The PCP Theorem for NP Over the Reals
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 \(\mathrm{NP}_{{\mathbb {C}}}\).
pcp theorem probabilistically checkable proofs real and complex number computations hilbert nullstellensatz decision problem 03d78 68q15 68q17 68q87