正规语言的无限结合

Question

Infinite union of regular language can be context free.

Is this statement true or false?

According to answer key, this is true! What I know is that infinite union or intersection is not closed under union/intersection.

Can anyone explain the procedure or logic behind this? How to know the infinite union / intersection for a particular language?

Answer 1

The statement is true, yes. It asks if such a union CAN be context-free, not if it always is. A verx simple example is taking the union of infinitely many times the same language; the result is just the original language, and if it was regular the result is, too. Or the union of all the { a^i } is the regular language a^*.

On the other hand, the infinite union can be uncomputable. Take a non-enumerable language L and the infinitely many (regular) singleton sets that contain exactly one word from this language. Their union is L and thus non-enumerable.