Parity Helps to Compute Majority
We study the complexity of computing symmetric and threshold functions by constant-depth circuits with Parity gates, also known as AC^0[oplus] circuits. Razborov [Alexander A. Razborov, 1987] and Smolensky [Roman Smolensky, 1987; Roman Smolensky, 1993] showed that Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/2(d-1)})}. By using a divide-and-conquer approach, it is easy to show that Majority can be computed with depth-d AC^0[oplus] circuits of size 2^{O~(n^{1/(d-1)})}. This gap between upper and lower bounds has stood for nearly three decades.
Somewhat surprisingly, we show that neither the upper bound nor the lower bound above is tight for large d. We show for d >= 5 that any symmetric function can be computed with depth-d AC^0[oplus] circuits of size exp(O~(n^{2/3 * 1/(d-4)})). Our upper bound extends to threshold functions (with a constant additive loss in the denominator of the double exponent). We improve the Razborov-Smolensky lower bound to show that for d >= 3 Majority requires depth-d AC^0[oplus] circuits of size 2^{Omega(n^{1/(2d-4)})}. For depths d <= 4, we are able to refine our techniques to get almost-optimal bounds: the depth-3 AC^0[oplus] circuit size of Majority is 2^{Theta~(n^{1/2})}, while its depth-4 AC^0[oplus] circuit size is 2^{Theta~(n^{1/4})}.
Computational Complexity
Boolean Circuits
Lower Bounds
Parity
Majority
Theory of computation
23:1-23:17
Regular Paper
This work was supported in part by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2014)/ERC Grant Agreement no. 615075.
This paper is the result of a collaboration that happened at the Simons Institute program on Lower Bounds in Computational Complexity, where all three authors were long-term visitors. We are grateful to the Simons Institute for their support.
Igor Carboni
Oliveira
Igor Carboni Oliveira
Department of Computer Science, University of Oxford, UK
Rahul
Santhanam
Rahul Santhanam
Department of Computer Science, University of Oxford, UK
Srikanth
Srinivasan
Srikanth Srinivasan
Department of Mathematics, IIT Bombay, India
10.4230/LIPIcs.CCC.2019.23
Miklós Ajtai. ∑₁¹-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983.
Miklós Ajtai and Michael Ben-Or. A Theorem on Probabilistic Constant Depth Computations. In Symposium on Theory of Computing (STOC), pages 471-474, 1984.
Josh Alman and Ryan Williams. Probabilistic Polynomials and Hamming Nearest Neighbors. In Symposium on Foundations of Computer Science (FOCS), pages 136-150, 2015.
Kazuyuki Amano. Bounds on the Size of Small Depth Circuits for Approximating Majority. In International Colloquium on Automata, Languages and Programming (ICALP), pages 59-70, 2009.
James Aspnes, Richard Beigel, Merrick L. Furst, and Steven Rudich. The Expressive Power of Voting Polynomials. Combinatorica, 14(2):135-148, 1994.
David A. Mix Barrington, Richard Beigel, and Steven Rudich. Representing Boolean Functions as Polynomials Modulo Composite Numbers. Computational Complexity, 4:367-382, 1994. URL: https://doi.org/10.1007/BF01263424.
https://doi.org/10.1007/BF01263424
Paul Beame. A switching lemma primer. Technical report, UW-CSE-95-07-01, 1994.
Ravi B. Boppana. Threshold Functions and Bounded Depth Monotone Circuits. In Symposium on Theory of Computing (STOC), pages 475-479, 1984. URL: https://doi.org/10.1145/800057.808716.
https://doi.org/10.1145/800057.808716
Eshan Chattopadhyay, Pooya Hatami, Shachar Lovett, and Avishay Tal. Pseudorandom Generators from the Second Fourier Level and Applications to AC⁰ with Parity Gates. In Innovations in Theoretical Computer Science Conference (ITCS), pages 22:1-22:15, 2019.
