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Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds

Authors Abhishek Bhowmick, Shachar Lovett

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  • nonclassical polynomials
  • polynomials
  • lower bounds
  • barrier

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Abstract

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

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Abhishek Bhowmick and Shachar Lovett. Nonclassical Polynomials as a Barrier to Polynomial Lower Bounds. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 72-87, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.CCC.2015.72

Author Details

Abhishek Bhowmick
Shachar Lovett

References

  1. Noga Alon, Oded Goldreich, Johan Hastad, and René Peralta. Simple constructions of almost k-wise independent random variables. Random Structures and Algorithms, 3(3):289-304, 1992. Google Scholar
  2. Boaz Barak, Anup Rao, Ronen Shaltiel, and Avi Wigderson. 2sourcee dispersers for sub-polynomial entropy and ramsey graphs beating the frankl-wilson construction. In Proceedings of the Thirty-eighth Annual ACM Symposium on Theory of Computing, STOC'06, pages 671-680, New York, NY, USA, 2006. ACM. Google Scholar
  3. D.A. Barrington and G. Tardos. A lower bound on the mod 6 degree of the or function. computational complexity, 7(2):99-108, 1998. Google Scholar
  4. David A. Barrington. Some problems involving Razborov-Smolensky polynomials. In Boolean function complexity (Durham, 1990), volume 169 of London Math. Soc. Lecture Note Ser., pages 109-128. Cambridge Univ. Press, Cambridge, 1992. Google Scholar
  5. David A. Mix Barrington, Richard Beigel, and Steven Rudich. Representing Boolean functions as polynomials modulo composite numbers. Comput. Complexity, 4(4):367-382, 1994. Special issue on circuit complexity (Barbados, 1992). Google Scholar
  6. Richard Beigel and John Gill. Counting classes: thresholds, parity, mods, and fewness. Theoret. Comput. Sci., 103(1):3-23, 1992. 7th Annual Symposium on Theoretical Aspects of Computer Science (STACS 90) (Rouen, 1990). Google Scholar
  7. Richard Beigel and Jun Tarui. On ACC. In 32nd Annual Symposium on Foundations of Computer Science (San Juan, PR, 1991), pages 783-792. IEEE Comput. Soc. Press, Los Alamitos, CA, 1991. Google Scholar
  8. Abhishek Bhowmick, Zeev Dvir, and Shachar Lovett. New Bounds for Matching Vector Families. SIAM J. Comput., 43(5):1654-1683, 2014. Google Scholar
  9. A. Bogdanov and E. Viola. Pseudorandom bits for polynomials. In Proc. 48^th IEEE Symp. on Foundations of Computer Science (FOCS'07), 2007. Google Scholar
  10. Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM J. Comput., 40(4):1154-1178, 2011. Google Scholar
  11. Zeev Dvir and Guangda Hu. Matching-vector families and ldcs over large modulo. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21-23, 2013. Proceedings, pages 513-526, 2013. Google Scholar
  12. Klim Efremenko. 3-query locally decodable codes of subexponential length. In Proceedings of the Forty-first Annual ACM Symposium on Theory of Computing, STOC'09, pages 39-44, New York, NY, USA, 2009. ACM. Google Scholar
  13. P. Frankl and R. M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981. Google Scholar
  14. Parikshit Gopalan. Constructing ramsey graphs from boolean function representations. Combinatorica, 34(2):173-206, April 2014. Google Scholar
  15. W.T. Gowers. A new proof of szemerédi’s theorem. Geometric and Functional Analysis GAFA, 11(3):465-588, 2001. Google Scholar
  16. Vince Grolmusz. Superpolynomial size set-systems with restricted intersections mod 6 and explicit Ramsey graphs. Combinatorica, 20(1):71-85, 2000. Google Scholar
  17. Shachar Lovett. Unconditional pseudorandom generators for low degree polynomials. Theory of Computing, 5(3):69-82, 2009. Google Scholar
  18. Edouard Lucas. Théorie des fonctions numériques simplement périodiques. American Journal of Mathematics, 1(2):pp. 184-196, 1878. Google Scholar
  19. Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM journal on computing, 22(4):838-856, 1993. Google Scholar
  20. Noam Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991. Google Scholar
  21. Noam Nisan and Avi Wigderson. Hardness vs. randomness. J. Comput. System Sci., 49(2):149-167, 1994. Google Scholar
  22. A. A. Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki, 41(4):598-607, 623, 1987. Google Scholar
  23. R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proc. 19^th ACM Symposium on Theory of Computing (STOC'87), pages 77-82, 1987. Google Scholar
  24. T. Tao and T. Ziegler. The inverse conjecture for the Gowers norm over finite fields in low characteristic. ArXiv e-prints, January 2011. Google Scholar
  25. Emanuele Viola. The sum of d small-bias generators fools polynomials of degree d. Computational Complexity, 18(2):209-217, 2009. Google Scholar
  26. Emanuele Viola and Avi Wigderson. Norms, xor lemmas, and lower bounds for polynomials and protocols. Theory of Computing, 4(7):137-168, 2008. Google Scholar
  27. Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. J. ACM, 55(1):1:1-1:16, February 2008. Google Scholar
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