DEPTH EFFICIENT NEURAL NETWORKS FOR DIVISION AND RELATED PROBLEMS

An artificial neural network (ANN) is commonly modeled by a threshold circuit, a network of interconnected processing units called linear th reshold gates. The depth of a circuit represents the number of unit de lays or the time for parallel computation. The size of a circuit is th e number of gates and measures the amount of hardware. It was known th at traditional logic circuits consisting of only unbounded fanin AND, OR, NOT gates would require at least OMEGA(log n/log log n) depth to c ompute common arithmetic functions such as the product or the quotient of two n-bit numbers, if the circuit size is polynomially bounded (in n). It is shown that ANN's can be much more powerful than traditional logic circuits, assuming that each threshold gate can be built with a cost that is comparable to that of AND/OR logic gates. In particular, the main results show that powering and division can be computed by p olynomial-size ANN's of depth 4, and multiple product can be computed by polynomial-size ANN's of depth 5. Moreover, using the techniques de veloped here, a previous result can be improved by showing that the so rting of n n-bit numbers can be carried out in a depth-3 polynomial si ze ANN. Furthermore, it is shown that the sorting network is optimal i n depth.