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P$$\neq$$NP over the nonstandard reals implies P$$\neq$$NP over $$\mathbb{R}$$. (English) Zbl 0822.68033
Summary: L. Blum, M. Shub and S. Smale [Bull. Am. Math. Soc. 21, No. 1, 1-46 (1989; Zbl 0681.03020)] showed the existence of a NP-complete problem over the real closed fields in the framework of their theory of computation over the reals. This allows to ask for the P$$\neq$$NP question over real close fields. Here we show that P$$\neq$$NP over a real closed extension of the reals implies P$$\neq$$NP over the reals. We also discuss the converse. This leads to define some subclasses of P/poly. Finally we show that the transfer result about P$$\neq$$NP is a part of a very abstract result.

##### MSC:
 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 12L15 Nonstandard arithmetic and field theory
##### Keywords:
NP-complete problem
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##### References:
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