Context Free Grammar for which a RegEx is impossible

问题

I'm trying to find out if its possible to have an example of a CFG for which it is impossible to give a Regular Expression which can accept the same language.

回答1:

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

回答2:

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: aⁿbⁿ (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.

来源：https://stackoverflow.com/questions/5120170/context-free-grammar-for-which-a-regex-is-impossible