Lower Bounds for Monotone Counting Circuits

TL;DR

This paper investigates the complexity of monotone counting circuits, demonstrating that counting can be exponentially easier than computing or deciding certain multilinear polynomials, and establishes general lower bounds on circuit size.

Contribution

It introduces the concept of counting circuits, shows exponential separations in complexity, and provides fundamental lower bounds for their size.

Findings

Counting can be exponentially easier than computing for some polynomials.

Deciding can be exponentially easier than counting for certain polynomials.

General lower bounds on the size of counting circuits are established.

Abstract

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.