Abstract

A Threshold Circuit consists of an acyclic digraph of unbounded fanin, where each node computes a threshold function or its negation. This paper investigates the computational power of Threshold Circuits. A surprising relationship is uncovered between Threshold Circuits and another class of unbounded fanin circuits which are denoted Finite Field ZP (n) Circuits, where each node computes either multiple sums or products of integers modulo a prime P (n). In particular, it is proved that all functions computed by Threshold Circuits of size S(n) n and depth D(n) can also be computed by ZP (n) Circuits of size O(S(n) log S(n)+nP (n) log P (n)) and depth O(D(n)). Furthermore, it is shown that all functions computed by ZP (n) Circuits of size S(n) and depth D(n) can be computed by Threshold Circuits of size O ( 1 2 (S(n) log P (n)) 1+) and depth O ( 1 5 D(n)). These are the main results of this paper. There are many useful and quite surprising consequences of this result. For example, integer reciprocal can be computed in size n O(1) and depth O(1). More generally, anyanalytic function with a convergent rational polynomial power series (such as sine, cosine, exponentiation, square root, and logarithm) can be computed within accuracy 2,nc, for any constant c, by Threshold Circuits of

Citations