Periodic symmetric functions, serial addition, and multiplication with neural networks

This paper investigates threshold based neural networks for periodic symmetric Boolean functions and some related operations. It is shown that any n-input variable periodic symmetric Boolean function can be implemented with a feedforward linear threshold-based neural network with size of O(log n) and depth also of O(log n), both measured in terms of neurons. The maximum weight and fan-in values are in the order of O(n). Under the same assumptions on weight and fan-in values, an asymptotic bound of O(log n) for both size and depth of the network is also derived for symmetric Boolean functions that can be decomposed into a constant number of periodic symmetric Boolean subfunctions. Based on this results neural networks for serial binary addition and multiplication of n-bit operands are also proposed. It is shown that the serial addition can be computed with polynomially bounded weights and a maximum fan-in in the order of O(log n) in O(n= log n) serial cycles, where a serial cycle comprises a neural gate and a latch. The implementation cost is in the order of O(log n), in terms of neural gates, and in the order of O(log2 n), in terms of latches. Finally, it is shown that the serial multiplication can be computed in O(n) serial cycles with O(log n) size neural gate network, and with O(n log n) latches. The maximum weight value in the network is in the order of O(n2) and the maximum fan-in is in the order of O(n log n).