On the Computation of Boolean Functions by Analog Circuits of Bounded Fan-In

We consider the complexity of computing Boolean functions by analog circuits of bounded fan-in, i.e., by circuits of gates computing real-valued functions, either exactly or as sign-representation. Sharp upper bounds are obtained for the complexity of the most difficult n-variable function over certain bases (sign-representation by arithmetic circuits and exact computation by piecewise linear circuits). Bounds are given for the computational power gained by adding discontinuous gate functions and nondeterminism. We also prove explicit nonlinear lower bounds for the formula size of analog circuits over bases containing addition, subtraction, multiplication, the sign function and all real constants. ] 1997 Academic Press