{"title":"单子集的 CFG 复杂性","authors":"Lance Fortnow, William Gasarch","doi":"arxiv-2405.20026","DOIUrl":null,"url":null,"abstract":"Let G be a context-free grammar (CFG) in Chomsky normal form. We take the\nnumber of rules in G to be the size of G. We also assume all CFGs are in\nChomsky normal form. We consider the question of, given a string w of length n, what is the\nsmallest CFG such that L(G)={w}? We show the following: 1) For all w, |w|=n, there is a CFG of size with O(n/log n) rules, such that\nL(G)={w}. 2) There exists a string w, |w|=n, such that every CFG G with L(G)={w} is of\nsize Omega(n/log n). We give two proofs of: one nonconstructive, the other\nconstructive.","PeriodicalId":501124,"journal":{"name":"arXiv - CS - Formal Languages and Automata Theory","volume":"103 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The CFG Complexity of Singleton Sets\",\"authors\":\"Lance Fortnow, William Gasarch\",\"doi\":\"arxiv-2405.20026\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let G be a context-free grammar (CFG) in Chomsky normal form. We take the\\nnumber of rules in G to be the size of G. We also assume all CFGs are in\\nChomsky normal form. We consider the question of, given a string w of length n, what is the\\nsmallest CFG such that L(G)={w}? We show the following: 1) For all w, |w|=n, there is a CFG of size with O(n/log n) rules, such that\\nL(G)={w}. 2) There exists a string w, |w|=n, such that every CFG G with L(G)={w} is of\\nsize Omega(n/log n). We give two proofs of: one nonconstructive, the other\\nconstructive.\",\"PeriodicalId\":501124,\"journal\":{\"name\":\"arXiv - CS - Formal Languages and Automata Theory\",\"volume\":\"103 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - CS - Formal Languages and Automata Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2405.20026\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - CS - Formal Languages and Automata Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.20026","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
让 G 成为乔姆斯基正则形式的无上下文语法 (CFG)。我们将 G 中的规则数视为 G 的大小。我们还假设所有 CFG 都是乔姆斯基正态形式。我们要考虑的问题是：给定长度为 n 的字符串 w，L(G)={w} 的最小 CFG 是什么？我们证明如下：1) 对于所有 w，|w|=n，存在一个大小为 O(n/log n) 规则的 CFG，使得 L(G)={w}.2) 存在一个字符串 w，|w|=n，使得具有 L(G)={w} 的 CFG G 的大小为 Omega(n/log n)。我们给出两个证明：一个是非结构性证明，另一个是结构性证明。
Let G be a context-free grammar (CFG) in Chomsky normal form. We take the
number of rules in G to be the size of G. We also assume all CFGs are in
Chomsky normal form. We consider the question of, given a string w of length n, what is the
smallest CFG such that L(G)={w}? We show the following: 1) For all w, |w|=n, there is a CFG of size with O(n/log n) rules, such that
L(G)={w}. 2) There exists a string w, |w|=n, such that every CFG G with L(G)={w} is of
size Omega(n/log n). We give two proofs of: one nonconstructive, the other
constructive.
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