Notes on Boolean Read-k and Multilinear Circuits
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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, as well as the same power as monotone arithmetic (+,x) circuits computing multilinear homogeneous polynomials. We also show that monotone read-1 circuits are not weaker than (semantically) multilinear non-monotone (AND,OR,NOT) circuits. Finally, we show that already monotone read-2 circuits can be exponentially smaller than monotone read-1 circuits.
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