{"title":"析取范式的灵敏度猜想","authors":"S. KarthikC., Sébastien Tavenas","doi":"10.4230/LIPIcs.FSTTCS.2016.15","DOIUrl":null,"url":null,"abstract":"The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. \nIn this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function $f$ admitting the Normalized Block property, $bs(f) \\leq 4s(f)^2$. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. \nRecently, Gopalan et al. [ITCS '16] showed that every Boolean function $f$ is uniquely specified by its values on a Hamming ball of radius at most $2s(f)$. We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.","PeriodicalId":175000,"journal":{"name":"Foundations of Software Technology and Theoretical Computer Science","volume":"60 2 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2016-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"10","resultStr":"{\"title\":\"On the Sensitivity Conjecture for Disjunctive Normal Forms\",\"authors\":\"S. KarthikC., Sébastien Tavenas\",\"doi\":\"10.4230/LIPIcs.FSTTCS.2016.15\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open. \\nIn this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function $f$ admitting the Normalized Block property, $bs(f) \\\\leq 4s(f)^2$. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property. \\nRecently, Gopalan et al. [ITCS '16] showed that every Boolean function $f$ is uniquely specified by its values on a Hamming ball of radius at most $2s(f)$. We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.\",\"PeriodicalId\":175000,\"journal\":{\"name\":\"Foundations of Software Technology and Theoretical Computer Science\",\"volume\":\"60 2 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2016-07-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"10\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Software Technology and Theoretical Computer Science\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.FSTTCS.2016.15\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Software Technology and Theoretical Computer Science","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.FSTTCS.2016.15","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the Sensitivity Conjecture for Disjunctive Normal Forms
The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function $f$, the maximum sensitivity $s(f)$, is polynomially related to its block sensitivity $bs(f)$, and hence to other major complexity measures. Despite major advances in the analysis of Boolean functions over the last decade, the problem remains widely open.
In this paper, we consider a restriction on the class of Boolean functions through a model of computation (DNF), and refer to the functions adhering to this restriction as admitting the Normalized Block property. We prove that for any function $f$ admitting the Normalized Block property, $bs(f) \leq 4s(f)^2$. We note that (almost) all the functions mentioned in literature that achieve a quadratic separation between sensitivity and block sensitivity admit the Normalized Block property.
Recently, Gopalan et al. [ITCS '16] showed that every Boolean function $f$ is uniquely specified by its values on a Hamming ball of radius at most $2s(f)$. We extend this result and also construct examples of Boolean functions which provide the matching lower bounds.