无数的常规语言
[英]Uncountably many regular languages
问题描述
Consider the following language S = {0, 00, 000, 0000, 00000,....}.
考虑以下语言 S = {0, 00, 000, 0000, 00000,....}。
Consider the power-set of S and let each element of the power-set of S be a regular language.考虑 S 的幂集，并让 S 的幂集的每个元素都是正则语言。 Since S is countably infinite its power set is uncountably infinite.因为 S 是可数无限的，所以它的幂集是不可数无限的。 Since each element of the power set of S is finite each element is a regular language, but this implies that there are uncountably many regular languages.由于 S 的幂集的每个元素都是有限的，每个元素都是一种正则语言，但这意味着存在不可数的正则语言。
I know that the above 'proof' is wrong but I don't understand why.我知道上面的“证明”是错误的，但我不明白为什么。 Where exactly does the proof break down.证明究竟在哪里失效。
1 个解决方案
解决方案1
1 2019-12-19 00:56:39
It's not true that every element of the powerset is finite. powerset 的每个元素都不是有限的。 For example, the powerset includes S itself.例如，powerset 包括 S 本身。 It's also not true that every element of the powerset is a regular language. powerset 的每个元素都是常规语言也不是真的。 For example, it includes the set {0^n |例如，它包括集合 {0^n | n is the code of a Turing machine that halts on empty input}. n 是在空输入时停止的图灵机的代码}。
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