为什么{a ^ nb ^ n | n> = 1}是不是普通语言？

Question

I was following this video, and wonder why {a ⁿ b ⁿ | n≥1} is not a regular language?

Answer 1

Regular language must be recognized by finite automaton. As n is not bounded by any constant the automaton cannot be finite.

Answer 2

If you take the definition of “regular language” as “recognized by a finite automaton”, let m be the number of states of such an automaton. If the automaton is to recognize a ¹ b ¹ , a ² b ² , …, a ^m+1 b ^m+1 , the states of the automaton cannot be the same after it has read a ¹ , a ² , …, a ^m+1 , leading to a contradiction.