Notes on Boolean Read-k and Multilinear Circuits

TL;DR

This paper investigates the computational power of read-k monotone Boolean circuits, showing that even read-1 circuits are as powerful as certain arithmetic and non-monotone circuits, and that read-2 circuits can be exponentially more efficient than read-1 circuits.

Contribution

It establishes the equivalence of read-1 circuits with various circuit models and demonstrates exponential size gaps between read-1 and read-2 circuits.

Findings

Read-1 circuits are as powerful as monotone arithmetic and non-monotone circuits for certain functions.

Read-2 circuits can be exponentially smaller than read-1 circuits.

Read-k circuits generalize several circuit models and show increased efficiency with higher k.

Abstract

A monotone Boolean (OR,AND) circuit computing a monotone Boolean function f is a read-k circuit if the polynomial produced (purely syntactically) by the arithmetic (+,x) version of the circuit has the property that for every prime implicant of f, the polynomial contains at least one monomial with the same set of variables, each appearing with degree at most k. Every monotone circuit is a read-k circuit for some k. We show that already read-1 (OR,AND) circuits are not weaker than monotone arithmetic constant-free (+,x) circuits computing multilinear polynomials, are not weaker than non-monotone multilinear (OR,AND,NOT) circuits computing monotone Boolean functions, and have the same power as tropical (min,+) circuits solving combinatorial minimization problems. Finally, we show that read-2 (OR,AND) circuits can be exponentially smaller than read-1 (OR,AND) circuits.