Foundations of Computational Mathematics

, Volume 15, Issue 3, pp 651–680 | Cite as

The PCP Theorem for NP Over the Reals

  • Martijn Baartse
  • Klaus MeerEmail author


In this paper we show that the PCP theorem holds as well in the real number computational model introduced by Blum, Shub, and Smale. More precisely, the real number counterpart \(\mathrm{NP}_{{\mathbb {R}}}\) of the classical Turing model class NP can be characterized as \(\mathrm{NP}_{{\mathbb {R}}}= \mathrm{PCP}_{{\mathbb {R}}}(O(\log {n}), O(1))\). Our proof structurally follows the one by Dinur for classical NP. However, a lot of minor and major changes are necessary due to the real numbers as underlying computational structure. The analogue result holds for the complex numbers and \(\mathrm{NP}_{{\mathbb {C}}}\).


Probabilistically checkable proofs Real and complex number computations PCP theorem Hilbert Nullstellensatz decision problem 

Mathematics Subject Classification

03D78 68Q15 68Q17 68Q87 



We thank the anonymous referees for their numerous suggestions which helped a lot to improve the first version of the paper. Both authors were supported by project ME 1424/7-1 and ME 1424/7-2 of the Deutsche Forschungsgemeinschaft (DFG). We gratefully acknowledge the support.


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Copyright information

© SFoCM 2014

Authors and Affiliations

  1. 1.Computer Science InstituteBTU Cottbus-SenftenbergCottbusGermany

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