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A Weak Version of the Blum, Shub & Smale Model
, 1994
"... We propose a weak version of the BlumShubSmale 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 ..."
Abstract

Cited by 34 (6 self)
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We propose a weak version of the BlumShubSmale 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 semialgebraic sets which is of independent interest. As an application, we generalize recent results of Siegelmann & 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: \Pi an efficient simulation of orderfree real Turing machines by probabilistic Turing machines in the full BlumShubSmale model; \Pi an efficient simulation of arithmetic circuits over the integers by boolean circuits; \Pi the strict inclusion of the real polynomial hierarchy in weak exponentia...
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"... We propose a weak version of the BlumShubSmale model of computation over the real numbers. In this weak model only a “moderate ” usage of multiplications and divisions is allowed. The class of languages recognizable in polynomial tine is shown to be the complexity class P/poly. This implies under ..."
Abstract
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We propose a weak version of the BlumShubSmale model of computation over the real numbers. In this weak model only a “moderate ” usage of multiplications and divisions is allowed. The class of languages recognizable in polynomial tine is shown to be the complexity class P/poly. This implies under a standard complexitytheoretic assumption that P#NP in the weak model, and that problems such as the real traveling salesman problem cannot be solved in polynomial time. As an application, we generalize recent results of H. Siegelmann €4 E. D. Sontag on recurrent neural networks, and of W. Maass on feedforward nets. 1