The approximate degree of DNF and CNF formulas
Pages 1194 - 1207
Abstract
The approximate degree of a Boolean function f∶{0,1}n→{0,1} is the minimum degree of a real polynomial p that approximates f pointwise: |f(x)−p(x)|≤1/3 for all x∈{0,1}n. For any δ>0, we construct DNF and CNF formulas of polynomial size with approximate degree Ω(n1−δ), essentially matching the trivial upper bound of n. This fully resolves the approximate degree of constant-depth circuits (AC0), a question that has seen extensive research over the past 10 years. Prior to our work, an Ω(n1−δ) lower bound was known only for AC0 circuits of depth that grows with 1/δ (Bun and Thaler, FOCS 2017). Furthermore, the DNF and CNF formulas that we construct are the simplest possible in that they have constant width.
Our result gives the first near-linear lower bounds on the bounded-error communication complexity of polynomial-size DNF and CNF formulas in the challenging k-party number-on-the-forehead model and two-party quantum model: Ω(n/4kk2)1−δ and Ω(n1−δ), respectively, where δ>0 is any constant. Our lower bounds are essentially optimal. Analogous to above, such lower bounds were previously known only for AC0 circuits of depth that grows with 1/δ.
References
[1]
Scott Aaronson. 2005. Limitations of Quantum Advice and One-Way Communication. Theory of Computing, 1, 1 (2005), 1–28. https://doi.org/10.4086/toc.2005.v001a001
[2]
Scott Aaronson and Yaoyun Shi. 2004. Quantum lower bounds for the collision and the element distinctness problems. J. ACM, 51, 4 (2004), 595–605. https://doi.org/10.1145/1008731.1008735
[3]
Miklós Ajtai, Henryk Iwaniec, János Komlós, János Pintz, and Endre Szemerédi. 1990. Construction of a Thin Set with small Fourier Coefficients. Bulletin of the London Mathematical Society, 22, 6 (1990), 583–590. https://doi.org/10.1112/blms/22.6.583
[4]
Andris Ambainis. 2005. Polynomial Degree and Lower Bounds in Quantum Complexity: Collision and Element Distinctness with Small Range. Theory of Computing, 1, 1 (2005), 37–46. https://doi.org/10.4086/toc.2005.v001a003
[5]
Andris Ambainis, Andrew M. Childs, Ben Reichardt, Robert Špalek, and Shengyu Zhang. 2010. Any AND-OR Formula of Size N can be Evaluated in time N^1/2+o(1) on a Quantum Computer. SIAM J. Comput., 39, 6 (2010), 2513–2530. https://doi.org/10.1137/080712167
[6]
James Aspnes, Richard Beigel, Merrick L. Furst, and Steven Rudich. 1994. The Expressive Power of Voting Polynomials. Combinatorica, 14, 2 (1994), 135–148. https://doi.org/10.1007/BF01215346
[7]
László Babai, Peter Frankl, and Janos Simon. 1986. Complexity classes in communication complexity theory. In Proceedings of the Twenty-Seventh Annual IEEE Symposium on Foundations of Computer Science (FOCS). 337–347. https://doi.org/10.1109/SFCS.1986.15
[8]
László Babai, Noam Nisan, and Mario Szegedy. 1992. Multiparty Protocols, Pseudorandom Generators for Logspace, and Time-Space Trade-Offs. J. Comput. Syst. Sci., 45, 2 (1992), 204–232. https://doi.org/10.1016/0022-0000(92)90047-M
[9]
Ziv Bar-Yossef, T. S. Jayram, Ravi Kumar, and D. Sivakumar. 2004. An information statistics approach to data stream and communication complexity. J. Comput. Syst. Sci., 68, 4 (2004), 702–732. https://doi.org/10.1016/j.jcss.2003.11.006
[10]