Xi Chen, Igor Carboni Oliveira, Rocco A. Servedio, and Li-Yang Tan. Near-optimal small-depth lower bounds for small distance connectivity. In Symposium on Theory of Computing (STOC), pages 612-625, 2016.
Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, Circuits, and the Polynomial-Time Hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984.
Frederic Green. A complex-number Fourier technique for lower bounds on the Mod-m degree. Computational Complexity, 9(1):16-38, 2000.
Kristoffer Hansen and Vladimir Podolskii. Exact Threshold Circuits. In Conference on Computational Complexity (CCC), pages 270-279, 2010.
Johan Håstad. Almost Optimal Lower Bounds for Small Depth Circuits. In Symposium on Theory of Computing (STOC), pages 6-20, 1986.
Peter Keevash and Benny Sudakov. Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices. SIAM J. Discrete Math., 18(4):713-727, 2005.
Swastik Kopparty. AC⁰ lower bounds and pseudorandomness, 2013. Lecture notes on `Topics in Complexity Theory and Pseudorandomness'. Can be found at: URL: http://sites.math.rutgers.edu/~sk1233/courses/topics-S13/lec4.pdf.
http://sites.math.rutgers.edu/~sk1233/courses/topics-S13/lec4.pdf
Swastik Kopparty and Srikanth Srinivasan. Certifying Polynomials for AC⁰[⊕] Circuits, with Applications to Lower Bounds and Circuit Compression. Theory of Computing, 14(1):1-24, 2018.
Nutan Limaye, Karteek Sreenivasiah, Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. The Coin Problem in Constant Depth: Sample Complexity and Parity gates. Electronic Colloquium on Computational Complexity (ECCC), 157, 2018.
Alexis Maciel and Denis Thérien. Threshold Circuits of Small Majority-Depth. Inf. Comput., 146(1):55-83, 1998. URL: https://doi.org/10.1006/inco.1998.2732.
https://doi.org/10.1006/inco.1998.2732
Zipei Nie and Anthony Y Wang. Hilbert functions and the finite degree Zariski closure in finite field combinatorial geometry. Journal of Combinatorial Theory, Series A, 134:196-220, 2015.
Ryan O'Donnell and Karl Wimmer. Approximation by DNF: Examples and Counterexamples. In International Colloquium on Automata, Languages and Programming (ICALP), pages 195-206, 2007.
Ramamohan Paturi, Michael E. Saks, and Francis Zane. Exponential lower bounds for depth three Boolean circuits. Computational Complexity, 9(1):1-15, 2000. URL: https://doi.org/10.1007/PL00001598.
https://doi.org/10.1007/PL00001598
Alexander A. Razborov. Lower bounds on the size of constant-depth networks over a complete basis with logical addition. Mathematicheskie Zametki, 41(4):598-607, 1987.
Benjamin Rossman and Srikanth Srinivasan. Separation of AC⁰[⊕] Formulas and Circuits. In International Colloquium on Automata, Languages, and Programming (ICALP), pages 50:1-50:13, 2017.
Roman Smolensky. Algebraic Methods in the Theory of Lower Bounds for Boolean Circuit Complexity. In Symposium on Theory of Computing (STOC), pages 77-82, 1987.
Roman Smolensky. On Representations by Low-Degree Polynomials. In Symposium on Foundations of Computer Science (FOCS), pages 130-138, 1993.
Mario Szegedy. Algebraic Methods in Lower Bounds for Computational Models. PhD thesis, University of Chicago, 1989.
Leslie G. Valiant. Short Monotone Formulae for the Majority Function. J. Algorithms, 5(3):363-366, 1984.
Victor K. Wei. Generalized Hamming weights for linear codes. IEEE Transactions on Information Theory, 37(5):1412-1418, 1991.
Igor C. Oliveira, Rahul Santhanam, and Srikanth Srinivasan
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