An arithmetic model of computation equivalent to threshold circuits

Abstract

We define a new structured and general model of computation: circuits using arbitrary fan-in arithmetic gates over the characteristic-two finite fields ( F 2 n ). These circuits have only one input and one output. We show how they correspond naturally to boolean computations with n inputs and n outputs. We show that if circuit sizes are polynomially related, then the arithmetic circuit depth and the threshold circuit depth to compute a given function differ by at most a constant factor. We use threshold functions with arbitrary weights; however, we show that when compared to the usual threshold model, the depth measure of this generalised model differs only by at most a constant factor (at polynomial size). The fan-in of our arithmetic model is also unbounded in the most generous sense: circuit size is measured as the number of Σ and Π-gates; there is no bound on the number of “wires”. We show that these results are provable for any reasonable correspondence between strings of n-bits and elements of F 2 n . And we find two such distinct characterizations. Thus, we show that arbitrary fan-in arithmetic computations over F 2 n constitute a precise abstraction of Boolean threshold computations with the pleasant property that various algebraic laws have been recovered.