单子集的 CFG 复杂性

arXiv - CS - Formal Languages and Automata Theory Pub Date : 2024-05-30 DOI:arxiv-2405.20026

Lance Fortnow, William Gasarch

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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。

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The CFG Complexity of Singleton Sets

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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arXiv - CS - Formal Languages and Automata Theory

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