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

✍ Scribed by Pascal Koiran

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 646 KB
Volume: 54
Category: Article
ISSN: 0022-0000
DOI: 10.1006/jcss.1997.1478

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✦ Synopsis

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 PÂpoly. 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.

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