On the power of small-depth computation

Foundations and Trends in Theoretical Computer Science

Volumn 5, Issue 1, 2009, Pages 1-72

  Viola, Emanuele ^a  

^a NORTHEASTERN UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

DEPTH 3; DEPTH COMPUTATION; FOUR SECTIONS; PARITY FUNCTIONS;

EID: 74549191898     PISSN: 1551305X     EISSN: 15513068     Source Type: Journal    
DOI: 10.1561/0400000033     Document Type: Article

References (81)

1. S. Aaronson and A. Wigderson, "Algebrization: A new barrier in complexity theory," in 40th Annual ACM Symposium on the Theory of Computing (STOC), pp. 731-740, 2008.
2. M. Agrawal and V. Vinay, "Arithmetic circuits: A chasm at depth four," in 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 67-75, 2008.
3. M. Ajtai, "S11 -formulae on finite structures," Annals of Pure and Applied Logic, vol.24, no.1, pp. 1-48, 1983.
4. N. Alon and R. Beigel, "Lower bounds for approximations by low degree polynomials over Zm," in 16th Annual Conference on Computational Complexity, pp. 184-187, IEEE, June 18-21 2001.
5. N. Alon, O. Goldreich, J. Ha°stad, and R. Peralta, "Simple constructions of almost k-wise independent random variables," Random Structures & Algorithms, vol.3, no.3, pp. 289-304, 1992.
6. N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn, and D. Ron, "Testing lowdegree polynomials over GF(2)," in 7th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), vol.2764 of Lecture Notes in Computer Science, pp. 188-199, Springer, 2003.
7. 0," SIAM Journal of Computing, vol.36, no.4, pp. 845-888, 2006.
8. S. Arora and B. Barak, Computational Complexity: A Modern Approach. Cambridge University Press, January 2007.
9. J. Aspnes, R. Beigel, M. Furst, and S. Rudich, "The expressive power of voting polynomials," Combinatorica, vol.14, no.2, pp. 135-148, 1994.
10. L. Babai, N. Nisan, and M. Szegedy, "Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs," Journal of Computer System and Sciences, vol.45, no.2, pp. 204-232, 1992.
11. T. Baker, J. Gill, and R. Solovay, "Relativizations of the P=?NP question," SIAM Journal of Computing, vol.4, no.4, pp. 431-442, 1975.
12. P. Beame, S. A. Cook, and H. J. Hoover, "Log depth circuits for division and related problems," SIAM Journal of Computing, vol.15, no.4, pp. 994-1003, 1986.
13. R. Beigel, "The polynomial method in circuit complexity," in 8th Annual Structure in Complexity Theory Conference, pp. 82-95, IEEE, 1993.
14. R. Beigel, "When do extra majority gates help? polylog(N) majority gates are equivalent to one," Computational Complexity, vol.4, no.4, pp. 314-324, 1994. Special issue devoted to the 4th Annual McGill Workshop on Complexity Theory.
15. I. Ben-Eliezer, R. Hod, and S. Lovett, "Random low degree polynomials are hard to approximate," Manuscript, 2008.
16. E. Ben-Sasson, M. Sudan, S. Vadhan, and A. Wigderson, "Randomness-efficient low degree tests and short PCPs via epsilon-biased sets," in 35th Annual Symposium on Theory of Computing (STOC), pp. 612-621, ACM, 2003.
17. N. Bhatnagar, P. Gopalan, and R. J. Lipton, "Symmetric polynomials over Zm and simultaneous communication protocols," Journal of Computer and System Sciences, vol.72, no.2, pp. 252-285, 2006.
18. A. Bogdanov and E. Viola, "Pseudorandom bits for polynomials," in 48th Annual Symposium on Foundations of Computer Science (FOCS), pp. 41-51, IEEE, 2007. To appear in SIAM Journal on Computing.
19. R. B. Boppana and M. Sipser, "The complexity of finite functions," in Handbook of Theoretical Computer Science, vol.A, pp. 757-804, Amsterdam: Elsevier, 1990.
20. J. Bourgain, "Estimation of certain exponential sums arising in complexity theory," Comptes Rendus Mathematique Academy of Sciences Paris, vol.340, no.9, pp. 627-631, 2005.
21. M. Braverman, "Poly-logarithmic independence fools AC0 circuits," in 24th Conference on Computational Complexity (CCC), IEEE, 2009.
22. J.-Y. Cai, "With probability one, a random oracle separates PSPACE from the polynomial-time hierarchy," Journal of Computer and System Sciences, vol.38, no.1, pp. 68-85, 1989.
23. J.-Y. Cai, F. Green, and T. Thierauf, "On the Correlation of Symmetric Functions," Mathematical Systems Theory, vol.29, no.3, pp. 245-258, 1996.
24. A. Chattopadhyay, "Discrepancy and the power of bottom fan-in in depththree circuits," in 48th Annual Symposium on Foundations of Computer Science (FOCS), pp. 449-458, IEEE, 2007.
