{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,4]],"date-time":"2026-04-04T05:59:26Z","timestamp":1775282366209,"version":"3.50.1"},"publisher-location":"Berlin, Heidelberg","reference-count":26,"publisher":"Springer Berlin Heidelberg","isbn-type":[{"value":"9783642229343","type":"print"},{"value":"9783642229350","type":"electronic"}],"license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2011]]},"DOI":"10.1007\/978-3-642-22935-0_54","type":"book-chapter","created":{"date-parts":[[2011,8,12]],"date-time":"2011-08-12T09:20:39Z","timestamp":1313140839000},"page":"640-651","source":"Crossref","is-referenced-by-count":3,"title":["Correlation Bounds for Poly-size $\\mbox{\\rm AC}^0$ Circuits with n 1\u2009\u2212\u2009o(1) Symmetric Gates"],"prefix":"10.1007","author":[{"given":"Shachar","family":"Lovett","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Srikanth","family":"Srinivasan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"54_CR1","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0168-0072(83)90038-6","volume":"24","author":"M. Ajtai","year":"1983","unstructured":"Ajtai, M.: $\\Sigma^1_1$ -formulae on finite structures. Annals of Pure and Applied Logic\u00a024, 1\u201348 (1983)","journal-title":"Annals of Pure and Applied Logic"},{"issue":"2","key":"54_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01215346","volume":"14","author":"J. Aspnes","year":"1994","unstructured":"Aspnes, J., Beigel, R., Furst, M., Rudich, S.: The expressive power of voting polynomials. Combinatorica\u00a014(2), 1\u201314 (1994)","journal-title":"Combinatorica"},{"issue":"2","key":"54_CR3","doi-asserted-by":"publisher","first-page":"204","DOI":"10.1016\/0022-0000(92)90047-M","volume":"45","author":"L. Babai","year":"1992","unstructured":"Babai, L., Nisan, N., Szegedy, M.: Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. Syst. Sci.\u00a045(2), 204\u2013232 (1992)","journal-title":"J. Comput. Syst. Sci."},{"key":"54_CR4","doi-asserted-by":"crossref","unstructured":"Barrington, D.A.M., Straubing, H.: Complex polynomials and circuit lower bounds for modular counting. In: Computational Complexity, pp. 314\u2013324 (1994)","DOI":"10.1007\/BF01263421"},{"key":"54_CR5","doi-asserted-by":"crossref","unstructured":"Beigel, R.: When do extra majority gates help? Polylog(n) majority gates are equivalent to one. In: Computational Complexity, pp. 314\u2013324 (1994)","DOI":"10.1007\/BF01263420"},{"key":"54_CR6","doi-asserted-by":"crossref","unstructured":"Beigel, R., Reingold, N., Spielman, D.: The perceptron strikes back. In: Proceedings of the 6th Conference on Structure in Complexity Theory, pp. 286\u2013291 (1991)","DOI":"10.1109\/SCT.1991.160270"},{"issue":"9","key":"54_CR7","doi-asserted-by":"publisher","first-page":"627","DOI":"10.1016\/j.crma.2005.03.008","volume":"340","author":"J. Bourgain","year":"2005","unstructured":"Bourgain, J.: Estimation of certain exponential sums arising in complexity theory. Comptes Rendus Mathematique\u00a0340(9), 627\u2013631 (2005)","journal-title":"Comptes Rendus Mathematique"},{"key":"54_CR8","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1007\/11523468_80","volume-title":"Automata, Languages and Programming","author":"A. Chattopadhyay","year":"2005","unstructured":"Chattopadhyay, A., Hansen, K.A.: Lower bounds for circuits with few modular and symmetric gates. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol.\u00a03580, pp. 994\u20131005. Springer, Heidelberg (2005)"},{"key":"54_CR9","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"994","DOI":"10.1007\/11523468_80","volume-title":"Automata, Languages and Programming","author":"A. Chattopadhyay","year":"2005","unstructured":"Chattopadhyay, A., Hansen, K.A.: Lower bounds for circuits with few modular and symmetric gates. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol.\u00a03580, pp. 994\u20131005. Springer, Heidelberg (2005)"},{"key":"54_CR10","doi-asserted-by":"crossref","unstructured":"Goldmann, M., Hastad, J., Razborov, A.: Majority gates vs. general weighted threshold gates. In: Proceedings of the 7th Structure in Complexity Theory Annual Conference (1992)","DOI":"10.1109\/SCT.1992.215375"},{"key":"54_CR11","doi-asserted-by":"crossref","unstructured":"Gopalan, P., Servedio, R.A.: Learning and lower bounds for AC0 with threshold gates. In: APPROX-RANDOM, pp. 588\u2013601 (2010)","DOI":"10.1007\/978-3-642-15369-3_44"},{"key":"54_CR12","unstructured":"Hansen, K.A.: Lower bounds for circuits with few modular gates using exponential sums. Electronic Colloquium on Computational Complexity (ECCC)\u00a013(079) (2006)"},{"key":"54_CR13","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1007\/978-3-540-28629-5_24","volume-title":"Mathematical Foundations of Computer Science 2004","author":"K.A. Hansen","year":"2004","unstructured":"Hansen, K.A., Miltersen, P.B.: Some meet-in-the-middle circuit lower bounds. In: Fiala, J., Koubek, V., Kratochv\u00edl, J. (eds.) MFCS 2004. LNCS, vol.