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泵引理和层次结构

[英]Pumping Lemma and Hierachy

原文 2023-01-06 20:38:45 4 1 theory/ pumping-lemma/ chomsky-hierarchy

I have a question involving the Pumping Lemma for Regular Languages and Pumping Lemma for Context-free Languages : Is it possible that there's a language which doesn't meet the criteria for the pumping-lemma for context-free languages but does meet the criteria for the pumping-lemma for regular languges?我有一个问题涉及常规语言的抽取引理和上下文无关语言的抽取引理：是否有可能有一种语言不符合上下文无关语言抽取引理的标准但确实符合以下标准常规语言的泵引理？

Or is there a hierachy similar to the Chomsky-Hierachy?或者是否存在类似于乔姆斯基层次结构的层次结构？

I'm just trying to understand that and pumping lemma in general我只是想了解这一点并总体上抽取引理

1 个解决方案

Is it possible that there's a language which doesn't meet the criteria for the pumping-lemma for context-free languages but does meet the criteria for the pumping-lemma for regular languges?是否有可能存在一种语言不符合上下文无关语言的泵引理标准但符合常规语言的泵引理标准的语言？

Let's consider the classic a^nb^n language.让我们考虑一下经典的 a^nb^n 语言。 aabb is in it, while abb is not. aabb 在其中，而 abb 不在其中。

We know it is a CFL.我们知道它是 CFL。 (S -> aSb | epsilon ) (S -> aSb | epsilon )

We can proof that is is not a regular language using the PL for CFL (cf. https://stackoverflow.com/a/2309755 )我们可以使用 CFL 的 PL 证明它不是常规语言（参见https://stackoverflow.com/a/2309755 ）

The PL for CFL is used to proof that a language is NOT CF. CFL 的 PL 用于证明语言不是 CF。 But the language IS CF (see above.).但是语言是 CF（见上文）。

Thus we can never use the PL for CFL for the language to proof that is it not CF.因此，我们永远不能使用 CFL 的 PL 作为语言来证明它不是 CF。

A regular language [...] must be a CFL itself and therefore should be able to meet the PL criteria for CFL or am I wrong?常规语言 [...] 本身必须是 CFL，因此应该能够满足 CFL 的 PL 标准，或者我错了吗？

Yes, any RL is also a CFL (and also a CSL and also a REL).是的，任何 RL 也是 CFL（也是 CSL 和 REL）。 You are wrong in your conclusion though.不过你的结论是错误的。

The PL is used to proof that a given language is NOT in the class. So we use the PL for RL to show that a language is not a RL, so "at most" CF. PL 用于证明给定语言不在 class 中。因此我们使用 PL 表示 RL 以表明语言不是 RL，因此“最多”CF。 And we use the PL for CFL to show that a language is not even a CFL, so "at most" context sensitive.我们使用 CFL 的 PL 来表明一种语言甚至不是 CFL，因此“至多”是上下文敏感的。

Is there a hierarchy similar to the Chomsky-Hierarchy?是否存在类似于乔姆斯基层次结构的层次结构？

Well if you can proof a language is not context free, it can also not be regular, as RL is a subset of CFL.好吧，如果你能证明一种语言不是上下文无关的，它也可能不是正则的，因为 RL 是 CFL 的一个子集。