A note on monotone real circuits

We show that if a Boolean function f:{0,1} n →{0,1} 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(sn k−2 ) O ( s n k − 2 ) which uses unary or binary monotone gates only. This partially solves an open problem presented in [2] . In fact, in size O(sn k−1 ) O ( s n k − 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.