# The PCP Theorem for NP Over the Reals

- 136 Downloads
- 4 Citations

## Abstract

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 \(\mathrm{NP}_{{\mathbb {C}}}\).

## Keywords

Probabilistically checkable proofs Real and complex number computations PCP theorem Hilbert Nullstellensatz decision problem## Mathematics Subject Classification

03D78 68Q15 68Q17 68Q87## Notes

### Acknowledgments

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.

## References

- 1.S. Arora and B. Barak, Computational Complexity: A Modern Approach, Cambridge University Press, 2009.Google Scholar
- 2.S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy, Proof verification and hardness of approximation problems, J. ACM 45 (3), (1998), 501–555. Extended abstract in Proc. of the 33rd Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 1992, 14–23.Google Scholar
- 3.S. Arora and S. Safra, Probabilistic checking proofs: a new characterization of \(NP\), J. ACM 45 (1), (1998), 70–122. Extended abstract in Proc. of the 33rd Annual Symposium on the Foundations of Computer Science, IEEE Computer Society, 1992, 2–13.Google Scholar
- 4.M. Baartse and K. Meer, Testing low degree trigonometric polynomials. Extended abstract in Proc. 9th International Computer Science Symposium in Russia CSR 2014, Moscow (N.K. Vereshchagin, E.A. Hirsch, S.O. Kuznetsov, and J.E. Pin, eds.), Springer LNCS, 2014.Google Scholar
- 5.M. Baartse and K. Meer, Topics in real and complex number complexity theory, in Recent Advances in Real Complexity and Computation (J.L. Montaña and L.M. Pardo, eds.), Contemporary Mathematics, vol. 604, American Mathematical Society (2013), 1–53.Google Scholar
- 6.M. Baartse and K. Meer, The PCP Theorem for NP over the Reals. Extended abstract in Proc. 30th Symposium on Theoretical Aspects of Computer Science STACS 2013 (N. Portier and T. Wilke, eds.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 20, Schloss Dagstuhl - Leibniz Zentrum für Informatik (2013), pp. 104–115, doi: 10.4230/LIPIcs.STACS.2013.104.
- 7.L. Blum, F. Cucker, M. Shub, and S. Smale, Complexity and Real Computation, Springer, 1998.Google Scholar
- 8.L. Blum, M. Shub, and S. Smale, On a theory of computation and complexity over the real numbers, NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc., vol. 21 (1989), 1–46.CrossRefzbMATHMathSciNetGoogle Scholar
- 9.F. Cucker and M. Matamala, On digital nondeterminism. Mathematical Systems Theory 29 (1996), 635–647.CrossRefzbMATHMathSciNetGoogle Scholar
- 10.F. Cucker, M. Karpinski, P. Koiran, T. Lickteig, and K. Werther, On Real Turing Machines that Toss Coins, in Proceedings 27th Annual ACM Symposium on the Theory of Computing STOC (1995), 335–342.Google Scholar
- 11.I. Dinur, The PCP theorem by gap amplification. J. ACM vol. 54 (3), Article 12 (2012). doi: 10.1145/1236457.1236459.
- 12.K. Friedl, Z. Hátsági and A. Shen, Low-Degree Tests, in Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms SODA (D.D. Sleator, ed.), 1994, 57–64.Google Scholar
- 13.O. Goldreich, Basic Facts about Expander Graphs, in Studies in Complexity and Cryptography (O. Goldreich, ed.), Springer Lecture Notes in Computer Science 6650, 2011, 451–464.Google Scholar
- 14.K. Meer, Almost Transparent Short Proofs for NP\(_{\mathbb{R}}\), Extended abstract in Proc. 18th International Symposium on Fundamentals of Computation Theory FCT 2011 (O. Owe, M. Steffen, and J.A. Telle, eds.), Lecture Notes in Computer Science 6914, 2011, 41–52.Google Scholar
- 15.K. Meer, Transparent long proofs: a first PCP theorem for NP\(_{\mathbb{R}}\), , Vol. 5, Nr. 3 (2005), 231–255.CrossRefzbMATHMathSciNetGoogle Scholar
- 16.O. Reingold, S. Vadhan, and A. Wigderson, Entropy waves, the zig-zag graph product, and new constant degree expanders, Annals of Mathematics, vol. 155 (2002), 157–187.CrossRefzbMATHMathSciNetGoogle Scholar