Notes on Boolean read-k and multilinear circuits

A monotone Boolean $(\vee ,\wedge )$ circuit computing a monotone Boolean function $f$ is a read-$k$ circuit if the polynomial produced (purely syntactically) by the arithmetic $(+,\times )$ 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 $⩽k$. Every monotone circuit is a read-$k$ circuit for some $k$. We show that already read-1 $(\vee ,\wedge )$ circuits are not weaker than monotone arithmetic constant-free $(+,\times )$ circuits computing multilinear polynomials, are not weaker than non-monotone multilinear $(\vee ,\wedge ,\neg )$ circuits computing monotone Boolean functions, and have the same power as tropical $(min,+)$ circuits solving $0/1$ minimization problems. Finally, we show that read-2 $(\vee ,\wedge )$ circuits can be exponentially smaller than read-1 $(\vee ,\wedge )$ circuits.