Advertisement

Interactive proofs and a Shamir-like result for real number computations

  • Martijn Baartse
  • Klaus MeerEmail author
Article
  • 17 Downloads

Abstract

We introduce and study interactive proofs in the framework of real number computations as introduced by Blum, Shub, and Smale. Ivanov and de Rougemont started this line of research showing that an analogue of Shamir’s result holds in the real additive Blum–Shub–Smale model of computation when only Boolean messages can be exchanged. Here, we introduce interactive proofs in the full BSS model in which also multiplications can be performed and reals can be exchanged. The ultimate goal is to give a Shamir-like characterization of the real counterpart \({{\rm IP}_\mathbb{R}}\) of classical IP. Whereas classically Shamir’s result implies IP  =  PSPACE  =  PAT  =  PAR, in our framework a major difficulty arises: In contrast to Turing complexity theory, the real number classes \({{\rm PAR}_\mathbb{R}}\) and \({{\rm PAT}_\mathbb{R}}\) differ and space resources considered separately are not meaningful. It is not obvious how to figure out whether at all \({{\rm IP}_\mathbb{R}}\) is characterized by one of the above classes—and if so by which.

We obtain two main results, an upper and a lower bound for the new class \({{\rm IP}_\mathbb{R}.}\) As upper bound we establish \({{{\rm IP}_\mathbb{R}} \subseteq {\rm MA\exists}_\mathbb{R}}\), where \({{\rm MA} \exists_\mathbb{R}}\) is a real complexity class introduced by Cucker and Briquel satisfying \({{\rm PAR}_\mathbb{R} \subsetneq {\rm MA}\exists_{\mathbb{R}} \subseteq {\rm PAT}_\mathbb{R}}\) and conjectured to be different from \({{\rm PAT}_\mathbb{R}}\). We then complement this result and prove a non-trivial lower bound for \({{\rm IP}_\mathbb{R}}\). More precisely, we design interactive real protocols verifying function values for a large class of functions introduced by Koiran and Perifel and denoted by UniformVPSPACE\({^{0}.}\) As a consequence, we show \({{\rm PAR}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}\), which in particular implies co-\({{\rm NP}_\mathbb{R} \subseteq {\rm IP}_\mathbb{R}}\), and \({{\rm P}_\mathbb{R}^{Res} \subseteq {\rm IP}_\mathbb{R}}\), where Res denotes certain multivariate Resultant polynomials.

Our proof techniques are guided by the question in how far Shamir’s classical proof can be used as well in the real number setting. Towards this aim results by Koiran and Perifel on UniformVPSPACE\({^{0}}\) are extremely helpful.

Keywords

Computational complexity Blum–Shub–Smale model Interactive proofs 

Subject classification

68Q15 68Q05 03D78 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We thank the anonymous referees for their careful reading of the paper and several useful comments to improve its presentation. The paper is an extended version of the two conference papers Baartse & Meer (2015b, 2016). Both authors were partially supported under projects ME 1424/7-1 and ME 1424/7-2 by the Deutsche Forschungsgemeinschaft DFG. We gratefully acknowledge the support.

