Context Free Grammar for which a RegEx is impossible

Question

我试图找出是否可能有一个CFG的例子，因为它不可能给出一个可以接受相同语言的正则表达式。

Answer 1

Since a regular machine/expression has only a limited (pre-defined) number of states, it cannot "remember" (infinitely) earlier parts of the input.

As such recognizing the following expression is impossible for a state-machine: a ⁿ b ⁿ (n∈ℕ)

You could make such a machine for n ≤ x, where x∈ℕ, but no state-machine can do it for every possible value from ℕ.

Answer 2

Any language which requires counting/remembering can't be expressed as a regular expression.

For example, a language which checks balanced parenthesis:

S -> { S } S

S -> ε