A note on monotone real circuits

We show that if a Boolean function f : {0, 1}n → {0, 1} can be computed by a monotone real circuit of size s using k-ary monotone gates then f can be computed by a monotone real circuit of size O(snk−2) which uses unary or binary monotone gates only. This partially solves an open problem presented in [2]. In fact, in size O(snk−1), the circuit uses only unary monotone gates and binary addition. We also show that if the monotone Karchmer-Wigerson game of f can be solved by a real communication protocol of size s then f can be computed by a monotone real circuit of the same size.