References

  1. S. Arora & B. Barak (2009). Computational Complexity: A Modern Approach. Cambridge University Press.Google Scholar
  2. Baartse, M., Meer, K.: The PCP theorem for NP over the reals. Foundations of Computational Mathematics 15(3), 651–680 (2015a)MathSciNetCrossRefGoogle Scholar
  3. M. Baartse & K. Meer (2015b). Some results on interactive proofs for real computations. In 11th conference Computability in Europe CiE 2015, Bucharest. Proceedings, A. Beckmann, V. Mitrana & M. Soskova, editors, volume 9136 of LNCS, 107–116. SpringerGoogle Scholar
  4. M. Baartse & K. Meer (2016). Real Interactive Proofs for VPSPACE. In 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22–26, 2016–Kraków, Poland, Piotr Faliszewski, Anca Muscholl & Rolf Niedermeier, editors, volume 58 of LIPIcs, 14:1–14:13. Schloss Dagstuhl-Leibniz-Zentrum für Informatik. http://www.dagstuhl.de/dagpub/978-3-95977-016-3
  5. Baartse, M., Meer, K.: An algebraic proof of the real number PCP theorem. Journal of Complexity 40, 34–77 (2017)MathSciNetCrossRefGoogle Scholar
  6. Basu, S., Zell, T.: Polynomial hierarchy, Betti numbers, and a real analogue of Toda's theorem. Foundations of Computational Mathematics 10(4), 429–454 (2010)MathSciNetCrossRefGoogle Scholar
  7. L. Blum, F. Cucker, M. Shub & S. Smale (1998). Complexity and real computation Springer, New York, xvi+453CrossRefGoogle Scholar
  8. L. Blum, M. Shub & S. Smale (1989). On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc (N.S.) 21(1), 1–46MathSciNetCrossRefGoogle Scholar
  9. Olivier Chapuis & Pascal Koiran (1999). Saturation and stability in the theory of computation over the reals. Ann. Pure Appl. Logic 99(1–3), 1–49. ISSN 0168–0072Google Scholar
  10. Cucker, F.: On the complexity of quantifier elimination: The structural approach. The Computer Journal 36(5), 400–408 (1993)MathSciNetCrossRefGoogle Scholar
  11. Cucker, F., Briquel, I.: A note on parallel and alternating time. Journal of Complexity 23, 594–602 (2007)MathSciNetCrossRefGoogle Scholar
  12. Goldwasser, S.: Interactive Proof Systems. In Computational Complexity Theory, Proc. of Symposia in Applied Mathematics. J. Hartmanis, editor 38, 108–128 (1989)MathSciNetGoogle Scholar
  13. S. Goldwasser & M. Sipser (1986). Private coins versus public coins in interactive proof systems. In Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28–30, 1986, Berkeley, California, USA, 59–68Google Scholar
  14. Ivanov, S., de Rougemont, M.: Interactive Protocols on the reals. Computational Complexity 8, 330–345 (1999)MathSciNetCrossRefGoogle Scholar
  15. Koiran, P., Perifel, S.: VPSPACE and a transfer theorem over the reals. Computational Complexity 18(4), 551–575 (2009)MathSciNetCrossRefGoogle Scholar
  16. Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. Journal of the ACM 39(4), 859–868 (1992)MathSciNetCrossRefGoogle Scholar
  17. G. Malod (2011). Succinct Algebraic Branching Programs Characterizing Non-Uniform Complexity Classes. In Proc. ,18th International Symposium on Fundamentals of Computation Theory FCT 2011, Oslo, Olaf Owe, Martin Steffen & Jan Arne Telle, editors, volume 6914 of Lecture Notes in Computer Science, 205–216. SpringerGoogle Scholar
  18. Michaux, C.: Une remarque à propos des machines sur \(\mathbb{R}\) introduites par Blum, Shub et Smale. C.R. Acad. Sci. Paris 309, 435–437 (1989)MathSciNetzbMATHGoogle Scholar
  19. Poizat, B.: Â la recherche de la définition de la complexité d'espace pour le calcul des polynômes à la manière de Valiant. Journal of Symbolic Logic 73(4), 1179–1201 (2008)MathSciNetCrossRefGoogle Scholar
  20. Renegar, J.: On the computational Complexity and Geometry of the first-order Theory of the Reals, I-III. Journal of Symbolic Computation 13, 255–352 (1992)MathSciNetCrossRefGoogle Scholar
  21. Shamir, A.: IP = PSPACE. Journal of the ACM 39(4), 869–877 (1992)MathSciNetCrossRefGoogle Scholar
  22. L.J. Stockmeyer & A.R. Meyer (1973). Word problems requiring exponential time. In Proceedings of the 5th Annual ACM Symposium on Theory of Computing, April 30-May 2, 1973, Austin, Texas, USA, 1–9Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Lehrstuhl Theoretische InformatikBrandenburgische Technische, Universität Cottbus-SenftenbergCottbusGermany

Personalised recommendations