On the complexity of monotone circuits for threshold symmetric Boolean functions

The complexity of implementation of a threshold symmetric n-place Boolean function with threshold k = O(1) via circuits over the basis {∨, ∧} is shown not to exceed 2 log2 k ⋅ n + o(n). Moreover, the complexity of a threshold-2 function is proved to be 2n + Θ( $\begin{array}{} \sqrt n \end{array} $ ), and the complexity of a threshold-3 function is shown to be 3n + O(log n), the corresponding lower bounds are put forward.