Robert Beals, Harry Buhrman, Richard Cleve, Michele Mosca, and R. de Wolf. 2001. Quantum lower bounds by polynomials. J. ACM, 48, 4 (2001), 778–797. https://doi.org/10.1145/502090.502097
[11]
Paul Beame and Trinh Huynh. 2012. Multiparty Communication Complexity and Threshold Circuit Size of AC^0. SIAM J. Comput., 41, 3 (2012), 484–518. https://doi.org/10.1137/100792779
[12]
Paul Beame, Toniann Pitassi, Nathan Segerlind, and Avi Wigderson. 2006. A Strong Direct Product Theorem for Corruption and the Multiparty Communication Complexity of Disjointness. Computational Complexity, 15, 4 (2006), 391–432. https://doi.org/10.1007/s00037-007-0220-2
[13]
Richard Beigel, Nick Reingold, and Daniel A. Spielman. 1995. PP Is Closed under Intersection. J. Comput. Syst. Sci., 50, 2 (1995), 191–202. https://doi.org/10.1006/jcss.1995.1017
[14]
Harry Buhrman, Richard Cleve, R. de Wolf, and Christof Zalka. 1999. Bounds for Small-Error and Zero-Error Quantum Algorithms. In Proceedings of the Fortieth Annual IEEE Symposium on Foundations of Computer Science (FOCS). 358–368. https://doi.org/10.1109/SFFCS.1999.814607
[15]
Harry Buhrman and R. de Wolf. 2001. Communication complexity lower bounds by polynomials. In Proceedings of the Sixteenth Annual IEEE Conference on Computational Complexity (CCC). 120–130. https://doi.org/10.1109/CCC.2001.933879
[16]
Harry Buhrman, Nikolai K. Vereshchagin, and R. de Wolf. 2007. On computation and communication with small bias. In Proceedings of the Twenty-Second Annual IEEE Conference on Computational Complexity (CCC). 24–32. https://doi.org/10.1109/CCC.2007.18
[17]
Mark Bun, Robin Kothari, and Justin Thaler. 2020. The Polynomial Method Strikes Back: Tight Quantum Query Bounds via Dual Polynomials. Theory Comput., 16 (2020), 1–71. https://doi.org/10.4086/toc.2020.v016a010
[18]
Mark Bun and Justin Thaler. 2015. Dual lower bounds for approximate degree and Markov–Bernstein inequalities. Inf. Comput., 243 (2015), 2–25. https://doi.org/10.1016/j.ic.2014.12.003
[19]
Mark Bun and Justin Thaler. 2015. Hardness Amplification and the Approximate Degree of Constant-Depth Circuits. In Proceedings of the Forty-Second International Colloquium on Automata, Languages and Programming (ICALP). 268–280. https://doi.org/10.1007/978-3-662-47672-7_22
[20]
Mark Bun and Justin Thaler. 2020. A Nearly Optimal Lower Bound on the Approximate Degree of ^0. SIAM J. Comput., 49, 4 (2020), https://doi.org/10.1137/17M1161737
[21]
Mark Bun and Justin Thaler. 2021. The Large-Error Approximate Degree of ^0. Theory of Computing, 17, 7 (2021), 1–46. https://doi.org/10.4086/toc.2021.v017a007
[22]
Ashok K. Chandra, Merrick L. Furst, and Richard J. Lipton. 1983. Multi-Party Protocols. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing (STOC). 94–99. https://doi.org/10.1145/800061.808737
[23]
Karthekeyan Chandrasekaran, Justin Thaler, Jonathan Ullman, and Andrew Wan. 2014. Faster private release of marginals on small databases. In Proceedings of the Fifth Conference on Innovations in Theoretical Computer Science (ITCS). 387–402. https://doi.org/10.1145/2554797.2554833
[24]
Arkadev Chattopadhyay and Anil Ada. 2008. Multiparty Communication Complexity of Disjointness. In Electronic Colloquium on Computational Complexity (ECCC). Report TR08-002
[25]