25. F. R. K. Chung and P. Tetali, "Communication complexity and quasi randomness," SIAM Journal on Discrete Mathematics, vol.6, no.1, pp. 110-123, 1993.
26. P. Clote and E. Kranakis, Boolean Functions and Computation Models. Springer, 2002.
27. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, 2006.
28. D. Dubhashi and A. Panconesi, Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, 2009.
29. Y. Freund, "Boosting a weak learning algorithm by majority," Information and Computation, vol.121, no.2, pp. 256-285, 1995.
30. M. L. Furst, J. B. Saxe, and M. Sipser, "Parity, circuits, and the polynomialtime hierarchy.," Mathematical Systems Theory, vol.17, no.1, pp. 13-27, 1984.
31. A. Ǵal and V. Trifonov, "On the correlation between parity and modular polynomials," in 31st International Symposium on Mathematical Foundations of Computer Science (MFCS), vol.4162 of Lecture Notes in Computer Science, pp. 387-398, Springer, 2006.
32. M. Goldmann, J. Ha°stad, and A. A. Razborov, "Majority gates vs. general weighted threshold gates," Computational Complexity, vol.22, pp. 277-300, 1992.
33. O. Goldreich, "Candidate one-way functions based on expander graphs," Technical Report, Electronic Colloquium on Computational Complexity, 2000.
34. O. Goldreich, Foundations of Cryptography: Volume 1, Basic Tools. Cambridge: Cambridge University Press, 2001.
35. O. Goldreich, N. Nisan, and A. Wigderson, "On Yaos XOR lemma," Techncial Report TR95-050, Electronic Colloquium on Computational Complexity, www.eccc.uni-trier.de/, March 1995.
36. T. Gowers, "A new proof of Szemeŕedis theorem for arithmetic progressions of length four," Geometric and Functional Analysis, vol.8, no.3, pp. 529-551, 1998.
37. T. Gowers, "A new proof of Szemeŕedis theorem," Geometric and Functional Analysis, vol.11, no.3, pp. 465-588, 2001.
38. A. Granville and Z. Rudnick, "Uniform distribution," in Equidistribution in Number Theory, An Introduction, vol.237 of NATO Science Series II: Mathematics, Physics and Chemsitry, pp. 1-13, Springer, 2007.
39. B. Green and T. Tao, "The distribution of polynomials over finite fields, with applications to the Gowers norms," arXiv:0711.3191v1, 2007.
40. B. Green and T. Tao, "An inverse theorem for the Gowers U3(G) norm," Proceedings of the Edinburgh Mathematical Society (Series 2), vol.51, no.01, pp. 73-153, 2008.
41. F. Green, "The correlation between parity and quadratic polynomials mod 3," Journal of Computer System and Sciences, vol.69, no.1, pp. 28-44, 2004.
42. F. Green and A. Roy, "Uniqueness of Optimal Mod 3 Circuits for Parity," in Dagstuhl Seminar Proceedings, Algebraic Methods in Computational Complexity, vol.07411, 2007.
43. F. Green, A. Roy, and H. Straubing, "Bounds on an exponential sum arising in Boolean circuit complexity," Comptes Rendus Mathematique Academy of Sciences Paris, vol.341, no.5, pp. 279-282, 2005.
44. V. Grolmusz, "Separating the communication complexities of MODm and MODp circuits," Journal of Computer System and Sciences, vol.51, no.2, pp. 307-313, 1995.
45. D. Gutfreund and E. Viola, "Fooling parity tests with parity gates," in 8th International Workshop on Randomization and Computation (RANDOM), pp. 381-392, Springer, 2004.
46. J. Ha°stad and M. Goldmann, "On the power of small-depth threshold circuits," Computational Complexity, vol.1, no.2, pp. 113-129, 1991.
47. A. Hajnal, W. Maass, P. Pudĺak, M. Szegedy, and G. Tuŕan, "Threshold circuits of bounded depth," Journal of Computer System and Sciences, vol.46, no.2, pp. 129-154, 1993.
48. K. A. Hansen, "Lower bounds for circuits with few modular gates using exponential sums," Electronic Colloquium on Computational Complexity, 2006. Technical Report TR06-079, www.eccc.uni-trier.de/.
49. J. Ha°stad, Computational Limitations of Small-depth Circuits. MIT Press, 1987.
50. A. Healy, "Randomness-efficient sampling within NC1," Computational Complexity, vol.17, pp. 3-37, April 2008.
51. A. Healy and E. Viola, "Constant-depth circuits for arithmetic in finite fields of characteristic two," in 23rd International Symposium on Theoretical Aspects of Computer Science (STACS), pp. 672-683, Springer, 2006.
52. R. ImPagliazzo and M. Naor, "Efficient cryptographic schemes provably as secure as subset sum," Journal of Cryptology, vol.9, pp. 199-216, Fall 1996.
53. R. ImPagliazzo and A. Wigderson, "P = BPP if E requires exponential circuits: Derandomizing the XOR lemma," in 29th Annual Symposium on Theory of Computing (STOC), pp. 220-229, ACM, 1997.
