On a model of plane switching circuits

We introduce a model of plane contact scheme which takes into account the possibility to carry out controlling actions at the contacts of circuits. For the Shannon function $L(n)$ which characterizes the minimal area needed to realize an arbitrary Boolean function in $n$ variables by these schemes we obtain estimates of the form
$$ {2^{n} \over \log _{2}36} \mathbin{\scriptstyle\lesssim} L(n) \mathbin{\scriptstyle\lesssim} 2^{n}. $$