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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
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