Notes on Boolean Read-k and Multilinear Circuits

A monotone Boolean (AND,OR) 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 a 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 monotone read-1 circuits have the same power as: tropical (min,+) circuits solving 0/1 minimization problems, monotone arithmetic (+,x) circuits computing multilinear homogeneous polynomials, as well as multilinear non-monotone (AND,OR,NOT) circuits computing monotone homogeneous Boolean functions. Finally, we show that already monotone read-2 circuits can be exponentially smaller than monotone read-1 circuits.