Jeff Kahn, Nathan Linial, and Alex Samorodnitsky. 1996. Inclusion-Exclusion: Exact and Approximate. Combinatorica, 16, 4 (1996), 465–477. https://doi.org/10.1007/BF01271266
[26]
Adam Tauman Kalai, Adam R. Klivans, Yishay Mansour, and Rocco A. Servedio. 2008. Agnostically learning halfspaces. SIAM J. Comput., 37, 6 (2008), 1777–1805. https://doi.org/10.1137/060649057
[27]
Bala Kalyanasundaram and Georg Schnitger. 1992. The Probabilistic Communication Complexity of Set Intersection. SIAM J. Discrete Math., 5, 4 (1992), 545–557. https://doi.org/10.1137/0405044
[28]
Hartmut Klauck, Robert Špalek, and R. de Wolf. 2007. Quantum and Classical Strong Direct Product Theorems and Optimal Time-Space Tradeoffs. SIAM J. Comput., 36, 5 (2007), 1472–1493. https://doi.org/10.1137/05063235X
[29]
Adam R. Klivans, Ryan O’Donnell, and Rocco A. Servedio. 2004. Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci., 68, 4 (2004), 808–840. https://doi.org/10.1016/j.jcss.2003.11.002
[30]
Adam R. Klivans and Rocco A. Servedio. 2004. Learning DNF in time 2^~ O(n^1/3). J. Comput. Syst. Sci., 68, 2 (2004), 303–318. https://doi.org/10.1016/j.jcss.2003.07.007
[31]
Matthias Krause and Pavel Pudlák. 1997. On the Computational Power of Depth-2 Circuits with Threshold and Modulo Gates. Theor. Comput. Sci., 174, 1–2 (1997), 137–156. https://doi.org/10.1016/S0304-3975(96)00019-9
[32]
Matthias Krause and Pavel Pudlák. 1998. Computing Boolean functions by polynomials and threshold circuits. Comput. Complex., 7, 4 (1998), 346–370. https://doi.org/10.1007/s000370050015
[33]
Troy Lee. 2009. A note on the sign degree of formulas. Available at arxiv:0909.4607
[34]
Troy Lee and Adi Shraibman. 2009. Disjointness is Hard in the Multiparty Number-on-the-Forehead Model. Computational Complexity, 18, 2 (2009), 309–336. https://doi.org/10.1007/s00037-009-0276-2
[35]
Nathan Linial and Noam Nisan. 1990. Approximate Inclusion-Exclusion. Combinatorica, 10, 4 (1990), 349–365. https://doi.org/10.1007/BF02128670
[36]
Nikhil S. Mande, Justin Thaler, and Shuchen Zhu. 2020. Improved Approximate Degree Bounds for k-Distinctness. In Proceedings of the 15th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC). 158, 2:1–2:22. https://doi.org/10.4230/LIPIcs.TQC.2020.2
[37]
Marvin L. Minsky and Seymour A. Papert. 1969. Perceptrons: An Introduction to Computational Geometry. MIT Press, Cambridge, Mass.
[38]
Noam Nisan and Mario Szegedy. 1994. On the degree of Boolean functions as real polynomials. Computational Complexity, 4 (1994), 301–313. https://doi.org/10.1007/BF01263419
[39]
Ryan O’Donnell and Rocco A. Servedio. 2010. New degree bounds for polynomial threshold functions. Combinatorica, 30, 3 (2010), 327–358. https://doi.org/10.1007/s00493-010-2173-3
[40]
Ramamohan Paturi and Michael E. Saks. 1994. Approximating Threshold Circuits by Rational Functions. Inf. Comput., 112, 2 (1994), 257–272. https://doi.org/10.1006/inco.1994.1059
[41]
Alexander A. Razborov. 1992. On the distributional complexity of disjointness. Theor. Comput. Sci., 106, 2 (1992), 385–390. https://doi.org/10.1016/0304-3975(92)90260-M
[42]
Alexander A. Razborov. 2002. Quantum communication complexity of symmetric predicates. Izvestiya of the Russian Academy of Sciences, Mathematics, 67 (2002), 145–159.