\u00a03153, pp. 334\u2013345. Springer, Heidelberg (2004)"},{"key":"54_CR14","first-page":"6","volume-title":"Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC 1986)","author":"J. Hastad","year":"1986","unstructured":"Hastad, J.: Almost optimal lower bounds for small depth circuits. In: Proceedings of the 18th Annual ACM Symposium on Theory of Computing (STOC 1986), pp. 6\u201320. ACM, New York (1986)"},{"key":"54_CR15","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/BF01272517","volume":"1","author":"J. H\u00e5stad","year":"1991","unstructured":"H\u00e5stad, J., Goldmann, M.: On the power of small-depth threshold circuits. Computational Complexity\u00a01, 113\u2013129 (1991)","journal-title":"Computational Complexity"},{"key":"54_CR16","doi-asserted-by":"publisher","DOI":"10.1016\/S0065-2458(08)60342-3","volume-title":"Communication Complexity","author":"E. Kushilevitz","year":"1997","unstructured":"Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)"},{"issue":"1","key":"54_CR17","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1007\/BF01375474","volume":"11","author":"N. Nisan","year":"1991","unstructured":"Nisan, N.: Pseudorandom bits for constant depth circuits. Combinatorica\u00a011(1), 63\u201370 (1991)","journal-title":"Combinatorica"},{"key":"54_CR18","doi-asserted-by":"crossref","unstructured":"Nisan, N.: The communication complexity of threshold gates. In: Proceedings of \u201cCombinatorics, Paul Erdos is Eighty\u201d, pp. 301\u2013315 (1994)","DOI":"10.1007\/BF01263419"},{"issue":"2","key":"54_CR19","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1016\/S0022-0000(05)80043-1","volume":"49","author":"N. Nisan","year":"1994","unstructured":"Nisan, N., Wigderson, A.: Hardness vs. randomness. J. Comput. Syst. Sci.\u00a049(2), 149\u2013167 (1994)","journal-title":"J. Comput. Syst. Sci."},{"issue":"4","key":"54_CR20","first-page":"333","volume":"41","author":"A.A. Razborov","year":"1987","unstructured":"Razborov, A.A.: Lower bounds for the size of circuits of bounded depth with basis { &,\u2009\u2295\u2009}. Math. Notes Acad. Sci. USSR\u00a041(4), 333\u2013338 (1987)","journal-title":"Math. Notes Acad. Sci. USSR"},{"key":"54_CR21","doi-asserted-by":"crossref","unstructured":"Alexander, A.R., Wigderson, A.: n \u03a9(log n) lower bounds on the size of depth-3 threshold circuits with and gates at the bottom. Inf. Process. Lett.\u00a045(6), 303\u2013307 (1993)","DOI":"10.1016\/0020-0190(93)90041-7"},{"issue":"3","key":"54_CR22","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1137\/0404038","volume":"4","author":"K.I. Siu","year":"1991","unstructured":"Siu, K.I., Bruck, J.: On the power of threshold circuits with small weights. SIAM Journal on Discrete Mathematics\u00a04(3), 423\u2013435 (1991)","journal-title":"SIAM Journal on Discrete Mathematics"},{"key":"54_CR23","first-page":"77","volume-title":"Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC 1987)","author":"R. Smolensky","year":"1987","unstructured":"Smolensky, R.: Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC 1987), pp. 77\u201382. ACM, New York (1987)"},{"issue":"5","key":"54_CR24","doi-asserted-by":"publisher","first-page":"1387","DOI":"10.1137\/050640941","volume":"36","author":"E. Viola","year":"2007","unstructured":"Viola, E.: Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates. SIAM J. Comput.\u00a036(5), 1387\u20131403 (2007)","journal-title":"SIAM J. Comput."},{"issue":"1","key":"54_CR25","doi-asserted-by":"publisher","first-page":"137","DOI":"10.4086\/toc.2008.v004a007","volume":"4","author":"E. Viola","year":"2008","unstructured":"Viola, E., Wigderson, A.: Norms, xor lemmas, and lower bounds for polynomials and protocols. Theory of Computing\u00a04(1), 137\u2013168 (2008)","journal-title":"Theory of Computing"},{"key":"54_CR26","doi-asserted-by":"crossref","unstructured":"Williams, R.: Non-uniform ACC circuit lower bounds. In: Conference on Computational Complexity (2011)","DOI":"10.1109\/CCC.2011.36"}],"container-title":["Lecture Notes in Computer Science","Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques"],"original-title":[],"link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-642-22935-0_54","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,11,30]],"date-time":"2021-11-30T08:26:27Z","timestamp":1638260787000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-642-22935-0_54"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"ISBN":["9783642229343","9783642229350"],"references-count":26,"URL":"https:\/\/doi.org\/10.1007\/978-3-642-22935-0_54","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011]]}}}