Obradovic and Parberry showed that any n-input k-ary function can be computed by a depth 4 unit-weight k-ary threshold circuit of size O(nkn). They also showed that any n-input k-ary symmetric function can be computed by a depth 6 unit-weight k-ary threshold circuit of size O(nk+1). In this paper, we improve upon and expand their results. The k-ary threshold circuits of nonunit weight and unit weight are considered. We show that any n-input k-ary function can be computed by a depth 2 k-ary threshold circuit of size O(kn-1). This means that depth 2 is optimal for computing some k-ary functions (e.g., a PARITY function). We also show that any n-input k-ary function can be computed by a depth 3 unit-weight k-ary threshold circuit of size O(kn). Next, we show that any n-input k-ary symmetric function can be computed by a depth 3 k-ary threshold circuit of size O(nk-1), and can be computed by a depth 3 unit-weight k-ary threshold circuit of size O(knk-1). Finally, we show that if the weights of the circuit are polynomially bounded, some k-ary symmetric functions cannot be computed by any depth 2 k-ary threshold circuit of polynomial-size.

In this paper, we show that a parity function with n variables can be computed by a threshold circuit of depth O((log n)/c) and size O((2clog n)/c), for all 1c [log(n+1)]-1. From this construction, we obtain an O(log n/log log n) upper bound for the depth of polylogarithmic size threshold circuits for parity functions. By using the result of Impagliazzo, Paturi and Saks[5], we also show an Ω (log n/log log n) lower bound for the depth of the threshold circuits. This is an answer to the open question posed in [11].