmetic circuits, establishing explicit polynomials with specific width and size properties, and demonstrating infinite hierarchies and hardness-randomness tradeoffs in identity testing.

Contribution

It introduces explicit polynomials that separate circuit widths and sizes, proving infinite hierarchies and hardness-randomness tradeoffs for identity testing in constant-width circuits.

Findings

Existence of explicit polynomials with width and size separation

Infinite hierarchies of monotone arithmetic circuits

Hardness-randomness tradeoffs for identity testing

Abstract

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the constant-depth hierarchies of monotone arithmetic circuits are infinite, both in the commutative and the noncommutative settings. 2. We prove hardness-randomness tradeoffs for identity testing constant-width commutative circuits analogous to [KI03,DSY08].