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 NP_R of the classical Turing model class NP can be characterized as NP_R = PCP_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 NP_C.
@InProceedings{baartse_et_al:LIPIcs.STACS.2013.104, author = {Baartse, Martijn and Meer, Klaus}, title = {{The PCP theorem for NP over the reals}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {104--115}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2013.104}, URN = {urn:nbn:de:0030-drops-39262}, doi = {10.4230/LIPIcs.STACS.2013.104}, annote = {Keywords: PCP, real number computation, systems of polynomials} }
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