54. K. Iwama and H. Morizumi, "An explicit lower bound of 5n - o(n) for Boolean circuits," in Mathematical Foundations of Computer Science (MFCS), pp. 353- 364, 2002.
55. A. R. Klivans, "On the derandomization of constant depth circuits," in 5th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), Springer 2001.
56. E. Kushilevitz and N. Nisan, Communication complexity. Cambridge: Cambridge University Press, 1997.
57. S. Lovett, "Unconditional pseudorandom generators for low degree polynomials," in 40th Annual Symposium on the Theory of Computing (STOC), pp. 557- 562, ACM, 2008.
58. S. Lovett, R. Meshulam, and A. Samorodnitsky, "Inverse conjecture for the Gowers norm is false," in 40th Annual Symposium on the Theory of Computing (STOC), pp. 547-556, ACM, 2008.
59. M. Luby, B. Veličkovíc, and A. Wigderson, "Deterministic approximate counting of depth-2 circuits," in 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), pp. 18-24, 1993.
60. J. Naor and M. Naor, "Small-bias probability spaces: Efficient Constructions and Applications," SIAM Journal of Computing, vol.22, no.4, pp. 838-856, 1993.
61. V. A. Nepomnjaščii, "Rudimentary predicates and Turing calculations," Soviet Mathematics-Doklady, vol.11, no.6, pp. 1462-1465, 1970.
62. N. Nisan, "Pseudorandom bits for constant depth circuits," Combinatorica, vol.11, no.1, pp. 63-70, 1991.
63. N. Nisan and A. Wigderson, "Hardness vs Randomness," Journal of Computer and Systems Sciences, vol.49, no.2, pp. 149-167, 1994.
64. R. Raz, "The BNS-Chung criterion for multi-party communication complexity," Computational Complexity, vol.9, no.2, pp. 113-122, 2000.
65. A. Razborov, "Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function," Matematicheskie Zametki, vol.41, no.4, pp. 598-607, 623, 1987.
66. A. Razborov and S. Rudich, "Natural Proofs," Journal of Computer and System Sciences, vol.55, pp. 24-35, August 1997.
67. A. Razborov and A. Wigderson, "nO(logn) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom," Information Processing Letters, vol.45, no.6, pp. 303-307, 1993.
68. B. Rossman, "On the constant-depth complexity of k-clique," in 40th Annual Symposium on the Theory of Computing (STOC), ACM, 2008.
69. A. Samorodnitsky, "Low-degree tests at large distances," in 39th Annual Symposium on Theory of Computing (STOC), pp. 506-515, ACM, 2007.
70. R. Shaltiel and E. Viola, "Hardness amplification proofs require majority," in 40th Annual Symposium on the Theory of Computing (STOC), pp. 589-598, ACM, 2008.
71. R. Smolensky, "Algebraic methods in the theory of lower bounds for Boolean circuit complexity," in 19th Annual Symposium on Theory of Computing, pp. 77-82, ACM, 1987.
72. L. G. Valiant, "Graph-theoretic arguments in low-level complexity," in 6th Symposium on Mathematical Foundations of Computer Science, vol.53 of Lecture Notes in Computer Science, pp. 162-176, Springer, 1977.
73. L. G. Valiant, "Exponential lower bounds for restricted monotone circuits," in 15th annual ACM symposium on Theory of computing (STOC), pp. 110-117, ACM, 1983.
74. E. Viola, "The complexity of hardness amplification and derandomization," PhD thesis, Harvard University, www.eccc.uni-trier.de/ 2006.
75. E. Viola, "New correlation bounds for GF(2) polynomials using Gowers uniformity," Electronic Colloquium on Computational Complexity, Technical Report TR06-097, www.eccc.uni-trier.de/ 2006.
76. E. Viola, "On Constructing Parallel Pseudorandom Generators from One-Way Functions," in 20th Annual Conference on Computational Complexity (CCC), pp. 183-197, IEEE, 2005.
77. E. Viola, "Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates," SIAM Journal on Computing, vol.36, no.5, pp. 1387-1403, 2007.
78. E. Viola, "Gems of theoretical computer science," Lecture notes of the class taught at Northeastern University. Available at http://www.ccs.neu.edu/home/viola/classes/gems-08/index.html, 2009.
79. E. Viola, "The sum of d small-bias generators fools polynomials of degree d," Computational Complexity, vol.18, no.2, pp. 209-217, 2009.
80. E. Viola and A. Wigderson, "Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols," Theory of Computing, vol.4, pp. 137- 168, 2008.
81. A. C.-C. Yao, "Separating the polynomial-time hierarchy by oracles," in Proceedings of the 26th annual symposium on Foundations of computer science, pp. 1-10, IEEE Press, 1985.