A Superpolynomial Lower Bound on the Size of Uniform Non-constant-depth Threshold Circuits for the Permanent

TL;DR

This paper proves that the permanent function cannot be computed by small, shallow threshold circuits with polynomial size, establishing a superpolynomial lower bound for such computational models.

Contribution

It establishes a superpolynomial lower bound on the size of uniform threshold circuits of depth o(log log n) for computing the permanent, a longstanding open problem.

Findings

Permanent cannot be computed by polynomial-size, shallow threshold circuits.

Depth o(log log n) threshold circuits are insufficient for computing the permanent.

Provides a new lower bound in circuit complexity theory.

Abstract

We show that the permanent cannot be computed by DLOGTIME-uniform threshold or arithmetic circuits of depth o(log log n) and polynomial size.