{"title":"宇称与二次多项式模3的关系","authors":"Frederic Green","doi":"10.1109/CCC.2002.1004341","DOIUrl":null,"url":null,"abstract":"We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2/sup /spl Omega/(n)/. This is the first result of this type for general mod subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a recent result of Alon and Beigel (2001). The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai et al. (1996). The exponential sum bounds are tight.","PeriodicalId":193513,"journal":{"name":"Proceedings 17th IEEE Annual Conference on Computational Complexity","volume":"45 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2002-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"23","resultStr":"{\"title\":\"The correlation between parity and quadratic polynomials mod 3\",\"authors\":\"Frederic Green\",\"doi\":\"10.1109/CCC.2002.1004341\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2/sup /spl Omega/(n)/. This is the first result of this type for general mod subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a recent result of Alon and Beigel (2001). The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai et al. (1996). The exponential sum bounds are tight.\",\"PeriodicalId\":193513,\"journal\":{\"name\":\"Proceedings 17th IEEE Annual Conference on Computational Complexity\",\"volume\":\"45 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2002-05-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"23\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings 17th IEEE Annual Conference on Computational Complexity\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/CCC.2002.1004341\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings 17th IEEE Annual Conference on Computational Complexity","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/CCC.2002.1004341","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
The correlation between parity and quadratic polynomials mod 3
We prove exponentially small upper bounds on the correlation between parity and quadratic polynomials mod 3. One corollary of this is that in order to compute parity, circuits consisting of a threshold gate at the top, mod 3 gates in the middle, and AND gates of fan-in two at the inputs must be of size 2/sup /spl Omega/(n)/. This is the first result of this type for general mod subcircuits with ANDs of fan-in greater than 1. This yields an exponential improvement over a recent result of Alon and Beigel (2001). The proof uses a novel inductive estimate of the relevant exponential sums introduced by Cai et al. (1996). The exponential sum bounds are tight.
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