[43]
Alexander A. Razborov and Alexander A. Sherstov. 2010. The sign-rank of ^0. SIAM J. Comput., 39, 5 (2010), 1833–1855. https://doi.org/10.1137/080744037
[44]
Alexander A. Sherstov. 2008. Communication Lower Bounds Using Dual Polynomials. Bulletin of the EATCS, 95 (2008), 59–93.
[45]
Alexander A. Sherstov. 2009. Approximate Inclusion-Exclusion for Arbitrary Symmetric Functions. Computational Complexity, 18, 2 (2009), 219–247. https://doi.org/10.1007/s00037-009-0274-4
[46]
Alexander A. Sherstov. 2009. Separating ^0 from depth-2 majority circuits. SIAM J. Comput., 38, 6 (2009), 2113–2129. https://doi.org/10.1137/08071421X
[47]
Alexander A. Sherstov. 2011. The pattern matrix method. SIAM J. Comput., 40, 6 (2011), 1969–2000. https://doi.org/10.1137/080733644
[48]
Alexander A. Sherstov. 2012. Strong Direct Product Theorems for Quantum Communication and Query Complexity. SIAM J. Comput., 41, 5 (2012), 1122–1165. https://doi.org/10.1137/110842661
[49]
Alexander A. Sherstov. 2013. The intersection of two halfspaces has high threshold degree. SIAM J. Comput., 42, 6 (2013), 2329–2374. https://doi.org/10.1137/100785260
[50]
Alexander A. Sherstov. 2013. Making polynomials robust to noise. Theory of Computing, 9 (2013), 593–615. https://doi.org/10.4086/toc.2013.v009a018
[51]
Alexander A. Sherstov. 2014. Communication lower bounds using directional derivatives. J. ACM, 61, 6 (2014), 1–71. https://doi.org/10.1145/2629334
[52]
Alexander A. Sherstov. 2016. The multiparty communication complexity of set disjointness. SIAM J. Comput., 45, 4 (2016), 1450–1489. https://doi.org/10.1137/120891587
[53]
Alexander A. Sherstov. 2018. Breaking the Minsky–Papert Barrier for Constant-Depth Circuits. SIAM J. Comput., 47, 5 (2018), 1809–1857. https://doi.org/10.1137/15M1015704
[54]
Alexander A. Sherstov. 2018. The Power of Asymmetry in Constant-Depth Circuits. SIAM J. Comput., 47, 6 (2018), 2362–2434. https://doi.org/10.1137/16M1064477
[55]
Alexander A. Sherstov. 2021. The hardest halfspace. Comput. Complex., 30, 11 (2021), 1–85. https://doi.org/10.1007/s00037-021-00211-4
[56]
Alexander A. Sherstov. 2022. The Approximate Degree of DNF and CNF Formulas. In Electronic Colloquium on Computational Complexity (ECCC).
[57]
Alexander A. Sherstov and Pei Wu. 2019. Near-optimal lower bounds on the threshold degree and sign-rank of ^0. In Proceedings of the Fifty-First Annual ACM Symposium on Theory of Computing (STOC). 401–412. https://doi.org/10.1145/3313276.3316408
[58]
Kai-Yeung Siu, Vwani P. Roychowdhury, and Thomas Kailath. 1994. Rational approximation techniques for analysis of neural networks. IEEE Transactions on Information Theory, 40, 2 (1994), 455–466. https://doi.org/10.1109/18.312168
[59]
Jun Tarui and Tatsuie Tsukiji. 1999. Learning DNF by Approximating Inclusion-Exclusion Formulae. In Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity (CCC). 215–221. https://doi.org/10.1109/CCC.1999.766279
[60]
Justin Thaler, Jonathan Ullman, and Salil P. Vadhan. 2012. Faster Algorithms for Privately Releasing Marginals. In Proceedings of the Thirty-Ninth International Colloquium on Automata, Languages and Programming (ICALP). 810–821. https://doi.org/10.1007/978-3-642-31594-7_68
[61]
Robert Špalek. 2008. A Dual Polynomial for OR. Available at. arxiv:0803.4516
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