Question

I was told that the language generated by the regular expression:

(a*b*)*

is regular.

However, my thinking goes against this, as follows. Can anyone please provide an explanation whether I’m thinking right or wrong?

My Thoughts

(a*b*) refers to a single sequence of any amount of a, followed by any amount of b (can be empty). And this single sequence (which can’t be changed) can be repeated 0 or more time. For example:

   a* = a  
   b* = bbbb  
-> (a*b*) = abbbb  
-> (a*b*)* = abbbbabbbbabbbb, ...

On the other hand, since aba is not an exact repetition of the sequence ab, it is not included in the language.

aaabaaabaaab  => is included in the language  
aba           => is not included in the language

Thus, the language consists of sequences that are an arbitrary-time repetition of a subsequence that is any amount of a followed by any amount of b. Therefore, the language is not regular since it requires a stack.

Answer 1

* is not +.

aba is in that language; it’s just an overly-complicated way to say “the set of all strings consisting of as and bs”.

EDIT: The repeating group doesn’t mean that the contents of the group must be repeated exactly; that would require a backreference. ((a*b*)?\1*)
Rather, it means that the group itself should be repeated, matching any string that it can match.