Question

I’m quite lost in identifying regular language.

I know that if R is a regular language then if A = RR, since is a concatenation of R therefore A is a regular language

But is B = {ww| w <- R} regular?

My first instinct was yes. Because it is also concatenation of R.. But since it is a subset of concatenation I feel I cannot prove it that way. Then I was thinking since w is a string of a regular language, which is a concatenation of singletons, then their concatenation… I get it that I’m quite off the track since if think that way, what isn’t? now I’m more inclined to say it is not. Because I really cannot find a regular expression for it. I wanted to try using pumping lemma but it is really hard to apply to this example.

Can anyone offer some suggest? even a right track for me to follow would be great?

Answer 1

Go ahead and try the pumping lemma. Start with a really simple regular expression, for example:

R = ab*

Since at this point you are trying to prove that it is not regular, all you need is one counterexample. So you can choose any R you’d like. (The above will work fine.)