A Weak Version of the Blum, Shub, and Smale Model

We propose a weak version of the Blum Shub Smale model of computation over the real numbers. In this weak model only a ``moderate'' usage of multiplications and divisions is allowed. The class of boolean languages recognizable in polynomial time is shown to be the complexity class P poly. The main tool is a result on the existence of small rational points in semi-algebraic sets which is of independent interest. As an application, we generalize recent results of Siegelmann and Sontag on recurrent neural networks, and of Maass on feedforward nets. A preliminary version of this paper was presented at the 1993 IEEE Symposium on Foundations of Computer Science. Additional results include: an efficient simulation of order-free real Turing machines by probabilistic Turing machines in the full Blum Shub Smale model; the strict inclusion of the real polynomial hierarchy in weak exponential time. ] 1997 Academic Press